Mariana Netto

Françoise Lamnabhi-Lagarrigue

Yijing Chen

Gilney Damm

Abdelkrim Benchaib

Françoise Lamnabhi

Miguel Jiménez Carrizosa

• "Feedback Linearization for the DC Voltage Control of a VSC-HVDC Termi

LAGARRIGUE, Dr. Gilney DAMM and Dr. Abdelkrim BENCHAIB, for their guidance, advices and assistance. It is my greatest pleasure to work with them.

Thanks a lot to Dr. Gilney DAMM for his unremittingly invaluable guidance and comprehensive assistance. He is always ready to help me whenever I had scientic or personal diculties. He usually shared his experience with me and gave me constant advice in research and career orientation. He never put undue pressure on me. It was really a great chance to work with him. Thanks a lot to Dr. Abdelkrim BENCHAIB for oering invaluable advice from the point view of industry. After every discussion with him, I acquired a greater depth of understanding of my thesis work. Thanks a lot to Dr. Françoise LAMNABHI-LAGARRIGUE who admitted me as a Ph.D. candidate in L2S. She helped me a lot to deal with all the administrative trivial things. She always tried her best to build a convivial ambiance for work. With her support and trust, my Ph.D. years have become a pleasant life journey. I would like to thank the members of the jury, Dr. Anuradha ANNASWAMY, Prof. Fouad GIRI, Prof. Alain GLUMINEAU, Prof. Riccardo MARINO and Prof.

Didier GEORGES, for their comments, questions and suggestions. Special thanks to Dr. Anuradha ANNASWAMY and Prof. Fouad GIRI, who spent a lot of time and eort to review my thesis and gave advice to improve my work. Thanks a lot to Dr. Anuradha ANNASWAMY who warmly hosted me during my visit to the active adaptive control laboratory (AACLAB) at Massachusetts Institute of Technology (MIT). She introduced me to the domain of adaptive control which is a powerful tool to solve the system uncertainties. Thanks to this visit, I have the opportunity to meet researchers in the Department of Mechanical Engineering and discussed with them to expand my knowledge.

I am indebted to the former and the current faculty, personnel, Ph.D. students from L2S, who constantly helped me. Special thanks to Jing DAI who used to be a post-doc in L2S. He helped me to correct many bad writing habits when I wrote my rst paper. He also gave me lots of comments and suggestions on my dissertation and other papers. I will always remember that even during the Christmas holiday, he still corrected my dissertation and gave me feedbacks. Many thanks to Miguel Jiménez Carrizosa for discussing with me. In fact, I did not take many courses in the eld of electrical engineering during my undergraduate and master study. Miguel shared his professional knowledge of electrical engineering. I also want to thank Fernando DORADO NAVAS, Mawulikplimi Komlan Crédo PANIAH, Van Cuong NGUYEN, Alessio Iovine, Alexandre Azevedo, Yujun HE • • • for their helpful discussions. I would like to thank Mme. Maryvonne GIRON and Mme. Laurence ANTUNES for helping me a lot to handle the visa paper work and renew of my "titre de séjour ".

I am grateful to my dear friends, Yuling ZHENG, Long CHEN, Chuan XU, Zheng CHEN • • • . I shall evermore cherish the memory that we played badminton together.

ii Last, my deepest gratitude is undoubtedly to my families who are always beside me and support me. I want to thank my father, my oldest friend, who always supported and encouraged me; my mother who always prayed for me and my sister who always comforted me and straightened me out when I encountered unhappy things. I'm so thankful that I have such wonderful families.

Abstract

Nowadays the world total electricity demand increases year by year while the existing alternating current (AC) transmission grids are operated close to their limits. As it is dicult to upgrade the existing AC grids, high voltage direct current (HVDC) is considered as an alternative solution to several related problems such as: the increase of transmission capability; the interconnection of remote and scattered generation from renewable energy sources (in particular oshore); the interconnection of dierent asynchronous zones. At present, over one hundred direct current (DC) transmission projects exist in the world, the vast majority for two-terminal HVDC transmission systems and only three for multi-terminal HVDC systems. The traditional two-terminal HVDC transmission system can only carry out point-to-point power transfer. As the economic development and the construction of the power grid require that the DC grid can achieve power exchanges among multiple power suppliers and multiple power consumers, multi-terminal voltage-sourced converter HVDC (MTDC) systems draw more and more attention. As a DC transmission network connecting more than two converter stations, an MTDC transmission system oers a larger transmission capacity than the AC network and also provides a more exible and ecient transmission method. Most studies in the eld of MTDC systems have involved an empirical control approach, namely, vector control approach consisting of several standard proportional-integral (PI) controllers. This control concept is mainly based on the assertion that the system state variables exhibit the performances with dierent dynamics. However, very rare relevant work has ever presented a theoretically detailed explanation on the validity and the implications of this assertion in the literature.

The research work in this dissertation was started with the intention of lling some gaps between the theory and the practice, in particular: 1) to investigate various control approaches for the purpose of improving the performance of MTDC systems; 2) to establish connections between existing empirical control design and theoretical analysis; 3) to improve the understanding of the multi-time-scale behavior of MTDC systems characterized by the presence of slow and fast transients in response to external disturbances.

The main contributions of this thesis work can be put into three areas, namely nonlinear control design of MTDC systems, analysis of MTDC system's dynamic behaviors and application of MTDC systems for frequency control of AC systems.

In the area of nonlinear control design of MTDC systems, based on dierent nonlinear control design techniques, new control schemes have been proposed with corresponding theoretical analysis. Besides, the developed control algorithms have been tested by numerical simulations, whose performances are evaluated in comparison to the performance of the conventional vector control method.

The main motivation for the topic on analysis of MTDC system's dynamic behav-iv iors is the desire to provide a rigorous theoretical demonstration on the assertion that part of system states' dynamics are much faster than the rest. As a consequence, it is possible to simplify the control design procedure. The contribution in this area consists of three parts: 1) control induced time-scale separation for a class of nonlinear systems; 2) analysis of time-scale separation for an MTDC system with master-slave control conguration; 3) analysis of time-scale separation for an MTDC system with droop control conguration. Theoretical analysis, mainly based on singular perturbation and Lyapunov theories, have been carried out for each of the aforementioned aspects and conrmed by various simulation studies. The study of the rst part has been mostly dedicated to propose a time-scale separation control method which can drive a class of nonlinear systems to exhibit a multi-time-scale performance. The theoretical results obtained from the rst part have been applied to investigate the dynamic behaviors of MTDC systems where the two main control congurations, i.e. master-slave and droop control congurations, are considered.

Empirical vector current control designs have been proposed for each control conguration. Theoretical explanations and fundamental analysis indicate that with the proposed empirical control algorithms, the inductor currents indeed exhibit a dynamic behavior with dierent dynamics. Furthermore, we have provided more details on how and why these empirical vector current control designs work as well as the rules of tuning the control parameters.

The nal contribution relates to the application of MTDC systems where frequency support strategy using MTDC systems has been introduced and analyzed.

The principle of the frequency control is to regulate the AC frequency by modulating each AC grid's scheduled (or prescribed) active power. A DC-voltage-based control scheme for the AC frequency regulation is proposed, which achieves the objective of sharing primary reserves between dierent AC areas interconnected via an MTDC system without using remote information communication. Frequency of the i th SAC at the PCC V rms,g i AC voltage magnitude of the i th SAC at the PCC v sg i ,dq dand qaxis AC voltage of the i th SAC at the PCC v sg i ,abc a-, band caxis AC voltage of the i th SAC at the PCC v cg i ,dq dand qaxis AC voltage of the i th SAC at the converter side v cg i ,abc a-, band caxis AC voltage of the i th SAC at the converter side i g i ,dq

List of Symbols

dand qaxis phase reactor current of the i th SAC i g i ,abc

a-, band caxis phase reactor current of the i th SAC P g i Active power of the i th SAC at the PCC Q g i Reactive power of the i th SAC at the PCC ω g i i th SAC system's pulsation δ g i Phase angle of the i th SAC connected VSC m g i Modulation ratio of the i th SAC connected VSC m g i ,dq

dand qaxis modulation index of the i th SAC connected VSC i cg i DC current owing from the i th SAC connected VSC R equi , g i Equivalent resistance representing the i th SAC connected VSC's loss L w j j th WAC connected VSC terminal's phase reactor inductance R w j j th WAC connected VSC terminal's phase reactor resistance f w j Frequency of the j th WAC at the PCC C f w j j th WAC connected VSC terminal's lter capacitance V rms,w j AC voltage magnitude of the j th WAC at the PCC v sw j ,dq dand qaxis AC voltage of the j th WAC at the PCC v sw j ,abc a-, band caxis AC voltage of the j th WAC at the PCC v cw j ,dq dand qaxis AC voltage of the j th WAC at the converter side v cw j ,abc a-, band caxis AC voltage of the j th WAC at the PCC I w j ,dq dand qaxis controlled current source of the j th WAC network i w j ,dq dand qaxis phase reactor current of the j th WAC i w j ,abc a-, band caxis phase reactor current of the j th WAC P w j Active power of the j th WAC at the PCC Q w j

Reactive power of the j th WAC at the PCC ω w j j th WAC system's pulsation δ w j Phase angle of the j th WAC connected VSC m w j Modulation ratio of the j th WAC connected VSC xiv Contents m w j ,dq dand qaxis modulation index of the j th WAC connected VSC i cw j DC current owing from the j th WAC connected VSC R equi , w j Equivalent resistance representing the j th WAC connected VSC '

u cg = [u cg 1 • • • u cg N ] T u cw DC voltage vector: u cw = [u cw 1 • • • u cw M ] T u ct DC voltage vector: u ct = [u ct 1 • • • u ct P ] T u c DC voltage vector: u c = [u cg u cw u ct ] T z State variables of the DC grid: z = [u c i c ] T u c,min Contents xv R ik
Resistance of the transmission line between the i th and the k th AC areas P m i i th AC area's mechanical power input J i i th AC area's moment of inertia D g i i th AC area's damping factor of the aggregated generator f i i th AC area's frequency f nom,i i th AC area's nominal frequency P l i i th AC area's load T sm,i i th AC area's time constant of the servomotor P s m i i th AC area's scheduled mechanical power input P nom,i i th AC area's nominal mechanical power input σ i i th AC area's generator droop List of Figures The response of i g i q for the changes of R L and i o g i q . (a) PI controller. The response of i L for the changes of R L and i o g i q . (a) PI controller. This thesis was devoted to the study of Multi-Terminal High Voltage Direct Current networks. The main contributions were in the eld of nonlinear automatic control, applied to power systems, power electronics and renewable energy sources.

In the current chapter, we rst give a general introduction to HVDC technology where its advantages and disadvantages are analyzed. Then, we present the related research work on the control of HVDC. Finally, we elaborate the objectives and the contributions of this dissertation.

History and background

Direct current (DC) technology was introduced at a very early stage in the eld of power systems. The rst long-distance electrical energy transmission line was built using DC technology between Miesbach and Munich in Germany in 1882. Since the power losses in the transmission lines are proportional to the square of the current owing through the conductors, electrical power is usually transmitted at a high voltage level so as to reduce these power losses. In the early days, due to technical limitations, it was dicult and uneconomic to convert the DC voltage between the low voltage level for consumer's use and the high voltage level for transmission. In alternating current (AC) system, thanks to the invention of AC transformer, it is possible to convert the AC voltage between dierent levels in a simple and economic way. In addition, a three-phase AC generator is superior to a DC generator in many aspects. For example, the AC generator is more ecient, has a much simpler structure with lower cost and requires easier maintenance. Because of these reasons, AC technology has become dominant in the areas of power generation, transmission and distribution. Although AC provides many benets, it still has some limitations in practice, mainly in the following aspects [START_REF] Siemens | High Voltage Direct Current-Proven Technology for Power Exchange[END_REF], Andersen 2002]. :

• The eciency of power transmission capacity is subject to the capacitive and inductive eects of underground/undersea cables or AC overhead transmission lines. For a long AC transmission system, the current that ows to charge the line (or cable) capacitance could be remarkable, causing a signicant amount of energy losses as heat in the wires.

• It is impossible to connect unsynchronized AC networks via an AC transmission link. The latter is only suitable for the connection between synchronized AC networks with the same frequency where the phase dierence between these synchronized AC networks is also restricted to certain allowable range.

• AC transmission technology is not a very good choice for integrating massive renewable energy sources into the existing electrical grid. In particular, since the renewable generation outputs are usually intermittent (for example, the energy production of wind farms strongly depends on the wind speed), the power variations seriously aect the AC voltage at the point of common coupling (PCC) when the renewable power occupies a signicant share of the total electricity generation.

• For oshore applications, what implies the use of cables instead of overhead lines, the well known limitations of using AC is very expensive while DC is not concerned by these problems.

People have been therefore making eorts to nd a satisfactory alternative to AC transmission technology. In the past decades, signicant advances have been achieved in the development of high power devices [START_REF] Long | [END_REF]]. This has provided benecial conditions for DC technology which could come again in sight in the application of power transmission after years of silence.

The invention of mercury-vapor valves which could convert AC current to DC current appeared in 1901 and made HVDC transmission possible [Rudervall 2000].

Several experiments with mercury-arc valves for DC power transmission have been carried out in America and Europe before 1940. In 1941, a commercial HVDC system was designed for supplying Berlin in Germany via a 115 km buried cable using mercury-arc valves. However, it was never put in operation because of the war. The rst commercial HVDC transmission used to connect the mainland of Sweden and the island of Gotland was put into service in 1954. Here, we also pay tribute to 1.2. Overview of HVDC technology 3 the father of HVDC, Uno Lamm, who made a signicant contribution towards the application of HVDC in practice [Wollard 1988, Lamm 1966]. Since then, several HVDC transmission systems have been built with mercury arc valves up to 1972, the installation of the last mercury-arc HVDC system in Manitoba, Canada. The solid state semiconductor valves (thyristor valves) rst appeared in 1970, and then soon replaced the mercury-arc valves [Karady 1973]. Thyristor valves have many advantages over mercury-arc valves. For example, it is easy to increase the voltage just by connecting thyristors in series. The rst HVDC system using thyristor valves were put into operation in 1972. Today, most of existing HVDC projects are still based on thyristor valves [Asplund 2007, Varley 2004]. Since mercury-arc valves and thyristors can not achieve the operation of turn-o by themselves, both of them need an external AC circuit to force the current to zero and hence, the converters based on these two valves are known as line-commutated converters (LCCs) [Daryabak 2014].

During the conversion process of LCCs, the converters need to consume the reactive power which is usually supplied by the AC lters or series capacitors embedded in the converter station [START_REF] Bahrman | [END_REF]]. In the late 1990's, capacitor-commutated converters (CCCs) were introduced where capacitors in series connected between the thyristors and the transformers are used to provide the reactive power. The rst CCC based HVDC system was built in 1998 used to connect Argentina and Brazil.

In fact, CCCs have not been widely applied in the construction of HVDC systems because of the advent of new power electronics in the mid 1990's. The development of high rated semiconductor devices such as integrated gate-commutated thyristors (IGCTs), insulated-gate bipolar transistors (IGBTs), gate turn-o thyristors (GTOs) etc. has marked the beginning of a new era in the area of HVDC systems [Andersen 2000, Poullain 2001, Flourentzou 2009, Andersen 2000]. The IGBT diers from thyristor in that it has the ability to turn o by itself, which can be used to make self-commutated converters. Converters built with IGBTs are known as voltage-sourced converters (VSCs). The rst complete VSC HVDC system was built in Gotland, Sweden, 1999 [Eriksson 2001]. Since then, many VSC HVDC systems have been put into operation [Asplund 2000[START_REF] Johansson | [END_REF], Stendius 2006, Dodds 2010]. The recent rst three-terminal VSC HVDC system operated by China Southern Power Grid has been put into service at the end of 2013, which is used to feed wind power generated from Nanao island into the mainland of Guangdong via a combination of underground cables, submarine cables and overheard lines [Li 2014, Fu 2014].

Overview of HVDC technology

The most essential process in an HVDC system is the conversion of the energy between AC and DC. The converter station located at the sending end is called the rectier (to convert the current from AC to DC) while the one at the receiving end is called the inverter (to convert the current from DC to AC). As mentioned in Section 1.1, there are two basic converter technologies in today's HVDC transmission Chapter 1. Introduction systems, LCC and VSC.

1.2.1 Line-commutated current source converters Conventional (or classical) HVDC systems are based on LCCs with thyristor valves which are used to achieve the operation of converting the current between AC and DC [Carlsson 2003, Carlsson 2002, Varley 2004, CIGRE 2014]. The essential component of LCCs is the three-phase or six-pulse bridge composed of six controlled switches or thyristor valves. In order to operate at the desired voltage rating, the thyristors are usually connected in series to build up a suitable thyristor valve. The drawback of the six-pulse bridge is that considerable AC current and DC voltage harmonics are produced due to each phase change every 60 • . To deal with this problem, two six-pulse bridges are connected in series to constitute a twelve-pulse bridge. In this manner, with each of the two six-pulse bridges connected to one DC rail, the 30 • phase displacement can be achieved so that some harmonics can be eliminated. In fact, the twelve-pulse bridge has become dominant in modern LCC HVDC systems.

Since LCC can not be turned o by itself, an external relatively strong AC voltage source is required to perform the commutation. As a result, it is impossible to change the direction of the current. In an LCC HVDC system, in order to change the power direction, it is necessary to change the polarity of the output voltage.

This leads to inherent diculty when a weak AC system or a passive network is connected to an LCC converter station. As a controlled solid-state semiconductor device, the thyristor can only be turned on by control action and hence, LCC has one degree of freedom, i.e. the ring angle, which represents the phase lag of AC current behind the voltage. The only way to regulate the DC voltage rating across a valve is by means of controlling the ring angle. More details of the control of ring angle are well documented in the literature [Arrillaga 1998, Kimbark 1971, CIGRE 2014].

Voltage-sourced converters

Voltage-sourced converter is built up with fully controlled semiconductor devices such as IGBT and GTO with the ability of turn-o compared to LCC [Flourentzou 2009, Dodds 2010, Wang 2011]. As a result, an external AC voltage is no longer required since the converter based on IGBTs (or GTOs) are selfcommutated. The additional controllability (the operation of turn-o ) gives many advantages to the VSC:

• Pulse-width modulation (PWM) can be applied to the operation of VSC, which works at higher frequencies than the line frequency and gives rise to high dynamic performance [Trzynadlowski 1996]. With PWM, the VSC can generate any voltage with desired phase angle and amplitude. Less harmonics are produced compared to LCC with PWM.

• Because the AC current always lags behind the voltage in the operation of LCC, the converter needs to consume the reactive power to keep the amplitude 1.2. Overview of HVDC technology 5 of AC voltage within acceptable range, while the VSC itself has no reactive power need. Therefore, it is exible to place the VSCs anywhere in the AC system.

• Unlike LCC, a strong AC voltage source is not indispensable for the VSC connected AC system and hence, there is no restriction on the inherent characteristics of the connected AC networks.

• Since the VSC possesses two degrees of freedom, this converter has the ability to control active and reactive power independently. This capability makes the VSC act close to an ideal generator in a transmission network.

• The bidirectional power transmission can be achieved with the VSC by changing the current direct without changing the output voltage polarity.

More details of the operation of VSCs are presented in Section 2.

LCC vs VSC

Although the capability of turn-o produces many benets for the VSC, it does not mean that VSC can replace LCC. Both of the two converters are applied in modern HVDC systems and they have their respective merits that meet dierent requirements [Marques 2011, Rudervall 2000]:

• In recent years, signicant advances and improvements in the thyristor technology have been achieved. In particular, the thyristor has a large capacity that can withstand high voltage (up to 10 kV) and high current (4 kA) ratings.

Therefore, there is no special restriction on the power range of HVDC with LCC since very high voltages can be achieved by connecting the thyristors in series. Besides, the inductive and capacitive elements of the transmission lines have no eect on the HVDC with LCC and hence, the power losses caused by these elements do not limit the transmission distance. For these reasons, LCC HVDC system is well established for long-distance bulk power transmission.

• VSC-HVDC technology is mainly used in medium-capacity power transmission based upon the considerations: 1) Unlike the thyristor, the IGBT has a limit capacity with weak overload capability; 2) Due to the use of PWM at high frequency, the switching losses of the VSC HVDC are higher than the LCC HVDC. 3) A VSC HVDC system usually uses a special cross-linked polyethylene (XLPE) cable which has a lower voltage capability compared to the mineral-insulated (MI) cable used in LCC HVDC system.

• A VSC HVDC system can be used to supply the electrical power to passive or weak AC networks with low short-circuit capacity. Since the semiconductor devices used in the VSC are self-commutating, for the connected AC system, a strong AC voltage source is not necessary.

• At present, the vast majority of existing HVDC links are composed of two converter stations, only carrying out point-to-point power transfer. As the economic development and the construction of the power grid require that the DC grid can achieve power exchanges between multiple power suppliers and multiple power consumers, multi-terminal HVDC system consisting of more than two converter stations have drawn more and more attention. VSC HVDC technology is more appropriate than LCC HVDC to build a multiterminal HVDC system because of the following reasons: 1) For the LCC HVDC, the only way of reversing the power direction is to change the DC voltage polarity. However, in a multi-terminal HVDC system, the reversal of one terminal's DC voltage polarity gives remarkable disturbances to other terminals and hence, it is essential to arrange the switching operation of each terminals [Reeve 1980]. 2) For the VSC HVDC, the power direction can be changed through the reversal of the current direction while keeping the DC voltage polarity unchanged. Therefore, one terminal's power direction can be changed without considering other terminal's power ow directions.

Advantages of HVDC over AC

As discussed in Section 1.1, the inherent drawbacks of AC encourage the development of DC technology. The rationale for the choice of HVDC instead of AC is often various and complex. The common applications in favor of HVDC are listed as follows:

• Long-distance transmission:

For an AC transmission, the capacitive eect of long overhead lines not only causes the additional power losses but also limits the transmission distance

while it has less impact on the HVDC transmission line. For these reasons, HVDC technology provides bulk power transmission solution over very long distances. From economic aspects, the costs of an HVDC transmission line are less that an AC line for the same distance but the construction of a converter station is expensive. A famous diagram as illustrated in Fig. 1.1 shows a typical cost analysis for AC and DC [SIEMENS 2014]. Above the break-even distance, the HVDC transmission line becomes an attractive solution with less costs in comparison to an AC transmission line with the same power capacity.

Moreover, DC overhead lines occupy much less space than AC transmission lines with the same power capacity. This reduces the visual impact and the usage of rights-of-way. Therefore, it is possible to realize the power delivery from remote power resources such as oshore wind farms or hydro-power plants using lower visual impact and fewer transmission lines with HVDC.

• Underground or submarine transmission:

From the environment aspects, it is hard to upgrade the electrical grid with overhead AC lines which require large transmission line corridors. It is recommended to use underground cables that can share rights-of-way with other utilities. As discussed above, the main drawback of underground or submarine cable systems with AC lines is the remarkable power losses and transmission distance limits caused by the capacitive eect. The VSC HVDC system with XLPE cables provides a more exible, ecient and reliable way to carry out the power transfer via long-distance underground or submarine transmission.

• Asynchronous connection:

HVDC technology is the only way that can connect AC networks with dierent frequencies. The advantage of this application is that a disturbance in one AC area does not aect another, and thus the HVDC link can play the role of a rewall to prevent cascading failure. In addtion, in an asynchronous connection, back-to-back conguration is often used without transmission lines to achieve power exchange.

• Renewable power transmission:

Literature review: related studies on control designs

The control design is always one of the most popular research topics in the area of VSC HVDC systems. A large number of studies have been devoted to the control of VSC HVDC systems from dierent points of view.

Control of a single VSC terminal

For any VSC HVDC system, the converter is the most basic component, whose operation has a great impact on the system's overall performance. Therefore, it is particularly important to develop appropriate control structures for the VSC. There are two most frequently discussed control methods, namely direct control method and vector control method. The direct control approach is very straightforward where the control inputs, the phase angle and the amplitude modulation ratio, are directly derived from the measurements of the currents and the voltages at the PCC [START_REF] Ooi | [END_REF], Ohnishi 1991[START_REF] Noguchi | [END_REF][START_REF] Sood | [END_REF]]. The major advantage of this control method is its simple structure easily implementable while the gravest drawback is its incapability of limiting the current through the converter to protect the semiconductor devices.

To overcome this shortcoming, the vector control method is widely used in the context of the control of VSC [START_REF] Lindberg | [END_REF][START_REF] Lindberg | [END_REF][START_REF] Blasko | [END_REF], Li 2010, Thomas 2001, Xu 2005[START_REF] Beccuti | [END_REF]] where the VSC is modeled in a synchronous reference dq frame. The main feature of this method is that it consists of two control loops, namely the fast inner current loop and the slow outer loop, where the slow outer loop provides the reference trajectories to the fast inner current one. Thanks 1.3. Literature review: related studies on control designs 9 to the fast inner current loop, this control structure has the ability to limit the current through the converter.

Studies on comparing the direct control method and the vector control method are carried out in [START_REF] Sood | [END_REF][START_REF] Moustafa | [END_REF]. The conventional vector control method is usually based on several standard PI controllers where a phaselocked loop (PLL) is usually applied to provide a synchronous reference frame [START_REF] Lindberg | [END_REF][START_REF] Lindberg | [END_REF][START_REF] Blasko | [END_REF]]. However, several limitations of pure PI control are presented in [START_REF] Du | [END_REF][START_REF] Dannehl | [END_REF], Durrant 2003], which show that the maximum transferable power capability with the conventional vector control method is far less than the system's maximum achievable capacity and the system overall performance is remarkably disturbed by the system uncertainties.

Thus, some modied vector control methods are developed. In [START_REF] Zhao | [END_REF]],

adaptive backstepping control technique is applied to the DC voltage outer loop and the inner current loop to ensure the global stability. In [Li 2010], an optimal control strategy using a direct current vector control technology is proposed where the current loop can be a combination of dierent technologies such as PID, adaptive control, etc. In [START_REF] Beccuti | [END_REF]], a nonlinear model predictive control method is applied to the inner current loop without the need of a PLL. Since the conventional vector control does not fully use the potential of the VSC, in particular, in the case of connecting to a weak AC system, several other control methods are developed.

In [Zhang 2010], power-synchronization control with a new synchronization method as an alternative to PLL is established, whose control concept is derived from the behaviors of synchronous generators. This method is eective when a very weak AC system is connected to the DC grid via a VSC [Zhang 2011a, Zhang 2011b]. In [Mariethoz 2014], a VSC HVDC decentralized model predictive control method is proposed with the purpose of achieving fast control of active and reactive power, improving power quality, etc. In [Xu 2007b], a new direct power control is proposed where the converter voltage is directly derived from the dened power ux.

This method is applied to the control of doubly fed induction generators (DFIG)

with good performance [START_REF] Zhi | [END_REF]]. Furthermore, an improved DPC is presented in [Zhi 2009]. In [Durrant 2006], both linear matrix inequality (LMI) and heuristic schemes like genetic algorithm are presented and a comparison is carried out between them.

Control of MTDC systems

As mentioned in the previous section, AC transmission lines are not feasible for carrying out power transfer from large oshore power plants due to the requirement of transmission lines over long distances. In order to connect those large-scale, oshore and scattered energy sources such as wind farms to the grid, MTDC systems provide a exible, fast and reversible control of power ow. However, the operation and the control of an MTDC system is still an open and challenging problem. For example, since the semiconductor devices in VSC are very sensitive to overvoltage, it is very important to restrict the DC voltage to an acceptable band.

Chapter 1. Introduction

Several research works have proposed dierent control structures to ensure the normal operation of an MTDC system. Master-slave control strategy (or constant DC voltage control strategy) are applied in [Ekanayake 2009, Zhang 2012, Lu 2003, Lu 2005] where one terminal called master terminal is arranged to control the DC voltage at a constant desired level while the remaining terminals control their respective active power or other variables. The main drawback of this control approach is that N-1 secure operation can not be ensured in case of outage of the master terminal. Besides, since only the master terminal is used to ensure the power balance of the DC grid and keep the DC voltage constant, it is needed a master terminal with large capacity.

To avoid the above problems, some other control strategies are introduced.

In [START_REF] Tokiwa | [END_REF][START_REF] Haileselassie | [END_REF], Nakajima 1999[START_REF] Mier | [END_REF]], a control strategy called voltage margin method is developed. With this approach, the operation of each converter is characterized by specic active power-DC voltage (P -V DC ) curve.

With the voltage margin method, one VSC terminal, for instance, VSC 1, is initially used to keep the DC voltage constant by adjusting its active power P 1 to counteract disturbances. When P 1 reaches its upper or lower limit, VSC 1 starts to operate in constant active power mode and the DC voltage will rise or decrease until it reaches another VSC terminal's DC voltage reference setting, for instance, VSC 2's DC voltage reference setting. Then, VSC 2 is responsible for the regulation of DC voltage according to its P -V DC characteristic. Furthermore, a modied two-stage voltage margin control is proposed in [Nakajima 1999] by considering each converter's inherent physical features.

Another control scheme, namely, DC voltage droop control method, whose control philosophy is to use more than one converter to regulate the DC voltage, is widely applied in the context of MTDC systems [Karlsson 2003[START_REF] Prieto-Araujo | [END_REF], Wang 2014a[START_REF] Chaudhuri | [END_REF], Bianchi 2011, Chen 2014, Haileselassie 2012b, Wang 2014b[START_REF] Eriksson | [END_REF][START_REF][END_REF][START_REF] Liang | [END_REF][START_REF] Rogersten | Applying powersynchronization control in a multi-terminal DC system[END_REF]. With this control approach, some or all converters are equipped with a droop controller which is characterized by P -V DC or I -V DC (current vs DC voltage) curve with the slope K droop called the droop gain. This control method has the advantage of 1)

sharing the duty of eliminating the power imbalance of the DC grid between several terminals; 2) taking actions only based on local information without remote communication. In [Pinto 2011], each of the aforementioned control strategies is tested and moreover, a performance comparison between them is also carried out.

Concerning the power relations between converter terminals, three DC voltage control and power dispatch modes are established in [START_REF] Xu | [END_REF]]. In the rst mode, one terminal, for instance, VSC 1, has priority over the other converters in carrying out the power transfer. When VSC achieves its active power's upper limit, another converter takes over the duty of transmitting the energy. In the second mode, two converters transmit the energy according to a power ratio which can be varied in real time. The third mode is the combination of the two mentioned modes. In [START_REF] Pinto | [END_REF]], a novel distributed direct-voltage control strategy is introduced. The main idea of this control approach is to provide a specic voltage 1.4. Motivations and contributions of the thesis 11 reference to each of the terminals which participate in the regulation of the DC voltage. In [Rodrigues 2013], an optimal power ow control is proposed where the convariance matrix adaptation (CMA) algorithm is applied to solve an optimal DC load-ow problem and then provide the DC voltage references to the distributed voltage control developed in [START_REF] Pinto | [END_REF]].

The use of MTDC systems for connecting oshore wind farms have been studied by a number of papers [Haileselassie 2012a, Li 2014, Karaagac 2014[START_REF] Haileselassie | [END_REF], Chen 2011[START_REF] Lie | Multi-terminal DC transmission systems for connecting large oshore wind farms[END_REF][START_REF] Prieto-Araujo | [END_REF][START_REF] Liang | [END_REF][START_REF] Haileselassie | [END_REF], Xu 2007a, Fu 2014] where the control structures are well established and the simulation results are also carried out.

The above mentioned articles in this section are entirely dedicated to the control design of VSC HVDC systems. There are also a large number of studies that focus on the modeling of the VSC HVDC systems [Cole 2010[START_REF] Beerten | [END_REF]], the protection system [START_REF] Kong | Development of a Novel Protection Device for Bipolar HVDC Transmission Lines[END_REF], Xiang 2015], the development of multilevel modular converters (MMC) which generates less harmonics and reduces the losses of the semiconductor devices [Shi 2015[START_REF] Adam | [END_REF][START_REF] Ilves | [END_REF], etc.

Motivations and contributions of the thesis

For years, people have been working on the development of a European supergrid that allows the massive integration of renewable energy sources to meet the ever increasing consumption of electricity. Due to some undesirable inherent characteristics of the renewable energy sources, such as the variability of the generation output, the conventional AC transmission lines are not quite suitable for the connection of large shares of renewable energy sources while VSC HVDC systems provide an alternative solution. Although numerous research studies on VSC HVDC systems have been carried out, there are still many challenges and it is not yet totally feasible to build such a supergrid.

Motivations

This dissertation is motivated by the challenges in the control design of MTDC systems and analysis of MTDC system's dynamic behavior.

Control methods

As a system with complex dynamics, any operation of the VSC HVDC system may give rise to both potential dangers of unexpected interaction and improvement of the system performance. Therefore, developing control structures which are capable to keep good performances like fast tracking or small overshoots, during disturbances is always a challenge for the control of VSC HVDC systems.

As presented in Section 1.3.1, many control methods have been developed. However, most of them lack the corresponding theoretical support and usually need signicant eorts to adjust the controller. For example, though the traditional vector Chapter 1. Introduction control method based on standard PI controllers has a simple structure, we need to tune the PI control gains by using other procedures [Bajracharya 2008]. Reference [Zhang 2011c] gives a stability analysis for power synchronization control method.

Nevertheless, this analysis is carried out by linearizing the system around the operating point and hence, the nonlinear behavior of the system can not be well exhibited.

Reference [START_REF] Dannehl | [END_REF]] also presents a study on the limitations of vector control method, but, similar to the case of power synchronization control method, this research work is done via linearization and it is only valid for the behavior within a small neighborhood of the operating point. It is worth considering that, when designing a control structure, we should not only focus on the actual results (or performances) of the control strategy but also on the theoretical explanation of the control operating principle.

Dynamic behaviors

As discussed in the section of literature review, vector control method, which is composed of two control loops, is the most widely used control approach for VSC HVDC systems due to its simple structure and its capability to limit the current through the converter. The rationale of this method is based on the assertion that the dynamics of the inductor currents are much faster than the dynamics of the capacitor voltages. As a result, the independent control designs of the two control loops are achievable. However, there exist very few relevant studies in the literature that have ever presented a detailed theoretical explanation on the validity and the implications of this assertion. It is not clear if the time-scale separation between the dynamics of the current and the voltage is an inherent characteristics or it exists only under certain conditions. For example, as demonstrated in [Kimball 2008],

for some DC-DC converters, the dynamic separation between the system variables exists only when the stability requirement is satised. If the time-scale separation exists in a high-order system, this feature has a great signicance to the analysis of system's behavior. Based on the time-scale separation, two (or more) subsystems of lower orders can usually be deduced from the original high-order system and then, we can use the behaviors of the low-order subsystems to approximate the behaviors of the high-order original system. This greatly reduces the complexity of analyzing the system's dynamic performance. Therefore, the verication of the existence of the time-scale separation is very desirable.

Contributions

The research work in this dissertation aims at lling some gaps between the theory and the practice, i.e. 1) to investigate various control approaches for the purpose of improving the performance of MTDC systems; 2) to establish connections between existing empirical control designs and theoretical analysis; 3) to increase understanding of the multi-time-scale behavior of MTDC systems characterized by the presence of slow and fast transients in response to external disturbances.

Motivations and contributions of the thesis 13

The main contributions of the research work reported in this thesis can be put into three aspects, namely nonlinear control designs of MTDC systems, analysis of MTDC system's dynamic behaviors and application of MTDC systems to provide frequency support for AC networks.

Nonlinear control designs

In the area of nonlinear control design of MTDC systems, three nonlinear control design tools, namely feedback linearization control, feedback linearization control with sliding mode control and passivity-based state feedback control, are applied to establish dierent control schemes with corresponding theoretical analysis. Besides, the developed control algorithms have been tested by detailed and realistic numerical simulations, whose performances are evaluated in comparison to the performance of the conventional vector current algorithm.

Analysis of dynamic performances

The main motivation for the research work in the area of analysis of MTDC system's dynamic behaviors is the desire to provide a rigorous theoretical demonstration to verify a common assertion used in the control design of VSC HVDC systems, that is, the dynamics of the inductor currents are much faster than the capacitor voltage.

The contribution in this area consists of three parts: 1) development of a general control theory that we named "Control induced time-scale separation" for a class of nonlinear systems; 2) analysis of control induced time-scale separation for an MTDC system with master-slave control conguration; 3) analysis of control induced timescale separation for an MTDC system with droop control conguration. Theoretical analysis, mainly based on singular perturbation and Lyapunov theories, have been carried out for each of the aforementioned aspects and claried with various simulation studies. The study of the rst part has been mainly performed to propose a time-scale separation control method which can drive a class of nonlinear systems to exhibit a multi-time-scale performance. The theoretical results obtained from the rst part have been applied to investigate the dynamic behaviors of MTDC systems under the two main control congurations, i.e. master-slave and droop control.

Empirical vector current control designs have been proposed for each control conguration. Theoretical explanations and fundamental analysis indicate that with the currently used empirical control algorithms, the inductor currents indeed perform a dynamic behavior with dierent dynamics. Furthermore, more details on how and why these empirical vector current control designs work have been also provided.

Frequency support

The nal contribution relates to analysis of the application of MTDC systems for frequency support strategy. A DC voltage based control scheme for the AC frequency regulation is proposed, achieving the objective of sharing primary reserves between dierent AC areas interconnected via MTDC system without using remote information though telecommunication. The frequency control principle is to regulate the AC frequency by modulating each AC grid's scheduled (or prescribed) input of active power.

Outline of the thesis

This dissertation is organized as follows: In Chapter 2 a brief introduction of VSC HVDC technology is presented. In particular, the possible operation modes of the VSC are discussed, which are essential for the design of a control system. In Chapter 3 the dynamic model of a multi-terminal VSC HVDC system is established based on a synchronous reference dq frame. This MTDC system consists of two dierent AC networks connected via a generic DC grid. In Chapter 4 two most used conventional control methods, namely direct control method and vector control method, are introduced. In addition, the merits and drawbacks of these two methods are also discussed. In Chapter 5 three dierent nonlinear control structures are proposed for VSC HVDC systems. With the help of numerical simulations, the advantages claimed for each of the proposed controllers are veried in comparison to the conventional vector current control method. In Chapter 6 the system dynamic behaviors are investigated. We rst demonstrate that there exists a control structure that can drive a class of nonlinear systems to exhibit arbitrary multi-timescale dynamics that were not natural to the system. Then, we apply the obtained theoretical results to the MTDC system under two control congurations, namely master-slave control conguration and droop control conguration. The theoretical analysis of the time-scale separation between the dynamics of the system states are also carried out for each of the aforementioned control conguration. In Chapter 7 the potential of VSC HVDC technology for frequency regulation is investigated where a DC-voltage-based control scheme is proposed to make the interconnected AC networks share their primary reserves. In Chapter 8 conclusions are drawn and future work is discussed.

Publications

Most of the materials in this thesis have been published in the following articles:

• "Nonlinear Control Design for a Multi-Terminal VSC-HVDC System", Yijing • "A detailed study on a DC-voltage-based control scheme using a multiterminal HVDC system for frequency control", Yijing Chen, Jing Dai, Gilney

Key components

A simplied representation of a VSC HVDC converter station is depicted in Fig.

2.1. The characteristics and purposes of the main components of a VSC HVDC system are introduced in this section [START_REF] Siemens | High Voltage Direct Current-Proven Technology for Power Exchange[END_REF][START_REF] Bahrman | [END_REF], Dodds 2010, Andersen 2002].

Transformer

A bank of transformers are installed to adapt the AC voltage level of the connected AC grid to the rated DC voltage transmission level. Since harmonic lters are located between the phase reactors and the transformers, harmonic currents have little or no eects on the transformers. Therefore, ordinary transformers can be used for VSC HVDC systems.

Harmonic lter

With VSCs, the generated harmonics are usually at high frequencies, which strongly depend on the PWM method and converter topology. In a VSC HVDC system, shunt lters are used to reduce such high-frequency harmonic currents owing into the AC system so that the transformers are not exposed to harmonic stress.

Phase reactor

The phase reactors are composed by large inductances with small resistances. On the one hand, the phase reactor can also reduce the harmonic distortions to smoothen the phase currents like the harmonic lters. On the other hand, they play an essential role in determining the dynamics of the converter on the AC side.

Converters

As the most important component in an HVDC system, the converters are used to perform the conversion between AC and DC. Each valve in the VSC bridge consists of series connected semiconductors (such as IGBT) to create a high-level DC voltage.

PWM technology is usually employed for the VSC to create the desired voltage and to improve the harmonic distortion of the converter.

DC capacitor

The main purpose of the DC capacitors is to maintain the DC voltage within its acceptable band by charging and discharging the capacitors. Besides, the DC capacitors can also reduce the voltage ripples generated by the converters.

DC cable

Solid-dielectric extrude polymer cables with prefabricated joints are used in VSC HVDC systems. This kind of cables are much lighter, more exible than mass-2.2. Operating principles of VSC impregnated or gas-lled conductors. In addition, there is no need to worry about the problem of leaking cable oil.

In this section, we just presented the components that have great eects on the system dynamic modeling and steady-state analysis. Hence, some other indispensable elements of VSC HVDC systems, such as high-side breakers, valve coolers etc, are not illustrated in Fig. 2.1.

Operating principles of VSC

The most common three-phase bidirectional VSC creates two voltage levels at the AC side of each phase with a six pulse bridge as shown in Fig. 2.2. Each valve in the six pulse bridge is made of a fully controllable semiconductor such as IGBT with a diode connected in anti-parallel. The operations of turning on and o of the self-commutated semiconductors result in a square waveform for each phase, which is relevant to the DC voltage u c . For example, the square waveform of the converter voltage in phase a can be expressed by

v ca = 1 2 u c m sin(ω e t + δ) + harmonics (2.1)
where m ∈ [0, 1] is the amplitude modulation ratio, ω e is the frequency of fundamental AC component and δ is the phase angle between the fundamental component of the converter AC voltage and the AC system voltage at the PCC. However, the two-level bridge in the square wave operation usually produces large unacceptable harmonics. Hence, the application of PWM technology can improve the harmonic distortion of the converter with less low-frequency harmonics and give a fast response [Trzynadlowski 1996]. By using PWM techniques such as sinusoidal PWM (SPWM), δ and m are fed to the pulse width modulator to determine the phase shift and the duration of ring pulses, respectively. Finally, the ring pulses are generated and then sent to the converter to switch the IGBTs [Stijn 2010[START_REF][END_REF]]. With PWM, the IGBTs are usually switched on and o many times in order to create the desired phase angle or amplitude.

Congurations of HVDC systems

Depending on the applications of HVDC, there are various congurations of HVDC systems.

Monopolar system

Monopolar HVDC link as illustrated in Fig. 2.3 is the simplest and least expensive conguration for HVDC link where a single conductor is used for power transmission while the current return path is earth, water or a metallic conductor. This type of HVDC link is mainly applicable for submarine cable transmission. Because of its low cost, it also has the advantage in connecting two remote AC systems through an overhead line with a distance of 300 to 3000 km and more. Back-to-back system Back-to-back HVDC link is usually a bipolar HVDC link without earth return as shown in Fig. 2.5. It is especially suitable for connection between two asynchronous AC systems with dierent system parameters (for example, dierent frequencies or dierent voltage level). In a back-to-back HVDC link, the rectier and the inverter are usually located in the same area or the same building. In addition, the length of the conductor line between the two converters can be kept as short as possible.

Therefore, the DC voltage transmission level can be freely chosen. In order to reduce the number of the semiconductors and the size of valve hall, the DC voltage is usually set to a low level and the DC current to a high level.

Multi-terminal system

Multi-terminal HVDC link is used to connect more than two converter terminals in parallel, series or hybrid. Its main advantage is that the power exchange can be fullled between multi-suppliers and multi-consumers. An example of three terminals connected in parallel is given by Fig. 2.6.

AC network connected to VSC terminal

The strength of the AC network connected to the converter station at the PCC has signicant impact on the performance of VSC HVDC systems [IEE 1997]. If the AC network gets most of the power from its own generators, the system electromotive force (EMF) can be kept within an acceptable band by the AC network itself in case of power variations in the DC grid. However, if the DC grid is the main source for the AC network or the AC network has a high impedance, the power variation in the DC grid may cause the system voltage and frequency to go beyond their safe operating ranges.

The strength of the AC network relative to DC power at the PCC is commonly described by the short circuit ratio (SCR) [Gavrilovic 1991], i.e. the ratio of the AC network three-phase short circuit power in MVA at the PCC and the rated DC power in MW. In general, three types of AC systems can be distinguished in terms of SCR. A system with SCR larger than 3 is called a high SCR system (or strong system) where an adequate inertia is available to maintain the frequency and amplitude of the AC voltage at the PCC irrespective of the change of active or reactive power. A system is said to be a low SCR system if the SCR is between 2 and 3. If a system has the SCR less than 2, we call it a very low SCR system (or weak system). For example, an isolated island or an oshore platform belongs to the family of weak systems. For this kind of system, any change in power ow at the PCC may lead to AC instability. To avoid this risk, an AC frequency and voltage controller need to be installed. It can be summarized that the interaction between AC and DC becomes more sensitive to disturbances as the SCR of the AC system becomes lower [Thallam 1992].

Control modes of VSC terminals

Consider a single line representation of the AC side of a VSC as illustrated in Fig. 2.7 where the reactance X includes the reactances of the transmission line, the phase reactors, etc. The AC voltages of the AC network at the PCC and the converter are denoted by V s = V s,rms ∠0 and V c = V c,rms ∠δ, respectively.

The transmitted active power and reactive power at the PCC in steady-state condition are given by

P s = V s,rms V c,rms X sin δ = P max sin δ Q s = V 2 s,rms X - V s,rms V c,rms X cos δ (2.2)
where P max = V s,rms V c,rms X is the maximum active power that can be transmitted through the line. In addition, the power angle δ can be deduced as δ = arcsin( P s P max )

(2.3) From (2.2), it can be seen that P s and Q s are determined by V s,rms , V c,rms and δ. Usually, V s,rms is considered to be xed at a constant level 1 and hence, P s and Q s are controlled by regulating the phase angle δ and the converter voltage V c,rms . From (2.1), it can be seen that V c,rms depends on the modulation index m. Therefore, P s and Q s are actually controlled by the modulation index m and the angle δ.

Based on the purposes of the converter, four possible control modes (control objectives) exist [Akhmatov 2014[START_REF] Moustafa | [END_REF]]:

• Mode 1: AC voltage control mode This control mode is particularly designed for the VSC connected to a weak AC system. Since the weak AC system with high impedance relative to the DC rated power is susceptible to AC/DC interactions, any disturbances either from the DC side of the converter or the AC side may lead to instability of the AC voltage (for example, voltage collapse, overvoltage, etc). Fortunately, fast AC voltage controller provides an eective support for the operation of the VSC which enables the weak AC system to operate at xed magnitude.

• Mode 2: Active power control mode

As described in (2.2), it shows that the active power can be controlled by the converter. An active power controller can be designed such that the active power follows its reference which is provided by a higher control level (or a central power dispatching station).

• Mode 3: Reactive power control mode

This control mode is used to regulate the reactive power at its reference. For example, if the reference input for the reactive power controller is set to zero, the reactive power can be kept at this value to obtain an unity-power factor.

1 In general, this voltage is kept by the Transmission System Operator (TSO).

This action is helpful in improving the power quality. Moreover, unlike LCC HVDC systems, the VSC itself does not need reactive power support. Actually, the VSC can work as a reactive power compensator to regulate the amplitude of the AC system voltage by controlling the reactive power at the PCC.

Because of the ability to fully control the power ow, the VSC has become a very important power ow controlling device [START_REF] Van Hertem | [END_REF] • Mode 4: DC voltage control mode • DC voltage and AC voltage control modes

• DC voltage and reactive power control modes

• Active and reactive power power control modes

• Active power and AC voltage control modes.

Chapter conclusions

In this chapter, we have introduced the key components of a VSC terminal, explained the operating principles of the converter and classied AC systems into three types in terms of SCR. According to various applications of VSC HVDC systems, four dierent congurations have been presented. Furthermore, four possible control modes for operating a VSC have been discussed. This chapter makes preparations for the system modeling in the following chapter. In this chapter, an average state-space model is developed for multi-terminal VSC HVDC systems. In the modeling, a synchronously rotating dq frame is chosen so that the three-phase quantities can be transformed into two-phase quantities.

Using basic physical laws, the dynamics of the converter's AC and DC sides are modeled by dierential equations. A generic topology for the DC network is considered and the properties of the DC grid are discussed. Furthermore, two most widely used control approaches are also presented.

Preliminary knowledge

There exist dierent approaches to modeling VSC HVDC systems. In the detailed modeling approach, every semiconductor device should be treated as an individual unit, where the type of modulation (sinusoidal PMW or optimized PMW), the converter switching level (two-level or multi-level) and harmonics with high frequency need to be analyzed. Such a method results in a very complex model which needs a 28 Chapter 3. Dynamic model of multi-terminal VSC HVDC systems lot of memory and CPU times to run simulations. Compared to the detailed modeling approach, the time-averaged modeling approach [Middlebrook 1976] seems much simpler and tractable while still enabling us to study the system behaviors related to the fundamental frequency components of the voltage and the current. In the time-averaged model, the VSC is viewed as a controllable AC voltage source from AC side whereas it is considered as a controllable current source from DC side as depicted in Fig. 2.2. By using the time-averaged modeling approach, there is no need to consider the dynamics of each individual semiconductor device and moreover, high frequency harmonics can also be neglected. This greatly simplies the modeling process, save simulation memory and accelerates the simulation speed.

Based on the above considerations, the time-averaged modeling approach is applied in this thesis work. As a result, according to (2.1), the converter voltage v c,abc and the DC voltage u c in the time-averaged model satisfy

v c,abc = 1 2 u c m abc (3.1)
where m a , m b and m c ∈ [0, 1] are the modulation indices of the phase a, phase b and phase c, respectively, determined by the amplitude modulation ratio m and the phase angle δ, according to the following expressions

m a = m sin(ω e t + δ) m b = m sin(ω e t -2 3 π + δ) m c = m sin(ω e t + 2 3 π + δ) (3.2)
In the remainder of this thesis, it is assumed that all VSC connected AC systems are under balanced network conditions. Thus, there is no negative sequence component of the three-phase voltage or current, and the three phases have equal voltage amplitudes with a phase angle of 120 • between themselves.

Clarke's and Park's transformations

Clarke's and Park's transformations are widely used in many studies on synchronous and asynchronous machines. Their basic principle is to transform a balanced threephase system into a two-phase system by simplifying the analysis of three-phase circuit. The dierence between both transformations is that Clarke's transformation is to project the three-phase quantities onto a stationary two-axis reference frame, whereas Park's transformation onto a rotating two-axis reference frame.

Clarke's transformation

The stationary two axes of Clarke's transformation are denoted by αand β-, which are orthogonal. In order to make the transformation invertible, the zerosequence component is usually added and then, Clarke's transformation matrix can 3.2. Clarke's and Park's transformations 29 be expressed by

x αβ0 = T αβ0 x abc = k c    1 -1 2 -1 2 0 √ 3 2 - √ 3 2 1 2 1 2 1 2    x abc (3.3)
where k c is a constant and x can represent the AC voltage, current, etc. The inverse Clarke's transformation is given by

T -1 αβ0 = 2 3 k -1 c    1 0 1 -1 2 √ 3 2 1 -1 2 - √ 3 2 1    (3.4)
Let us take an example. Consider a balanced three-phase system whose voltages and currents in a-, b-, and c-axis are given by

v a = √ 2V rms cos(ωt + δ a ) v b = √ 2V rms cos(ωt -2 3 π + δ a ) v c = √ 2V rms cos(ωt + 2 3 π + δ a ) (3.5) and i a = √ 2I rms cos(ωt + δ a + φ) i b = √ 2I rms cos(ωt -2 3 π + δ a + φ) i c = √ 2I rms cos(ωt + 2 3 π + δ a + φ) (3.6)
where

• V rms and I rms are the root mean square values of the AC voltage and current, respectively.

• ω = 2πf is the angular frequency in rad/s and f is the frequency of the AC system.

• δ a is the phase of v a and φ is the phase angle between the voltage and the current.

Applying Clarke's transformation to v abc and i abc , we get

v α = 3 2 k c √ 2V rms cos(ωt + δ a ) v β = 3 2 k c √ 2V rms sin(ωt + δ a ) (3.7) and i α = 3 2 k c √ 2I rms cos(ωt + δ a + φ) i β = 3 2 k c √ 2I rms sin(ωt + δ a + φ) (3.8)
v d = 3 2 k c √ 2V rms cos(ωt + δ a -θ) v q = 3 2 k c √ 2V rms sin(ωt + δ a -θ) (3.11) and i d = 3 2 k c √ 2I rms cos(ωt + δ a + φ -θ) i q = 3 2 k c √ 2I rms sin(ωt + δ a + φ -θ) (3.12)
By setting the rotation angle θ to ωt, v dq become

v d = 3 2 k c √ 2V rms cos δ a v q = 3 2 k c √ 2V rms sin δ a (3.13)
which are two non time-varying constants.

Actually, the three-phase abc frame can be directly transformed into the twophase dq frame by

x dq0 = T dq0 x abc = k c    cos θ cos(θ -2π 3 ) cos(θ + 2π 3 ) -sin θ -sin(θ -2π 3 ) -sin(θ + 2π 3 ) √ 2 2 √ 2 2 √ 2 2    x abc (3.14)
and the inverse is given by

x abc = T -1 dq0 x dqo = 2 3 k -1 c    cos θ -sin θ √ 2 2 cos(θ -2π 3 ) -sin(θ -2π 3 ) √ 2 2 cos(θ + 2π 3 ) -sin(θ + 2π 3 ) √ 2 2    x dq0 (3.15) If k c = 2 3
, the active power in dq frame has the identical magnitude as in abc frame,

i.e. v a i a + v b i b + v c i c = v d i d + v q i q . If k c = 2 3
, as seen in (3.11), the amplitudes of the AC voltage in abc frame and in dq frame are the same. In the remainder of this 3.3. Three-phase synchronous reference frame phase-locked loop 31 thesis, k c is equal to 2 3 . In this case, the active power P and the reactive power Q in dq frame satisfy

P = 3 2 (v d i d + v q i q ) Q= 3 2 (v q i d -v d i q ) (3.16)
By comparing the expressions of the AC voltage in abc frame (3.5) and in dq frame (3.13), it can be seen that the application of Park's transformation reduces the complexity of the AC voltage equations, i.e. three time-varying quantities are transformed into two non time-varying quantities. This action favors the analysis of the electrical system and the development of control system. Therefore, an equivalent continuous-time averaged state-space model established in a synchronously rotating dq reference frame will be presented for multi-terminal VSC HVDC systems in the next section.

Three-phase synchronous reference frame phaselocked loop

As discussed in Section 2.4, a strong AC network itself has the ability to maintain the AC voltage at the PCC. Hence, when a VSC is connected to a strong AC system, it must be synchronized to the frequency of the strong AC system to avoid the risk of instability. This can be achieved with the help of three-phase synchronous reference frame phase-locked loop (SRF-PLL) technology, which is the most widely used technique to synthesize the phase and frequency in electrical systems [Chung 2000].

In the SRF-PLL, the three-phase voltages are transformed into dq reference frame by applying Park's transformation. Then, the rotation angle θ can be controlled via a feedback path. The SRF-PLL system consists of three major parts: the phase detecting device, loop lter, voltage-controlled oscillator (VOC), as illustrated in Fig. 3.1. The phase detecting device is used to generate a voltage, which is designed as a function of the phase dierence between the reference input θ and the PLL output θ, i.e. θ = θ -θ. The dynamics of the SRF-PLL system is governed by the loop lter which also enables to eliminate high frequency signals. The design of loop lter is very important, which should consider the trade-o between the lter performance and the system stability. Usually, a typical proportional-integral (PI) lter is chosen. By convention, the active power on the AC side and the current through the phase reactor are positive if they ow from the AC side to the DC side via the VSC. The DC current on the DC side is positive if it is injected into the DC grid from the converter. For example, if the active power P g 1 (or the DC current i cg 1 ) of SAC 1 is negative, then SAC 1 absorbs the power from the DC grid as a consumer.

Otherwise, it provides the power as a supplier.

Strong AC system side

The conguration of the i th SAC connection terminal is presented in Fig. 3.3 where the currents i g i ,abc ow through the phase reactor made up of an aggregated resistance R g i and an aggregated inductance L g i . Since the SAC enables to control its AC voltage at the PCC in case of disturbances, the SAC can be modeled by an ideal three-phase AC source with constant parameters. Consequently, the AC voltage of the i th SAC, i.e. v sg i ,abc , can be always maintained at xed frequency f g i and amplitude V g i ,rms .

According to Kirchho 's circuit laws, the dynamics of the currents i g i ,abc can be expressed by

L g i di g i ,abc dt = v sg i ,abc -v cg i ,abc -R g i i g i ,abc
(3.17)

where v cg i ,abc are the converter voltages. Due to the application of PWM, the relation between v cg i ,abc and the DC voltage u cg i can be provided by the amplitude modulation ratio m g i and the phase angle δ g i according to (3.1) and (3.2)

v cg i ,abc = 1 2 u cg i m g i ,abc (3.18)
where

m g i a = m g i sin(ω g i t + δ g i ) m g i b = m g i sin(ω g i t -2 3 π + δ g i ) m g i c = m g i sin(ω g i t + 2 3 π + δ g i ) (3.19)
with ω g i = 2πf g i .

Substituting (3.18) and (3.19) into (3.17) and then applying Park's transformation (3.14), the dynamics of i g i ,abc in the dq reference frame are described by

di g i d dt = - R g i L g i i g i d + ω g i i g i q + v sg i d L g i - u cg i 2L g i m g i d di g i q dt = - R g i L g i i g i q -ω g i i g i d + v sg i q L g i - u cg i 2L g i m g i q (3.20)
It can be seen that the modulation indices m g i ,abc are turned into m g i d and m g i q which satisfy

m g i = m 2 g i d + m 2 g i d ≤ 1 δ g i = arctan( m g i d m g i q ) (3.21)
For the sake of simplicity, the rotating angle θ g i of Park's transformation is chosen such that the d-axis is aligned to the phase a of AC voltage, which results in v sg i d = V g i ,rms and v sg i q = 0. As a result, from (3.16), the instantaneous active power P g i and reactive power Q g i at the PCC are given by:

P g i = 3 2 v sg i d i g i d Q g i = - 3 2 v sg i d i g i q (3.22)
34 Chapter 3. Dynamic model of multi-terminal VSC HVDC systems By virtue of the active power balance on both sides of the converter, the DC current i cg i can be then deduced as

i cg i = P g i u cg i (3.23)
while the losses of the phase reactor and the VSC are neglected [START_REF] Du | [END_REF], Lee 2000].

Weak AC system side

In this thesis work, we choose VSC HVDC linked wind farm as a study case of weak AC system connection [START_REF] Chaudhary | [END_REF]]. The schematic diagram of wind farm with VSC HVDC integration is illustrated in Fig. 3.4. Wind energy is converted into electrical power by wind turbine generators. Actually, the generation capacity of a single wind turbine is small. For example, the capacity of a large wind turbine is about 5MW. Hence, a wind farm (or wind power plant) usually consists of tens or hundreds of distributed wind turn generators which are connected to the collector bus. Since the voltage operating level of each single wind turbine generator is very low, typically 690V, step-up transformers are needed to increase the AC voltage level of wind turbine generator to the collector bus voltage level. Because system impedance can be viewed as aggregated impedance, which contains the impedances of generators, transformers, transmission lines, etc [Gavrilovic 1991], the use of the step-up transformers causes a high impedance leading to the wind farm with very low SCR. Hence, it is reasonable to consider the VSC connected wind farm as a good study example of weak AC network connection. A simplied schematic diagram of the j th VSC connected wind farm is shown in Fig. 3.5. An aggregated output is used to represent all individual wind turbine generator's outputs. In addition, we consider that every wind turbine is based on doubly-fed induction generator (DFIG) 1 [Xu 2007a]. Therefore, the wind farm can be modeled as a controlled source described by I w j ,abc [Pena 1996[START_REF] Lie | Multi-terminal DC transmission systems for connecting large oshore wind farms[END_REF].

Due to the use of step-up transformers, some high-order harmonics are produced.

1 One of the most common technology used for large turbines. Similar to the modeling of strong AC system connection, the dynamics of the current i w j ,abc owing into the phase reactor represented by R w j and L w j satisfy

L w j di w j ,abc dt = v sw j ,abc -v cw j ,abc -R w j i w j ,abc (3.24)
where the converter voltages v cw j ,abc are the functions of the DC voltage u cw j , the amplitude modulation ratio m w j and the phase angle δ w j according to (3.1) and

(3.2), expressed by v cw j ,abc = 1 2 u cw j m w j ,abc

(3.25)
where m w j a = m w j sin(ω

w j t + δ w j ) m w j b = m w j sin(ω w j t -2 3 π + δ w j ) m w j c = m w j sin(ω w j t + 2 3 π + δ w j ) (3.26)
with ω w j = 2πf w j . As discussed in Section 2.4, for the WAC connection, the VSC is responsible for assuring the AC voltage at the PCC to operate at xed frequency and amplitude. So there is no need to synchronize the rotor frequency with the general grid frequency f w j (usually 50 or 60 Hz), we can freely set ω w j . For example, we can employ an independent voltage controlled oscillator as an alternative without using PLL technique [Zakaria [START_REF] Moustafa | [END_REF].

The basic equation for the AC voltage v sw j ,abc at the PCC is given by C f w j dv sw j ,abc dt = I w j ,abc -i w j ,abc

(3.27)

After using Park's transformation, the AC side of the VSC connected wind farm in the dq frame is described by di

w j d dt = - R w j L w j i w j d + ω w j i w j q + v sw j d L w j - u cw j 2L w j m w j d di w j q dt = - R w j L w j i w j q -ω w j i w j d + v sw j q L w j - u cw j 2L w j m w j q (3.28)
and

dv sw j d dt = ω w j v sw j q + 1 C f w j (I w j d -i w j d ) dv sw j q dt = -ω w j v sw j d + 1 C f w j (I w j q -i w j q ) (3.29)
where the modulation indices m w j d and m w j q must satisfy m w j = m 2 w j d + m 2 w j d ≤ 1 δ w j = arctan( m w j q m w j q )

(3.30)

In addition, according to (3.16), the active and the reactive power at the PCC are given by

P w j = 3 2 (v sw j d i w j d + v sw j q i w j q ) Q w j = 3 2 (v sw j q i w j d -v sw j d i w j q ) (3.31)
Similarly as in Section 3.4.1, the DC current i cw j can be deduced as As illustrated in Fig. 3.6, the green arrow means that the branch circuit current is fed into the node, whereas the violet one represents the branch circuit current discharges from the node. An example of such DC grid is presented in Fig. 3.7, which consists of two SAC converter nodes (u cg 1,2 ), two WAC converter nodes (u cw 1,2 ), three intermediate nodes (u ct 1,2,3 ) and seven transmission branches

(i c k , k = 1, • • • , 7).
It can be seen that the second intermediate node is connected to three transmission branches (l 2 , l 5 and l 6 ). Moreover, it has one incoming current (i c 6 ) and two outgoing currents (i c 1 and i c 5 ).

Chapter 3. Dynamic model of multi-terminal VSC HVDC systems

According to basic circuit laws, the dynamic equations for the DC network are In order to better understand the properties of the DC network, we analyze its topology structure with the help of graph theory [Parthasarathy 1994]. In this thesis, we study a class of DC networks which can be represented by a weakly

C g i du cg i dt = i cg i + I + cg i -I - cg i C w j du cw j dt = i cw j + I + cw j -I - cw j C t h du ct h dt = I + ct h -I - ct h L c k di c k dt = U + c k -U - c k -R c k i c k
connected directed graph G without self-loops. This graph is labeled by G = (V, E). V = {V 1 , V 2 , V 3 } is the set of the vertices where V 1 = {v 1 , • • • , v N }, V 2 = {v N +1 , • • • , v N +M } and V 3 = {v N +M +1 , • • • , v N +M +P } correspond
H is of the form H =               l 1 l 2 l 3 l 4 l 5 l 6 l 7 u cg 1 1 0 0 0 0 0 0 u cg 2 0 1 0 0 0 0 0 u cw 1 0 0 -1 0 0 0 -1 u cw 2 0 0 0 -1 0 0 0 u ct 1 -1 0 0 0 1 0 1 u ct 2 0 -1 0 0 -1 1 0 u ct 3 0 0 1 1 0 -1 0               (3.35)
As a result, the dynamics of the DC grid (3.33) can be rewritten in matrix expression form

ż = Az + ϑ (3.36)
3.4. Multi-terminal HVDC system model 39 with the following denitions

z u c i c T ∈ R N +M +P +L u c u cg u cw u ct T u cg u cg 1 • • • u cg N T u cw u cw 1 • • • u cw M T u ct u ct 1 • • • u ct P T i c i c 1 • • • i c L T ϑ i cg 1 C g 1 • • • i cg N C g N i cw 1 C w 1 • • • i cw M C w M 0 (P +L) T (3.37)
The matrix A is of the form

A = 0 (N +M +P )×(N +M +P ) C -1 H -L -1 H T -L -1 R (3.38)
where C ∈ R (N +M +P )×(N +M +P ) and L, R ∈ R L×L are the capacitor, inductance and resistance matrices, respectively, which are given by

C = diag(C g 1 • • • C g N C w 1 • • • C w M C t 1 • • • C t P ) L = diag(L c 1 • • • L c L ) R = diag(R c 1 • • • R c L )
In general, the incidence matrix H of the weakly connected directed graph G without self-loops has the following features

• Since the directed graph G is weakly connected, the numbers of the vertices and the edges must satisfy L ≥ (N + M + P ) -1.

• Every edge (transmission line) can only connect two vertices (nodes) and hence, each column of H has only two non-zero elements, i.e. 1 and -1.

Based on these characteristics, we have the following results whose proofs are referred in [Parthasarathy 1994, Bondy 1976].

Lemma 3.4.1. The vectors H(1, :), H(2, :), ..., H(N + M + P, :)2 are linearly dependent. In addition, we have

N +M +P i=1 H(i, :) = 0 T L . The rank of H, i.e. rank(H), is equal to N + M + P -1.
Lemma 3.4.2. If any one row is removed from H, for example, the l th row, i.e.

H(l, :), we obtain a reduced incidence matrix denoted by R(H) l whose rank is still equal to N + M + P -1. It means that any (N + M + P -1) row vectors of H are linearly independent.

Domain of interest

It is worthwhile to keep in mind that taking into account physical considerations [START_REF][END_REF]], the feasible region of the state variables is not boundless [Stijn 2010, Haileselassie 2012a]. Therefore, in this thesis work, we restrict the state variables to some domains of interest. All DC voltage u cg i , u cw j and u ct h are considered on the domain D uc [u c,min , u c,max ] ∈ R where u c,min is positive. The current i g i ,dq and i w j ,dq are dened on D i g i ,dq ∈ R and D i w j ,dq ∈ R, respectively. All branch currents i c k are on the domain D ic . Consequently, the domains D (•) are dened as the domains of interest, which are bounded.

Conventional control methods

It is essential for any VSC HVDC system the ability to counteract disturbances with fast response, good transient and steady-state performances. This ability is strongly dependent on the employed control system and hence, a study of control system design for MTDC systems is necessary. Many researches have been devoted to the control design of VSC HVDC systems. In this section, we briey introduce two conventional control methods, direct and vector control methods, which are widely used and discussed.

Direct control method

Direct control strategy [Ohnishi 1991[START_REF] Noguchi | [END_REF]] is realized by measuring and comparing the controlled outputs to their references in such a way that the control variables, i.e. the phase angle δ and the amplitude modulation ratio m, can be directly obtained by means of PI control technique, and then sent to the pulse width modulator. Figure 3.8 gives the block diagrams of direct control method applied to the four control modes (as discussed in Section 2.4.2) of the i th VSC connected strong grid [START_REF] Sood | [END_REF]]. It can be seen that the errors denoted by (•) between the measured values and the references denoted by (•) o are sent to PI controllers as the inputs to generate the control variables of the converter. It is interesting to remark that the choice of these PI controllers is completely arbitrary or empirical, without vigorous mathematical proofs for their design. Nevertheless, the direct control method has the following advantages:

• The design concept is quite simple and the controller is easy to be implemented. The control variables are given by

δ g i = K g i ,p 1 (P o g i -P g i ) + K g i ,i 1 (P o g i -P g i ) m g i = K g i ,p 2 (Q o g i -Q g i ) + K g i ,i 2 (Q o g i -Q g i ) δ g i = K g i ,p 3 (u o cg i -u cg i ) + K g i ,i 3 (u o cg i -u cg i ) m g i = K g i ,p 4 (V o g i ,rms -V g i ,rms ) + K g i ,i 4 (V o g i ,rms -V g i ,rms ) (3.39)
where the control gains of the PI controllers, K g i ,p (•) and K g i ,i (•) , are positive.

It is shown that the formulas of δ g i and m g i are indeed expressed in a fairly simple way. • As seen in the control blocks, there are no other control loops. The direct control method provides a direct and eective way to control both active power (or DC voltage) and reactive power (or AC voltage) without using any other intermediate variable such as the AC current.

• Since the control variables are deduced directly by using the measurements, there is no need to use Clarke's or Park's transformation and hence, no PLL is required in the control algorithm. This avoids an unfavorable inuence on the performance of the system due to the non-linearity caused by PLL.

However, the most serious problem of the direct control method is that it has no ability to limit the current owing into the converter, which may damage the converter in case of over-current. In addition, because of cross-coupling between the control variables as seen in (2.2), the active power and the reactive power can not be controlled independently [START_REF] Sood | [END_REF]]. For example, consider that the i th SAC connection terminal operates in active and reactive control modes. At an instant t = t 1 , if P o g i is subjected to a step change, the control variable δ g i varies correspondingly. Because of the coupling relation described by (2.2), the reactive power is readily inuenced [Zakaria [START_REF] Moustafa | [END_REF].

Vector control method

Vector control method [START_REF] Lindberg | [END_REF][START_REF] Lindberg | [END_REF][START_REF] Blasko | [END_REF]] is the most widely used control approach in today's electric power system applications. This

τ = C g i u 2 rate 2S rate (3.40)
which is the ratio of the energy stored in the capacitor at the rated DC voltage u rate to the rated apparent power S rate of the converter. The time constant is usually chosen no less than 5 ms by taking into account practical constraints [Du 2003].

The main objective of the inner current loop is to design v cg i d and v cg i q such that i g i d and i g i q follow their respective reference trajectories i * g i d and i *

g i q .
As illustrated in Fig. 3.9, two PI controllers are involved and then, v cg i ,dq are given by

v cg i d = K g i ,p d (i g i d -i * g i d ) + K g i ,i d (i g i d -i * g i d ) + v sg i d + ω g i L g i i g i q v cg i q = K g i ,pq (i g i q -i * g i q ) + K g i ,iq (i g i q -i * g i q ) + v sg i q -ω g i L g i i g i d (3.41)
Depending on the application (or control mode), the reference i * g i d can be provided by a DC voltage outer loop or an active power outer loop as while the reference i * g i q are derived from a reactive outer loop or an AC voltage outer loop as

i * g i d = K g i ,p P (P o g i -P g i ) + K g i ,i P (P o g i -P g i ) i * g i d = K g i ,pu (u o cg i -u cg i ) + K g i ,iu (u o cg i -u cg i ) (3.42)
i * g i q = K g i ,p Q (Q o g i -Q g i ) + K g i ,i Q (Q o g i -Q g i ) i * g i q = K g i ,p V (V o g i ,rms -V g i ,rms ) + K g i ,i V (V o g i ,rms -V g i ,rms ) (3.43)
To obtain the best possible performance of the system, modulus optimum and symmetrical optimum techniques are usually applied to tune the control gains of the PI controllers [Bajracharya 2008].

Compared to the direct control method, the vector control method has superior advantages:

• The active power and the reactive power can be controlled independently by using the inner current loop with feed-forward compensations to remove the coupling terms.

• This method inherently has the ability to protect the converter against overcurrent in case of disturbances due to the inner current control loop.

• It provides faster response and better performance than the direct control method in case of variations in references or some other disturbances.

The vector control structure is basically composed by several standard PI controllers, whose implementation and design are simple. Nevertheless, the use of such pure PI controllers results in some limitations [START_REF] Dannehl | [END_REF], Durrant 2003]. On the one hand, the performance of the vector control scheme is very sensible to the 44 Chapter 3. Dynamic model of multi-terminal VSC HVDC systems system uncertainties. On the other hand, in order to apply Park's transformation, a PLL is needed to provide a synchronous reference frame. The dynamics of the PLL and AC lters increase the diculty in the VSC control when the SCR of the connected AC system decreases.

Chapter conclusions

In this chapter, we choose the averaged modeling approach to develop the dynamical model for a multi-terminal VSC HVDC system. The modeling work mainly focuses on three parts, i.e. the VSC connected strong AC networks, the VSC connected wind farms and the DC network. According to the features of the VSC connected AC systems, we have dierent considerations:

• For the VSC connected strong AC networks, the SRF-PLL is usually required to detect the phase and frequency of the electrical circuit such that the VSC can be synchronized to its connected strong AC system whereas there is no need for the VSC connected weak AC network.

• For a strong AC system, we consider that its AC voltage at the PCC can be kept by TSO at xed frequency and amplitude. This is contrary to the case of a weak AC system, which with high impedance is very sensible to any disturbances (for example, the change of power ow), it is necessary in that case to consider the dynamics of the weak AC system's AC voltage at the PCC.

For the sake of convenience, Park's transformation is applied to transform the system from the abc frame into the dq reference frame, which greatly simplies the expression of the dynamic equations. For the DC network, we map it to a weakly connected directed graph and then use a incidence matrix to describe its topology.

It is much easier to analyze the DC grid in matrix form since there exist quite a lot of results on matrix and graph studies.

Finally, a detailed averaged state-space model for the multi-terminal VSC HVDC system described by (3.20), (3.28), (3.29) and (3.36) is established. We summarize this model as follows

• The dimension of the full scale system is 3N + 5M + P + L.

• The state variables are i g i ,dq , i w j ,dq , v sw j ,dq , u cg i , u cw j , u ct h and i c k for i ∈ N , j ∈ M, h ∈ P and k ∈ L 3 .
• The control inputs are the modulation indices, i.e. m g i ,dq and m w j ,dq for i ∈ N and j ∈ M.

• The external parameters I w j ,dq , for j ∈ M, are considered constant.

Chapter 4

Control methods based on nonlinear control design tools Although the vector control method occupies a predominant position in the eld of control design for VSC HVDC systems, the limitations of this approach also draw our attention [START_REF] Dannehl | [END_REF]]. In particular, in [START_REF] Dannehl | [END_REF]], it indicates that the traditional vector current control is very sensitive to system uncertainties.

In [Durrant 2003], a detailed analysis of a VSC connected to a weak AC system is carried out. It is shown that the dynamics of PLL and AC lter increase the diculty in the VSC control when the SCR of the connected AC system decreases.

In order to improve the performance of VSC HVDC systems, developing new control structures has been a very popular research topic.

In this chapter, we propose new control structures by means of dierent nonlinear control design tools for VSC HVDC systems.

Feedback linearization control

Feedback linearization is well-known in the eld of nonlinear control design. The principle of this technique is to transform the nonlinear system fully or partly to a linear one and thereafter, linear control, robust control or some other control techniques can be applied. As presented in [Lee 2000[START_REF] Kim | [END_REF]], feedback linearization is applied to three-phase converters. However, only a single three-phase converter station is considered in their study case. Reference [START_REF] Moharana | [END_REF]] proposes a nonlinear controller based on input-output feedback linearizaiton and sliding mode control. However, in [START_REF] Moharana | [END_REF]], the stability of the zero dynamics of the system is not analyzed, which determines whether the feedback linearization is applicable for the control design.

In this section, according to the control objectives of HVDC systems, we develop dierent nonlinear controllers by making full use of feedback linearization where the corresponding zero dynamics are also analyzed.

Before designing the exact control structures to VSC HVDC systems, we give a brief introduction to feedback linearizaiton for multi-input and multi-output (MIMO) nonlinear systems [Khalil 1996, Isidori 1995[START_REF] Slotine | [END_REF], Hedrick 2005].

Theoretical results

Consider a class of MIMO systems

ẋ= f (x) + j=m j=1 g j (x)u j y y 1 • • • y m T = h 1 (x) • • • h m (x) T (4.1)
where the control input vector u = u i • • • u m ∈ R m has the same dimension as the output vector y. The functions f , g j and h j , i = 1, • • • , m, are suciently smooth in an open set D ⊂ R m . The mappings f : D → R n and g i : D → R are the vector elds on D. The derivative of the i th output y i is given by

ẏi = L f h i (x) + j=m j=1 L g j h i (x)u j (4.2)
where

L f h i (x) ∂h i ∂x f (x) L g j h i (x) ∂h i ∂x g j (x) (4.3)
are the Lie derivatives of h i (x) with respect to f and g i . If all L g j h i (x) = 0, we continue to dierentiate y i until some u j explicitly appears in the γ th i derivative of y i as follows y

(γ i ) i = L γ i f h i (x) + j=m j=1 L g j (L γ i -1 f h i )u j (4.4)
where L g j (L γ i -1

f h i ) = ∂(L γ i -1 f h i ) ∂x g j = 0 for some j ∈ {1, , • • • , m}. In this case, γ i
is called the relative degree (see Denition 13.2 in [Khalil 1996]) of the i th output y i = h i (x). After performing the above procedure for each output, we obtain m equations in a similar form as (4.4), which can be expressed in matrix form

y γ = l γ + Eu (4.5)
where the decoupling matrix E ∈ R m×m and the vectors y γ , l γ and u are given by

E(x)=    L g 1 (L γ 1 -1 f h 1 ) • • • L gm (L γ 1 -1 f h 1 ) • • • • • • • • • L g 1 (L γm-1 f h m ) • • • L gm (L γm-1 f h m )    y γ = y (γ 1 ) 1 • • • y (γm) m T l γ = L γ 1 f h 1 • • • L γm f h m T u= u 1 • • • u m T (4.6)
If the decoupling matrix E is non-singular in D, then the control vector u can be designed as

u = E -1 (v -l γ ) (4.7) where v = v 1 • • • v m T is the additional input vector. Substituting (4.7) into (4.5) yields    y (γ 1 ) 1 • • • y (γm) m    =   v 1 • • • v m   (4.8)
which results in a decoupled set of equations. The additional input v can be designed using any linear method or other techniques. For example, if the control objective is to make y follow the reference

y d = y d1 • • • y dm T , then v can be designed as   v 1 • • • v m   =    -c 1,0 (y 1 -y d1 ) -• • • -c 1,γ 1 -1 (y (γ 1 -1) 1 -y (γ 1 -1) d1 ) • • • -c m,0 (y m -y dm ) -• • • -c m,γm-1 (y (γm-1) m -y (γm-1) dm )    (4.9)
where the coecients c (•) are chosen such that for all j ∈ {1, • • • , m}

C j =        0 1 0 • • • 0 0 0 1 • • • 0 • • • • • • • • • • • • • • • 0 0 0 • • • 1 -c j,0 -c j,1 -c j,2 • • • -c j,γ j -1        ∈ R γ j ×γ j (4.10)
are Hurwitz. In addition, with the integrated tracking error e e

1 • • • e m T = y -y d , v is then designed as   v 1 • • • v m   =    -c 1,0 (y 1 -y d1 ) -• • • -c 1,γ 1 -1 (y (γ 1 -1) 1 -y (γ 1 -1) d1 ) -k I1 e I1 • • • -c m,0 (y m -y dm ) -• • • -c m,γm-1 (y (γm-1) m -y (γm-1) dm ) -k Im e Im    (4.11)
where e Ij satises ėIj = e j = y j -y dj (4.12)

Again, the control gains must be chosen such that for all j ∈ {1, • • • , m}

C j =        0 1 0 • • • 0 0 0 1 • • • 0 • • • • • • • • • • • • • • • 0 0 0 • • • 1 -k Ij -c j,0 -c j,1 • • • -c j,γ j -1        ∈ R (γ j +1)×(γ j +1) (4.13)
are Hurwitz. The term (-k Ij e Ij ) is used to eliminate the tracking error so as to improve the robustness of the system.

If the relative degree of the whole system (4.1)

γ = m j=1 γ j (4.14)
is smaller than n, there exist internal dynamics of order n-γ which are unobservable.

We perform the change of variables

z = T (x)   η --- ξ          φ 1 (x) • • • φ n-r (x) - - - ξ        (4.15)
and ξ is taken as

ξ   ξ 1 • • • ξ m   , ξ j    y j • • • y (γ j -1) j    (4.16) where φ k , k = 1, • • • , n -γ, are chosen to ensure that the transformation T is a dieomorphism on D. In the new variable (η, ξ), the system (4.1) becomes η= Ψ 1 (η, ξ) + Ψ 2 (η, ξ)u ξj = A j ξ j + B j v j (4.17)
where

A j =        0 1 0 • • • 0 0 0 1 • • • 0 • • • • • • • • • • • • • • • 0 0 0 • • • 1 0 0 0 • • • 0        ∈ R γ j ×γ j , B j =        0 0 • • • 0 1        ∈ R γ j (4.18)
When the output y is identically equal to the reference y d , the system (4.17) is According to (4.7), ū and ξ are given by

degenerated into η= Ψ 1 (η, ξ) + Ψ 2 (η, ξ)ū (4.19)
ū = -E -1 l γ , ξ     ξ1 ξ2 • • • ξm     , ξj =     y dj 0 • • • 0     (4.20)
Actually, the autonomous system (4.19) is called the zero dynamics. Since ξ can be stabilized at its equilibrium ξ by designing a suitable expression for the additional input v, the stability of the system (4.17) totally depends on the stability of the zero dynamics (4.19). The system (4.1) is said to be minimum phase if the equilibrium point of the zero dynamics is asymptotically stable. Moreover, the input-output linearization is applicable to the system (4.1) when it is minimum phase.

Application to a VSC HVDC link connected to a resistive load 4.1.2.1 Control design

A simple VSC HVDC link is considered as shown in Fig. 4.1 where a resistive load R L is connected to the i th SAC terminal. As presented in Section 3.4.1, the system can be modeled by

di g i d dt = - R g i L g i i g i d + ω g i i g i q + v sg i d L g i - u cg i 2L g i m g i d di g i q dt = - R g i L g i i lq -ω g i i g i d + v sg i q L g i - u cg i 2L g i m g i q du cg i dt = 1 C g i (i cg i -i L ) (4.21)
According to (3.23), i cg i is deduced as

i cg i = 3 2 v sg i d i g i d + v sg i q i g i q u cg i (4.22)
and i L is given by i

L = u cg i R L (4.23)
52 Chapter 4. Control methods based on nonlinear control design tools

Since there are two control inputs m g i ,dq , it is possible to control two outputs. To ensure the normal operation of the system, the DC voltage transmission level must be always kept within its acceptable range. As a results, u cg i is chosen as one output, which is required to track the reference u o cg i . For the purpose of good power quality, the reactive power Q g i should be well-behaved. From (3.22), it is clear that i g i d and i g i q regulate the active and reactive power, respectively. Consequently, i g i q is chosen as the other output, which is expected to follow the reference i o g i q given by i o g i q = -2 3

Q o g i v sg i d (4.24)
where Q o g i is the desired reactive power. For example, in order to get the unitary power factor, Q o g i is usually set to zero.

By dening x 1 x 2 x 3 i g i d i g i q u cg i , the system (4.21) can be rewritten as

ẋ = f x (x) + g d m g i d + g q m g i q (4.25)
where f (x), g d and g q are given by

f x =     - Rg i Lg i i g i d + ω g i i g i q + v sg i d Lg i - Rg i Lg i i lq -ω g i i g i d + vsg i q Lg i 1 Cg i ( 3 2 v sg i d i g i d +vsg i q ig i q ucg i - ucg i R L )     (4.26)
and

g d =    - ucg i 2Lg i 0 0    , g q =    0 - ucg i 2Lg i 0    (4.27)
The two outputs in this case are dened as

y y 1 y 2 T = i g i q u cg i T (4.28)
which are needed to be regulated at

y o y o 1 y o 2 T = i o g i q u o cg i T .
It is evident that γ 1 and γ 2 , the relative degrees of u cg i and i g i q , are equal to 2 and 1, respectively. The relative degree of the whole system is 3 which is equal to the dimension of the system (4.21). This means that the system is feedback linearizable and there is no unobservable dynamics. The decoupling matrix E 1 is given by

E 1 = L g d (i g i q ) L gq (i g i q ) L g d L fx (u cg i ) L gq L fx (u cg i ) (4.29) where L g d (i g i q )= 0 L gq (i g i q )= - ucg i 2Lg i L g d L fx (u cg i )= -1 Cg i 3 4 v sg i d Lg i L gq L fx (u cg i )= -1 Cg i 3 4 vsg i q Lg i (4.30)

Feedback linearization control 53

Since we only consider the case when u cg i is positive, E is invertible in the domain of interest. Therefore, a nonlinear feedback control can be developed as

m g i d m g i q = E -1 1 v 1 -L fx (i g i q ) v 2 -L 2 fx (u cg i ) (4.31)
where

L fx (i g i q )= - Rg i Lg i i g i d + ω g i i g i q + v sg i d Lg i L 2 fx (u cg i )= 1 C g i { 3 2 v sg i d u cg i (- R g i L g i i g i d + ω g i i g i q + v sg i d L g i ) + 3 2 v sg i q u cg i (- R g i L g i i g i q -ω g i i g i d + v sg i q L g i )} -[ 3 2 v sg i d i g i d + v sg i q i g i q u 2 cg i + 1 R load ] du cg i dt } (4.32)
The additional control inputs v 1 and v 2 are yet to be designed.

Consider the transformation

z   z 1 z 2 z 3   = T x =   i g i q u cg i ucg i   (4.33)
which is a dieomorphism on the domains of interest and then, the original system (4.21) can be converted into the following expression

ż1 = v 1 ż2 = z 3 ż3 = v 2 (4.34)
We dene the tracking errors as

e 1 = z 1 -i o g i q e 2 = z 2 -u o cg i (4.35)
To achieve zero steady-state errors, the integrated tracking errors

ėI1 = e 1 ėI2 = e 2 (4.36)
are considered, which result in an augmented system described by

ėI1 = e 1 ė1 = v 1 ėI2 = e 2 ė2 = z 3 ż3 = v 2 (4.37)
Applying linear control technique, v 1 and v 2 are designed as The control gains are chosen such that matrices C 1 and C 2 given by

v 1 = -k I1 e I1 -c 1,0 e 1 v 2 = -k I2 e I2 -c 2,0 e 2 -c 2,1 ż3
C 1 = 0 1 -k I1 -c 1,0 , C 2 =   0 1 0 0 0 1 -k I2 -c 2,0 -c 2,1   (4.39)
are Hurwitz. Now the controller composed by (4.31) and (4.38) has been developed.

Simulation studies

To evaluate the performance of the nonlinear controller, numerical simulations are carried out by using SimPowerSystems toolbox of MATLAB/Simulink as depicted in Fig. 4.2. Detailed parameters of the VSC link are provided in Table 4.1. The DC voltage reference u o cg i is set to 150 V and to get the unitary power factor, the reference of q-axis i o g i q is equal to 0 A. At the start of the simulation, only Switch 1 is closed and the other two switches are open and hence we have R L = R L1 . The VSC HVDC link initially operates in a steady-state condition given by Table 4.2. Figures 4.3-4.5 present the tracking performance of the system that responds to the step change imposed on the DC voltage reference u o cg i while i o g i q is always set to zero. At t = 0.8, u o cg i is changed from 150 V to 151.5 V and then, at t = 1.8

L g i 2.1 mH Resistance R g i 0.142 Ω R L 1,2,3
150 Ω Table 4.2: Initial values of the system variables.

Variables Values

R L 150 Ω u cg i 150 V i g i q 0 A i L 1 A i g i d 2.46 A
controllers have similar performance, i.e. they make u cg i track its reference u o cg i with fast response and good transient performance. In addition, i g i q as displayed in Fig. 4.4 is always regulated at zero irrespective of the change in u o cg i . As illustrated in Fig. 4.5, every time u cg i achieves a new steady state, i L1 also arrives at a corresponding steady state since in steady-state condition, i L1 and u cg i satisfy To test the robustness of both controllers in case of the variations in R L and the step change in i o g i q , a sequence of events is applied to the system, as listed in Table 4.3. The simulation results are illustrated in where the performances of the conventional PI and feedback nonlinear controller are compared. Due to the operations of Switch 2 and Switch 3, R L is changed from 150 Ω to 75 Ω at t = 1 s, and then it is reduced to 50 Ω at t = 1.5 s.

u cg i = R L i L1 .
As shown in Fig. 4.6, the trajectories of u cg i are always kept within the safe operating domain [0.9u o cg i , 1.1u o cg i ] under both controllers, but the response of u cg i has better performance using feedback nonlinear controller. It can be seen that u cg i has a faster response and lower crest value under the feedback nonlinear controller (see the red curve). Although both approaches are able to regulate i g i q at zero before t = 2 s, for the PI controller, i g i q is disturbed with remarkable transient because of the unexpected change in R L (see the blue curves in . At t = 2 s, i o g i q is set to 6.53 A. As presented in Fig. 4.7, both trajectories of i g i q start to increase and then quickly converge to the new reference. During the q-axis current's increasing period, we nd that signicant overshoots appear in the trajectories of u cg i , i L1 , i L2 and i L3 under the PI controller (see the blue curve in Fig. 4.6) whereas there is little or negligible inuence on the responses of u cg i , i L1 , i L2 and i L3 under the feedback nonlinear controller. This phenomenon fully shows that the PI controller can not keep the DC voltage unaected in case of the change of i g i q while the feedback nonlinear controller provides a much better decoupling characteristics of the DC voltage and q-axis current control. Moreover, Feedback nonlinear controller. According to Section 3, the full state-space model of the system as depicted in 62 Chapter 4. Control methods based on nonlinear control design tools Fig. 4.12 is expressed as

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 1 1.2 1.4 1.6 1.8 2 Time (s) i L2 (A) (a) (b)
di g 1 d dt = - R g 1 L g 1 i g 1 d + ω g 1 i g 1 q + v sg 1 d L g 1 - u cg 1 2L g 1 m g 1 d di g 1 q dt = - R g 1 L g 1 i g 1 q -ω g 1 i g 1 d + v sg 1 q L g 1 - u cg 1 2L g 1 m g 1 q du cg 1 dt = 1 C g 1 ( 3 2 v sg 1 d i g 1 d + v sg 1 q i g 1 q u cg 1 + i c 1 ) di c 1 dt = 1 L c 1 (-R c 1 i c 1 + u cw 1 -u cg 1 )
du cw 1 dt = 1 C w 1 ( 3 2 v sw 1 d i w 1 d + v sw 1 q i w 1 q u cw 1 -i c 1 ) dv sw 1 d dt = ω w 1 v sw 1 q + 1 C f w 1 (I w 1 d -i w 1 d ) dv sw 1 q dt = -ω w 1 v sw 1 d + 1 C f w 1 (I w 1 q -i w 1 q ) di w 1 d dt = - R w 1 L w 1 i w 1 d + ω w 1 i w 1 q + v sw 1 d L w 1 - u cw 1 2L w 1 m w 1 d di w 1 q dt = - R w 1 L w 1 i w 1 q -ω w 1 i w 1 d + v sw 1 q L w 1 - u cw 1 2L w 1 m w 1 q (4.40)
For the SAC terminal, the converter is used to maintain u cg 1 at a set point u o cg 1 . In addition, i g 1 q should be regulated at -2 3

Q o g 1 v sg 1 d
in such a way that the reactive power Q g 1 can be kept at its set point Q o g 1 . Hence, the SAC converter station operates in DC voltage and reactive power control modes. For the WAC terminal, the converter is dedicated to control the AC voltage v sw 1 ,dq at the set points v o sw 1 ,dq so that the WAC can operate at constant magnitude V sw j ,rms satisfying

V sw 1 ,rms = (v o sw 1 d ) 2 + (v o sw 1 q ) 2 (4.41)
Meanwhile, the power P w 1 generated by the wind farm can be totally transmitted to the SAC via the DC grid [START_REF] Lie | Multi-terminal DC transmission systems for connecting large oshore wind farms[END_REF]]. Therefore, the controlled output vector is dened as y y 1 y 2 y 3 y 4

T = i g 1 q u cg 1 v sw 1 d v sw 1 q T
and then, the control structure is designed to make the output vector y track the prescribed point

y o = i o g 1 q u o cg 1 v o sw 1 d v o sw 1 q T . By dening x i g 1 d i g 1 q u cg 1 i c 1 u cw 1 v sw 1 d v sw 1 q i w 1 d i w 1 q T
, the system (4.40) can be formalized as

ẋ = f (x) + g 1 m g 1 d + +g 2 m g 1 q + g 3 m w 1 d + g 4 m w 1 q (4.42)
with the trivial expressions for f and g 1,2,3,4 . Calculating the derivatives of the 4.1. Feedback linearization control 63 output variables, we obtain y

(1)

1 = L f (i g 1 q ) +L g 1 (i g 1 q )m g 1 d + L g 2 (i g 1 q )m g 1 q + L g 3 (i g 1 q )m w 1 d + L g 3 (i g 1 q )m w 1 q y (2) 2 = L 2 f (u cg 1 ) + L g 1 L f (u cg 1 )m g 1 d + L g 2 L f (u cg 1 )m g 1 q +L g 3 L f (u cg 1 )m w 1 d + L g 4 L f (u cg 1 )m w 1 q y (2) 3 = L 2 f (v sw 1 d ) + L g 1 L f (v sw 1 d )m g 1 d + L g 2 L f (v sw 1 d )m g 1 q +L g 3 L f (v sw 1 d )m w 1 d + L g 4 L f (v sw 1 d )m w 1 q y (2) 4 = L 2 f (v sw 1 q ) + L g 1 L f (v sw 1 q )m g 1 d + L g 2 L f (v sw 1 q )m g 1 q +L g 3 L f (v sw 1 q )m w 1 d + L g 4 L f (v sw 1 q )m w 1 q (4.43) where L f (i g 1 q )= - R g 1 L g 1 i g 1 q -ω g 1 i g 1 d + v sg 1 q L g 1 L g 2 (i g 1 q )= - u cg 1 2L g 1 , L g 1 (i g 1 q ) = L g 3 (i g 1 q ) = L g 4 (i g 1 q ) = 0 L 2 f (u cg 1 )= 1 C g 1 { 3 2 v sg 1 d u cg 1 (- R g 1 L g 1 i g 1 d + ω g 1 i g 1 q + v sg 1 d L g 1 ) + 3 2 v sg 1 q u cg 1 (- R g 1 L g 1 i g 1 q -ω g 1 i g 1 d + v sg 1 q L g 1 )} -( 3 2 v sg 1 d i g 1 d + v sg 1 q i g 1 q u 2 cg 1 ) du cg 1 dt + di c 1 dt } L g 1 L f (u cg 1 )= - 1 C g 1 3 4 v sg 1 d L g 1 , L g 2 L f (u cg 1 ) = - 1 C g 1 3 4 v sg 1 q L g 1 L g 3 L f (u cg 1 )= L g 4 L f (u cg 1 ) = 0 L 2 f (v sw 1 d )= ω w 1 vsw 1 q + İw 1 d C f w 1 - 1 C f w 1 (ω w 1 i w 1 q - R w 1 L w 1 i w 1 d + v sw 1 d L w 1 ) L g 1 L f (v sw 1 d )= L g 2 L f (v sw 1 d ) = L g 4 L f (v sw 1 d ) = 0 L g 3 L f (v sw 1 d )= u cw 1 2L w 1 C f w 1 L 2 f (v sw 1 q )= İw 1 q C f w 1 -ω w 1 vsw 1 d - 1 C f w 1 ( v sw 1 q L w 1 -ω w 1 i w 1 d - R w 1 L w 1 i w 1 q ) L g 1 L f (v sw 1 d )= L g 2 L f (v sw 1 d ) = L g 3 L f (v sw 1 d ) = 0 L g 4 L f (v sw 1 d )= u cw 1 2L w 1 C f w 1 (4.44)
Therefore, the relative degrees of y 1 , y 2 , y 3 and y 4 are 1, 2, 2, 2, respectively. The system (4.40) has relative degree 7 in R 9 , which is smaller than 9, and hence, we have internal dynamics of order 2. To characterize the zero dynamics of the system, we restrict x to 45) This process leads to

Z * = {x ∈ R 9 | i g 1 q ≡ i o g 1 q , u cg 1 ≡ u o cg 1 , v sw 1 d ≡ v o sw 1 d , v sw 1 q ≡ v o sw 1 q } (4.
di c 1 dt = 1 L c 1 (-R c 1 i c 1 + u cw 1 -u o cg 1 ) du cw 1 dt = 1 C w 1 ( 3 2 v o sw 1 d I w 1 d + v o sw 1 q I w 1 q u cw 1 -i c 1 ) (4.
J =     - R c 1 L c 1 1 L c 1 - 1 C w 1 - 3 2 v o sw 1 d I w 1 d + v o sw 1 q I w 1 q C w 1 ū2 cw 1     (4.47)
where īc 1 ūcw 1 T can be obtained by solving the following equations

0= -R c 1 īc 1 + ūcw 1 -ūo cg 1 0= 3 2 v o sw 1 d I w 1 d + v o sw 1 q I w 1 q ūcw 1 -īc 1 (4.48)
The characteristic polynomial p(t) of the matrix J is given by

p(t) det(tI -J) = t 2 + α 1 t + α 0 (4.49)
where

α 1 = R c 1 L c 1 + 3 2 v o sw 1 d I w 1 d + v o sw 1 q I w 1 q C w 1 ū2 cw 1 α 0 = 3 2 v o sw 1 d I w 1 d + v o sw 1 q I w 1 q C w 1 ū2 cw 1 • R c 1 L c 1 + 1 C w 1 L c 1 (4.50) Since 3 2 (v o sw 1 d I w 1 d + v o sw 1 q I w 1 q
), the active power generated by the wind farm, is non-negative, the coecients α 0 and α 1 are positive. Therefore, the real part of all eigenvalues of the matrix J is negative. The zero dynamics have an asymptotically stable equilibrium point at īc 1 ūcw 1 T , which shows that the system described by (4.40) is minimum phase.

Based on the above mentioned equations, the decoupling matrix E 2 is given by

E 2 =     L g 1 (i g 1 q ) L g 2 (i g 1 q ) L g 3 (i g 1 q ) L g 4 (i g 1 q ) L g 1 L f (u cg 1 ) L g 2 L f (u cg 1 ) L g 3 L f (u cg 1 ) L g 4 L f (u cg 1 ) L g 1 L f (v sw 1 d ) L g 2 L f (v sw 1 d ) L g 3 L f (v sw 1 d ) L g 4 L f (v sw 1 d ) L g 1 L f (v sw 1 q ) L g 2 L f (v sw 1 q ) L g 3 L f (v sw 1 q ) L g 4 L f (v sw 1 q )     (4.51)
which is non-singular for positive u cw 1 and u cg 1 . According to (4.31), the control variables are given by

    m g 1 d m g 1 q m w 1 d m w 1 q     = E -1 2      v 1 -L f (i g i q ) v 2 -L 2 f (u cg i ) v 3 -L 2 f (v sw 1 d ) v 4 -L 2 f (v sw 1 q )      (4.52)
Applying the (4.11) to the additional inputs v 1,2,3,4 leads to

v 1 = -k I1 e I1 -c 1,0 (y 1 -i o g 1 d ) v 2 = -k I2 e I2 -c 2,0 (y 2 -u o cg 1 ) -c 2,1 ( ẏ2 -uo cg 1 ) v 3 = -k I3 e I3 -c 3,0 (y 3 -v o sw 1 d ) -c 3,1 ( ẏ3 -vo sw 1 d ) v 4 = -k I4 e I4 -c 4,0 (y 4 -v o sw 1 q ) -c 4,1 ( ẏ4 -vo sw 1 q ) (4.53)
4.1. Feedback linearization control 65 with the integrated tracking errors

ėI1 = y 1 -i o g 1 d ėI2 = y 2 -u o cg 1 ėI3 = y 3 -v o sw 1 d ėI4 = y 4 -v o sw 1 q (4.54)
The control gains must be chosen such that the matrices C j , j = 1, 2, 3, 4, are Hurwitz where

C 1 = 0 1 -k I1 -c 1,0 , C 2 =   0 1 0 0 0 1 -k I2 -c 2,0 -c 2,1   C 3 =   0 1 0 0 0 1 -k I3 -c 3,0 -c 3,1   , C 4 =   0 1 0 0 0 1 -k I4 -c 4,0 -c 4,1   (4.55)

Simulation studies

Simulation studies of the VSC HVDC link depicted in Fig. 4.12 are carried out where the values of the system parameters are provided in Table 4.4. The operation of this two-terminal HVDC system is arranged as follows:

• The DC voltage of the SAC terminal is required to be maintained at u cg 1 = u o cg 1 = 150 V and the reactive power

Q g 1 is regulated at Q g 1 = Q o g 1 = 0 Var.
• The frequency of the WAC's AC voltage at the PCC must be kept at 50 Hz. In addition, V sw 1 ,rms is initially set to 40.82 V. At t = 1 s and t = 1.5 s, V sw 1 ,rms is changed to 60.23 V and 63.59 V respectively. According to (4.41), this can be achieved by varying the values of v o sw 1 .dq as listed in Table 4.5.

In order to evaluate the performance of the nonlinear controller (4.52), two control strategies are tested by comparison as presented in Table 4.6. The HVDC system starts from a steady-state condition where The arrangement of VSC operation for the MTDC system is set as follows:

u cg 1 = u o cg 1 = 150 V, i g 1 q = i o g 1 q = 0 A, v sw 1 d = v o sw 1 d = 40.82 V and v sw 1 q = v o sw 1 q = 0 V.
• The 1 st SAC converter station chosen as the master terminal is used to maintain the DC voltage at the constant level u o cg 1 while the remaining SAC con- verter stations operate in active control mode to control P gρ at P o gρ1 . Additionally, all reactive power Q g i should be kept at their respective references Q o g i .

• Each WAC converter station must ensure that the magnitude and the frequency of the AC voltage at the PCC, V sw,rms and f w j , are kept constant. This can be fullled by regulating v sw j ,dq at their references v o sw j ,dq .

As mentioned in Section 4.1.3,

Q g i can be kept at Q o g i by regulating i g i q at i o g i q = - 2 3 Q o g i v sw j d . Likewise, we make i g i d track the reference i o g i d = 2 3 P o g i v sw j d
for the purpose of controlling P g i at P o g i . According to the above arrangement, it is natural to choose the output vector as

y i g 1 q u cg 1 i gρd i gρq v sw j d v sw j q T ∈ R 2N +2M (4.56)
Obviously, the reference vector is set to

y o i o g 1 q u o cg 1 i o gρd i o gρq v o sw j d v o sw j q T ∈ R 2N +2M (4.57)
Through simple calculations, it is not dicult to get that the relative degrees of u cg 1 , i g 1 q , i gρd , i gρq , v sw j d and v sw j q are 2, 1, 1, 1, 2, 2, respectively, and hence, the order of the internal dynamics is N -1 + M + P + L.

Before designing the exact control algorithm, we need the following feasibility assumption.

Assumption 4.1.1.

0= - R g i L g i īg i d + ω g i īg i q + v sg i d L g i - ūcg i 2L g i mg i d 0= - R g i L g i īg i q -ω g i īg i d + v sg i q L g i - ūcg i 2L g i mg i q 0= - R w j L w j īw j d + ω w j īw j q + vsw j d L w j - ūcw j 2L w j mw j d 0= - R w j L w j īw j q -ω w j īw j d + vsw j q L w j - ūcw j 2L w j mw j q 0= ω w j vsw j q + 1 C f w j (I w j d -īw j d ) 0= -ω w j vsw j d + 1 C f w j (I w j q -īw j q ) 0= Az + θ (4.58)
where

ūcg 1 = u o cg 1 , īg 1 q = - 2Q o g 1 3v sg 1 d , īgρd = 2P o gρ 3v sg 1 d , īgρq = - 2Q o gρ 3v sg 1 d and vsw j ,dq = v o sw j ,dq .
By applying the theoretical results in Section 4.1.1, the following nonlinear feedback control algorithm

m g 1 d m g 1 q =    0 - u cg 1 2L g 1 - 3v sg 1 d 4C g 1 L g 1 - 3v sg 1 q 4C g 1 L g 1    -1 v g 1 q -L f (i g 1 q ) v cg 1 -L 2 f (u cg 1 ) m gρd m gρq =    - u cgρ 2L gρ 0 0 - u cgρ 2L gρ    -1 v gρd -L f (i gρd ) v gρq -L f (i gρq ) m w j d m w j q =    u cw j 2L w j C f w j 0 0 u cw j 2L w j C f w j    -1 v w j d -L 2 f (v sw j d ) v w j q -L 2 f (v sw j q ) (4.59)
2 Unless otherwise stated, the notation

L f (i g 1 q )= - R g 1 L g 1 i g 1 q -ω g 1 i g 1 d + v sg 1 q L g 1 L 2 f (u cg 1 )= 1 C g 1 { 3 2 v sg 1 d u cg 1 (- R g 1 L g 1 i g 1 d + ω g 1 i g 1 q + v sg 1 d L g 1 ) + 3 2 v sg 1 q u cg 1 (- R g 1 L g 1 i g 1 q -ω g 1 i g 1 d + v sg 1 q L g 1 )} -( 3 2 v sg 1 d i g 1 d + v sg 1 q i g 1 q u 2 cg 1 ) du cg 1 dt + H(1, :) di c dt } L f (i gρd )= - R gρ L gρ i gρd + ω gρ i gρq + v sgρd L gρ L f (i gρq )= - R gρ L gρ i gρq -ω gρ i gρd + v sgρq L gρ L 2 f (v sw j d )= ω w j vsw j q + İw j d C f w j - 1 C f w j (ω w j i w j q - R w j L w j i w j d + v sw j d L w j ) L 2 f (v sw j q )= İw j q C f w j -ω w j vsw j d - 1 C f w j ( v sw j q L w j -ω w j i w j d - R w j L w j i w j q ) (4.60)
transforms the original system described by (3.20), (3.28), (3.29) and (3.36) into the normal form (4.17). Moreover, these additional inputs v (•) are designed as

v g 1 q = -c g 1 q,0 (i g 1 q -i o g 1 q ) -k Ig 1 q e Ig 1 q v cg 1 = -c cg 1 ,0 (u cg 1 -u o cg 1 ) -c cg 1 ,1 ( ucg 1 -uo cg 1 ) -k Icg 1 e Icg 1 v gρd = -c gρd,0 (i gρd -i o gρd ) -k Igρd e Igρd v gρq = -c gρq,0 (i gρq -i o gρq ) -k Igρq e Igρq v w j d = -c w j d,0 (v sw j d -v o sw j d ) -c w j d,1 ( vsw j d -vo sw j d ) -k Iw j d e Iw j d v w j q = -
c w j q,0 (v sw j q -v o sw j q ) -c w j q,1 ( vsw j q -vo sw j q ) -k Iw j q e Iw j q (4.61)

with the integrated tracking errors

ėIg 1 q = i g 1 q -i o g 1 q , ėIcg 1 = u cg 1 -u o cg 1 ėIgρd = i gρd -i o gρd , ėIgρq = i gρq -i o gρq e Iw j d = v sw j d -v o sw j d , e Iw j d = v sw j q -v o sw j q (4.62)
where the control gains c (•) and k I(•) must be chosen to satisfy (4.13).

When the output vector y is identically equal to the reference y o , the zero dynamics of the system is deduced as

żr = A r z r + ϑ r + ϑ ucg 1 f zero (z r ) (4.63)
z r u cg,r u cw u ct i c T ∈ R N +M +P +L-1 u cg,r u cg 2 • • • u cg N T ϑ r i cg 2 C g 2 • • • i cg N C g N i cw 1 C w 1 • • • i cw M C w M 0 (P +L) T ϑ ucg 1 0 N +M +P -1 H (1,1) u o cg 1 • • • H (1,L) u cg T i cgρ 3 2 (v sgρd i o gρd + v sgρq i o gρq ) u cgρ i cw j 3 2 (v o sw j d I w j d + v o sw j q I w j q ) u cw j (4.64)
The matrix A r is the submatrix formed by deleting the 1st row and 1st column of the matrix A, which is still of the form

A r = 0 (N +M +P -1)×(N +M +P -1) C -1 r H r -L -1 H T r -L -1 R (4.65) where C r = diag(C g 2 • • • C g N C w 1 • • • C w M C p 1 • • • C p P )
and the matrix H r is the submatrix of H by removing the rst row of H.

To check the stability of the equilibrium point of the zero dynamics, we linearize (4.63) around its equilibrium point zr and then get the Jacobian matrix

J = ∂f zero ∂z r | zr=zr = -D C -1 r H r -L -1 H T r -L -1 R (4.66) where D = diag(d k ), k = 1, • • • , N + M + P -1, is a diagonal matrix given by                d (ρ-1) = P o gρ ū2 cgρ , ρ ∈ N -1 d (j+N -1) = P o w j C w j ū2 w j , j ∈ M d (h+N +M -1) =0. h ∈ P P o w j = 3 2 (v o sw j d I w j d + v o sw j q I w j q )
Lemma 4.1.2. Consider the MTDC system described by (3.20), (3.28), (3.29) and

(3.36). If the prescribed references

u o cg 1 , Q o g 1 , P o gρ , Q o gρ , v o sw j d
and v o sw j q and the system parameters are set such that the Jacobian matrix J is Hurwitz, the equilibrium point of the zero dynamics (4.63) is asymptotically stable and hence the MTDC system is minimum phase. In order to prove Lemma 4.1.3, we need some properties of complex matrix as referred in [Horn 1985]. We recall the following basic facts: Denition 4.1.4. A matrix Ψ ∈ C n×n is said to be semi-positive denite if Re(x H Ψx) is non-negative for every non-zero column vector x ∈ C n×1 . Moreover, Ψ ∈ C n×n is said to be positive denite if Re(x H Ψx) is positive for every non-zero column vector x ∈ C n×1 . The set of the positive denite matrix Ψ is denoted as

P + C .
Lemma 4.1.5. Proof. Let us assume that there exists a particular eigenvalue of J denoted by λ = α + jβ ∈ C, whose real part is non-negative, i.e. α ≥ 0. Then, λ satises det(λI -J) = 0

Alternatively, it can be expressed as

det( λI + D -C -1 r H r L -1 r H T r λI + L -1 R ) = 0 (4.67) We dene Φ 1 λI + L -1 R = Λ 1 + jΛ 2 where Λ 1,2 are expressed by Λ 1 = diag(α + R c 1 L c 1 , • • • , α + R c L L c L ) ∈ R L×L Λ 2 = diag(β, • • • , β) ∈ R L×L
Since the Hermitian part of Φ 1 is equal to H(Φ 1 ) = Λ 1 , which is positive denite, the complex matrix Φ 1 is also positive denite (Lemma 4.1.6). Consequently, Φ 1 must be invertible (Lemma 4.1.9) and det(Φ 1 ) = 0. Then, Eq. (4.67) is equal to

det( λI + D -C -1 r H r L -1 r H T r λI + L -1 R ) =det(λI + D + C -1 r H r Φ -1 1 L -1 H T r )det(Φ 1 ) =det(λC r + C r D + H r (λL + R) -1 H T r )det(Φ 1 )det(C -1 r ) Again, we dene Φ 2 λC r + C r D + H r (λL + R) -1 H T r , λC r + C r D = Λ 3 + jΛ 4 and (λL + R) -1 = Λ 5 + jΛ 6 with the notations Λ 3 = αC r -C r D diag(σ 1 , • • • , σ N +M +P -1 )                σ (ρ-1) =C gρ (α + P o gρ ū2 cgρ ), ρ ∈ N -1 σ (j+N -1) =C w j (α + P o w j C w j ū2 w j ), j ∈ M σ (h+N +M -1) =C p h α. h ∈ P Λ 4 = βC r Λ 5 = diag( αL c k + R c k (αL c k + R c k ) 2 + (βL c k ) 2 ) ∈ R L×L Λ 6 = diag( -βL c k (αL c k + R c k ) 2 + (βL c k ) 2 ) ∈ R L×L (4.68)
Now, Φ 2 can be rewritten as

Φ 2 = Φ 3 + j(Λ 4 + H r Λ 6 H T r ) Φ 3 = Λ 3 + H f H T f (4.69)
where H f are given by H

f = H r Λ 1 2 5 . Since Λ 1 2
5 is a full rank matrix, i.e. rank(Λ

1 2 5 ) = L, then rank(H f H T f ) = rank(H f ) = rank(H r ) = N + M + P -1 (see Lemma 3.4.2). As a result, H f H T
f is invertible and positive. Because P o gρ and P o w j are non-negative, obviously, Λ 3 is semi-positive denite and furthermore, by applying Lemma 4.1.8, Φ 3 is positive denite.

As presented in (4.69), we know that H(Φ 2 ) = Φ 3 and then, according to Lemma 4.1.6 and Lemma 4.1.9, Φ 2 is positive denite and invertible. Therefore, the following result is obtained:

det(λI -J) = det(Φ 2 ) det(Φ 1 ) det(C -1 r ) = 0
for α ≥ 0. This leads to a contradiction to (4.67). Thus, all eigenvalues of the Jacobian matrix Φ must have negative real parts, i.e. α < 0. Hence, J is a Hurwitz matrix. As a result, the origin of the zero dynamics (4.63) is locally asymptotically stable. Finally, the MTDC system is minimum phase. The proof is completed.

78 Chapter 4. Control methods based on nonlinear control design tools The performance of the feedback nonlinear controller is evaluated under dierent scenarios and also compared to the performance of the same MTDC system using PI control technique. Initially, the MTDC system operates in the steady state given by Table 4.8. At t = 0.5 s, i o g 2 d is changed to a new reference value with an increase of 50% and then, at t = 1 s and t = 1.5 s, new reference values of v sw 1 q and v sw 2 q are introduced, v o sw 1 q = v o sw 1 q = 20.41 V.

u cg 1 = u o cg 1 = 150 V i g 2 d = i o g 2 d = 6.53 A i g 1 q = i o g 1 q = 0 A i g 2 q = i o g 2 q = 0 A WAC 1 WAC 2 v sw 1 d = v o sw 1 d = 40.82 V v sw 2 d = v o sw 2 d = 40.82 V v sw 1 q = v o sw 1 q = 0 V v sw 2 q = v o sw 2 q = 0 V
The simulation results are plotted in .23 demonstrates that both controllers can keep the DC voltage at u o cg 1 = 150 V. Compared to the PI controller, the feedback nonlinear controller gives the response of u cg 1 a faster convergence but a larger undershoot as v o sw 1 q and v o sw 2 q are subjected to a step change at t = 1 s and t = 1.5 s respectively, since there is usually a compromise between the settling time and undershoot (or overshoot).

Figure 4.24 clearly shows that the step changes of i o g 2 d , v o sw 1 q and v o sw 2 q greatly disturb the performance of i g 1 q under the PI controller while having negligible eects on i g 1 q under the feedback nonlinear controller. This phenomenon indicates that the feedback nonlinear controller has the benet of providing a better decoupling characteristics of the DC voltage and q-axis current control.

Feedback linearization with sliding mode control

Control design

As presented in (4.31), the developed nonlinear controller strongly depends on the system model, which requires the exact informations on the system parameters. For the purpose of good robustness, sliding mode control is used to deal with the case when the system parameters, L g i , R g i , C g i and R L , are poorly known.

By applying the transformation T x in (4.33) and considering the integrated tracking errors (4.36), the original system (4.21) becomes

ėI1 = e 1 ė1 = L fx (i g i q ) + L g d (i g i q )m g i q h 1 + φ 1 • m g i q ėI2 = e 2 ė2 = z 3 ż3 = L 2 fx (u cg i ) + L g d L fx (u cg i )m g i d + L gq L fx (u cg i )m g i q h 2 + φ 2d • m g i d + φ 2q • m g i q (4.70)
where h 1,2 and φ 1,2 are nonlinear continuous functions on e I 1,2 , e 1,2 and e 3 with unknowns parameters. Let us impose two sliding surfaces as

s 1 = a 0 e I 1 + e 1 s 2 = b 0 e I 2 + b 1 e 2 + z 3 (4.71)
86 Chapter 4. Control methods based on nonlinear control design tools where a 0 is positive and b 0,1 are chosen such that the matrix

B 0 = 0 1 -b 0 -b 1 (4.72)
is Hurwitz. The derivatives of the sliding surfaces are given by

ṡ1 = a 0 e 1 + h 1 + φ 1 • m g i q ṡ2 = b 0 e 2 + b 1 z 3 + h 2 + φ 2d • m g i d + φ 2q • m g i q (4.73) Let us consider the Lyapunov function V 1 = s 2 1 + s 2 2 and then, the derivative of V 1 is deduced as V1 = W 1 + W 2 W 1 = s 1 (a 0 e 1 + h 1 + φ 1 • m g i q ) W 2 = s 2 (b 0 e 2 + b 1 z 3 + h 2 + φ 2d • m g i d + φ 2q • m g i q ) (4.74)
From (4.30), it is clear that φ 1 and φ 2d are always negative and moreover, φ 1 and φ 2d are continuous and invertible for positive u cg i .

Assumption 4.2.1. In the domain of interest as described in Section 3.4.4, h 1 , h 2 , φ 1 and φ 2,dq satisfy the inequalities

|h 1 | ≤ h 1,max , 0 ≤ φ 1,min ≤ |φ 1 | ≤ φ 1,max |h 2 | ≤ h 2,max , 0 ≤ φ 2d,min ≤ |φ 2d | ≤ φ 2d,max 0 ≤ φ 2q,min ≤ |φ 2q | ≤ φ 2q,max (4.75)
when the system parameters are bounded.

With Assumption 4.2.1, we can then dene the control law for m g i q as m g i q = β q 1 φ 1,min sgn(s 1 )

(4.76)
where β q are designed as

β q = a 0 |e 1 | + h 1,max + β 0 , β 0 > 0 (4.77)
The control law for m g i d is developed as

m g i d = β d 1 φ 2d,min sgn(s 2 ) (4.78)
where

β d = b 0 |e 2 | + b 1 |z 3 | + h 2,max + φ 2q,max φ 1,min β q + β 1 , β 1 > 0 (4.79)
Applying (4.76) to W 1 , we obtain

W 1 = s 1 (a 0 e 1 + h 1 + β q φ 1 φ 1,min sgn(s 1 )) (4.80)
The sign of W 1 can be analyzed as follows:

• If s 1 > 0, then sgn(s 1 ) = 1 and recalling that φ 1 is negative and φ 1,min is positive, we have

φ 1 φ 1,min ≤ -1 ṡ1 = a 0 e 1 + h 1 + (a 0 |e 1 | + h 1,max + β 0 ) φ 1 φ 1,min < 0 • If s 1 < 0, sgn(s 1 ) = -1 leads to ṡ1 = a 0 e 1 + h 1 -(a 0 |e 1 | + h 1,max + β 0 ) φ 1 φ 1,min > 0
As a result, W 1 = s 1 ṡ1 is negative for all s 1 = 0.

Substituting (4.76) and (4.78) into (4.74) yields

W 2 = s 2 (b 0 e 2 + b 1 z 3 + h 2 + φ 2d • m g i d + φ 2q • m g i q ) = s 2 (b 0 e 2 + b 1 z 3 + h 2 + β q φ 2q φ 1,min sgn(s 1 ) + β d φ 2d φ 2d,min sgn(s 2 )) (4.81)
Recalling (4.75) and the design of β d in (4.79), we have the following results:

• If s 2 > 0, sgn(s 2 ) = 1 and then, φ 2d φ 2d,min ≤ -1 (4.82) ṡ2 = b 0 e 2 + b 1 z 3 + h 2 + β q φ 2q φ 1,min sgn(s 1 ) + β d φ 2d φ 2d,min < 0 (4.83)
• If s 2 < 0, sgn(s 2 ) = -1 and then, we have

ṡ2 = b 0 e 2 + b 1 z 3 + h 2 + β q φ 2q φ 1,min sgn(s 1 ) -β d φ 2d φ 2d,min > 0 (4.84)
Therefore, W 2 = s 2 ṡ2 is also negative for all s 2 = 0. From the above analysis, the controller dened by (4.76) and (4.78) will lead to the manifolds s 1 = 0 and s 2 = 0.

On these two manifolds, the behavior of the system is totally governed by

ėI1 = e 1 ė1 = -a 0 e I1 ėI2 = e 2 ė2 = -b 0 e I 2 -b 1 e 2 (4.85)
whose equilibrium point is asymptotically stable.

Since the use of sign function sgn(•) usually causes the chattering problem [Khalil 1996], in order to deal with this issue, a high-slope saturation function sat(

m g i d = -β d 1 φ 2d,min sat( s 2 µ 2
)

m g i q = -β q 1 φ 1,min sat( s 1 µ 1 ) (4.87)
with positive µ 1 and µ 2 .

Simulation studies

To assess the performance of the feedback linearization controller with sliding mode control, numerical simulations are realized for a VSC HVDC link connected to a resistive load by using MATLAB/Simulink. The system parameters are the same as the case in Section 4.1.2.2.

The system starts from a steady-state condition, and then a sequence of events is applied to the resistive load R as shown in Fig. 4.33. The control objectives are to keep the DC voltage at u o cg i = 150 V and to get the unitrary power factor, i.e. i o g i q = 0 in spite of the variations in the resistive load. In , we show the behaviors of the state variables. As shown in the plots, both u cg i and i g i q are well controlled at u o cg i = 150 V and i o g i q = 0 respectively no matter how the resistive load changes. Besides, as illustrated in Figs. 4.37 and 4.38, the control modulation variables m g I ,dq always satisfy the physical constraint m 2 g i d + m 2 g i q ≤ 1. 92 Chapter 4. Control methods based on nonlinear control design tools

Passivity-based control

As presented in Section 4.1.4, when the output is dened as (4.56), in order ensure the asymptotic stability of the equilibrium point of the zero dynamics (4.63), the

prescribed references u o cg 1 , Q o g 1 , P o gρ , Q o gρ , v o sw j d and v o
sw j q and the system parameters must be chosen such that the Jacobian matrix J in (4.66) is Hurwitz. It implies that these prescribed references can not be set arbitrarily. To overcome this shortcoming, we want to nd a specic output y s such that the stability of the equilibrium point of the zero dynamics is irrelevant to the choice of the prescribed references. In this section, we will demonstrate that this can be achieved by means of passivity theory.

In fact, there have been several studies on the passivity-based control (PBC)

design for a class of switched power converters which can be described in port-

Hamiltonian form [Perez 2004[START_REF] Hernandez-Gomez | [END_REF], Escobar 1999]. Apart from the prior research work, we rst design a passive output derived from a storage function and then, further investigate the corresponding zero dynamics. Since some variables to be controlled belong to the state variables of the zero dynamics and the convergence rate of the zero dynamics are determined by the system inherent characteristics, additional control inputs are introduced to modify the dynamics and improve the performance of those desired controllable variables of the zero dynamics.

Recall the dynamics of the DC grid (3.33) where i cg i and i cw j are obtained from (3.23) and (3.32) by neglecting the losses of the phase reactor and the VSC. In fact, taking into account these losses and according to the power balance on both sides of the converter station, i cg i also satises

3 2 (v cg i d i g i d + v cg i q i g i q )
Active power on the AC side of the converter

= u 2 cg i R equi,g i loss of the VSC + u cg i i cg i
Active power injected to the DC grid (4.88)

where the equivalent resistance R equi,(•) is used to represent the loss of the converter.

According to (3.18), the converter voltages v cg i ,dq are expressed as

v cg i ,dq = 1 2 u cg i m g i ,dq (4.89)
and then, substituting (4.89) into (4.88), we deduce

i cg i = 3 4 (m g i d i g i d + m g i q i g i q ) - u cg i R equi,g i (4.90)
A similar procedure is done to obtain a new formulation for i cw j as

i cw j = 3 4 (m w j d i w j d + m w j q i w j q ) - u cw j R equi,
ϑ p = • • • 3(m g i d i g i d + m g i q i g i q ) 4C g i • • • 3(m w j d i w j d + m w j q i w j q ) 4C w j • • • 0 (P +L) T A p = -D p C -1 H -L -1 H T -L -1 R D p = C -1 R -1 equi ∈ R (N +M +P )×(N +M +P ) R equi = diag(• • • , R equi,g i , • • • R equi,w j , • • • , 0, • • • , 0) (4.93)
Let us consider the dynamics of v sw j ,dq in (3.29). Similar to the modication of the dynamics of the DC grid, here we use the equivalent resistance R f w j to represent the losses of the lter and hence, v sw j ,dq are modeled by

dv sw j d dt = ω w j v sw j q + 1 C f w j (I w j d -i w j d - v sw j d R f w j ) dv sw j q dt = -ω w j v sw j d + 1 C f w j (I w j q -i w j q - v sw j q R f w j ) (4.94)
In this section, the passivity-based state feedback control design is developed based on the above new DC grid and AC voltage model.

Steady-state analysis

The same master-slave control conguration is used in this section as in Section 4.1.4. In addition, in order to make all SAC systems operate with unity power factor, we set

Q o g i = 0 (4.95)
Similar to Assumption 4.1.1, to ensure the operating feasibility of the MTDC system, we take the following assumption. where

0= - R g i L g i īg i d + ω g i īg i q + v sg i d L g i - ūcg i 2L g i mg i d 0= - R g i L g i īg i q -ω g i īg i d + v sg i q L g i - ūcg i 2L g i mg i q 0= - R w j L w j īw j d + ω w j īw j q + vsw j d L w j - ūcw j 2L w j mw j d 0= - R w j L w j īw j q -ω w j īw j d + vsw j q L w j - ūcw j 2L w j mw j q 0= ω w j vsw j q + 1 C f w j (I w j d -īw j d - vsw j d R f w j ) 0= -ω w j vsw j d + 1 C f w j (I w j q -īw j q - vsw j q R f w j ) 0= A p z + θp
ūcg 1 = u o cg 1 , īg 1 q = - 2Q o g 1 3v sg 1 d = 0, īgρd = 2P o gρ 3v sg 1 d , īgρq = - 2Q o gρ 3v sg 1 d = 0 and vsw j ,dq = v o sw j ,dq .
Under Assumption 4.3.1, the steady-state values of i w j ,dq and the control variables are deduced as

īw j d = I w j d + C f w j ω w j v o sw j q - v o sw j d R f w j īw j q = I w j q -C f w j ω w j v o sw j d - v o sw j q R f w j (4.97)
and

mg i d = 2 u cg i (-R g i īg i d + L g i ω g i īg i q + v sg i d ) mg i q = 2 u cg i (-R g i īg i q -L g i ω g i īg i d + v sg i q ) mw j d = 2 u cw j (-R w j īw j d + L w j ω w j īw j q + v o sw j d ) mw j q = 2 u cw j (-R w j īw j q -L w j ω w j īw j d + v o sw j q ) (4.98)
Substituting (4.97) and (4.98) into (4.96), we nd that the DC grid in steady state satises

(v sg i d īg i d + v sg i q īg i q ) -R g i ( ī2 g i d + ī2 g i q ) + 2 3 [H(i, :) īc ūcg i - ū2 cg i R equi,g i ] = 0 (v o sw j d īw j d + v o sw j q īw j q ) -R w j ( ī2 w j d + ī2 w j q ) + 2 3 [H(N + j, :) īc ūcw j - ū2 cw j R equi,w j ] = 0 H(N + M + h, :) īc = 0 H(:, k) T ūc = 0 (4.99)
Note that the above expression (4.99) contains several quadratic terms. It implies that the MTDC system has more than one equilibrium point that corresponds to the prescribed references [Lee 2003, Zonetti 2014]. But, not all of these equilibrium points are achievable. As presented in [Lee 2003], there are two possible steady-state values for the d-axis current. However, the larger one is physically impossible, making the converter operate beyond its ability. The choice of the reasonable equilibrium point is determined by the system physical characteristics, the feasibility of the modulation technique (see (3.21) and (3.30)), etc. But this issue is beyond the scope of this thesis. Here, we make the following assumption on the equilibrium point.

ig i d = - R g i L g i ĩg i d + ω g i ĩg i q - 1 2L g i ( mg i d ũcg i + mg i d u cg i ) ig i q = - R g i L g i ĩg i q -ω g i ĩg i d - 1 2L g i ( mg i q ũcg i + mg i q u cg i ) iw j d = - R w j L w j ĩw j d + ω w j ĩw j q + ṽsw j d L w j - 1 2L w j ( mw j d ũcw j + mw j d u cw j ) iw j q = - R w j L w j ĩw j q -ω w j ĩw j d + ṽsw j q L w j - 1 2L w j
( mw j q ũcw j + mw j q u cw j )

vsw j d = ω w j ṽsw j q - 1 C f w j ( ĩw j d + ṽsw j d R f w j ) vsw j q = -ω w j ṽsw j d - 1 C f w j ( ĩw j q + ṽsw j q R f w j ) ż= A p z + θp (4.100)
where θp is given by

                   θp (i) = 3 4C g i ( mg i d ĩg i d + mg i d i g i d + mg i q ĩg i q + mg i q i g i q ), i ∈ N θp (N + j) = 3 4C w j
( mw j d ĩw j d + mw j d i w j d + mw j q ĩw j q + mw j q i w j q ), j ∈ M

θp (N + M + h) = 0. h ∈ P θp (N + M + P + k) = 0. k ∈ L
In addition, setting mg i ,dq = mw j ,dq = 0, we get the unforced system of (4.100) as

ig i d = - R g i L g i ĩg i d + ω g i ĩg i q - 1 2L g i mg i d ũcg i ig i q = - R g i L g i ĩg i q -ω g i ĩg i d - 1 2L g i mg i q ũcg i iw j d = - R w j L w j ĩw j d + ω w j ĩw j q + ṽsw j d L w j - 1 2L w j mw j d ũcw j iw j q = - R w j L w j ĩw j q -ω w j ĩw j d + ṽsw j q L w j - 1 2L w j mw j q ũcw j vsw j d = ω w j ṽsw j q - 1 C f w j ( ĩw j d + ṽsw j d R f w j ) vsw j q = -ω w j ṽsw j d - 1 C f w j ( ĩw j q + ṽsw j q R f w j ) ż= A p z + θ p (4.
             θp (i) = 3 4C g i ( mg i d ĩg i d + mg i q ĩg i q ), i ∈ N θp (N + j) = 3 4C w j ( mw j d ĩw j d + mw j q ĩw j q ), j ∈ M θp (N + M + h) = θp (N + M + P + k) = 0. h ∈ P, k ∈ L Take V = 1 2 N i=1 L g i [( ĩg i d ) 2 + ( ĩg i q ) 2 ] + 1 2 M j=1 L w j [( ĩw j d ) 2 + ( ĩw j q ) 2 ] + 1 2 M j=1 C f w j [(ṽ sw j d ) 2 + (ṽ sw j q ) 2 ] + 1 2 • 2 3 zT Λz (4.102)
as a storage function candidate with

Λ = C 0 (N +M +P )×L 0 L×(N +M +P ) L
The function V is positive denite and its derivative along the trajectories of (4.100)

satises V = -N i=1 R g i [( ĩg i d ) 2 + ( ĩg i q ) 2 ] -M j=1 R w j [( ĩw j d ) 2 + ( ĩw j q ) 2 ] - 2 3 N i=1 (ũ cg i ) 2 R equi,g i - 2 3 M j=1 (ũ cw j ) 2 R equi,w j -M j=1 [ (ṽ sw j d ) 2 + (ṽ sw j q ) 2 R f w j ] - 2 3 L k=1 R c k ( ĩc k ) 2 + 1 2 N i=1 [ mg i d (ũ cg i i g i d -u cg i ĩg i d ) + mg i q (ũ cg i i g i q -u cg i ĩg i q )] + 1 2 M j=1
[ mw j d (ũ cw j i w j d -u cw j ĩw j d ) + mw j q (ũ cw j i w j q -u cw j ĩw j q )] (4.103)

Taking the output vector y p as

y p = [y pg y pw ] T ∈ R 2N +2M y pg = [y pg 1 d y pg 1 q • • • y pg i d y pg i q • • • y pg N d y pg N q ] T ∈ R 2N y pw = [y pw 1 d y pw 1 q • • • y pw j d y pw j q • • • y pw M d y pw M q ] T ∈ R 2M y pg i d = 1 2 (ũ cg i i g i d -u cg i ĩg i d ); y pg i q = 1 2 (ũ cg i i g i q -u cg i ĩg i q ) y pw j d = 1 2 (ũ cw j i w j d -u cw j ĩw j d ); y pw j q = 1 2 (ũ cw j i w j q -u cw j ĩw j q ) (4.104)
V then satises

V = L f V + L g V u ≤ y T p u (4.105) 4.3. Passivity-based control 97 with L g V = y T p L f V = -N i=1 R g i [( ĩg i d ) 2 + ( ĩg i q ) 2 ] -M j=1 R w j [( ĩw j d ) 2 + ( ĩw j q ) 2 ] - 2 3 N i=1 (ũ cg i ) 2 R equi,g i - 2 3 M j=1 (ũ cw j ) 2 R equi,w j -M j=1 [ (ṽ sw j d ) 2 + (ṽ sw j q ) 2 R f w j ] - 2 3 L k=1 R c k ( ĩc k ) 2 u= [u g u w ] T ∈ R 2N +2M u g = [m g 1 d m g 1 q • • • m g N d m g N q ] T ∈ R 2N u w = [m w 1 d m w 1 q • • • m w M d m w M q ] T ∈ R 2M (4.106)
It can be seen that the MTDC system has the Kalman-Yacubovitch-Popov (KYP)

property and it is passive with the passive output vector y p .

Control design

Since the steady-state values of all DC voltages are positive, ūc(•) > 0, here we change the output vector to

y= [y g y w ] T ∈ R 2N +2M y g = [y g 1 d y g 1 q • • • y g i d y g i q • • • y g N d y g N q ] T ∈ R 2N y w = [y w 1 d y w 1 q • • • y w j d y w j q • • • y w M d y w M q ] T ∈ R 2M y g i d = α g i d u cg i -i g i d ;
y g i q = α g i q u cg i -i g i q y w j d = α w j d u cw j -i w j d ; y w j q = α w j q u cw j -i w j q α g i ,dq = īg i ,dq ūcg i ; α w j ,dq = īw j ,dq ūcw j (4.107)

Comparing y p in (4.104) and y in (4.107), it is easy to nd that y g i ,dq = 2 ūcg i y pg i ,dq ; y w j ,dq = 2 ūcw j y pw j ,dq

(4.108)
Accordingly, the reference of y is y o = 0 (2N +2M )×1 . Let us restrict y ≡ y o and then, by performing the change of variables i g i ,dq = α g i ,dq u cg i -y g i ,dq ; i w j ,dq = α w j ,dq u cw j -y w j ,dq (4.109)

we obtain the following equations

0 = ẏg i d = a 1g i m g i d + a 2g i m g i q + f g i d 0 = ẏg i q = a 3g i m g i d + a 4g i m g i q + f g i q 0 = ẏw j d = a 1w j m w j d + a 2w j m w j q + f w j d 0 = ẏw j q = a 3w j m w j d + a 4w j m w j q + f w j q (4.110)
98 Chapter 4. Control methods based on nonlinear control design tools where

a 1g i = α g i d C g i (α g i d u cg i -y g i d ) + u cg i 2L g i a 2g i = α g i d C g i (α g i q u cg i -y g i q ) a 3g i = α g i q C g i (α g i d u cg i -y g i d ) a 4g i = α g i q C g i (α g i q u cg i -y g i q ) + u cg i 2L g i a 1w j = α w j d C w j (α w j d u cw j -y w j d ) + u cw j 2L w j a 2w j = α w j d C w j (α w j q u cw j -y w j q ) a 3w j = α w j q C w j (α w j d u cw j -y w j d ) a 4w j = α w j q C w j (α w j q u cw j -y w j q ) + u cw j 2L w j f g i d = α g i d C g i [- u cg i R equi,g i + H(i, :)i c ] + R g i L g i (α g i d u cg i -y g i d ) -ω g i (α g i q u cg i -y g i q ) - v sg i d L g i f g i q = α g i q C g i [- u cg i R equi,g i + H(i, :)i c ] + R g i L g i (α g i q u cg i -y g i q ) +ω g i (α g i d u cg i -y g i d ) - v sg i q L g i f w j d = α w j d C w j [- u cw j R equi,w j + H(N + j, :)i c ] + R w j L w j (α w j d u cw j -y w j d )
-ω w j (α w j q u cw j -y w j q ) -v sw j d L w j

f w j q = α w j q C w j [- u cw j R equi,w j + H(N + j, :)i c ] + R w j L w j (α w j q u cw j -y w j q ) +ω w j (α w j d u cw j -y w j d ) - v sw j q L w j (4.111) Let us denote E g i = a 1g i a 2g i a 3g i a 4g i ; E w j = a 1w j a 2w j a 3w j a 4w j ; (4.112)
Lemma 4.3.3. The decoupling matrices E g i and E w j are nonsingular at y g i d =

y g i q = 0 and y w j d = y w j q = 0.

Proof. The determinants of E g i and E w j are equal to

∆ g i = det(E g i ) = u cg i 2L g i [ u cg i 2L g i + 3 4 α g i d C g i (α g i d u cg i -y g i d ) + 3 4 α g i q C g i (α g i q u cg i -y g i q )] ∆ w j = det(E w j ) = u cw j 2L w j [ u cw j 2L w j + 3 4 α w j d C w j (α w j d u cw j -y w j d ) + 3 4 α w j q C w j (α w j q u cw j -y w j q )]
Setting y g i d = y g i q = 0 and y w j d = y w j q = 0 yields

∆ g i 0 = ∆ g i | y g i d =yg i q =0 = u cg i 2L g i [ u cg i 2L g i + 3 4 (α g i d ) 2 C g i u cg i + 3 4 (α g i q ) 2 C g i u cg i ] ∆ w j 0 = ∆ w j | y w j d =yw j q =0 ] = u cw j 2L w j [ u cw j 2L w j + 3 4 (α w j d ) 2 C w j u cw j + 3 4 (α w j q ) 2 C w j u cw j ] (4.113)
Because u cg i and u cw j are positive, ∆ g i 0 = 0 and ∆ w j 0 = 0 are always positive.

Since ∆ g i (∆ w j ) are continuously dierentiable for (α g i d , α g i q , y g i d , y g i q ) ∈ B α g i d ×B αg i q ×B y g i d ×B yg i q ((α w j d , α w j q , y w j d , y w j q ) ∈ B α w j d ×B αw j q ×B y w j d ×B yw j q ) where these B (•) ⊂ R are open connected sets, based on Lemma 4.3.3, we have the following lemma. Lemma 4.3.4. There exist a series of connected sets S (•) ⊂ B (•) such that the decoupling matrices E g i and E w j are nonsigular for (α g i d , α g i q , y g i d , y g i q ) ∈ S α g i d × S αg i q × S y g i d × S yg i q and (α w j d , α w j q , y w j d , y w j q ) ∈ S α w j d × S αw j q × S y w j d × S yw j q . In addition, S y g i ,dq and S y w j ,dq contain the origin. S α g i d , S αg i q , S α w j d and S αw j q contain īg i d ūcg i , īg i q ūcg i , īw j d ūcw j and īw j q ūcw j , respectively.

Since the solution of the state variables is restricted to (4.110) when the output vector y is identically equal to y o , the control modulation indices must satisfy

m g i d = 1 ∆ g i (a 4g i f g i d -a 2g i f g i q )| y g i d =yg i q =0 m g i q = 1 ∆ g i (a 4g i f g i d -a 2g i f g i q )| y g i d =yg i q =0 m w j d = 1 ∆ w j (a 4w j f w j d -a 2w j f w j q )| y w j d =yw j q =0 m w j q = 1 ∆ w j (a 4w j f w j d -a 2w j f w j q )| y w j d =yw j q =0 (
dv sw j d dt = ω w j v sw j q + 1 C f w j (I w j d -α w j d u cw j - v sw j d R f w j ) dv sw j q dt = -ω w j v sw j d + 1 C f w j (I w j q -α w j q u cw j - v sw j q R f w j ) C g i du cg i dt = 1 ∆ g i 0 1 2L g I [- u cg i R equi,g i + H(i, :)i c ] - 1 ∆ g i 0 3 4 [ R g i L g i u cg i (α 2 g i d + α 2 g i q ) - 1 L g i (v sg i d α g i d + v sg i q α g i q )] C w j du cw j dt = 1 ∆ w j 0 1 2L w j [- u cw j R equi,w j + H(N + j, :)i c ] - 1 ∆ w j 0 3 4 [ R w j L w j u cw j (α 2 w j d + α 2 w j q ) - 1 L w j (v sw j d α w j d + v sw j q α w j q )] C t h du ct h dt = H(N + M + h, :)i c L c k di ck dt = -R c k i c K -H(:, k) T u c (4.115) With W = 1 2 M j=1 C f w j [(ṽ sw j d ) 2 + (ṽ sw j q ) 2 ] + 1 2 M j=1 C w j k w j (ũ cw j ) 2 + 1 2 P h=1 C t h 2 3 (ũ ct h ) 2 + 1 2 L k=1 2 3 L c k ( ĩc k ) 2 + 1 2 N i=1 C g i k g i (ũ cg i ) 2 (4.116)
as a Lyapunov function candidate for the zero dynamics (4.115) where

k w j = { 1 2L w j + 3 4 [(α w j d ) 2 + (α w j q ) 2 ]} 4 3 L w j k g i = { 1 2L g i + 3 4 [(α g i d ) 2 + (α g i q ) 2 ]} 4 3 L g i (4.117)
The derivative of W along the trajectories of (4.115) is given by

Ẇ = -M j=1 [ 2 3R equi,w j + R w j (α 2 w j d + α 2 w j q )](ũ w j ) 2 -N i=1 [ 2 3R equi,g i + R g i (α 2 g i d + α 2 g i q )](ũ g i ) 2 - 2 3 R c k ( ĩc k ) 2 -M j=1 [ (ṽ sw j d ) 2 + (ṽ sw j q ) 2 R f w j ] ≤ 0 (4.118)
which shows that the MTDC system with the output vector y in (4.107) is weakly minimum phase. Furthermore, we have the following result.

Lemma 4.3.5. The equilibrium point of the zero dynamics (4.115) is asymptotically stable.

Proof. This lemma can be proved using Barbalat's lemma. Since Ẇ is non-positive, we then have W (t) ≤ W (t = 0), which means that all variables (•) in W are bounded.

Passivity-based control 101

Based on (4.118), Ẅ can be computed, which is composed only of bounded terms. Hence, Ẅ is also bounded. As a consequence, Ẇ is uniformly continuous in time. Applying Barbalat's lemma, we have Ẇ → 0 in (4.118) as t → ∞, which implies that ũcw j → 0; ũcg i → 0; ĩc k → 0; ṽsw j d → 0; ṽsw j q → 0 (4.119) and ucw j → 0; ucg i → 0; ic k → 0; vsw j d → 0; vsw j q → 0 (4.120)

Combining (4.115), (4.119) and (4.120), we can obtain

H T ũc → 0 (4.121)
with unknown ũct h . Since Lemma 3.4.1 states that rank(H ) = N + M + P -1 ≥ P , ũct h → 0 is the only solution of (4.121). As a result, only the trivial solution exists for Ẇ = 0 and then the lemma is proved.

Based on Lemma 4.3.5, we can also get that only the trivial solution of the unforced system (4.101) can stay in the set {y = y o = 0 (2N +2M )×1 } where y is dened as (4.107). Finally, we can conclude that the error system (4.100) is zerostate observable.

According to Theorem 3.2 in [Byrnes 1991], the control variables m(•) can be developed as

mg i d mg i q = - φ g i d (y g i d )
φ g i q (y g i q )

; mw j d mw j q = -φ w j d (y w j d ) φ w j d (y w j q )

(4.122)

where φ (•) are any smooth functions such that φ (•) (0) = 0 and y g i d φ g i d (y g i d ) > 0; y g i q φ g i q (y g i q ) > 0 y w j d φ w j d (y w j d ) > 0; y w j q φ w j q (y w j q ) > 0 for each nonzero y (•) . Then, m (•) can be deduced as

m g i d m g i q = mg i d mg i q - φ g i d (y g i d ) φ g i q (y g i q ) ; m w j d m w j q = mw j d mw j q - φ w j d (y w j d ) φ w j d (y w j q ) (4.123)
In fact, the choice of φ (•) is free. In this thesis, we choose the easiest way to design the control variable as

m g i d = -c g i d,0 • y g i d -k Ig i d • e Iy g i d m g i q = -c g i q,0 • y g i q -k Ig i q • e Iyg i q m w j d = -c w j d,0 • y w j d -k Iw j d • e Iy w j d m w j q = -c w j q,0 • y w j q -k Iw j q • e Iyw j q
(4.124)

with positive control gains c (•) and k I(•) and the integrated tracking errors ėIy g i d = y g i d ; ėIyg i q = y g i q ėIy w j d = y w j d ; ėIyw j q = y w j q 102Chapter 4. u cg i and i w j ,dq u cw j perfectly follow α g i ,dq and α w j ,dq , respectively). Therefore, by controlling the output variable at zero, we can indirectly drive u cg 1 , i gρd , i g i q and the AC voltage v sw j ,dq to the prescribed references (•) o given by (4.57). From Assumption 4.3.1, we get īg i q = 0 and hence α g i q = īg i q ūcg i = 0 can be directly obtained. However, there is still a need to deduce α g i d and α w j ,dq by solving (4.96). Due to the uncertainties on the system parameters or unmodeled elements in the power system, it is usually dicult to get the accurate values of α g i d and α w j ,dq and then steady-state errors may exist. Furthermore, the convergence rate of the uncontrolled zero dynamics totally depends on the system inherent characteristics, which would be very slow.

Therefore, we expect to regulate the behaviors of the states of the zero dynamics and to achieve good tracking performance, i.e. to make u cg 1 , i gρd , i g i q and the AC voltage v sw j ,dq follow their respective prescribed values (•) o by designing α g i d and α w j ,dq . In the following part, we consider the case that īgρd = 0. If P o gρd are set to zero for some SAC terminals, then the corresponding α gρd = īg i d

ūcg i = 2P o gρ 3ū cgρ v sgρd = 0
can be directly deduced without further design.

As mentioned in the previous section, the control goal is to make u cg 1 , i gρd , i g i q and v sw j ,dq track their respective references. Thus, we substitute α g i q = 0,

dv sw j d dt = ω w j v sw j q + 1 C f w j (I w j d -α * w j d u cw j - v sw j d R f w j ) dv sw j q dt = -ω w j v sw j d + 1 C f w j (I w j q -α * w j q u cw j - v sw j q R f w j ) C g 1 du cg 1 dt = 1 2L g 1 1 2L g 1 + 3 4 (α * g 1 d ) 2 {[- u cg 1 R equi,g 1 + H(1, :)i c ] - 3 2 [R g 1 u cg 1 (α * g 1 d ) 2 -v sg 1 d α * g 1 d ]} C gρ di gρd dt = 1 2L gρ 1 2L gρ + 3 4 (α * gρd ) 2 {[- i gρd R equi,gρ + α * gρd H(i, :)i c ] - 3 2 [R gρ i gρd (α * gρd ) 2 -α * gρd (v sgρd α * gρd )]} C w j du cw j dt = 1 2L w j 1 2L w j + 3 4 [(α * w j d ) 2 + (α * w j q ) 2 ] {[- u cw j R equi,w j + H(N + j, :)i c ] - 3 2 [R w j u cw j ((α * w j d ) 2 + (α * w j q ) 2 ) -(v sw j d α * w j d + v sw j q α * w j q )]} C t h du ct h dt = H(N + M + h, :)i c L c k di ck dt = -R c k i c K -H(:, k) T u c (4.125)
Now, the transformed zero dynamics (4.125) is considered as a new system where α g i d and α w j ,dq are replaced by α * g i d and α * w j ,dq to emphasize that they are consid- ered as control inputs. It can be seen that the number of control variables (α * g i d and α * w j ,dq ) equals the number of controlled variables (u cg 1 , i gρd and v sw j ,dq ) and moreover, α * w j ,dq are directly collocated with v sw j ,dq . We can use two steps to design the control variables.

4.3.4.1

Step 1: design of α * w j ,dq

We rst design α * w j ,dq to regulate v sw j ,dq while α * .

Feedback linearization control

Here, the input-output feedback linearzation method is applied to design α * w j ,dq . It is natural to choose the output

y w = [y w 1 • • • y w M ] T y w j = [v sw j d v sw j q ] T (4.126)
104Chapter 4. Control methods based on nonlinear control design tools

Then, following the input-output feedback linearization procedure, the control law for α * w j ,dq is deduced as

α * w j d = 1 u cw j [I w j d - v sw j d R f w j + C f w j ω w j v sw j q -C f w j v α w j d ] α * w j q = 1 u cw j [I w j q - v sw j q R f w j -C f w j ω w j v sw j d -C f w j v αw j q ] (4.127)
where

ėIα w j d = e v sw j d v sw j d -v o sw j d v α w j d = -c α w j d • e v sw j d -k Iα w j d
• e Iα w j d ėIαw j q = e vsw j q v sw j q -v o sw j q v αw j q = -c αw j q • e vsw j q -k Iαw j q • e Iαw j q with positive control gains c α w j ,dq and k Iα w j ,dq .

If the tracking errors e α w j ,dq approach zero, we then take α * w j ,dq = ᾱ * w j ,dq where ᾱ *

w j d = 1 u cw j [I w j d - v o sw j d R f w j + C f w j ω w j v o sw j q + k Iα w j d ēIα w j d ] ᾱ * w j q = 1 u cw j [I w j q - v o sw j q R f w j -C f w j ω w j v o sw j d + k Iαw j q ēIαw j q ] (4.128)
which are calculated by restricting v sw j ,dq = v o sw j ,dq and

0 = ω w j v o sw j q + 1 C f w j (I w j d -ᾱ * w j d u cw j - v o sw j d R f w j ) 0 = -ω w j v o sw j d + 1 C f w j (I w j q -ᾱ * w j q u cw j - v o sw j q R f w j )
Recall that the steady-state values of ᾱ * w j ,dq satisfy

ᾱw j d = īw j d ūcw j = 1 ūcw j [I w j d - vsw j d R f w j + C f w j ω w j v o sw j q + k Iα w j d ēIα w j d ] ᾱw j q = īw j q ūcw j = 1 ūcw j [I w j q - vsw j q R f w j -C f w j ω w j v o sw j d + k Iαw j q ēIαw j q ]
and hence, ᾱ * w j ,dq can be rewritten as ᾱ *

w j d = 1 u cw j īw j d ᾱ * w j q = 1 u cw j īw j q (4.129)
In addition, we can also obtain

∂ ᾱ * w j d ∂u cw j = - 1 (u cw j ) 2 īw j d ∂ ᾱ * w j q ∂u cw j = - 1 (u cw j ) 2 īw j q (4.130)

Passivity-based control 105

Then, the zero dynamics of the system (4.125) are characterized by

C g 1 du cg 1 dt = 1 2L g 1 1 2L g 1 + 3 4 (α * g 1 d ) 2 {[- u cg 1 R equi,g 1 + H(1, :)i c ] - 3 2 [R g 1 u cg 1 (α * g 1 d ) 2 -v sg 1 d α * g 1 d ]} C gρ di gρd dt = 1 2L gρ 1 2L gρ + 3 4 (α * gρd ) 2 {[- i gρd R equi,gρ + α * gρd H(ρ, :)i c ] - 3 2 [R gρ i gρd (α * gρd ) 2 -α * gρd (v sgρd α * gρd )]} C w j du cw j dt = 1 2L w j 1 2L w j + 3 4 [(ᾱ * w j d ) 2 + ( ᾱ * w j q ) 2 ] {[- u cw j R equi,w j + H(N + j, :)i c ] - 3 2 [R w j u cw j (( ᾱ * w j d ) 2 + ( ᾱ * w j q ) 2 ) -(v o sw j d ᾱ * w j d + v o sw j q ᾱ * w j q )]} C t h du ct h dt = H(N + M + h, :)i c L c k di ck dt = -R c k i c K -H(:, k) T u c (4.131)
The stability of the equilibrium point of the zero dynamics (4.131) is veried by linearizating (4.131) around the equilibrium point and we can obtain the following Jacobian matrix

J 1 = -Σ 1 C -1 Σ 2 Σ 3 H -L -1 H T Σ -1 3 -L -1 R
where Σ 1,2,3 are given by

Σ 1 = diag(σ 1 1 , • • • , σ 1 N +M +P ) Σ 2 = diag(σ 2 1 , • • • , σ 2 N +M +P ) Σ 3 = diag(σ 3 1 , • • • , σ 3 L )
106Chapter 4. Control methods based on nonlinear control design tools with the elements

                                                         σ 1 1 = 1 2L g 1 C g 1 1 2L g 1 + 3 4 (α * g 1 d ) 2 [ 1 R equi,g 1 + 3 2 R g 1 (α * g i d ) 2 ], σ 1 ρ = 1 2L gρ C gρ 1 2L gρ + 3 4 (α * gρd ) 2 [ 1 R equi,gρ + 3 2 R gρ (α * gρd ) 2 ], ρ ∈ N -1 σ 1 N +j = 1 2L w j C w j 1 2L w j + 3 4 [(ᾱ w j d ) 2 + ( ᾱw j q ) 2 ] { 1 R equi,w j + R w j ((ᾱ w j d ) 2 + ( ᾱw j q ) 2 ) + 3 2 1 (ū w j ) 2 [(v o sw j d īw j d + v o sw j q īw j q ) -2R w j (( īw j d ) 2 + ( īw j q ) 2 )], j ∈ M σ 1 (N +M +h) = 0. h ∈ P                                                σ 2 1 = 1 2L g 1 1 2L g 1 + 3 4 (α * g 1 d ) 2 , σ 2 ρ = 1 2L gρ 1 2L gρ + 3 4 (α * gρd ) 2 , ρ ∈ N -1 σ 2 N +j = 1 2L w j 1 2L w j + 3 4 [(ᾱ w j d ) 2 + ( ᾱw j q ) 2 ] , j ∈ M σ 2 (N +M +h) = 1. h ∈ P            σ 3 1 = 1, σ 3 ρ = α * gρd , ρ ∈ N -1 σ 3 N +j = 1, j ∈ M σ 3 (N +M +h) = 1. h ∈ P
The terms R w j (( īw j d ) 2 + ( īw j d ) 2 ) and R w j (( īw j d ) + ( īw j d )) represent the losses and the decrease of AC voltage caused by the reactor phases. An optimized HVDC transmission system should have lower losses than AC lines for the same power capacity. The losses in the converter station are only about 0.6 % of the transmitted power [START_REF] Abb | Lower losses[END_REF]]. As a consequence, it is reasonable to consider that v o sw j d īw j d + v o sw j q īw j q -2R w j (( īw j d ) 2 + ( īw j q ) 2 )> 0 v o sw j d + v o sw j q -2R w j ( īw j d + īw j q )> 0 4.3. Passivity-based control 107 Lemma 4.3.6. The system described by (4.125) with control variables α * w j ,dq and output y w in (4.126) is minimum phase . Proof. To demonstrate this lemma, we only need to prove that the Jacobian matrix J 1 is Hurwitz. Similar to demonstrate Lemma 4.1.3, we also assume that there exists an eigenvalue of J 1 denoted by λ = α + β ∈ C whose real part is non-negative, i.e. α ≥ 0. By denition of eigenvalues, we have

det(λI -J 1 ) = det( λI + Σ 1 -C -1 Σ 2 Σ 3 H L -1 H T Σ -1 3 λI + L -1 R ) = 0 (4.132)
As described in the demonstration of Lemma 4.1.3 in Section 4.1.4.1, Φ 1 λI + L -1 R = Λ 1 + jΛ 2 are positive denite where Λ 1,2 are given by

Λ 1 = diag(α + R c 1 L c 1 , • • • , α + R c L L c L ) ∈ R L×L Λ 2 = diag(β, • • • , β) ∈ R L×L Therefore, det(λI -J 1 ) is equivalent to det(λI -J 1 ) =det(λI + Σ 1 + C -1 Σ 2 Σ 3 HΦ -1 1 L -1 H T Σ -1 3 )det(Φ 1 ) =det(C -1 Σ 2 Σ 3 Σ -1 3 )det(λΣ -1 3 Σ -1 2 CΣ 3 + Σ -1 3 Σ -1 2 CΣ 1 Σ 3 + HΦ -1 1 L -1 H T )det(Φ 1 )
Since C and Σ 1,2,3 are diagonal, the above expression can be rewritten as

det(λI -J 1 ) =det(C -1 Σ 2 )det(λΣ -1 2 C + Σ -1 2 CΣ 1 + HΦ -1 1 L -1 H T )det(Φ 1 )
We introduce

Φ 2 = λCΣ -1 2 + Σ 1 CΣ -1 2 + HΦ -1 1 L -1 H T λCΣ -1 2 + Σ 1 CΣ -1 2 = Λ 3 + jΛ 4 Φ -1 1 L -1 = (λI + R) -1 = Λ 5 + jΛ 6
Recalling the expressions of C, Σ 1 and Σ 2 (Σ 1 and Σ 2 are positive denite), it is evident that Λ 5 and Λ 3 are semi-positive denite where Λ 3 is given by

Λ 3 = diag(σ 1 , • • • , σ N +M +P )
and Λ 5 is expressed in (4.68). Now, Φ 2 can be rewritten as

Φ 2 = Φ 3 + j(Λ 4 + (Σ 3 H)Λ 6 (Σ 3 H) T ) Φ 3 = Λ 1 3 + Λ 1 3 + H f H T f (4.133)
108Chapter 4. Control methods based on nonlinear control design tools where Λ 1 3 is a diagonal matrix whose rst element on the main diagonal is σ 1 > 0 and all other elements are zero. Λ 1 3 and H f are given by

Λ 1 3 = Λ 3 -Λ 1 3 H f = HΛ 1 2 5 Let us dene Φ 4 = Λ 1 3 + H f H T f which is semi-positive denite. Then the determinant of Φ 4 is calculated as det(Φ 4 ) = σ 1 det(R(H f ) 1 R(H f ) T 1 ) + det(H f H T f )
Recall that R(H f ) 1 is the matrix obtained by removing the rst row of H f . Since Λ 1 2

5 is a full rank matrix, i.e. rank(Λ

1 2 5 ) = L, then rank(H f H T f ) = rank(H f ) = rank(H ) = N + M + P -1 and as a results, det(H f H T f ) = 0. According to Lemma 3.4.2, the rank of R(H f ) 1 is N + M + P -1. Consequently, R(H f ) 1 R(H f ) T 1 is
invertible and positive denite. Because of the positive σ 1 , we then have

det(Φ 4 ) = σ 1 det(R(H f ) 1 R(H f ) T 1 ) + det(H f H T f ) = σ 1 det(R(H f ) 1 R(H f ) T 1 ) > 0
and hence, the semi-denite matrix Φ 4 must be positive denite. Recalling that Φ 3 = Φ 4 + Λ 1 3 as derived from (4.133), Φ 3 is also positive denite (see Lemma 4.1.8). Since the hermitian part of Φ 2 , i.e. H(Φ 2 ), is Φ 3 which is positive denite, we get the result that Φ 2 is also positive denite (see Lemma 4.1.7). Therefore, we deduce that det(λI -J 1 ) =det(Φ 2 )det(Φ 1 )det(C -1 Σ 2 ) = 0 for α ≥ 0. This leads to a contradiction to (4.132). Thus, we must have that the real part of all eigenvalues of J 1 is negative, i.e. α < 0. Finally, we prove that J 1 is Hurwitz and as a result, the equilibrium point of the zero dynamics is asymptotically stable. The proof is completed.

Adaptive control

Let us revisit the zero dynamics (4.125) which is a multiple-input and multipleoutput (MIMO) system with the control variables α * w j ,dq . In the previous part, by using the feedback linearization technique, α * w j ,dq is designed as (4.127) which strongly depend on the system parameters, R f w j C f w j , ω j and I w j ,dq . However, due to the ageing of the facilities, the system parameters are usually slowly varying.

Passivity-based control 109

In particular, R f w j used to model the losses of the lter is very hard to obtain its exact value. Several studies have been carried out to deal with the uncertainties of the system parameters [START_REF] Ruan | [END_REF][START_REF] Hernandez-Gomez | [END_REF][START_REF] Durrant | [END_REF]]. In Section 4.2, the sliding mode control method is also applied to make the system more robust. In this part, the adaptive control method is proposed to adjust the system parameters for enhancing the dynamic performance of the system [START_REF] Narendra | [END_REF]].

We recall the dynamics of v sw j ,dq given by

dv sw j d dt = ω w j v sw j q + 1 C f w j I w j d -α * w j d u cw j 1 C f w j - v sw j d R f w j 1 C f w j dv sw j q dt = -ω w j v sw j d + 1 C f w j I w j q -α * w j q u cw j 1 C f w j - v sw j q R f w j 1 C f w j (4.134)
with the uncertain C f w j , ω w j , R f w j and I w j ,dq . Our task here is to design α * w j ,dq to make v sw j ,dq track their respective references v o sw j ,dq . Then, the output tracking errors are given by

e v sw j d = v sw j d -v o sw j d e vsw j q = v sw j q -v o sw j q (4.135)
Augmenting (4.134) with the integrated output tracking errors ėIv sw j d = e v sw j d ėIvsw j q = e vsw j q (4.136) results in the following extend open-loop system

     ėIv sw j d vsw j d ėIvsw j q vsw j q      =         0 1 0 0 0 - 1 C f w j R f w j 0 ω w j 0 0 0 1 0 -ω w j 0 - 1 C f w j R f w j         A P j      e Iv sw j d v sw j d e Ivsw j q v sw j q      +         0 I w j d C f w j 0 I w j d C f w j         +         0 0 - 1 C fwj 0 0 0 0 - 1 C fwj         B P j u cw j α * w j d u cw j α * w j q +      -v o sw j d 0 -v o sw j q 0     
    ėIv sw j d,m vsw j d,m ėIvsw j q,m vsw j q,m      =      0 1 0 0 -k Iα w j d -c α w j d 0 0 0 0 0 1 0 0 -k Iαw j q -c αw j q      Am j      e Iv sw j d,m v sw j d,m e Ivsw j q,m v sw j q,m      +      0 c α w j d v o sw d 0 c αw j q v o swq      +      -v o sw j d 0 -v o sw j q 0      (4.138)
It is evident that A m j is Hurwitz. If all system parameters are known, α w j ,dq can be developed as

α * w j d = C f w j u cw j [ I w j d C f w j - v sw j d C f w j R f w j + ω w j v sw j q +c α w j d • e v sw j d + k Iα w j d • e Iα w j d ] α * w j q = C f w j u cw j [ I w j q C f w j - v sw j q C f w j R f w j +c α w j d • e v sw j d + k Iα w j d • e Iα w j d ] (4.139)
so that the closed-loop of the system (4.137) becomes

     ėIv sw j d vsw j d ėIvsw j q vsw j q      = A m j      e Iv sw j d v sw j d e Ivsw j q v sw j q      +      0 c α w j d v o sw d 0 c αw j q v o swq      +      -v o sw j d 0 -v o sw j q 0      (4.140)
which exhibits the same behavior as the reference model (4.138). However, in the current study case, since the system parameters are unknown, the controller is designed as

α * w j d = 1 u cw j [ Îw j d - v sw j d Rfw j + Ĉfw j ωw j v sw j q + Ĉfw j (c α w j d • e v sw j d + k Iα w j d • e Iα w j d )] α * w j q = 1 u cw j [ Îw j q - v sw j q Rfw j -Ĉfw j ωw j v sw j d + Ĉfw j (c αw j q • e vsw j q + k Iαw j q • e Iαw j q )] (4.141)
which is based on the estimate of the parameter (•) denoted by (•). Due to the nonlinear terms 1 Rfw j and Ĉfw j ωw j , the parameter adjustment of the controller (4.141) is a nonlinear parameterized adaptive control problem. For the sake of simplicity, we perform the change of variables ψ 1 j = I w j d ; ψ 2 j = I w j q ; ψ 3 j = 1 R f w j ; ψ 4 j = C f w j ω w j ;

α * w j d = 1 u cw j [ ψ1 j -ψ3 j v sw j d + ψ4 j v sw j q -Ĉfw j v α w j d ] α * w j q = 1 u cw j [ ψ2 j -ψ3 j v sw j q -ψ4 j v sw j d -Ĉfw j v αw j q ] (4.143)
Now, the parameter adjustment of the controller becomes a linear parameter adaptive control problem. Substituting (4.143) into the zero dynamics (4.125), we get

     ėIv sw j d vsw j d ėIvsw j q vsw j q      =          1 0 0 0 0 Ĉfw j C f w j 0 0 0 0 1 0 0 0 0 Ĉfw j C f w j          A m j      e Iv sw j d v sw j d e Ivsw j q v sw j q      +          0 Ĉfw j C f w j c α w j d v o sw d 0 Ĉfw j C f w j c αw j q v o swq          +         0 0 0 0 1 C f w j 0 - 1 C f w j v sw j d 1 C f w j v sw j q 0 0 0 0 0 1 C f w j - 1 C f w j v sw j q - 1 C f w j v sw j d              ψ1 j ψ2 j ψ3 j ψ4 j      +      -v o sw j d 0 -v o sw j q 0      (4.144)
which can be rewritten as

     ėIv sw j d ėv sw j d ėIvsw j q ėvsw j q      =          1 0 0 0 0 Ĉfw j C f w j 0 0 0 0 1 0 0 0 0 Ĉfw j C f w j          A m j     
e Iv sw j d e v sw j d e Ivsw j q e vsw j q

     +         0 0 0 0 1 C f w j 0 - 1 C f w j v sw j d 1 C f w j v sw j q 0 0 0 0 0 1 C f w j - 1 C f w j v sw j q - 1 C f w j v sw j d              ψ1 j ψ2 j ψ3 j ψ4 j      (4.145)
where ψ(•) = ψ (•) -ψ(•) denotes the error between the parameter's estimated value and its actual value. Although the exact value of C f w j is unknown, we can conrm that C f w j is positive. The classical adaptive law can be given by Ĉfw j = sgn(C f w j ) = 1 ψ1 j = -c 1 j e v sw j d ψ2 j = -c 2 j e vsw j q ψ3 j = c 3 j (e v sw j d v sw j d + e vsw j q v sw j q ) ψ4 j = c 4 j (e vsw j q v sw j d -e v sw j d v sw j q ) (4.146) 112Chapter 4. Control methods based on nonlinear control design tools where the adaptive gains c 1,2,3,4 j are positive. We consider a quadratic positive denite function as

V = 1 2 C f w j (e 2 v sw j d + e 2 v sw j d ) + 1 2 k Iα w j d e 2 Iv sw j d + 1 2 k Iαw j q e 2 Ivsw j q + 1 2c 1 j ψ2 1 j + 1 2c 2 j ψ2 2 j + 1 2c 3 j ψ2 3 j + 1 2c 4 j ψ2 4 j (4.147)
whose derivative is deduced as

V =-c α w j d e 2
v sw j d -c αw j q e 2 vsw j q ≤ 0 (4.148)

Since V is non-positive, we have V (t) ≤ V (t = 0) and hence e v sw j ,dq , e Iv sw j ,dq and ψ1,2,3,4 j are bounded. Furthermore, V can be calculated, which contains only bounded terms. Consequently, V is also bounded and then V is uniformly continuous. Based on Barbalat's lemma, we get V → 0 as t → ∞. It reveals that e v sw j ,dq → 0 as t → ∞. Finally, the adaptive controller is developed, which is composed by (4.143) and (4.146).

4.3.4.2

Step 2: design of α * g i d

From the previous section, we get that, when v sw j ,dq converge to v o sw j dq , the system behavior is governed by the zero dynamics (4.131) whose state variables, u cg 1 , i gρ , u cw j , u ct h and i c , are uncontrolled. Again, we want to regulate these states via α * g i d . In this part, the control algorithm for α * g i d is based on (4.131) where α * g i d are considered as control variables and ᾱ * w j ,dq are considered equal to their steady-state values ᾱw j ,dq = īw j ,dq ūcw j .

We rst consider using the following simple proportional controllers for α *

g i d α * g 1 d = -k P ucg 1 ũcg 1 + ᾱg 1 d α * gρd = -k P i gρd ĩgρd + ᾱgρd (4.149)
with positive control gains k P ucg 1 and k P i gρd . From (4.149), it can be seen that when ũcg 1 and ĩgρd converge to zero, α * g i d also converge to their steady-state values.

Passivity-based control 113

Substituting (4.149) into (4.131) results in the following closed-loop system

C g 1 du cg 1 dt = 1 2L g 1 1 2L g 1 + 3 4 (-k P ucg 1 ũcg 1 + ᾱg 1 d ) 2 {[- u cg 1 R equi,g 1 + H(1, :)i c ] - 3 2 [R g 1 u cg 1 (-k P ucg 1 ũcg 1 + ᾱg 1 d ) 2 -v sg 1 d (-k P ucg 1 ũcg 1 + ᾱg 1 d )]} C gρ di gρd dt = 1 2L gρ 1 2L gρ + 3 4 (-k P i gρd ĩgρd + ᾱgρd ) 2 {[- i gρd R equi,gρ +(-k P i gρd ĩgρd + ᾱgρd )H(ρ, :)i c ] + 3 2 [(v sρd -R gρ i gρd )(-k P i gρd ĩgρd + ᾱgρd ) 2 ]} C w j du cw j dt = 1 2L w j 1 2L w j + 3 4 [(ᾱ * w j d ) 2 + ( ᾱ * w j q ) 2 ] {[- u cw j R equi,w j + H(N + j, :)i c ] - 3 2 [R w j u cw j ((ᾱ * w j d ) 2 + ( ᾱ * w j q ) 2 ) -(v o sw j d ᾱ * w j d + v o sw j q ᾱ * w j q )] C t h du ct h dt = H(N + M + h, :)i c L c k di ck dt = -R c k i c K -H(:, k) T u c (4.150)
To check the stability of the above closed-loop system, we linearize the system (4.150) around the equilibrium point and then get the following Jacobian matrix

J 2 = -Θ 1 C -1 Θ 2 Θ 3 H -L -1 H T Θ -1 3 Θ 4 -L -1 R
where Θ 1,2,3,4 are given by

Θ 1 = diag(θ 1 1 , • • • , θ 1 N +M +P ) Θ 2 = diag(σ 2 1 , • • • , σ 2 N +M +P ) Θ 3 = diag(σ 3 1 , • • • , σ 3 L ) Θ 4 = diag(σ 4 1 , • • • , σ 4 L ) (4.
                                                       θ 1 1 = 1 2L g 1 C g 1 1 2L g 1 + 3 4 (ᾱ g 1 d ) 2 [ 1 R equi,g 1 + 3 2 R g 1 ( ᾱg 1 d ) 2 + 3 2 (v sg 1 d -2R g 1 īg 1 d )k P ucg 1 ], θ 1 ρ = 1 2L gρ C gρ 1 2L gρ + 3 4 (ᾱ gρd ) 2 [ 1 R equi,gρ + 3 2 R gρ (ᾱ gρd ) 2 + k P i gρd [ 3 2 • 2(v sgρd -R gρ īgρd )ᾱ gρd + H(ρ, :) īc ], ρ ∈ N -1 θ 1 N +j = 1 2L w j C w j 1 2L w j + 3 4 [(ᾱ w j d ) 2 + ( ᾱw j q ) 2 ] { 1 R equi,w j + 3 2 R w j ((α * w j d ) 2 + (α * w j q ) 2 ), j ∈ M θ 1 (N +M +h) = 0. h ∈ P                                                θ 2 1 = 1 2L g 1 1 2L g 1 + 3 4 (ᾱ g 1 d ) 2 , θ 2 ρ = 1 2L gρ 1 2L gρ + 3 4 (ᾱ gρd ) 2 , ρ ∈ N -1 θ 2 N +j = 1 2L w j 1 2L w j + 3 4 [(α * w j d ) 2 + (α * w j q ) 2 ] , j ∈ M θ 2 (N +M +h) = 1. h ∈ P            θ 3 1 = 1, θ 3 ρ = ᾱgρd , ρ ∈ N -1 θ 3 N +j = 1, j ∈ M θ 3 (N +M +h) = 1. h ∈ P            θ 4 1 = 1, θ 4 ρ = 1 + k P i gρd ūcgρ , ρ ∈ N -1 θ 4 N +j = 1, j ∈ M θ 4 (N +M +h) = 1. h ∈ P (4.152)
When the system is in steady state, we have

- i gρd R equi,gρ + ᾱgρd H(ρ, :)i c - 3 2 [R gρ i gρd (ᾱ gρd ) 2 -ᾱgρd (v sgρd ᾱgρd )] = 0 which leads to H(ρ, :) īc = ūcgρ R equi,gρ + 3 2 (R gρ i gρd -v sgρd ) ᾱgρd (4.153) Substituting (4.153) into θ 1 ρ in (4.152) yields θ 1 ρ = 1 2L gρ C gρ 1 2L gρ + 3 4 ( ᾱgρd ) 2 { 1 R equi,gρ + 3 2 R gρ (ᾱ gρd ) 2 +k P i gρd [ 3 2 (v sgρd -R gρ īgρd )ᾱ gρd + ūcgρ R equi,gρ ]} (4.154)
To ensure the stability of the closed-loop system (4.150), the control gains must render the Jacobian matrix J 2 Hurwitz.

Lemma 4.3.7. By choosing k P i gρd such that

1 + k P i gρd ūcgρ > 0 1 R equi,gρ + 3 2 R gρ (ᾱ gρd ) 2 + k P i gρd [ 3 2 (v sgρd -R gρ īgρd )ᾱ gρd + ūcgρ R equi,gρ ]> 0 (4.155)
J 2 is Hurwitz.

Proof. When k P ig ρ d satises (4.155), then Θ 4 becomes positive dene. By the definition of Θ 1,2 (see (4.151)), both of them are diagonal and positive denite. By following the similar procedure used in the proof of Lemma 4.3.6, we can prove that J 2 is Hurwitz.

The above description clearly shows that the simple proportional controller (4.149) can stabilize the zero dynamics (4.131). To get good tracking performance, we use the PI controllers to replace the proportional controllers as

ėIucg 1 = ũcg 1 α * g 1 d = -k P ucg 1 ũcg 1 -k Iucg 1 e
u cg 1 = u o cg 1 = 150 V i g 2 d = i o g 2 d = 13.1 A i g 1 q = i o g 1 q = 0 A i g 2 q = i o g 2 q = 0 A WAC 1 WAC 2 v sw 1 d = v o sw 1 d = 40.82 V v sw 2 d = v o sw 2 d = 40.82 V v sw 1 q = v o sw 1 q = 0 V v sw 2 q = v o sw 2 q = 0 V 4.3.

Simulation studies

In this section, we rst apply the passive control scheme which is composed of (4.124), (4.127) and (4.149) to the MTDC system described by Fig. 4.21 and then apply the adaptive passive control scheme which consists of (4.124), (4.143), (4.146) and (4.149) to the same MTDC system.

Performance evaluation of the passive control strategy

The proposed passive control scheme composed of (4.124), (4.127) and (4.149) is applied to the MTDC system described by Fig. 4.21 in Section 4.1.4 where two WACs and two SACs are considered. To evaluate the performance of the proposed control scheme, several scenarios are studied in the simulations.

Scenario 1: DC voltage regulation and power reversal operation

Initially, the MTDC system works in the steady state as illustrated in Table 4.9. At t = 0.5 s, i o g 2 d is set to -6.53 A from 13.1 A. It means that the 2 nd SAC connected VSC is required to operate in the inversion mode as a power consumer from the rectication mode as a power supplier. Then, at t = 1 s, a new reference value of u cg 1 is given with an increase of 5%.

The simulation results are shown in Figs. 4.39-4.44. As shown in Fig. 4.39, both controllers can make u cg 1 always track its reference u o cg 1 . However, when the power reversal happens in the 2 nd SAC connected VSC terminal, the trajectory of u cg 1 under the feedback nonlinear controller drops very fast and has an unacceptable undershoot. Compared to the feedback nonlinear controller, the passivity-based controller gives a much better performance with a faster convergence and keeps u cg 1 within its acceptable region. As illustrated in Figs. 4.40 and 4.41, the changes of u o cg 1 and i o g 2 d have negligible eects on the performance of i g 1 q (and i g 2 q ) under feedback nonlinear controller while remarkable overshoot (and undershoot) appears when the passivity-based controller is used. These phenomena imply that the feedback nonlinear controller always provides a better decoupling characteristics of the DC voltage (or the AC daxis current) and the AC qaxis current control than the passivity-based controller.

Scenario 2: AC voltage regulation

To evaluate the capability of the passivity-based controller in terms of AC voltage regulation, the step changes of v o sw 1 d and v o sw 2 q are considered in this scenario. Sim- ilar to Scenario 1, the MTDC system initially operates in the steady state provided by Table 4.9. At t = 0.5 s and t = 1 s, v o sw 1 d and v o sw 2 q are changed to 61.24 V and 20.41 V respectively. The simulation results are plotted in Figs. 4.45-4.49. The transient response of u cg 1 is illustrated in Fig. 4.45. It is evident that the performance of u cg 1 under the passivity-based controller is much better than the feedback nonlinear controller. Less oscillations, smaller overshoots and faster response convergence are found in the trajectory of u cg 1 under the passivity-based controller.

Similar to the results in Scenario 1, clearly show that the feedback nonlinear controller gives a better decoupling characteristics of the AC daxis and qaxis voltage control.

Performance evaluation of the adaptive passive control strategy

Simulations are carried out to compare the performance between the adaptive controller and the nominal controller where R f w 1 and R f w 2 are unknown. For the nominal controller, R f w 1 and R f w 2 are considered equal to their nominal values R f w 1 ,nom = 5 Ω and R f w 2 ,nom = 8 Ω respectively. The MTDC system initially operates in the steady state provided by Table 4.9. Then, a sequence of events applied to R f w 1 and R f w 2 are depicted in Figs.

Chapter conclusions

This chapter has presented dierent control strategies for VSC HVDC systems. The input-output feedback linearization technique is rst applied to the control design.

Comparing to the conventional PI controller, the feedback nonlinear controller gives a better decoupling characteristics of the system state control. Since the performance of the feedback nonlinear system is relevant to the system structure and parameters, in order to make the system be more robust, sliding mode control is used combined with feedback linearization. Due to the use of sliding mode control, the uncertainties of the system parameters and other possible exogenous inputs are taken into account. As stated in Section 4.1.4, if it is natural to choose the controlled variables as the output (see (4.56)), the input-output feedback linearization is applicable only when the system is (weakly) minimum phase (see Lemma 4.1.2).

This restricts the choice of those prescribed references. A common assumption used in the vector current control design (see Section 3.5.2) of VSCs is that the reactor phase currents have much faster dynamics than the capacitor DC voltage. Hence, the two loops, i.e. the inner and the outer loop, can be designed independently. Interestingly, no relevant work has ever mathematically explained the validity and the implications of this hypothesis. The main idea of this chapter is to provide a detailed theoretical analysis, inspired by a long stand practice that consists of empirical design of two control loops for the VSC terminals. Furthermore, experience has shown that such loops, when heuristically tuned, often display very dierent dynamics. Here, we acknowledge this practice and give 132 Chapter 5. Control induced time-scale separation explanation and fundamental analysis on why and how this empirical control practice works, and how to rigorously design the algorithms. In particular, we formalize this intuition from practitioners into a control scheme that articially induces two dierent time scales to the closed-loop system. From the theoretical analysis, the time-scale separation between the system state variables are clearly illustrated by means of singular perturbation theory. In this way, the controller allows the fullscale system to be divided into a boundary-layer and a reduced model. These two models are the key point in the designing of the full control scheme, and they greatly reduce the complexity of the task. Furthermore, in this chapter, our analysis allows to determine strict attraction regions based on well determined control gains, such that there is a clear trade-o between system performance (actuators constraints and as a consequence cost) and the size of the region of attraction of the controllers.

5.1 Control induced time-scale separation for a class of nonlinear systems

Introduction and motivation

This section aims at providing a multi-layered control architecture for a class of underactuated nonlinear systems.

The control of such systems, for example several robotic systems and even the well known pendulum-cart problem [START_REF] Spong | [END_REF], Spong 1995, Ortega 2002], is classical. Nevertheless, it is rather difcult to propose systematic control design approaches for them. One of the main reasons for this diculty comes from the interconnection between subsystems. In fact, most analytical approaches based on rigorous mathematical techniques rely on trajectory design. However, the trajectories are usually dicult to develop due to the subsystems' interaction. Furthermore, if additional subsystems are added or control goals are required, the existing control algorithms need to be re-designed, which involves extra re-engineering eorts. This motivates the development of a more exible and more functional control structure.

Inspired by "plug and play philosophy" [START_REF] Pernebo | Plug and play in control loop design[END_REF], Bendtsen 2013, Riverso 2013], we desire to build a control structure that consists of a "fundamental" control module and various "outer" control modules as shown in Fig. 5.1. In this way, when a new objective, such as Objective 1 (see Fig. tures [START_REF] Simon | [END_REF]]. In a mathematical sense, dynamical systems that exhibit multitime-scale behavior can be described as singularly perturbed systems. The most prominent feature of singularly perturbed systems is that the derivatives of their partial states are multiplied by a small positive parameter ε which is used to quantify the time-scale decomposition [Del Vecchio 2013, Chen 2013, Yang 2011, Petersen 2013[START_REF] Teel | [END_REF]]. Based on singular perturbation theory, lower-order subsystems in dierent time scales can be derived to approximate the behavior of the original system.

The main results of singular perturbation theory have existed for several decades [Kokotovic 1968, Kokotovic 1976, Kokotovic 1999, Khalil 1996] and they have been widely applied in the context of the dynamic modeling and analysis of power systems [Kokotovic 1980, Winkelman 1980, Xu 1998[START_REF] Peponides | Singular perturbations and time scales in nonlinear models of power systems[END_REF][START_REF] Martinez | [END_REF], Xin 2010[START_REF] Pekarek | [END_REF], Xu 2000]. In [Kokotovic 1980], an iterative approach for time-scale separation is proposed, which improves the accuracy of the lower-order subsystems. Furthermore, an application of this iterative separation method is presented in [Winkelman 1980] where a 20 th order model of a three-machine power system is analyzed as a two-time-scale and a four-time-scale systems. Reference [Xu 1998] investigates the reformulation of a full model of generators and their controllers, which demonstrates that singular perturbation method can provide an eective approach to simulating the system behavior. Recently, singular perturbation theory is also employed in analyzing some power electronics. In [Kimball 2008],

singular perturbation theory is applied to analyze DC-DC converters and a rigorous demonstration of the time-scale separation is also given. In those prior works, the considered power systems or power converters can be usually modeled by a standard form of singularly perturbed system, which means that ε can be directly obtained from the system model. However, many nonlinear systems are not expressed in the standard form. It means that ε cannot be deduced directly from the system parameters. In this case, high-gain technique, which has the ability to reduce some eects of disturbances, provides a design approach such that the multi-time-scale dynamics can be created by the control gains. There are many studies devoted to the use of high-gain feedback applied to control problems [Khalil 1987, Marino 1985, Young 1977, Marino 1988, Krishnamurthy 2007, Krishnamurthy 2013]. Our inter-134 Chapter 5. Control induced time-scale separation est here is the linkage between the high-gain technique and the singularly perturbed system. In general, a nonlinear system with a high-gain control can be analyzed as a singular perturbed system [Marino 1985, Krishnamurthy 2013].

In this thesis, we further investigate the use of high-gain and singular perturbation techniques to develop a systematic control method for a class of nonlinear systems with multiple inputs and multiple outputs. The control system is desired to separate the original system into dierent time-scale subsystems, with the aim of creating a "plug and play" characteristic. Here we dene a class of nonlinear systems in the form

ẋ= f 1 (x, y) + g 1 (x, y)u ẏ= f 2 (x, y) (5.1)
where x ∈ D x ⊂ R n and y ∈ D y ⊂ R m are the state variables and u ∈ D u ⊂ R n is the control input. f 1,2 and g 1 are continuously dierentiable in their arguments for (t, x, y) ∈ [0, t f ] × D x × D y . This type of nonlinear system is similar to underactuated mechanical system [START_REF] Spong | [END_REF] where the system's degree of freedom is larger than the actuated degree of freedom. In addition, part of the state variables, i.e. x, is collocated with the input u. We rst divide the full system into two subsystems, i.e. a driving subsystem and a driven subsystem represented by the dynamics of

x and y in (5.1), respectively and then, we develop a subcontroller such that the driving subsystem quickly enters an arbitrary manifold x * yet to be designed. During the design process of such subcontroller, the dynamics of x * are neglected. The fundamental control module is then established, which is composed of the driving subsystem and its subcontroller. The interaction between the fundamental and the outer control modules is given by x * . Dierent outer control modules could provide dierent corresponding manifolds. It is true that not all x * from the outer modules can be accepted by the fundamental control module. Therefore, in this work, a rigorous theoretical analysis based on singular perturbation theory is carried out, from which conditions on the design of the fundamental and the outer control modules are derived to ensure the system stability. We further investigate the importance of the time-scale separation between the fundamental and the outer control modules based on Lyapunov theory.

Problem formulation

Considering the nonlinear system (5.1), we assume that it can be expressed as

ẋ1 = f 11 (x 1 , x 2 , y 1 , y 2 ) + g 11 (x 1 , x 2 , y 1 , y 2 )u ẋ2 = f 12 (x 1 , x 2 , y 1 , y 2 ) + g 12 (x 1 , x 2 , y 1 , y 2 )u (5.2) ẏ1 = f 21 (x 1 , x 2 , y 1 , y 2 ) ẏ2 = f 22 (x 1 , x 2 , y 1 , y 2 ) (5.3)
where the system states x and y are partitioned into two vector parts,

x 1 ∈ D x 1 ⊂ R p , x 2 ∈ D x 2 ⊂ R n-p , y 1 ∈ D y 1 ⊂ R m+p-n and y 2 ∈ D y 2 ⊂ R n-p .
p is a nonnegative integer and satises n -m ≤ p ≤ n. In addition, D x and D y satisfy D x = 5.1. Control induced time-scale separation for a class of nonlinear systems 135

{(x 1 , x 2 ) | x 1 ∈ D x 1 and x 2 ∈ D x 2 } and D y = {(y 1 , y 2 ) | y 1 ∈ D y 1 and y 2 ∈ D y 2 }.
The functions f ij and g 1j with i, j = 1, 2 are continuously dierentiable in their arguments. We dene the control output as

s [x 1 y 2 ] T ∈ R n (5.4) Let s o , i.e. [x o 1 y o 2 ]
T , be a prescribed setpoint. The main purpose of this work is to establish a exible control scheme that regulates s at the prescribed setpoint s o and at the same time, ensures the system stability. Here, we rst introduce the following assumptions to describe the characteristics of the system (5.2)-( 5.3).

Assumption 5.1.1. For a given

s o = [x o 1 y o 2 ]
T , there exist corresponding steady values of x1 , x2 , ȳ1 , ȳ2 and ū such that

0= f 11 (x 1 , x2 , ȳ1 , ȳ2 ) + g 11 (x 1 , x2 , ȳ1 , ȳ2 )ū 0= f 12 (x 1 , x2 , ȳ1 , ȳ2 ) + g 12 (x 1 , x2 , ȳ1 , ȳ2 )ū 0= f 21 (x 1 , x2 , ȳ1 , ȳ2 ) 0= f 22 (x 1 , x2 , ȳ1 , ȳ2 ) x o 1 = x1 y o 2 = ȳ2 (5.5) where x1 ∈ D x 1 , x2 ∈ D x 2 , ȳ1 ∈ D x 1 and ȳ2 ∈ D y 2 . We denote P1 = [x 1 x2 ȳ1 ȳ2 ] T
as the equilibrium point.

Under Assumption 5.1.1, the existence of the equilibrium point of the system (5.2)-( 5.3) is ensured, which implies the physical feasibility of the system.

Assumption 5.1.2. f ij , g 1j and their partial derivatives up to the rst order are continuously dierentiable and bounded in [0,

t f ] × D x 1 × D x 2 × D y 1 × D y 2 where i, j = 1, 2.
Assumption 5.1.3. For (x, y) ∈ D x × D y , g 1 = [g 11 g 12 ] T is non-singular and g -1 1 is continuously dierentiable.

With Assumptions 5.1.2-5.1.3, the functions f ij , g 1j and g -1 1 are locally Lipschitz, as continuous functions are locally bounded and hence their gradients are locally bounded as well.

Assumption 5.1.4. Consider the following subsystem with Assumption 5.1.1

ẏ1 = f 21 (x o 1 , x 2 , y 1 , y 2 ) ẏ2 = f 22 (x o 1 , x 2 , y 1 , y 2 ) (5.6)
where x 2 can be viewed as the auxiliary control input. It is obvious that P2 = [ȳ 1 ȳ2 ] T is an equilibrium of the subsystem (5.6) as x 2 = x2 . We suppose that information of

x 2 = h 2 (y 1 , y 2 , x o 1 , y o 2 ) ∈ D
x 2 is available such that: 1) The function h 2 yet to be designed has continuous rst partial derivatives with respect to its arguments. In addition, when y 2 = y o 2 , we have

h 2 (ȳ 1 , ȳ2 , x o 1 , y o 2 ) = x2 136 Chapter 5. Control induced time-scale separation 2) P2 = [ȳ 1 ȳ2 ]
T is an equilibrium of the following closed-loop system

ẏ1 = f 21 (x o 1 , h 2 , y 1 , y 2 ) ẏ2 = f 22 (x o 1 , h 2 , y 1 , y 2 ) (5.7)
Moreover, P2 is exponentially stable in a region (y 1 , y 2 ) ∈ B y 1 × B y 2 where B y 1 ⊂ D y 1 and B y 2 ⊂ D y 2 .

To get a better understanding of Assumption 5.1.2-5.1.4, we consider the following example

ẋ1 ẋ2 = x 1 x 2 + x 1 y 1 -2 x 2 1 + x 2 y 1 -1 + x 1 0 y 1 x 1 x 2 u 1 u 2 ẏ1 = x 1 y 1 -x 2 s= [x 2 y 1 ] T (5.8) where x 1,2 ∈ [0.5, 1.5], y 1 ∈ [0.5, 2] and s o = [1 1] T .
Assumption 5.1.2 is satised due to the boundedness of x 1,2 and y 1,2 . Since (x 1 , x2 , ȳ1 , ȳ2 , ū1 , ū2 ) = (1, 1, 1, 1, 0, -1) that satises (5.8), Assumption 5.1.1 is also veried. The inverse matrix of g 1 can be deduced as:

g -1 1 = 1 x 1 0 -y 1 x 2 1 x 2 1 x 1 x 2
which is continuously dierentiable for x 1,2 ∈ [0.5, 1.5] and y 1 ∈ [0.5, 2]. Thus, Assumption 5.1.3 is satised. When x 2 = 1, the subsystem of (5.8) becomes: ẏ1 = x 1 y 1 -1

(5.9) In this way, x 1 is the auxiliary control input of (5.9). Then, x 1 can be designed as:

x 1 = h 2 = -(y 1 -1) + 1 y 1
such that the closed-loop system of (5.9) becomes:

ẏ1 = -(y 1 -1)
whose equilibrium ȳ = 1 is exponentially stable. Finally, Assumption 5.1.4 is veried.

Control design

In general, control algorithms are strongly related to the system control objectives.

Once these control objectives are changed, the existing controller usually has to be decommissioned and then, new control algorithms need to be re-engineered from scratch. For example, considering the studied system (5.2)-(5.3), there exist two dierent outputs, i.e.

s 1 = [x 1 1 y 1 2 ] T ∈ R n and s 2 = [x 2 1 y 2 2 ] T ∈ R n where x 1 1 ∈ R p 1 , x 2
1 ∈ R p 2 and p 1 = p 2 . We apply input-output feedback linearization methodology 5.1. Control induced time-scale separation for a class of nonlinear systems 137 to design the control scheme. When the system is required to track the reference from s 1 to s 2 , the entire control scheme must be redesigned since the corresponding Lie derivatives are totally changed. Here, we aim at establishing a more exible control scheme such that dierent control objectives can be achieved based on the existing control structure.

The described system (5.2)-( 5.3) has fewer control inputs (n) than state variables (n + m). In addition, u has an explicit eect on x and they have the same degree of freedom. The above aspects prompt us to divide the full system into two interconnected subsystems. The state variable x collocated with the control input u is called the driving state. We denote the dynamics of x 1,2 described by (5.2) as the driving subsystem. As seen in ( 5.3), x 1 and x 2 can be viewed as the control inputs, which implies that u indirectly inuences the behavior of y through x. Hence, y and (5.3) is called the driven state and the driven subsystem. Such solution to the division of the full system into two parts is inspired by underactuated mechanical systems [START_REF] Spong | [END_REF]].

We suppose that the proposed control scheme can make the system (5.2)-( 5.3) exhibit a multi-time-scale behavior where the driving subsystem has much faster dynamics than the driven subsystem. Based on this assumption, the two subsystems can be then designed independently.

Design of the driving subsystem

Step 5.1.5. For a given x 0 ∈ D x , let us impose the following autonomous system ẋ = -A 1 (x -x 0 ) + r 1 (x, x 0 )

(5.10)

where A 1 = diag(a 1i ) ∈ R n×n yet to be designed. We denote a 1 = min(a 11 , • • • , a 1n ).

The system (5.10) is designed as 1. r 1 (x, x 0 ) is a continuously dierentiable function and r 1 (x 0 , x 0 ) = 0. As a consequence, for a positive γ 1 , there exists b 1 > 0 such that, when

x ∈ B x 1 = {x ∈ D x | ||x -x 0 || < b 1 }, we have ||r 1 (x, x 0 )|| < γ 1 ||x -x 0 || (5.11)
2. a 1 is assigned to satisfy

γ 1 + c 1 ≤ a 1
where c 1 is a positive parameter.

Lemma 5.1.6. The equilibrium point x = x 0 of (5.10) is exponentially stable in

B x 1 .
Proof. We choose the Lyapunov function as

V 1 (x -x 0 ) = 1 2 (x -x 0 ) T (x -x 0 )
Chapter 5. Control induced time-scale separation

The system (5.18) can be rewritten as

ε ẋ= -A 1ε (x -x * ) + εr 1 (x, x * ) ẏ= f 2 (x, y) (5.19)
where A 1ε = diag(εa 1i ). Denoting a 1iε = εa 1i , we then have a 1iε ≥ 1 1 , for all i = 1, • • • , n. When ε is small enough, our problem (5.19) becomes a standard singular perturbation problem. By setting ε = 0, the driving subsystem is degenerated into the algebraic loop, i.e. A 1ε (x -x * ) = 0, from which we can get the isolated root x = x * . We call x * the quasi-steady state of x. By recalling that x * 1 = x o 1 in (5.15) and x * 2 = h 2 in (5.17), we substitute x = x * into the driven subsystem and then

obtain ẏ = f 2 (x o 1 , h 2 (y, x o 1 , y o 2 ), y)
(5.20)

Due to Assumption 5.1.4, the equilibrium point P2 of the reduced model (5.20) is exponentially stable in B y 1 . In addition, we denote the solution of (5.20) as y re .

In order to analyze the closed-loop system (5.19) more conveniently, we introduce the following new variables

x1 = x 1 -x * 1 ; x2 = x 2 -x * 2 ; x = [x 1 x2 ] T ỹ1 = y 1 -ȳ1 ; ỹ2 = y 2 -ȳ2 ; ỹ = [ỹ 1 ỹ2 ] T
In these new variables, the system (5.19) is rewritten as

ε ẋ= -A 1ε x + εr 1 (x + x * , x * ) -ε dx * dt ẏ= f 2 (x + x * , ỹ + ȳ) (5.21) with x * = [x o 1 h 2 (ỹ + ȳ)]
T . The origin is the equilibrium point of the system (5.21). Moreover, there exist maps such that D x → Dx and D y → Dỹ . The (asymptotic) stability of the origin of the error system (5.21) implies the (asymptotic) stability of the equilibrium P1 of the system (5.18).

Let us introduce a new time scale τ which satises

ε dx dt = dx dτ
In the τ time scale, the variables t, ỹ and x * are considered slowly varying. We rewrite the dynamics of x (in (5.21)) in the τ time scale

dx dτ = -A 1ε x + εr 1 (x + x * , x * ) -ε[0 ∂h 2 ∂ ỹ f 2 ] T
When setting ε = 0 and freezing t, ỹ and x * at their initial values, we get the boundary-layer system

dx dτ = -A 1ε x (5.22)
1 See the denition of a1 in (5.10).

Control induced time-scale separation for a class of nonlinear systems 141

whose equilibrium is exponentially stable, uniformly in (t, ỹ). Substituting x = 0, the reduced model of (5.21) is deduced as

ẏ = f 2 (x o 1 , h 2 (ỹ + ȳ, x o 1 , y o 2 ), ỹ + ȳ)
(5.23)

The above system has an exponentially stable equilibrium at the origin for all ỹ(t 0 ) ∈ By 1 where By 1 can be deduced from B y 1 based on Assumption 5.1.4. Now, we can state the following result.

Theorem 5.1.9. Consider the nonlinear system (5.19) with Assumptions 5.1.1-5.1.4 where x ∈ D x and y ∈ D y . For any given A 1ε , there exists a positive ε * and a region R xy ⊂ D x × D y such that for all 0 < ε < ε * and (x(t 0 ), y(t 0 )) ∈ R xy , the trajectories of the state variables can be approximated by

x(t) = x re (t) + x(t/ε) + O(ε) y(t) = y re (t) + O(ε)
where

x re (t) = [x o 1 h 2 (y re (t), x o 1 , y o 2 )].
Proof. As presented in the previous parts:

• The reduced model (5.20) has a unique solution y re that converges to P2 .

• The equilibrium of the boundary-layer model (5.22) is exponentially stable, uniformly in (t, ỹ).

Theorem 5.1.9 is then proved by direct apprehension of Theorem 11.1 in [Khalil 1996].

From the above description, x exhibits a multi-time-scale behavior. During the initial interval, x quickly converges to its manifold x * , which represents a fast dynamic. After the decay of the fast transient x(t/ε), x remains close to x * in the future time. It is notable that the small positive parameter ε in the singularly perturbed system (5.18) is not derived from the physical system, but is created by the subcontroller u = h 1 (x, y, x * ). It is shown that, even though the system has no inherent multi-time-scale dynamics, we can articially create them by the proposed control strategy.

Stability analysis

Theorem 5.1.9 guarantees the existence of the proposed time-scale separation controller. In this section, we carry out a detailed stability analysis, from which sucient conditions on the developed controller can be derived by means of Lyapunov theory.

In order to facilitate the analysis, we rewrite the system (5.21)

ε ẋ1 = -A 1 1ε x1 + εr 11 ε ẋ2 = -A 2 1ε x2 + εr 12 -ε d∆ h 2 ỹ dt ẏ= ∆ f 2 x + φ(ỹ)
0 0 A 2 1ε = A 1ε ; [r 11 r 12 ] T = r 1 φ(ỹ) = f 2 (x o 1 , h 2 (ỹ + ȳ, x o 1 , y o 2 ), ỹ + ȳ) f 2 (x + x * , ỹ + ȳ) = f 2 (x + x * , ỹ + ȳ)-f 2 (x * , ỹ + ȳ) ∆ f 2 x +φ h 2 (ỹ + ȳ, x o 1 , y o 2 ) = h 2 (ỹ + ȳ, x o 1 , y o 2 )-h 2 (ȳ, x o 1 , y o 2 ) ∆ h 2 ỹ + h2 h2 = h 2 (ȳ, x o 1 , y o 2 )
It can be seen that the terms ε d∆ h 2 ỹ dt and ∆ f 2 x in (5.24) reect the dynamic interactions between the driving and the driven subsystems. Moreover, ∆ f 2 x and ∆ h 2

ỹ satisfy ||∆ f 2 x|| → 0 when ||x|| → 0 ||∆ h 2 ỹ|| → 0 when ||ỹ|| → 0 Since h 2 (ỹ + ȳ, x o 1 , y o 2 ) and f 2 (x + x * , ỹ + ȳ) are continuously dierentiable, there exist positive L 1x and L 1ỹ such that ||∆ f 2 x||≤ L 1x ||x|| ||∆ h 2 ỹ||≤ L 1ỹ ||ỹ|| (5.25) for x ∈ Ω 1x ⊂ Dx and ỹ ∈ Ω 1ỹ ⊂ Dỹ .
Since the reduced model (5.23) is exponentially stable in By 1 ⊂ Dỹ , there exists a Lyapunov function V 0 satisfying

c 1 ||ỹ|| 2 ≤ V 2 (ỹ)≤ c 2 ||ỹ|| 2 ∂V 2 ∂ ỹ φ(ỹ)≤ -c 3 ||ỹ|| 2 || ∂V 2 ∂ ỹ ||≤ c 4 ||ỹ|| (5.26) for positive constants c i , i = 1, • • • , 4.
According to Taylor's theorem, φ can be rewritten as

φ(ỹ) = A φ ỹ + g φ (ỹ)ỹ (5.27)
which is called the Peano form of the remainder where We take

A φ = ∂φ ∂ ỹ | ỹ=0 lim ỹ→0 g φ (ỹ) = 0
V = dV 1 (x) + (1 -d)V 2 (ỹ)
as a Lyapunov function candidate for the closed-loop system (5.24) where d ∈ (0, 1).

The rst part of V , V 1 (x), is derived from the boundary-layer model (5.22). According to Lemma 5.1.6, the origin of the system

ẋ = -A 1 x + r 1 (x)
is exponentially stable in Bx 1 which corresponds to B x 1 . The second term V 2 is obtained from the reduced model. The composition of V (x, ỹ) in this way provides us a easier way to search the feasible region of ε.

The derivative of V along the trajectories (5.24) can be then deduced as

V =d[-x T A 1 x + xT r 1 -xT 2 ∂∆ h 2 ỹ ∂ ỹ (∆ f 2 x + φ)] + (1 -d) ∂V 2 ∂ ỹ (∆ f 2 x + φ) (5.30)
Applying the inequalities (5.12) and (5.26) to (5.30), we have

V ≤ -dc 1 ||x|| 2 -(1 -d)c 3 ||ỹ|| 2 + (1 -d) ∂V 2 ∂ ỹ ∆ f 2 x -dx T 2 ∂∆ h 2 ỹ ∂ ỹ (∆ f 2 x + A φ ỹ + g φ ỹ) ≤ -dc 1 ||x|| 2 -(1 -d)c 3 ||ỹ|| 2 + (1 -d)|| ∂V 2 ∂ ỹ || ||∆ f 2 x|| + d||x 2 || || ∂∆ h 2 ỹ ∂ ỹ || [ ||∆ f 2 x|| + ( ||g φ || + ||A φ || )||ỹ|| ] (5.31) When x ∈ Bx 1 ∩ Ω 1x , ỹ ∈ By 2 ∩ By 1 ∩ Ω 1ỹ ∩ Ω 2ỹ
V ≤ -dc 1 ||x|| 2 -(1 -d)c 3 ||ỹ|| 2 + (1 -d)c 4 L 1x ||ỹ|| ||x|| + dL 1ỹ L 1φ ||x 2 || ||x|| + dL 1ỹ (L 1φ + σ max (A φ ))||x 2 || ||ỹ|| ≤ -dc 1 ||x|| 2 -(1 -d)c 3 ||ỹ|| 2 + (1 -d)c 4 L 1x ||ỹ|| ||x|| + dL 1ỹ L 1φ ||x|| 2 + dL 1ỹ (L 1φ + σ max (A φ ))||x|| ||ỹ|| = -d(a 1 -γ 1 -L 1ỹ L 1φ )||x|| 2 -(1 -d)c 3 ||ỹ|| 2 + k 1 ||x|| ||ỹ|| (5.32)
where

k 1 = (1 -d)c 4 L 1x + dL 1ỹ (L 1φ + σ max (A φ )). Considering Young's inequality for the product ||x|| ||ỹ|| ≤ 1 2m ||x|| 2 + m 2 ||ỹ|| 2 (5.33)
where m is positive, the derivative of V satises

V ≤ -k 2 ||x|| 2 -k 3 ||ỹ|| 2 (5.34)
where

k 2 = d(a 1 -γ 1 -L 1ỹ L 1φ ) - k 1 2m k 3 = (1 -d)c 3 - mk 1 2
Theorem 5.1.10. Consider the nonlinear system (5.2)-(5.4) with Assumption 5.1.1-5.1.4. Select the control gain matrix A 1 of the subcontroller (5.13) such that k 2 and k 3 are positive. Then, the equilibrium point (x o 1 , x2 , ȳ1 , y o 2 ) of the system (5.2)-(5.4) is asymptotically stable under the proposed time-scale separation control scheme (5.13), (5.15) and (5.17). Moreover, we can obtain an estimation of the region of attraction

R = {(x, ỹ) ∈ Dx × Dỹ |V (x, ỹ) < C} where C = min x∈R x, ỹ∈R ỹ V (x, ỹ) with R x = Bx 1 ∩ Ω 1x and R ỹ = By 1 ∩ By 2 ∩ Ω 1ỹ ∩ Ω 2ỹ .
Proof. The proof is simple. For any xed c 3 , we can always nd an m such that k 3

is positive. Once m is chosen, k 2 is positive by choosing a 1 > a * 1 = k 1 2dm (γ 1 + L 1ỹ L 1φ ) (5.35) and as a consequence, ε < ε * = 1 a * 1 .
For all (x, ỹ) ∈ R, it follows that for all solutions starting from R, V is nonpositive. Hence, R is the region of attraction [Khalil 1996] and the equilibrium point of the system (5.2)-(5.4) is stable. Since (x, ỹ) = (0, 0) is the only invariant set such that V = 0 by solving (5.34), all solutions starting from R converge to the equilibrium point. Finally, the asymptotic stability is established and a feasible region (5.35) for a 1 ( and then for ε) is also found.

Control induced time-scale separation for a class of nonlinear systems 145

We note that the feasible region (5.35) of a 1 strongly depends on the choice of subcontroller x * 2 = h 2 (y, x o 1 , y o 2 ). If we want to stabilize the reduced model (5.23) in a larger region, we may need a larger L 1φ , c 3 and c 4 . Alternatively, if we want to accelerate the speed of the convergence of the reduced model, it is necessary to increase σ max (A 1φ ). Such scenarios may result in a larger a * 1 . However, sometimes, a very large a 1 may cause a large peak value of the control input u, which is not desired. Thus, it is very important to consider associated factors when we choose a 1 .

Study cases

In order to get a better understanding of the proposed control algorithm, two examples are studied in this section.

Example 1

Consider the following nonlinear system We rst use the input-output feedback linearization method to design the control structure and then apply the proposed time-scale separation control scheme.

ẋ1 = x 1 x 2 + x 3 u ẋ2 = x 1 + x 3 ẋ3 = x 2 + x 3 (5.
Feedback linearization control: When x 3 is the output, the system (5.36) is feedback linearizable since the relative degree of y = x 3 is three. Then, a controller based on input-output feedback linearization method can be developed

u = 1 L g L 2 f (y) (-L 3 f (y) + v) (5.37) where v = -b 0 (x 3 -x o 3 ) -b 1 L 1 f (y) -b 2 L 2 f (y) L 1 f (y) = x 2 + x 3 L 2 f (y) = x 1 + x 2 + 2x 3 L 3 f (y) = x 1 x 2 + x 1 + 2x 2 + 3x 3 L g L 2 f (y) = x 3
The controller (5.37) is tested by numerical simulations as illustrated in Fig. 5.2.

The parameter values are listed in Table 5.1. The simulation results clearly show that the controller (5.37) derived from the input-output feedback linearization method can make the system track its output trajectory.

Next, we consider that Objective 2 is to be achieved, i.e. make x 2 track its reference y o = 2. In this case, the relative degree of y = x 2 is two, smaller than three. Thus, the zero dynamics exist and satisfy

ẋ3 = x 3 + y o (5.38)
As seen in (5.38), since the zero dynamics are unstable, the system (5.36) is not feedback linearizable [Isidori 1995].

From the above description, when the control objective is changed, the inputoutput feedback linearization method is not practicable and we must re-design the entire control algorithm to achieve the new control objective. In the following section, we demonstrate that the proposed time-scale separation control method can realize these two objectives by just modifying the outer control modules. Time-scale separation control: Since x 1 is directly collocated with the control input u, x 1 is considered as the driving state while x 2 and y are the driven states. The task of the fundamental control module is to make x 1 quickly enter the manifold x * 1 yet to be designed. Here, let us impose the following autonomous

system ẋ1 = x 1 x 2 + x 3 u = -k 1 (x 1 -x * 1 ) (5.39)
where r 1 = 0. Then u can be obtained by solving (5.39)

u = 1 x 3 [-k 1 (x 1 -x * 1 ) -x 1 x 2 ]
(5.40)

Note that the dynamics of x * 1 is neglected in u. When x 1 quickly enters its manifold x * 1 , we get the reduced model

ẋ2 = x * 1 + x 3 ẋ3 = x 2 + x 3 (5.41)
where x * 1 is considered as an additional control input for the reduced model (5.41).

In order to achieve Objective 1, we only need that the reduced model converges to

x o 3 . Re-applying time-scale separation to (5.41), we design x * 1 such that x 2 quickly enters its manifold x * 2 yet to be designed.

ẋ2 = x * 1 + x 3 = -k 2 (x 2 -x * 2 ) Then, x * 1 is obtained as x * 1 = -k 2 (x 2 -x * 2 ) -x 3 (5.
k 1 k 2 k 3 Set 1 4 2 2.25 Set 2 20 2 2.25
Finally, x * 2 can be designed as

x * 2 = -k 3 (x 3 -x o 3 ) -x 3 (5.44)
such that the equilibrium of the sub-reduced model (5.43) is exponentially stable.

In this case, the controller is composed by (5.40), (5.42) and (5.44). The system exhibits a three-time-scale behavior under the proposed controller.

Simulations are carried out with two sets of control gains as listed in Table 5.2

where k 1 in Set 2 is larger than in Set 1. Simulation results are presented in Fig. During the design process, the time-scale separation algorithm is applied twice and x 1 and x 2 exhibit three-time-scale and two-time-scale behaviors, respectively.

This example reveals that we can divide a full-scale system into multi-time-scale subsystems by re-using the proposed control approach such that these subsystems can be designed independently. Now, when Objective 2 is to be achieved, we just need to design a new x * 1 based on the reduced model (5.41) as

x * 1 = -k 2 (x 2 -x o 2 ) -(k 3 + 1)(x 3 + x o 2 ) + x o 2 which can make x 2 in the reduced model converge to x o 2 .
Simulations are also carried out with two sets of control gains as listed in Table 5.3. As illustrated in Fig. 5.4, we get some similar results as in the case of Objective 1. The actual values of the state variables could be better approximated by their respective quasi-steady-state values with a larger k 1 .

As mentioned in the previous section, although a larger k 1 can make the system give a fast response, it may also lead to a larger peak value of the control input during the transient. As depicted in 5.3(d)-5.4(d), the crest value of u with k 1 = 20 is larger than that with k 1 = 4. The second example is expressed as

ẋ1 = x 1 x 2 + x 3 u ẋ2 = x 1 + x 3 ẋ3 = x 2 -x 3 (5.45)
which is required to achieve the same control objectives as in Example 1.

Feedback linearization control: When x 3 is dened as the output in Objective 1, (5.45) is feedback linearizable and hence, a controller by means of input/output feedback linearization is developed as

u = 1 L g L 2 f (x 3 ) (-L 3 f (x 3 ) + v) (5.46) where v = -b 0 (x 3 -x o 3 ) -b 1 L 1 f (x 3 ) -b 2 L 2 f (x 3 ) L 1 f (x 3 ) = x 2 -x 3 L 2 f (x 3 ) = x 1 x 2 + 2x 3 L 3 f (x 3 ) = x 1 x 2 -x 1 + 2x 2 -3x 3 L g L 2 f (x 3 ) = x 3
When Objective 2 is to be achieved, the relative degree of y = x 2 is also 2 as in Example 1. However, the zero dynamics here are expressed by

ẋ3 = -x 3 + x o 2 (5.47)
whose equilibrium is exponentially stable. Hence, input/output feedback linearization can be applied. The controller is developed as

u = 1 L g L f (x 2 ) (-L 2 f (x 2 ) + v) (5.48) where v = -b 0 (x 2 -x o 2 ) -b 1 L 1 f (x 2 ) L 1 f (x 2 ) = x 1 + x 3 L 2 f (x 2 ) = x 1 x 2 + x 2 -x 3 L g L f (x 2 ) = x 3
It is clear that, when the new control objective (Objective 2) is to be achieved, the existing control law (5.46) becomes invalid and should be rejected. The control design must be re-engineered to develop a new control algorithm (5.48). For a system of very high order, it is very time-consuming to re-design the control structure. However, the time-scale separation control scheme proposed in this section can eectively avoid a control re-design from scratch.

ẋ1 = x 1 x 2 + x 3 u = -k 1 (x 1 -x * 1 )
(5.49)

and then u can be obtained by solving (5.49)

u = 1 x 3 [-k 1 (x 1 -x * 1 ) -x 1 x 2 ]
When x 1 quickly enters its manifold x * 1 , the following reduced model is generated

ẋ2 = x * 1 + x 3 ẋ3 = x 2 -x 3
In order to attain the control objective 1, the same procedure is used to design x * 1 as in Example 1

x * 1 = -k 2 (x 2 -x * 2 ) -x 3 where x * 2 satises x * 2 = -k 3 (x 3 -x o 3 ) + x 3
If the control objective 2 is to be achieved, we just need to develop a new x * 1 as

x * 1 = -k 2 (x 2 -x o 2 )
for the fundamental module without decommissioning the entire existing control structure.

From this example, the advantage of the proposed control structure is explicitly presented. In order to make the system achieve dierent control objectives, we only need redesign the subcontroller of the reduced model and provide the new manifold

x * to the fundamental control module.

In the following sections, we will apply the obtained theoretical results to explain why the vector control structure for VSC HVDC systems can consist of two control loops with dierent dynamics and to demonstrate how the system exhibit a multitime-scale performance.

Control induced time-scale separation for MTDC systems using master-slave control conguration

In the present section, based on the previous theoretical results, we give a detailed description on how the MTDC system exhibits multiple time-scale dynamics using master-slave control conguration. Furthermore, a theoretical stability study is also performed where the stability limitations on this control conguration are also analyzed.

5.2. Control induced time-scale separation for MTDC systems using master-slave control conguration 153

Control design

We consider each VSC terminal as an individual unit, which is of the form

ẋ1[i] = f 1[i] (x 1[i] ) + g 1[i] (x 1[i] , u cg i )u 1[i] ẋ2[j] = f 2[j] (x 2[j] , y 2[j] ) + g 2[j] (x 2[j] , y 2[j] , u cw j )u 2[j]
(5.50)

ẏ2[j] = f 3[j] + I w [j]
(5.51)

where we use the following denitions

x 1[i] i g i d i g i q T ; x 2[j] i w j d i w j q T u 1[i] m g i d m g i q T ; u 2[j] m w j d m w j q T y 2[j] v sw j d v sw j q T ; I w [j] I w j d I w j q T (5.52) The expressions of f 1[i] , f 2,3[j] , g 1[i]
and g 2[j] are obviously obtained from the MTDC system described by (3.20), (3.28) and (3.29), which are continuous dierentiable in their respective arguments for i g i ,dq

∈ D i g i ,dq , i w j ,dq ∈ D i w j ,dq , v sw j ,dq ∈ D v sw j ,dq , u c(•) ∈ D uc and m g i ,dq , m w j ,idq ∈ [-1, 1].
For convenience, we also introduce

x [x 1[1] • • • x 1[N ] x 2[1] • • • x 2[M ] ] T = [x 1[i] x 2[j] ] T u [u 1[1] • • • u 1[N ] u 2[1] • • • u 2[M ] ] T = [u 1[i] u 2[j] ] T
Therefore, the overall model of the MTDC system described in Chapter 3 can be expressed by a general form (5.50), (5.51) and (3.36). Now, we are ready to develop the control algorithm by following the procedure presented in Section 5.1 as

•
Step 1: We divide the full-scale system into two subsystems, i.e. the driving subsystem (5.50) and the driven subsystem (5.51) and (3.36), where the driving subsystem is directly collocated with the control inputs u 1[i] and u 2[j] .

•

Step 2: Assume that the proposed control algorithm is explicitly designed with the purpose of making the driving subsystem have much faster dynamics than the driven subsystem. Based on the dynamic separation, we then obtain a boundary-layer and a reduced model where two subcontrollers are designed to control both models, respectively.

• Step 3: Substituting the two subcontrollers into the original model, a standard singularly perturbed system can be deduced. Applying singular perturbation theory, we derive sucient conditions on the proposed control algorithm to ensure the asymptotic stability of the MTDC system.

Note that, in

Step 2, we assume that the state variables in the driving subsystem have faster dynamics than those in the driven subsystem. In fact, this assumption is only used to help us design the control system. In the section of theoretical analysis, we will demonstrate that the controller based on this assumption indeed makes the system exhibit a dynamic behavior characterized by the presence of fast and slow transients.

154 Chapter 5. Control induced time-scale separation

Design of the driving subsystem

We assume that there exists a reference trajectory for the driving subsystem as

x * [x * 1[1] • • • x * 1[N ] x * 2[1] • • • x * 2[M ] ] T
which is yet to be designed. The only information on x * is that it is a function of the state variables of the driven subsystem with slow dynamic. We then design the

subcontrollers u 1[i] = h 1[i] and u 2[j] = h 2[j]
such that the closed-loop of the driven subsystem becomes

ẋ1[i] = v 1[i] ẋ2[j] = v 2[j]
(5.53)

where we introduce the auxiliary inputs v 1[i] and v 2[j] . Since g 1[i] and g 2[j] are invertible, by combining (5.50) and (5.53), the subcontroller of the driving system is designed as

h 1[i] = g -1 1[i] [v 1[i] -f 1[i] ] h 2[j] = g -1 2[j] [v 2[j] -f 2[j] ]
(5.54)

The design of v 1[i] , v 2[j] ∈ R 2 are free. We only need them so that the above autonomous system (5.53) at least locally exponentially converges to x * when x * is constant. To illustrate our control scheme more clearly, we design

ėIx 1[i] = x 1[i] -x * 1[i] v 1[i] = -K 1[i] (x 1[i] -x * 1[i] ) -Γ 1[i] e Ix 1[i] ėIx 2[j] = x 2[j] -x * 2[j] v 2[j] = -K 2[j] (x 2[j] -x * 2[j] ) -Γ 2[j] e Ix 2[j]
(5.55)

where the control gain matrices are of the form

K 1[i] = diag(k 11 [i] , k 12 [i] ); K 2[j] = diag(k 21 [j] , k 22 [j] ) Γ 1[i] = diag(γ 11 [i] , γ 12 [i] ); Γ 2[j] = diag(γ 21 [j] , γ 22 [j] )
(5.56) whose diagonal elements are positive.

Design of the driven system

With the aid of the subcontroller (5.54), we assume that x quickly enters the manifold x * and then substitute x = x * into the driven subsystem (5.51) and (3.36), leading to the following reduced model

ẏ2[j] = f 2[j] (x * 2[j] , y 2[j] ) ż= Az + ϑ(x * 1[i] , x * 2[j] )
(5.57) with (N + 3M + P + L) th order.

According to the master-slave control operation dened in Section 4.1.4 and the design procedure presented in Section 5.1.3.2, we then set

i * gρd = i o gρd ; i * g i q = i o g i q
(5.58)

Control induced time-scale separation for MTDC systems using master-slave control conguration 155

As a consequence, some components of the vector x * are directly given by the prescribed reference values. But we still need to design i * g 1 d and i * w j ,dq which are considered as the control inputs of the reduced model (5.57). The control goal now is to make u cg 1 and v sw j ,dq in (5.57) track the prescribed output reference, i.e.

so u o cg 1 v o sw 1 ,dq • • • v o sw N ,dq
(5.59) via i * g 1 d and i * w j ,dq . Input-output feedback linearization is therefore applied for the reduced model as

2 du cg 1 dt = 1 C g 1 ( 3 2 v sg 1 d i * g 1 d + v sg 1 q i o g 1 q u cg 1 -H(1, :)i c ) = v cg 1 dv sw j d dt = ω w j v sw j q + 1 C f w j (I w j d -i * w j d ) = v w j d dv sw j q dt = -ω w j v sw j d + 1 C f w j (I w j q -i * w j q ) = v w j q
Taking into account the integrated tracking error, the additional inputs v (•) are designed as

ėIcg 1 = ũcg 1 u cg 1 -u o cg 1 v cg 1 = -k g 1 d ũg 1 -k g 2 d e Icg 1 ėIw j d = ṽsw j d v sw j d -v o sw j d v w j d = -k w 1j d ṽsw j d -k w 2j d e Iw j d
ėIw j q = ṽsw j q v sw j q -v o sw j q v w j q = -k w 1j q ṽsw j q -k w 2j q e Iw j q (5.60) with positive control gains k g 1 d , k g 2 d , k w 1j ,dq and k w 2j ,dq . Then, by solving (5.60), i * g 1 d and i * g j ,dq are computed as

i * g 1 d = 2u cg 1 3v sg 1 d (C g 1 v cg 1 + H(1, :)i c ) - v sg 1 q v sg 1 d i o g 1 q i * w j d = C f w j ω w j v sw j q -C f w j v w j d + I w j d i * w j q = -C f w j ω w j v sw j d -C f w j v w j q + I w j q
(5.61)

Finally, the full control scheme is established, composed by (5.54), (5.58) and (5.61).

For each VSC terminal, the controller u [k] just relies on local information without the knowledge of other terminals. This can prevent the possible time delay of information exchange from the collective model.

The manifold x * is deduced from the lower-order reduced model (5.57) instead of the full-scale model (5.50), (5.51) and (3.36), and it greatly simplies the control 2 Recall the denition of H in Section 3.4.3.

156

Chapter 5. Control induced time-scale separation design. Clearly, x * is a function of the state variables of the driven system, i.e. y 2[j] and z. It is worthwhile to note that using a reduced model will introduce errors in the system. Such errors may cause stability problems which will be studied in the next section.

Theoretical analysis

Before stating the main results of this section, we recall that, to ensure the physical feasibility of the MTDC system, Assumption 4.1.1 still holds and the equilibrium point is denoted as

P1 = [• • • īg i d īg i q • • • īw j d īw j q • • • vsw j d vsw j q • • • z] T (5.62)
In this section, we will demonstrate how the proposed controller divides the MTDC system into dierent time scales and furthermore, the stability of the equilibrium point P1 will be also studied.

Two-time-scale dynamics

The aim of this section is to show how the MTDC system exhibits a multi-time-scale behavior. To establish this result, we substitute the control algorithm (5.54), (5.58) and (5.61) into the original system consisting of (5.50), (5.51) and (3.36), resulting in the following closed-loop system

ėIx 1[i] = x 1[i] -x * 1[i] ẋ1[i] = -K 1[i] (x 1[i] -x * 1[i] ) -Γ 1[i] e Ix 1[i] ėIx 2[j] = x 2[j] -x * 2[j] ẋ2[j] = -K 2[j] (x 2[j] -x * 2[j] ) -Γ 2[j] e Ix 2[j] ėIw j d = v sw j d -v o sw j d ėIw j q = v sw j q -v o sw j q ėIcg 1 = ũcg 1 ẏ2[j] = f 3[j] + I w [j]
ż= Az + ϑ

(5.63) with (9M + 5N + P + L) th order. The equilibrium point of the above closed-loop system is denoted as P2 .

Let us introduce a small scalar parameter ε satisfying

ε • k min = 1
where k min = min(k

11 [i] , k 12 [i] , k 21 [j] , k 22 [j]
). We then rewrite the dynamics of x 1[i] and x 2[j] in (5.63) as

ε ẋ1[i] = -K1[i] (x 1[i] -x * 1[i] ) -εΓ 1[i] e Ix 1[i] ε ẋ2[j] = -K2[j] (x 2[j] -x * 2[j] ) -εΓ 2[j] e Ix 2[j]
(5.64) 5.2. Control induced time-scale separation for MTDC systems using master-slave control conguration 157

where the new control gain matrices are dened as

K1[i] = diag( k11 [i] , k12 [i] ) diag(εk 11 [i] , εk 12 [i] ) K2[j] = diag( k21 [j] , k22 [j] ) diag(εk 21 [j] , εk 22 [j] ) k11 [i] , k12 [i] , k21 [j] , k22 [j] ≥ 1
Now, as an alternative expression, the closed-loop system (5.63) can be rewritten as

ėIx 1[i] = x 1[i] -x * 1[i] ε ẋ1[i] = -K1[i] (x 1[i] -x * 1[i] ) -εΓ 1[i] e Ix 1[i] ėIx 2[j] = x 2[j] -x * 2[j] ε ẋ2[j] = -K2[j] (x 2[j] -x * 2[j] ) -εΓ 2[j] e Ix 2[j] ėIw j d = v sw j d -v o sw j d ėIw j q = v sw j q -v o sw j q ėIcg 1 = ũcg 1 ẏ2[j] = f 3[j] + I w [j] ż= Az + ϑ (5.65)
which is of a standard singularly perturbed form when ε is small enough. We call x * as the quasi-steady state of x. It is worth noticing that, the small parameter ε, which is used to qualify the time-scale separation, is not derived from the physical system parameters but is arbitrarily created by the designed subcontroller (5.54).

For convenience, we introduce the following new variables

x1[i] = x 1[i] -x * 1[i] x2[j] = x 2[j] -x * 2[j]
that shift the quasi-steady state of x to the origin. In these new variables, the driving subsystem (5.64) becomes

ε ẋ1[i] = -K1[i] x1[i] -εΓ 1[i] e Ix 1[i] -ε dx * 1[i] dt ε ẋ2[j] = -K2[j] x2[j] -εΓ 2[j] e Ix 2[j] -ε dx * 2[j]

dt

(5.66)

Now we dene t as the time-scale variable for slow dynamics and introduce a new time variable τ for fast dynamics as

ε dx dt = dx dτ , x = [x 1[1] • • • x1[N] x2[1] • • • x2[M] ] T (5.67)
We then express (5.66) in the τ time scale as As ε → 0,a boundary-layer system is deduced from (5.68), given by

dx 1[i] dτ = -K1[i] x1[i] -εΓ 1[i] e Ix 1[i] -ε dx * 1[i] dt dx 2[j] dτ = -K2[j] x2[j] -εΓ 2[j] e Ix 2[j] -ε dx * 2[j]
dx 1[i] dτ = -K1[i] x1[i] dx 2[j] dτ = -K2[j] x2[j]
(5.69) whose equilibrium point is globally exponentially stable. Meanwhile, when x enters its manifold x * , the closed-loop system (5.65) is degenerated into a (5M + N + P + L + 1) th order reduced model expressed by

ėIw j d = v sw j d -v o sw j d ėIw j q = v sw j q -v o sw j q ėIcg 1 = ũcg 1 ẏ2[j] = [v w j d v w j q ] T ż= Az + ϑ (5.70)
where ϑ is given by

                   ϑ i = 3 2 v sg i d i * g i d + v sg i q i * g i q C g i (ũ cg i + ūcg i ) , i ∈ N ϑ N +j = 3 2 (ṽ sw j d + v o sw j d )i * w j d + (ṽ sw j q + vo sw j q )i * w j q C w j (ũ cw j + ūcw j ) , j ∈ M ϑ N +M +h = 0, h ∈ P ϑ N +M +P +k = 0, k ∈ L (5.71)
Remember that i * g i ,dq and i * w j ,dq are given by (5.58) and (5.61). We denote the solu- tion of the reduced model (5.70) as (y re 2[j] (t), z re (t)) and then, x re = x * (y re 2[j] (t), z re (t)) describes behavior of x * when (y 2[j] (t), z(t)) = (y re 2[j] (t), z re (t)). As a result, according to Theorem 5.1.9 in Section 5.1.4.1, we are ready to state the following result:

Theorem 5.2.1. Consider the system (5.63). For xed control parameters

K1[i] , K2[j] , k g 1,2 d , k w 1j
,dq and k w 2j ,dq , there exists a region R xyz and a positive constant

ε * (or k * min ) such that for all 0 < ε < ε * (or k * min < k min ), if the system (5.63) starts from R xyz , we have y 2[j] (t)= y re 2[j] (t) + O(ε) z(t)= z re (t) + O(ε) x(t)= x re (t) + x(τ ) + O(ε) (5.72)
where x(τ ) is the solution of the boundary-layer model (5.69).

Theorem 5.2.1 clearly indicates that the state variables of the driving system, i.e. all dq currents, exhibit a two-time-scale behavior. They start with a fast transient x(τ ) and then remain close to x re (t) which presents a slow transient. If we want a remarkable time-scale separation, a smaller ε (or a lager k min ) should be chosen. In the τ time scale, all y [j] and z seem slowly varying. 5.2. Control induced time-scale separation for MTDC systems using master-slave control conguration 159

Stability analysis

The previous section shows that the exact solution of the original system (5.63) can be approximated by the solutions of the boundary-layer and the reduced model. In this section, we carry out a detailed stability analysis.

To simplify the analysis, we perform the change of variables

ỹ2[j] = y 2[j] -ȳ2[j] z = z - z
and then, rewrite the system (5.63) in these new variables as

ėIx 1[i] = x1[i] ẋ1[i] = -K 1[i] x1[i] -Γ 1[i] e Ix 1[i] - dx * 1[i] dt ėIx 2[j] = x2[j] ẋ2[j] = -K 2[j] x2[j] -Γ 2[j] e Ix 2[j] - dx * 2[j] dt ėIw j d = ṽsw j d ėIw j q = ṽsw j q ėIcg 1 = ũcg 1 ẏ2[j] = v w[j] + B [j] x2[j] ż= Az + ϑ * + Az + θ (5.73)
where

B [j] = - 1 C f w j I 2×2 ; v w[j] [v w j d v w j q ] T θ [ θ1 • • • θM+N 0 P +L ] T ϑ * [ϑ * 1 • • • ϑ * M +N 0 P +L ] T (5.74)
with the elements

         θi = 3 2 v sg i d ĩg i d + v gs i q ĩg i q C g i (ũ cg i + ūcg i ) , i ∈ N θN+j = 3 2 (ṽ sw j d + vsw j d ) ĩw j d + (ṽ sw j q + vsw j q ) ĩw j q C cw j (ũ cw j + ūcw j ) . j ∈ M          ϑ * i = 3 2 v sg i d i * g i d + v gs i q i * g i q C g i (ũ cg i + ūcg i ) , i ∈ N ϑ * N +j = 3 2 (ṽ sw j d + vsw j d )i * w j d + (ṽ sw j q + vsw j q )i * w j q C w j (ũ cw j + ūcw j ) . j ∈ M (5.75)
It is important to remark that since x * is the control signal of the slow subsystem, its derivative would be very small compared to the derivative of x.

We rewrite the rst four equations of (5.73) in a novel matrix form

κ = A κ κ + θ κ (5.76) where κ = [• • • e Ix 1[i] x1[i] • • • e Ix 2[j] x2[j] • • • ] T θ κ = [• • • 0 2×1 dx * 1[i] dt • • • 0 2×1 dx * 2[j] dt • • • ] T A κ = diag(• • • A x 1 [i] • • • A x 2 [j] ) A x 1 [i] = 0 2×2 I 2×2 -Γ 1[i] -K 1[i] ; A x 2 [j] = 0 2×2 I 2×2 -Γ 2[j] -K 2[j]
The last ve equations of (5.73) can be expressed in the following form

ζ= A ζ ζ + θζ żr = A r zr + ϑ * r + A r zr + θr + θic + θic (5.77)
where

ζ= [e Icg 1 ũcg 1 • • • e Iw j d ṽsw j d • • • e Iw j q ṽsw j q • • • ] T θζ = [0 θ1 • • • 0 - 1 C f w j ĩw j d • • • 0 - 1 C f w j ĩw j q • • • ] T θic = [0 N +M +P -1 H(1, 1) L c 1 ūcg 1 • • • H(1, L) L c L ūcg 1 ] T θic = [0 N +M +P -1 H(1, 1) L c 1 ũcg 1 • • • H(1, L) L c L ũcg 1 ] T A ζ = diag(A g 1 A w 1 d • • • A w M d A w 1 q • • • A w M q ) A g 1 = 0 1 -k g 1 d -k g 2 d ; A w j ,dq = 0 1 -k w 1j ,dq -k w 2j ,dq (5.78)
The vectors θr , ϑ * r and zr are the sub-vectors formed by deleting the rst component of θ, ϑ * and z (see (5.75)), respectively. The matrix A r is given by (4.65) in Section 4.1.4 as

A r = 0 (N +M +P -1)×(N +M +P -1) C -1 r H r -L -1 H T r -L -1 R (5.79)
Now, the original closed-loop system (5.63) is transformed into the new form composed of (5.76) and (5.77). Note that the above error system (5.76) and (5.77) has an equilibrium at the origin, the asymptotic stability of which would imply the asymptotic stability of P2 of the original system (5.63). If x1[i] and x2[j] quickly con- verge to zero, the error system (5.76) and ( 5 5.2. Control induced time-scale separation for MTDC systems using master-slave control conguration 161 Lemma 5.2.2. Consider the system (5.73). If the prescribed references

u o cg 1 , Q o g 1 , P o gρ , Q o gρ , v o sw j d
and v o sw j q and the system parameters are set such that the Jacobian matrix of the internal dynamics (5.81) at the origin

J = -D C -1 r H r -L -1 H T r -L -1 R (5.82) is Hurwitz, where D = diag(d k ), k = 1, • • • , N + M + P -1, is a diagonal matrix given by                d (ρ-1) = P o gρ ū2 cgρ , ρ ∈ N -1 d (j+N -1) = P o w j C w j ū2 w j , j ∈ M d (h+N +M ) =0. h ∈ P P o w j = 3 2 (v o sw j d I w j d + v o sw j q I w j q )
then, there exist control matrices

K 1[i] , K 2[j] , Γ 1[i] and Γ 2[j]
such that the origin of the system (5.73) is locally exponentially stable. As a consequence, the equilibrium point P1 of the MTDC system (5.50), (5.51) and (3.36) is locally asymptotically stable under the control algorithm (5.54), (5.58) and (5.61).

Proof. According to Taylor's theorem, the reduced model (5.77) can be expressed in the Peano form of the remainder as

ζ żr = A ζ 0 A * J ζ zr + φ 1 (ζ, zr ) ζ zr + 0 θ * + A r zr + θic f p (5.83)
where θ * and φ(z r , ζ)

satisfy θ * =[ īg 2 C g 2 • • • īg N C g N īw 1 C w 1 • • • īw M C w M 0 T (P +L) ] T θ * +A r zr + θic = 0 lim ||[ζ zr] T ||→0 ||φ 1 (ζ, zr )|| = 0
For any positive L φ 1 , there exists a positive b φ 1 such that:

||φ 1 || ≤ L φ 1 , ∀ [ζ zr ] T ∈ B φ 1 where B φ 1 = { [ζ zr ] T | || [ζ zr ] T || < b φ 1 }.
We denote H A ζ 0 A * J and hence, H is Hurwitz since A ζ and J are Hurwitz. For any symmetric matrix Q r > 0, there exists P r > 0 satisfying the Lyapunov equation

P r H + H T P r = -Q r . Let V r = [ζ zr ]P r [ζ zr ] T
Proof. We have proved that, when P o gρ , ρ = 2, • • • , N and P o w j , j = 1, • • • , M are non-negative, the Jacobian matrix J is Hurwitz (see the demonstration of Lemma 4.1.3 in Section 4.1.4)

Plug-and-play operations

Consider that the MTDC system (5.50), (5.51) and (3.36) with the proposed control algorithm (5.54), (5.58) and (5.61) initially operates in a steady-state condition. If it is required to add or remove a VSC terminal, we propose a plug-and-play operation.

Adding one terminal

we rst consider plugging in a WAC connected VSC terminal at t 1 , whose AC side is characterized by x 2[M +1] , y 2[M +1] and u 2 [M +1] , i.e. i w M +1 ,dq , v sw M +1 ,dq and m w M +1 ,dq . On the DC side, a WAC converter node is associated to the plugged WAC connected VSC terminal, characterized by u cw M +1 and C w M +1 . In particular, the (M + 1) th WAC converter node is physically coupled to the DC network by a branch, which is used to connect the plugged node to other existing nodes. The extra branch is

characterized by i c L+1 , R c L+1 and L c L+1 . The new WAC connected VSC terminal is equipped with local controller u 2[M +1] with the control gain matrices K 2[M +1] , Γ 2[M +1] , A w M +1 ,dq .
Here we need to check whether this new terminal could be added to the existing MTDC system without aecting the existing stability of the system. The procedure is summarized as follows:

Step 1: Modify the DC network model (3.36) and then compute the new equilibrium and the new Jacobian matrix J. If J is Hurwitz, go to the next step. Otherwise, we stop here and declare that the new WAC connected VSC terminal can not be added.

Step 2: When the new Jacobian matrix J is still Hurwitz, we dene a desired region of attraction R for the new system with N + M + 1 terminals and then the feasible region for k min could be estimated from Condition (5.87). Therefore, it is possible to extend the MTDC system by adding the new terminal.

Removing one terminal

We consider removing a WAC connected VSC terminal. For example, the k th one is expected to be disconnected at t 2 . Meanwhile, the branches used to connect this terminal node to other nodes are removed as well. Then, a new modied MTDC system is generated. We rst check if the corresponding Jacobian matrix J of the new MTDC system is Hurwitz. If this is not the case, we can not remove the k th terminal. When the new J is Hurwitz, we then estimate the new region of attraction R . If the state of the MTDC system at t 1 is still in R , the unplugging of the terminal is allowable. Otherwise, the operation of removal may trigger the instability of the system. 5.2. Control induced time-scale separation for MTDC systems using master-slave control conguration 165

Simulation studies

In this section, we apply the proposed control scheme to an MTDC system composed of a WAC connected VSC terminal, two SAC connected VSC terminals and two transmission branches as shown in Fig. 5.5. System parameters are listed in Tables Figure 5.5: An MTDC system with a WAC and two SAC terminals.

5.4-5.5. The base quantities used in the per-unit system are given by Table 5.6.

The 1 st SAC connected VSC terminal is used to control the DC-bus voltage at u o cg 1 = 1 p.u.. The 2 nd SAC connected terminal guarantees a constant power output P o g 2 = 0.2 p.u. injected to the DC grid. Both SAC terminals are required to have the unitary power factor, i.e. i 0 g 1,2 q = 0 p.u. The WAC terminal is used to control its AC voltage at the prescribed values of v o sw 1 d = 1 p.u. and v o sw 1 q = 0 p.u.. To clarify the theoretical analysis, several scenarios are considered in this section. In addition, all simulations are carried out with the same control gains of i w 1 d and i w 1 q whose trajectories display two-time-scale behaviors. In contrast to the two states i w 1 d and i w 1 q , whose initial conditions are their prescribed initial values, i.e. i w 1 d 0 = 0.3 p.u. and i w 1 q 0 = 0.02 p.u., their quasi-steady states i re w 1 d and i re w 1 q do not start from the same initial conditions but from i re w 1 d 0 = 0.4638 p.u. and i re w 1 q 0 = -0.202 p.u. since the initial values of i re w 1 ,dq are determined by the initial values of v re sw 1 ,dq as mentioned in Section 5.2.2.1. As a consequence, large discrepancies may exist between i w 1 ,dq and i re w 1 ,dq at the initial instant.

As illustrated in Fig. 5.8, the trajectories of i w 1 ,dq start with very fast transients from the initial values i w 1 ,dq (t 0 ) to i re w 1 ,dq during the boundary-layer interval. After the decay of the transients, i w 1 ,dq remain close to i re w 1 ,dq and subsequently, both i w 1 ,dq and i re w 1 ,dq converge to their steady values, i.e. īw 1 d = 0.4667 p.u. and īw 1 q = -0.2528 p.u. as shown in Figs. 5.7(c)-5.7(d). A similar two-time-scale behavior can be found in the response of i g 1 d . Unlike the dq currents, v w 1 ,dq , u g 1,2 and u w 1 start from the same initial values as v re sw 1 ,dq , u re g 1,2 and u re w 1 . In addition, they uniformly converge to v re sw 1 ,dq , u re g 1,2 and u re w 1 , respectively. Simulation results (not presented here) show that while keeping k g 1 d , k w 11 d , k w 11 q , k g 2 d , k w 21 d and k w 22 q unchanged, we can reduce the errors between the dq currents and their quasi-steady states by increasing the control gain matrices K 1

[1] , K 2[1] and K 2[1] .

Scenario 2: Variations of the controlled current source

Variations in the controlled current source are considered in this scenario. Simulation results are illustrated in Fig. 5.9-5.10. This scenario reects the unpredictable variation of power produced by a wind farm. P Iw in Fig. 5.10(c) represents the generated power from the controlled current source, which is equivalent to P w 1 , i.e.

the active power of the WAC terminal at the PCC. The DC-bus voltage is always controlled to the set value of 1 p.u. by the 1 st SAC terminal irrespective of variations of P Iw as depicted in Fig. 5.9(a). In this study case, the 1 st SAC terminal operates as the slack bus to tolerate the power imbalance caused by P Iw and in addition, this terminal is responsible to keep the voltage of the DC network constant, in spite of the uncontrollable variations of the wind power generation. Therefore, P g 1 varies with P Iw while the active power of the 2 nd SAC terminal (P g 2 ) remains around its setpoint (0.2 p.u.). Moreover, the DC voltages of other nodes are also kept around needed to maintain the AC voltage. Finally, a similar conclusion can be drawn as in Scenario 1, that is, the behaviors of the state variables can be well approximated by the solutions of their reduced model.

Scenario 3: Unplugging and plugging a terminal

In this study case, we evaluate the performance of the control scheme when considering the unplugging of the 2 nd SAC connected VSC terminal and then plugging it back. Such unplugging and plugging operations can be highlighted as depicted in Fig. 5.12(d). At t = 2 s, the 2 nd SAC terminal is disconnected and at the same time, the branch 2 is also removed from the MTDC system. Consequently, the WAC terminal now is the only power supplier for the 1 st SAC terminal. As seen in Fig. 5.12(d), during t ∈ [2, 4] s, P g 1 is almost equal to P w 1 . At t = 4 s, we re-use the branch 2 to connect the 2 nd SAC terminal to the DC network. Now, the 2 nd SAC terminal and the WAC terminal feed the power to the 1 st SAC terminal together. In usually characterized by power vs DC voltage (P vs U ) or current vs DC voltage (I vs U ) curves, becomes an alternative method, in particular for a large network.

In a DC voltage droop control scheme, more than one terminal is in charge of regulating the DC voltage and hence, the burden caused by power imbalance could be shared between several terminals. Additionally, this control method takes actions only based on local information without relying on remote communication. In droop control, the droop gains specify the system operation in steady-state condition and also have a great eect on the system transient performance. Dierent methodologies of choosing droop gains have been investigated. In [Prieto-Araujo 2011], a criterion for tuning the droop gains based on the performance specications is proposed. In [Bianchi 2011], the droop gains are obtained by solving a constraint convex optimization problem. Reference [START_REF] Chaudhuri | [END_REF]] develops an adaptive droop control scheme according to each terminal's available spare capacity. However, prior studies are usually under the unproven assertion that the dynamics of the inductor currents are much faster than the dynamics of the DC network. Therefore, the currents through the converters are assumed to equal their references and then, the dynamic interaction between AC and DC sides of the converters can be neglected.

Similar to the study on master-slave control conguration in Section 5.2, in this section, we acknowledge this assertion, then explore and explain the droop control induced time-scale separation. To achieve this aim, we will carry out a full theoretical analysis of the time-scale separation between the system state variables, mainly based on the theoretical results in Section 5.1. Furthermore, we will also demonstrate that the control gains have dierent eects on the system performance.

For example, the droop gains play a major role in regulating the DC grid dynamics.

Another objective of this section is to study the eects of the control gains, especially the droop gains, on the system stability.

VSC operation

The greatest dierence between master-slave and droop control conguration is that the latter involves more terminals in the regulation of the DC voltage. This dierence 5.3. Control induced time-scale separation for MTDC systems using droop control conguration 175 leads to dierent VSC operation in droop control conguration, which is summarized as follows:

• Same to master-slave control conguration, to ensure that the power generated by the wind farms can be totally delivered via the DC grid, all WAC connected VSC terminals are still used to maintain their AC voltage magnitudes and frequencies constant at the PCCs, achieved by regulating v sw j ,dq at the prescribed values v o sw j ,dq .

• Every SAC terminal is equipped with the DC droop controller such that the duty of eliminating the power imbalance of the DC grid is shared between the SAC terminals. Moreover, the reactive powers of all the SAC terminals at the PCCs, i.e. Q g i , are required to track the reference values Q o g i .

Droop control structure

The control design of droop control conguration is similar to that of master-slave control conguration in Section 5.2.1, which consists of two control loops, a fast inner current and a slow outer control loop, where the dynamics of state variables in the fast control loop are much faster than those in the slow control loop. But, in droop control structure, a droop law is implemented in the slow outer control loop of all SAC terminals, which generates the reference to the fast inner control loop.

Design of the fast control loop

As mentioned in Section 5.2.1.1, the control inputs m (•),dq are directly collocated with the dq currents i (•),dq . Hence, a sub-controller can be developed such that i (•),dq quickly converge to their reference trajectories i * (•),dq yet to be designed.

We rst dene the dq current tracking errors by e i g i ,dq = i g i ,dq -i * g i ,dq

e i w j ,dq = i w j ,dq -i * w j ,dq

With the introduction of the integrated tracking errors e Ii g i ,dq and e Ii w j ,dq , an aug-Chapter 5. Control induced time-scale separation mented dq current subsystem is then generated by

de Ii g i d dt = e i g i d de i g i d dt = - R g i L g i i g i d + ω g i i g i q + v sg i d L g i - m g i d u cg i 2L g i - di * g i d dt de Iig i q dt = e ig i q de ig i q dt = - R g i L g i i g i q -ω g i i g i d + v sg i q L g i - m g i q u cg i 2L g i - di * g i q dt de Ii w i d dt = e i w i d de i w j d dt = - R w j L w j i w j d + ω w j i w j q + v sw j d L w j - m w j d u cw j 2L w j - di * w j d dt de Iiw i q dt = e iw i q de iw j q dt = - R w j L w j i w j q -ω w j i w j d + v sw j q L w j - m w j q u cw j 2L w j - di * w j q dt (5.89)
In practice, the dynamics of i * g i ,dq and i * w j ,dq are usually neglected and considered as slow compared to i g i ,dq and i w j ,dq . As a result, the modulation indices can then be designed as

m g i d = 2L g i u cg i (- R g i L g i i g i d + ω g i i g i q + v sg i d L g i + k 1g i d e i g i d + k 2g i d e Ii g i d ) m g i q = 2L g i u cg i (- R g i L g i i g i q -ω g i i g i d + v sg i q L g i + k 1g i q e ig i q + k 2g i q e Iig i q ) m w j d = 2L w j u cw j (- R w j L w j i w j d + ω w j i w j q + v sw j d L w j + k 1w i d e i w i d + k 2w i d e Ii w i d )
m w j q = 2L w j u cw j (-R w j L w j i w j q -ω w j i w j d + v sw j q L w j + k 1w i q e iw i q + k 2w i q e Iiw i q )

(5.90)

with positive control gains k 1g i ,dq , k 2g i ,dq , k 1w j ,dq and k 2w j ,dq .

Design of the slow control loop

The main task of the slow control loop is to provide the slowly varying dq current references to the fast control loop. Most importantly, the dq current references need to be designed such that all control objectives are achieved. Since the operations of SAC terminals and WAC terminals are dierent, the design principles of i * g i ,dq and i * w j ,dq are also dierent.

Design of i * w j ,dq

We assume that i w j ,dq can quickly converge to their reference trajectories i * w j dq under the sub-controller (5.90). From there on, the AC voltage dynamics (3.29) of the wind 5.3. Control induced time-scale separation for MTDC systems using droop control conguration 177 farm at the PCC become

dv sw j d dt = ω w j v sw j q + 1 C f w j (I w j d -i * w j d ) dv sw j q dt = -ω w j v sw j d + 1 C f w j (I w j q -i * w j q )
(5.91)

where i * w j dq can be viewed as the control inputs. We call (5.91) the reduced model of (3.29) since the dynamics of i w j ,dq are neglected and i w j ,dq are replaced by i * w j ,dq .

The control objective of WAC terminals is to keep v sw j ,dq at their reference values v o sw j ,dq in spite of variations in the power productions of the wind farms. We then want to develop i * w j ,dq in such a way that v sw j ,dq in (5.91) can be controlled at v o sw j ,dq .

Similar to the design procedure of the fast control loop, we dene the output tracking errors of voltage at the PCC as e v sw j ,dq = v sw j ,dq -v o sw j ,dq

(5.92)

We then augment (5.91) by taking the integrated tracking errors into account

de Iv sw j d dt = e v sw j d de v sw j d dt = 1 C f w j (I w j d + C f w j ω w j v sw j q -i * w j d ) - dv o sw j d dt de Ivsw j q dt = e vsw j q de v sw j dq dt = 1 C f w j (I w j q -C f w j ω w j v sw j d -i * w j q ) - dv o sw j q dt (5.93)
Since the prescribed values v o sw j ,dq are constant, we have dv o sw j ,dq dt = 0. Finally, i * w j ,dq can be designed as i * w j d = I w j d + C f w j (k 1d j e v sw j d + k 2d j e Iv sw j d + ω w j v sw j q ) i * w j q = I w j q + C f w j (k 1q j e vsw j q + k 2q j e Ivsw j q -ω w j v sw j d )

(5.94) with positive k 1,dq j and k 2,dq j such that the equilibrium of the augmented model (5.93) is exponentially stable.

Design of i * g i ,dq

As mentioned before, a droop law is implemented in the slow outer control loop of all SAC terminal so that all SAC terminals participate in the regulation of the DC voltage in case of power imbalance. The droop control is usually expressed by DC voltage versus active power characteristic [Haileselassie 2012b].

P g i = P o g i + K d i (u o cg i -u cg i ) (5.95)
where u o cg i and P o g i refer to the reference values of DC voltage and active power respectively. The droop gain K d i is the ratio of the change in DC voltage to the change in active power. According to (5.95), the varying reference for i g i d can be expressed by

i * g i d = i o g i d -k d i (u cg i -u o cg i ) (5.96)
where i o g i d is obtained from (3.22) and expressed as

i o g i d = 2P o g i 3v sg i d
In addition, the relation between K d i and k d i is given by

k d i = 2 3 1 v sg i d K d i
The reference of i g i q is directly provided by the prescribed reference value

i * g i q = i o g i q = - 2 3 Q o g i v sg i d
(5.97)

Theoretical analysis

As the master-slave control structure in Section 5.2.1, the developed droop control design concept is also mainly based on the assumption that a dynamic separation in time scales is imposed between the fast and slow control loops. As a consequence, z and v sw j ,dq can be considered as constant in the fast control loop, while the dq currents can be replaced by their references in the slow control loop.

In this section, we give a detailed theoretical analysis to demonstrate the validity of the time-scale separation assumption in droop control conguration and furthermore, to investigate the possible limitations on the droop gains. dt = e ig i q di g i q dt = -k 1g i q e ig i q -k 2g i q e Iig i q de Ii w j d dt = e i w j d di w j d dt = -k 1w j d e i w j d -k 2w j d e Ii w j d de Iiw j q dt = e iw j q di w j q dt = -k 1w j q e iw j q -k 2w j q e Iiw j q de Iv sw j d dt = e v sw j d dv sw j d dt = -k 1d j e v sw j d -k 2d j e Iv sw j d -e i w j d C f w j de Ivsw j q dt = e vsw j q dv sw j q dt = -k 1q j e vsw j q -k 2q j e Ivsw j q -e iw j q C f w j ż= Az + ϑ (5.98) It will be more convenient in the following analysis to divide ϑ into two parts as ϑ = ϑ g + ϑ w

Equilibrium analysis

Recalling i cg i in (3.23), i cw i in (3.32), i * w j ,dq in (5.94) and i * g i ,dq in (5.96) and (5.97), the new variables ϑ g and ϑ w are given by

ϑ g = ϑ g 1 + ϑ g 2 + ϑ g 3 , ϑ w = ϑ w 1 + ϑ w 2 + ϑ w 3 + ϑ w 4 with ϑ g 1 (i)=      3 2C g i v sg i d i o g i d + v sg i q i o g i q u cg i , i ∈ N 0. otherwise ϑ g 2 (i)=      - 3 2C g i v sg i d k d i u cg i -u o cg i u cg i , i ∈ N 0. otherwise ϑ g 3 (i)=    3 2C g i v sg i d e i g i d + v sg i q e ig i q u cg i , i ∈ N 0.
ϑ w 1 (j + N )=      3 2C w j v o sw j d I w j d + v o sw j q I w j q u cw j , j ∈ M 0. otherwise ϑ w 2 (j + N )=                      3 2C w j v o
sw j d C f w j ω w j e vsw j q + e v sw j d I w j d u cw j + e vsw j q I w j q -v o sw j q C f w j ω w j e v sw j d u cw j , j ∈ M 0.

otherwise

ϑ w 3 (j + N )=                      3 2C w j v sw j d (k 1d j e v sw j d + k 2d j e Iv sw j d ) u cw j + v sw j q (k 1q j e vsw j q + k 2q j e Ivsw j q ) u cw j , j ∈ M 0. otherwise ϑ w 4 (j + N )=      3 2C w j v sw j d e i w j d + v sw j q e iw j q u cw j , j ∈ M 0.
otherwise It is obvious that, if the dq currents quickly enter their respective manifolds, i.e. e i g i ,dq → 0 and e i w j ,dq → 0, ϑ g 3 and ϑ w 4 converge to zero. Moreover, ϑ w 2 and ϑ w 3 also go to zero as e v sw j ,dq → 0. These results are helpful for the theoretical analysis in the next part.

When the closed-loop system (5.98) is in the steady-state condition, the following algebraic equations

Az + θg + θw = 0

(5.99) must hold5 . Since P o g i = 3/2(v sg i d i o g i d + v sg i q i o g i q ), θg and θw can be then expressed as

θg (i)=        P 0 g i + 3 2 k d i v sg i d u o cg i C g i ūg i - 3v sg i d 2C g i k d i , i ∈ N 0. otherwise θw (j + N )=      P o w j C w j u cw j , j ∈ M 0. otherwise (5.100)
where we introduced the new variable

P o w j = 3/2(v o sw j d I w j d + v o sw j q I w j q )
5.3. Control induced time-scale separation for MTDC systems using droop control conguration 181

From the algebraic equation (5.99), it is shown that the prescribed values (•) o , the DC grid topology and the droop gains k d i determine the steady-state values of the system variables and the distribution of power sharing together in case of power imbalance. The control gains k 1g i ,dq , k 2g i ,dq , k 1w j ,dq , k 2w j ,dq , k 1,dq j and k 2,dq j have no eect on the steady state of the system. However, as (4.58) in Section 4.1.4 and (4.96) in Section 4.3.2, solving equation (5.99) to get the exact steady-state value is not easy and is not considered in this section. More details about this issue can be referred in [Wang 2014a, Haileselassie 2012b].

To ensure the operating feasibility of the MTDC system, we assume that īg i dq , īw j dq , ūcg i , ūcw j , ūct h and īc k exist and belong to their respective safe operating domains. The equilibrium of the closed-loop system is denoted by S.

In stead-state condition, From the droop law (5.95), we have

Pg i = P o g i + K d i (u o cg i -ūcg i ) (5.101)
Consider that the MTDC system initially (t = t 0 ) operates in a steady-state condition, if the production of the wind farms increases at the instant t 1 , this makes the SAC terminals absorb more power from the DC grid. When the MTDC system achieves a new steady state at t 2 , we have Pg i (t 2 ) < Pg i (t 0 ) (for example, Pg i (t 0 ) = -100 MW and Pg i (t 2 ) = -200 MW). According to (5.101), we can deduce that the DC voltage also rises to a new level and that ūg i (t 2 ) > ūg i (t 0 ). It can be summarized that u cg i will rise if more power is injected into the DC grid, and vice versa. This statement will be illustrated by numerical simulations in Section 5.3.4.

Multi-time-scale dynamics

In this section, a theoretical analysis to describe the dynamic separation in timescales is carried out. We present that two dierent dynamics are created by the designed control algorithm in Section 5.3.2. In particular, the two time-scales are quantied by the fast control gains k 1g i ,dq , k 2g i ,dq , k 1w j ,dq and k 2w j ,dq .

In order to make the analysis convenient, we perform a change of variables.

Chapter 5. Control induced time-scale separation

Then, the rst eight equations of the closed-loop system (5.98) can be rewritten as

de Ii g i d dt = e i g i d de i g i d dt = -k 1g i d e i g i d -k 2g i d e Ii g i d - di * g i d dt de Iig i q dt = e ig i q de ig i q dt = -k 1g i q e ig i q -k 2g i q e Iig i q - di * g i q dt de Ii w j d dt = e i w j d de i w j d dt = -k 1w j d e i w j d -k 2w j d e Ii w j d - di * w j d dt de Iiw j q dt
= e iw j q de iw j q dt = -k 1w j q e iw j q -k 2w j q e Iiw j q -di * w j q dt

(5.102)

This shifts the quasi-steady states of the dq currents to the origin. Denote k 1 = min(k 1g i d , k 2g i q , k 1w i d , k 2w i q ) and then introduce a new variable ε satisfying εk 1 = 1. dt = e ig i q ε de ig i q dt = -k1g i q e ig i q -εk 2g i q e Iig i q -ε di * = e iw j q ε de iw j q dt = -k1w j q e iw j q -εk 2w j q e Iiw j q -ε di * w j q dt (5.103)

where

k1g i d = εk 1g i d ≥ 1; k1g i q = εk 1g i q ≥ 1 k1w j d = εk 1w j d ≥ 1; k1w j q = εk 1w j q ≥ 1
We dene a new time variable τ as ε dy dt = dy dτ

(5.104) 5.3. Control induced time-scale separation for MTDC systems using droop control conguration 183

Substituting (5.104) into (5.103) and then setting ε = 0, we obtain the following autonomous system de i g i d dτ = -k1g i d e i g i d de ig i q dτ = -k1g i q e ig i q de i w j d dτ = -k1w j d e i w j d de iw j q dτ = -k1w j q e iw j q

(5.105)

which is called the boundary-layer model of the system (5.98). When ε → 0, it seems that the time variable t and the slow state variables are frozen at their initial values. By using Lyapunov analysis, the origin of the boundary system (5.105) is exponentially stable. When the dq currents quickly converge to their reference trajectories during the initial interval, the full-scale system (5.98) is degenerated into the following reduced model

de Iv sw j d dt = e v sw j d dv sw j d dt = -k 1d j e v sw j d -k 2d j e Iv sw j d de Ivsw j q dt
= e vsw j q dv sw j q dt = -k 1q j e vsw j q -k 2q j e Ivsw j q dz dt = Az + ϑ g + ϑ w

(5.106) where ϑ g = ϑ g 1 + ϑ g 2 and ϑ w = ϑ w 1 + ϑ w 2 + ϑ w 3 . In addition, the solution of the reduced model (5.106) is denoted by (•) re . Now, we can state the rst result of this section.

Theorem 5.3.1. Consider the system (5.98) where all state variables are restrict to their respective safe operating domains for t ∈ [t 0 , t 1 ]. There exists a positive constant ε * such that for all 0 < ε = 1/k 1 < ε * , the system (5.98) has a unique solution on [t 0 , t 1 ], and the trajectories of the state variables can be approximated by

i g i ,dq -i * g i ,dq (z re ) -e i g i ,dq = O(ε) i w j ,dq -i * w j ,dq (z re , v re sw j ,dq ) -e i w j ,dq = O(ε) v sw j ,dq -v re sw j ,dq = O(ε) z -z re = O(ε) (5.107)
held uniformly for t ∈ [t 0 , t 1 ] where êi g i ,dq and êi w j ,dq are the solution of the boundary-layer model (5.105) e i g i ,dq = e i g i ,dq (t 0 ) exp(-k1g i ,dq τ ) e i w j ,dq = e i w j ,dq (t 0 ) exp(-k1w j ,dq τ ) Therefore, the approximations (5.107) can be deduced from the theoretical results in Section 5.2.2.1.

In terms of (5.107), the dq currents exhibit a two-time-scale behavior by presenting a slow and a fast transients. It is the dq current tracking errors, e i g i ,dq and e i w j ,dq , that characterize the part of fast dynamics. It is shown that i (•),dq start with a fast transient which exactly corresponds to the solution of boundary-layer model (5.105) during the initial interval. After the exponential decay of the fast transients e i g i ,dq ( e j w i ,dq ), i (•),dq remain close to their respective manifolds i * (•),dq in the future time.

As expressed in (5.108), the control gains k 1g i ,dq , k 2g i ,dq , k 1w j ,dq and k 2w j ,dq play a major role in regulating the fast transient performance and hence we call them the fast control gains. The control gains k 1,dq j , k 2,dq j and k d i , dominate the slow transient performance, and they are consequently called the slow control gains.

Stability analysis

For the sake of simplicity, we introduce a new variable e z = z -z (5.109) 5.3. Control induced time-scale separation for MTDC systems using droop control conguration 185

Now, the closed-loop system (5.98) becomes

de Ii g i d dt = e i g i d de i g i d dt = -k 1g i d e i g i d -k 2g i d e Ii g i d - di * g i d dt de Iig i q dt = e ig i q de ig i q dt = -k 1g i q e ig i q -k 2g i q e Iig i q - di * g i q dt de Ii w j d dt = e i w j d de i w j d dt = -k 1w j d e i w j d -k 2w j d e Ii w j d - di * w j d dt de Iiw j q dt
= e iw j q de iw j q dt = -k 1w j q e iw j q -k 2w j q e Iiw j q -di *

w j q dt de Iv sw j d dt = e v sw j d de v sw j d dt = -k 1d j e v sw j d -k 2d j e Iv sw j d - e i w j d
C f w j de Ivsw j q dt = e vsw j q de vsw j q dt = -k 1q j e vsw j q -k 2q j e Ivsw j q -e iw j q C f w j de z dt = Ae z + Az + ϑ g + ϑ w

( 5.110) where the equilibrium of the original closed-loop system, S, is shifted to the origin.

We call (5.110) the error system of the closed-loop system. In terms of those tracking errors, the varying reference trajectories (5.94) can be expressed by i * w j d =C f w j ω w j (v o sw j q + e vsw j q ) + I w j d +k 1d j e v sw j d + k 2d j e Iv sw j d i * w j q =-C f w j ω w j (v o sw j d + e v sw j d ) + I w j q +k 1q j e vsw j q + k 2q j e Ivsw j q (5.111) Since I w j ,dq are considered constant or slowly varying and then dI w j ,dq dt = 0, the derivatives of the dq references are given by di *

g i d dt = -k d i de ug i dt di * g i q dt = 0 di * w j d dt = C f w j ω w j de vsw j q dt + k 1d j de v sw j d dt + k 2d j e v sw j d di * w j q dt = -C f w j ω w j de v sw j d dt + k 1q j de vsw j q dt + k 2q j e vsw j q
(5.112) 186 Chapter 5. Control induced time-scale separation Note that the asymptotic stability of the origin of the error system (5.110) would imply the asymptotic stability of the equilibrium S of the original closed-loop system (5.98). Hence, we investigate the stability of the error system (5.110) instead of the original closed-loop system (5.98).

If e i g i dq and e i w j dq converge to zero, a new reduced model is deduced from the error system as

de Iv sw j d dt = e v sw j d de v sw j d dt = -k 1d j e v sw j d -k 2d j e Iv sw j d de Ivsw j q dt
= e vsw j q de vsw j q dt = -k 1q j e vsw j q -k 2q j e Ivsw j q de z dt = Ae z + Az + ϑ g + ϑ w

(5.113)

where ϑ g = ϑ g 1 + ϑ g 2 and ϑ w = ϑ w 1 + ϑ w 2 . The reduced model (5.113) can be divided into two parts, the external dynamics represented by e v sw j ,dq and e Iv sw j ,dq and the internal dynamics represented by e z . As the external variables converge to zero, i.e. e v sw j ,dq → 0 and e Iv sw j ,dq → 0, the behavior of the reduced model (5.113) is governed by the internal subsystem

de z dt = Ae z + Az + ϑ g + ϑ w f zero (5.114)
where ϑ g = ϑ g and ϑ w = ϑ w 1 . We call (5.114) the zero dynamics of the reduced model (5.113).

In the reduced model (5.113), we remark that the dynamics of the external variables are controlled by k 1,dq j and k 2,dq j while k d i have a great eect on the dynamics of the internal variables (or the dynamics of the DC grid). It can be summarized that the control gains have dierent impacts on the system performance.

Lemma 5.3.2. Consider the reduced model (5.113). Fix positive control gains k 1,2d j and k 1,2q j , ∀j ∈ M. Select the droop gains k d i that satisfy the following conditions:

• ∀i ∈ N , k d i is chosen such that k d i ≥ k d i min
(5.115)

where

P o g i + 3 2 v sg i d k d i min u o cg i = 0
• There exists at least one SAC terminal whose droop gain satises

k dq > k d q min
(5.116)

where q ∈ N .

Control induced time-scale separation for MTDC systems using droop control conguration 187

Then, the origin of the reduced model (5.113) is locally asymptotically stable.

Proof. A similar approach used in the proof of Lemma 4.1.3 is applied here. For the system described by (5.113), if the zero dynamics (5.114) is locally asymptotically stable, then the origin of the system (5.113) is locally asymptotically stable.

Therefore, to establish the claim, we rst study the stability of the zero dynamics (5.114). This is done by linearizing the zero dynamics (5.114) around the origin.

The Jacobian matrix [

∂f zero ∂e z ]| ez=0 is expressed by Φ [ ∂f zero ∂e z ]| ez=0 = D C -1 H -L -1 H T -L -1 R (5.117) where D = diag(d k ), k = 1, • • • , N + M + P , is a diagonal matrix given by                  d i = - P o g i + 3 2 v sg i d k d i u o g i C g i ū2 g i , i ∈ N d (j+N ) = - P o w j C w j ū2 w j , j ∈ M d (h+N +M ) =0. h ∈ P
Let us assume that there exists a particular eigenvalue of Φ denoted by λ = α+jβ ∈ C, whose real part is non-negative, i.e. α ≥ 0. Then, λ satises det(λI -Φ) = 0

Alternatively, it can be expressed as

det( λI -D -C -1 H L -1 H T λI + L -1 R ) = 0 (5.118) We dene Φ 1 λI + L -1 R = Λ 1 + jΛ 2 where Λ 1,2 are expressed by Λ 1 = diag(α + R c 1 L c 1 , • • • , α + R c L L c L ) ∈ R L×L Λ 2 = diag(β, • • • , β) ∈ R L×L
Since the Hermitian part of Φ 1 is equal to H(Φ 1 ) = Λ 1 , which is positive denite, the complex matrix Φ 1 is also positive denite (Lemma 4.1.6). Consequently, Φ 1 must be invertible (Lemma 4.1.9) and then, Eq. (5.118) becomes det(

λI -D -C -1 H L -1 H T λI + L -1 R ) =det(λI -D + C -1 HΦ -1 1 L -1 H T )det(Φ 1 ) =det(λC -CD + H(λL + R) -1 H T )det(Φ 1 )det(C -1 )

Chapter 5. Control induced time-scale separation

To simplify the notations, we introduce the following variables e i g,dq e i g 1 ,dq e Ii g 1 ,dq where

F i g,dq = diag(F i i g,dq ); F i i g,dq = 1 0 0 k 2g i ,dq F i w,dq = diag(F j i w,dq ); F j i w,dq = 1 0 0 k 2w j ,dq F v w,dq = diag(F j v w,dq ) F j v w,dq =     1 + 1 k 1,dq j 1 1 k 1,dq j + k 2,dq j + k 2,dq j k 1,dq j    
It is clear that F i g,dq , F i w,dq and F v w,dq are positive denite. Then, the derivative of W along the trajectories (5.110) can be calculated by dW dt = D 1 + D 2 where

D 1 = Γ 1 + Γ 2 D 2 = e T i gd F i gd dη i gd dt + e T igq F igq dη igq dt + e T i wd F i wd dη i wd dt +e T iwq F iwq dη iwq dt + e T vwq F vwq η vwq + e T v wd F v wd η v wd +e T z F ez ( fzero • e z + ϑ g 3 + ϑ w 2 + ϑ w 3 + ϑ w 4 ) with η i g,dq i * g 1 ,dq 0 • • • i * g N ,dq 0 T η i w,dq i * w 1 ,dq 0 • • • i * w M ,dq 0 T η v w,dq e i w 1 ,dq C f w 1 0 • • • e i w N ,dq C f w N 0 T 5.3.
Γ 1 = -M j=1 (k 1d j e 2 v sw j d + k 2d j e 2 Iv w j d + k 1q j e 2 vsw j q +k 2q j e 2 Ivw j q ) -e T z G ez e z Γ 2 = -N i=1 (k 1g i d e 2 i g i d + k 1g i q e 2 ig i q ) -M j=1 (k 1w j d e 2 i w j d + k 1w i q e 2 iw i q ) G i g,dq = diag(Gi i g,dq ), G i i g,dq = k 1g i ,dq 0 0 0 G i w,dq = diag(G j i w,dq ), G j i w,dq = k 1w j ,dq 0 0 0 G v w,dq = diag(G j v w,dq ), G j v w,dq = k 1,dq j 0 0 k 2,
b i k , for k = 1, • • • , L + 3, i ∈ N , and c j l , for l = 1, • • • , 8, j ∈ M such that D 2 satises the inequality D 2 ≤ N i=1 |e i g i d |(b i 1 |e i g i d | + b i 2 |e ig i q | + b i 3 |e ucg i | + L t=1 b i t+3 |e ic t |) + M j=1 (c j 1 |e i w j d | + c j 2 |e iw j q |) •(c j 3 |e v sw j d | + c j 4 |e Iv sw j d | + c j 5 |e vsw j q | + c j 6 |e Ivsw j q | +c j 7 |e i w j d | + c j 8 |e iw j q |) (
Γ 3 = N i=1 b i 1 2 ν i 1 |e i g i d | 2 + b i 2 2ν i 2 |e ig i q | 2 + M j=1 d j 1 2 κ j 1 |e i w j d | 2 + d j 2 2 κ j 2 |e iw j q | 2 (5.130) and Γ 4 = N i=1 b i 3 2ν i 3 |e ucg i | 2 + L t=1 b i t+3 2ν i t+3 |e ic t | 2 + M j=1 d j 3 2κ j 3 |e v sw j d | 2 + d j 4 2κ j 4 |e Iv sw j d | 2 + d j 5 2κ j 5 |e vsw j q | 2 + d j 6 2κ j 6 |e Ivsw j q | 2 (5.131)
where ν (•) and κ (•) have the same role as µ, and can be chosen arbitrarily.

In order to make the derivative of W negative except at the origin, we rst determine the values of droop gains k d i , k 1,2d j , k 1,2q j and the size of region of attraction Figure 5.14: Simulation results with the control gains in Set 1 (1). level is now about 1.05 p.u. which deviates from the setpoints u o cg 1 ,2 = 1.0 p.u.. As seen in Fig. 5.16(c), i w 1 d arrives around 0.6 p.u. to response to the change of I w 1 d . On the other hand, v sw 1 d is always well controlled around its setpoint v o sw 1 d = 1.0 p.u. irrespective of the variation in I w 1 d after a short transient as depicted in Fig.

5.17(d).

By contrast to the increase in I w 1 d , at t = 4 s, I w 2 d drops from 0.5 p.u. to 0.3 p.u.. The simulation results are summarized in Fig. 5.18. Since the generated power from the wind farm 2 decreases, i w 2 d starts to decrease and then converges to a new steady state of about 0.3 p.u. as shown in Fig. 5.18(c). From Figs. 5.18(a) -5.18(b), i g 1 d is changed from -0.5 p.u. to -0.4 p.u. and i g 2 d varies from -0.6 p.u. to -0.5 p.u. This means less power is received by the two SAC terminals because less power is injected into the DC grid. Additionally, in order to comply with the droop law (5.95), the DC voltages also drop and then get to a new steady state (∼ 1.0 p.u.) as displayed in Figs. 5.18(d) -5.18(f ).

To evaluate the capability in terms of AC voltage regulation, at t = 6 s, a new setpoint v o sw 1 d = 0.9 p.u. is sent to the 1 st WAC terminal. Now v sw 1 d is required to be stabilized around this new reference point. The transient response of v sw 1 d is illustrated in Fig. 5.19(a). It is found that v sw 1 d and v re sw 1 d quickly converge to the new setpoint with an acceptable undershoot. Since I w 1 d is unchanged during the interval [6,7] s as depicted in Fig. 5.19(b), the change of v o sw 1 d has no eect on the steady state of i w 1 d . According to (3.31), less power ows through the 1 st WAC terminal due to the decrease of the AC voltage at the PCC, which implies that the total transmitted power reduces. Therefore, both SAC terminals get less power than before and then, i g 1 d and i g 2 d start to decrease until converging to new steady states as seen in Figs. 5.19(c) and 5.19(d). Similar to the results in Figs. The responses of each terminal's active power at the PCC are plotted in Fig. 5.20. It is clear that the two SAC terminals participate in balancing the active power of the DC grid. When the power injection grows, both SAC terminals share the incremental power and then absorb more power from the grid. Conversely, if

Conclusions

Vector current control has been widely applied in the context of VSC control, whose principle is based on the assumption (or empirical practice) of time-scale separation between the system state variables. However, this hypothesis (or experience) has been rarely theoretically studied. This chapter has bridged the gap between the theory and the practice. As presented in the previous sections, the developed controllers are used to guarantee the operation of the system states around their prescribed references which are given by a higher control level. In this chapter, we will propose a possible method based on frequency control for computing the active power references. The main idea of this approach is to make the non-synchronous systems interconnected via an MTDC system share their primary reserves by modifying their scheduled power references. A DC-voltage-based control scheme, only needing local information, is introduced to regulate the AC frequency, for which a sucient condition on tuning the control gains are derived to ensure the system stability.

Introduction and motivation

In an AC system, the frequency, as an indication of power balance, is common everywhere on the time scale of a few milliseconds. With this common frequency, all generating units within the system can sense a power imbalance and adjust their power output to counter this disturbance. This mechanism of restoring power balance is commonly called primary frequency control, and the region of variation of generators' output is referred to as primary reserve. With the current practice of transferring a scheduled power among the AC areas through HVDC links, the frequencies of the areas are independent of each other, and thus generators in one area are not sensible to other areas' power imbalances. The advantage of this practice is that a disturbance in one area does not aect another, and thus the HVDC link can play the role of a rewall to prevent cascading failure. However, in the case of a severe power imbalance, the generating units within the aected area may not be strong enough to restore the power balance in time, and the resulting frequency excursion may be so large that invasive and expensive corrective measures (e.g. load shedding) have to be triggered. Such undesirable scenarios may be avoided if the HVDC system can be controlled in real time in such a way that the interconnected AC areas can share their primary reserves.

References [Hamzei-nejad 1986[START_REF] Dai | [END_REF] proposed frequency controllers that modify the active power transferred by an HVDC link. These proposed controllers require other AC areas' frequency measurements. Due to the time-delay of information communication, the controllers may destabilize the overall system. To become independent of remote information, [Rashed 2008] proposes a controller, named the DC-voltage-based controller, which takes actions only based on local information, requiring no communication among the AC areas. The theoretical stability proof in [Rashed 2008] is only valid for the special case of identical parameters of the AC areas, which is unrealistic in practice. In addition, the choice of the control gain in [Rashed 2008] was also rather empirical, and no proper approach to tuning the control gain was given.

In this work, we study a more general case without any restrictions on AC area parameters. A rigorous stability analysis is given to ensure safety and reliability.

Moreover, we investigate the feasible region for the control gains based on Lyapunov theory and LMI techniques [START_REF] Boyd | [END_REF]].

Modeling

The work in this chapter is based on the following consideration:

• We do not distinguish the dierence between the SAC and WAC connected VSC terminals. Every converter station is considered to be a black box. Since the control of the VSC terminal has dynamics in the ms range (see the simulation results in the previous sections), much faster than the problem of frequency control, the dynamics of the converter stations are neglected. We assume that any active power references generated by the frequency control can be perfectly tracked by the converter stations.

• The dynamics of the DC grid is neglected where each DC circuit branch is simply represented by an aggregated resistance.

As depicted in Fig. 6.1, we consider a simplied MTDC system consisting of a DC grid, N AC areas, and N VSC converters where P dc i represent active power injections to the HVDC grid. By convention, if P dc i is negative, AC area i absorbs 6.2. Modeling 211 power from the HVDC grid as a consumer. Otherwise, AC area i provides power as a supplier.

Figure 6.1: Diagram of an MTDC system with N AC areas.

DC grid

Each DC voltage is denoted as V i . Thanks to VSC converter and PWM technology, all converters can independently control their V i . The power injected to the DC grid from AC area i, P dc i , satises Ohm's law

P dc i = N k=1 V i (V i -V k ) R ik (6.1)
where the aggregated resistance R ik represents the transmission line between AC areas i and k. Obviously R ik = R ki . If AC areas i and k are not connected directly, R ik is innity.

AC areas

Each AC area is modeled as an aggregated generator and a load. The equation of motion of the generator is

2πJ i df i dt = P m i -P l i -P dc i 2πf nom,i -2πD g i (f i -f nom,i ) (6.2)
where P m i , J i and D g i are the mechanical power input, the moment of inertia and the damping factor of the aggregated generator of area i, respectively. f i is the frequency of AC area i and f nom,i is its nominal value. P l i is the power consumed by area i's load. In addition, J i , D g i and f nom,i are considered as known parameters.

Chapter 6. Frequency control using MTDC systems

The primary frequency control is realized by the speed governor of the generator, which modies P m i in response to a deviation of f i from f nom,i . This dynamics is modeled as

T sm,i dP m i dt = P s m i -P m i -

P nom,i σ i f -f nom,i f nom,i (6.3)
where T sm,i is the time constant of the servomotor, P s m i is the scheduled value for P m i , P nom,i is the nominal power of the generator and σ i is the generator droop.

By combining (6.2) and ( 6.3), the whole system is represented as

2πJ i df i dt = P m i -P l i -P dc i 2πf nom,i -2πD g i (f i -f nom,i )
T sm,i dP m i dt = P s m i -P m i - 6.4) for i = 1, • • • , N . f i and P m i are the state variables. By substituting (6.1) into (6.4), V i explicitly appear in the system model, and they are considered as the control inputs.

P nom,i σ i f i -f nom,i f nom,i (

Reference operating point

The reference operating point dened here corresponds to the steady state of the system, which is qualied by specic values of input parameters P l i , P s m i and the variables f i , P m i , P dc i , V i . We here use the same notation (•) o as in the previous section to denote the reference values of the system variables.

We usually expect that the system would operate in nominal condition and hence, the reference values of f o i and P o m i are naturally set to their nominal values, i.e.

f o i -f nom,i f nom,i P dc,o i = N k=1 V o i (V o i -V o k ) R ik (6.5)
V o i can be obtained by solving (6.5). Since the above algebraic equations are in quadratic formula, nding the proper solution for V o i can be turned into an opti- mization problem. Here, we will not focus on this issue.

Control strategy

Consider that the MTDC system initially operates at the reference operating point.

Then, one of its AC areas is subjected to a disturbance that takes form of a step load . It is true that this load change can be satised by the primary frequency control of its own area (P m i ). However, in case of a large disturbance, if we rely only on P m i , the resulting frequency deviation may be so large that it endangers the correct operation of the system. Thus, our objective is to improve the transient frequency prole by calling for other area's primary reserves so that the frequency excursions in area i are less pronounced.

Control law

The control objective is to regulate each converter's DC voltage V i only based on local measurements f i , i.e. V i is a function of f i only. Therefore, a DC voltage based controller is designed as

V i = V o i + α i (f i -f o i ) (6.6)
with positive gain α i > 0.

Under this controller, due to the physical coupling of the HVDC grid (6.1), the resulting power injected to DC grid P dc i can be regulated by controlling V i . This method leads our whole system to achieve the objective of sharing every area's primary reserve. However, we are not sure whether f i and P m i are ultimately bounded around the reference operating point under the control law (6.6) with an arbitrary positive control gain α i .

Choice of control gains

In this part, we give a detailed analysis for choosing the control gain α i to guarantee ultimate boundedness of f i and P m i under (6.6).

To simplify our problem, at rst, we shift the reference values to the origin by introducing the following new variables

P dc i = P dc i -P dc,o i Pli = P l i -P o l i Pmi = P m i -P o mi V i = V i -V o i (6.7)
Recalling (6.1), P dc i can be expressed in these new variables as

P dc i = N k =i ( V 2 + 2 V i V o i -V i V k -V i V o k -V k V o i ) R ik (6.8)
An error system of the original system (6.4) is generated in terms of the new variables by applying (6.5) and (6.8) ḟi =-a 1i fi + a 2i Pmi -a 2i Pli -

a 2i • N k =i ( V 2 + 2 V i V o i -V i V k -V i V o k -V k V o i ) R ik Ṗmi =-a 4i fi -a 5i Pmi (6.9)
where a 1i = D g i J i ,a 2i = 1 4π 2 J i f nom,i

, a 4i = P nom,i T sm,i σ i f nom,i , a 5i = 1 T smi . It is worthwhile to note that a 1i , a 2i , a 4i , a 5i are strictly positive, and it can be seen that the control input V i explicitly appears in the system (6.9).

As seen in (6.9), Pli is viewed as a nonvanishing perturbation which is uniformly bounded if a load demand imbalance exists, i.e. P l i -P o l i = 0. As a result, the system (6.9) can be considered as a perturbation of the nominal system ḟi =-a 1i fi + a 2i Pmi -

a 2i • N k =i V 2 i + 2 V i V o i -V i V k -V i V o k -V k V o i R ik Ṗmi =-a 4i fi -a 5i Pmi (6.10)
If the origin of the nominal system (6.10) is asymptotically stable under the control law, then the perturbed system (6.9) is ultimately bounded around the origin [Khalil 1996]. Thus, the problem becomes to investigate a feasible region for α i such that the origin of the nominal system (6.10) is asymptotically stable under (6.6).

In the following part, two methods are devised to search a feasible region for α i by means of analyzing the stability of the nonlinear system (6.10).

First approach -linearization of injected DC power ow

The rst approach consists in analyzing the stability of the nonlinear system via linearization. Let us linearize (6.8) around the reference operating point resulting in

P dc i = N k =i 2V o i -V o k R ik V i - N k =i V o i R ik V k = N k =i V o i -V o k R ik V i + N k =i V o i R ik V i - N k =i V o i R ik V k (6.11)
To simplify the notation, we dene

f = f1 • • • fN T Pm = Pm1 • • • PmN T P dc = P dc 1 • • • P dc N T V = V 1 • • • V N T (6.12)
and then write (6.11) compactly in vector form as

P dc = diag(V o i )L V + diag(V o i )diag( N k =i 1 R ik ) V -diag( N k =i V o k R ik ) V
(6.13)

Control strategy 215

where L ∈ R N ×N is the weighted Laplacian matrix describing the topology of HVDC grid, dened as

[L] ik =      - 1 R ik , k = i N j =i 1 R ij , k = i
With the new notations

V = diag(V o i ); R = diag( N k =i 1 R ik ); V R = diag( N k =i V o k R ik )
the nominal system (6.10) can be expressed in matrix form as

ḟ Ṗm = -A 1 A 2 -A 4 -A 5 f Pm - A 2 (V L + V R -V R ) 0 V (6.14)
where A 1 = diag(a 1i ), A 2 = diag(a 2i ), A 4 = diag(a 4i ), A 5 = diag(a 5i ), A α = diag(α i ). In addition, according to the control algorithm given by (6.6), Ṽ in (6. where A is given by

A = -A 1 -A 2 (V L + V R -V R )A α A 2 -A 4 -A 5
A Lyapunov-based method is used to nd a feasible region for A α . Taking W = f T A 4 f + P T m A 2 Pm (6.17) as a Lyapunov function candidate, the derivative of W along the trajectories of (6.16) is

Ẇ = -f T F α f -2 P T m A 5 A 2 Pm (6.18)
where F α is a symmetric matrix, expressed as

F α = 2A 1 A 4 + A α (LV + RV -V R )A 2 A 4 + A 4 A 2 (V L + V R -V R )A α
If there is a region for A α such that F α is positive denite, the origin of the closedloop system (6.16) is then asymptotically stable. Thus, our problem becomes to nd 216 Chapter 6. Frequency control using MTDC systems a feasible region for α such that F α > 0. This leads to a linear matrix inequality (LMI) problem. The objective is now to nd a diagonal A α such that the following inequalities

A α > 0 2A 1 A 4 + A α (LV + RV -V R )A 2 A 4 + A 4 A 2 (V L + V R -V R )A α > 0 (6.19)
hold. The above LMI problem can be solved for typical values. As seen in (6.19), it is shown that the feasible region for A α is determined by all the AC areas parameters and the reference operating values of the system variables.

Second approach -nonlinear approach

It is well known that the approximation of the nonlinear system (6.10) by its linearized system (6.14) is valid only in a neighbourhood of the origin. Thus, to study the global behaviour of the system under the control law, another approach based on Lyapunov theory is devised to study the stability of the nonlinear system (6.10).

We rewrite P dc i (6.8) in a vector form as

P dc i = diag( V i )L V + diag( V i )LV o + diag(V o i )L V (6.20)
where

V o = V o 1 • • • V o N T
The matrix form of the system (6.10) is given by

ḟ Ṗm = -A 1 A 2 -A 4 -A 5 f Pm - A 2 0 • (diag( V i )L V + diag( V i )LV o + diag(V o i )L V ) (6.21)
Substituting V i = α i fi and diag( V i ) = A α diag( fi ) into the above expression, then the closed-loop system is obtained as

ḟ Ṗm = -A 1 A 2 -A 4 -A 5 f Pm - H 1 ( f ) 0 0 0 f Pm - H 2 ( f ) 0 (6.22)
where H 1 ( f ) and H 2 ( f ) take the form

H 1 ( f ) = A 2 A α diag( fi )LA α + A 2 diag(V o i )LA α H 2 ( f ) = A 2 A α diag( fi )LV o
To investigate the stability of the closed-loop system (6.22), the same Lyapunov

Ẇ = -2 f T A 1 A 4 f -f T A 4 H 1 f -f T H T 1 A 4 f -f T A 4 H 2 -H T 2 A 4 f -2 P T m A 5 A 2 Pm where f T A 4 H 1 f = N i=1 a 4i a 2i (α i f 2 i + V o i fi ) N k =i α i fi -α k fk R ik f T A 4 H 2 = N i=1 a 4i a 2i α i f 2 i N k =i V o i -V o k R ik
To facilitate our analysis, we rewrite Ẇ as

Ẇ = -M 1 -M 2 -2 P T m A 5 A 2 Pm (6.24)
where M 1 and M 2 are given by

M 1 = f T A 1 A 4 f + f T A 4 H 1 f + f T H T 1 A 4 f M 2 = f T A 1 A 4 f + f T A 4 H 2 + H T 2 A 4 f
It is possible to rewrite M 1 and M 2 in a total sum square form as

M 1 = N i=1 a 4i a 1i f 2 i + 2a 2i (α i f 2 i + V o i fi ) • N k =i α i fi -α k fk R ik M 2 = N i=1 a 4i f 2 i a 1i + 2a 2i α i N k =i V o i -V o k R ik
As a consequence, we get

M 1 + M 2 = N i=1 a 4i 2a 1i f 2 i + 2a 2i N k =i V o i -V o k R ik α i f 2 i +2a 2i N k =i 1 R ik α 2 i f 3 i -2a 2i α i N k =i α k x k R ik f 2 i +2a 2i N k =i 1 R ik V o i α i f 2 i -N i=1 2a 4i a 2i V o i N k =i α k fk fi R ik (6.25)
Applying arithmetic geometric mean inequality (6.26) to M 1 + M 2 in (6.25) yields (6.27) 218 Chapter 6. Frequency control using MTDC systems

α k fk fi R ik ≤ 1 2 α k f 2 i R ik + α k f 2 k R ik
M 1 + M 2 ≥ N i=1 a 4i 2a 1i f 2 i + 2a 2i N k =i V o i -V o k R ik α i f 2 i +2a 2i N k =i 1 R ik α 2 i f 3 i -2a 2i α i N k =i α k fk R ik f 2 i +2a 2i N k =i 1 R ik V o i α i f 2 i -N i=1 a 4i a 2i V o i N k =i α k f 2 i R ik + α k f 2 k R ik
The last term of (6.27) can be written as

N i=1 a 4i a 2i V o i N k =i α k f 2 i R ik + α k f 2 k R ik = N i=1 a 4i a 2i V o i N k =i α k R ik + α i N k =i a 4k a 2k V o k a 4i R ki f 2 i (6.28) Therefore, M 1 + M 2 satises M 1 + M 2 ≥ N i=1 a 4i 2a 1i + 2a 2i N k =i V o i -V o k R ik α i + 2a 2i N k =i 1 R ik α 2 i fi -2a 2i α i N k =i α k fk R ik + 2a 2i N k =i 1 R ik V o i α i -a 2i V o i N k =i α k R ik -α i N k =i a 4k a 2k V o k a 4i R ki f 2 i (6.29)
Finally, a feasible region for α i is found, denoted by

Ω α i = α i ∈ R 2a 1i + 2a 2i N k =i V o i -V o k R ik α i +2a 2i N k =i 1 R ik α 2 i fi -2a 2i α i N k =i α k fk R ik +2a 2i N k =i 1 R ik V o i α i -a 2i V o i N k =i α k R ik -α i N k =i a 4k a 2k V o k a 4i R ki ≥ 0 (6.30)
Therefore, for any α i ∈ Ω α i , we have Ẇ < 0, ∀[ f , Pm ] = 0 and as a conclusion, the origin of the nonlinear system (6.10) is asymptotically stable. In a way, once α i are determined, we can get an estimation of the region of attraction Ω c for the state variables

Ω c = [ f , Pm ] ∈ R 2N k =i -α i fi + α k fk R ik ≤ a 1i α i a 2i - N k =i V o i -V o k R ik - N k =i 1 R ik V o i - V o i 2α i N k =i α k R ik - 1 2a 4i a 2i N k =i a 4k a 2k V o k R ik , i = 1, • • • N.
(6.31) 6.3.3 Algorithm for the denition and verication of control gains and the associated region of attraction

It is not easy to use (6.30) and (6.31) directly to get a feasible region due to their complicated expressions. Though a feasible region can be easily found by the linear method (6.19), we can only approximate the nonlinear system (6.10) by its linearization in a small neighbourhood of the origin, of which we do not know the size. Thus, 6.4. Simulation studies 219 when determining the gains α i , we need to combine these two methods together. In general, the feasible region obtained by linear method is larger than the nonlinear method. At rst, we use linear method to get a feasible value for α i , then we put it into the nonlinear result (6.30) and (6.31) to check if this value satises the nonlinear method's condition such that the system is stable when the state variables are in their domain of interest. For example, the domain of interest of fi is dened by fi ∈ [-1, 1]. If this is not the case, we choose a smaller value for α i and verify (6.30) and (6.31) again. We repeat this procedure until a proper α * i is obtained.

The feasible region of the control gain α given by this thesis is determined by the choice of Lyapunov function. The one used here has two advantages. First, as seen in (6.18), there is no cross term ( f T Pm ), which greatly simplies the calculation.

Second, the feasible region thus obtained is not so small. However, the feasible region of the control gain is only a sucient condition, which guarantees ultimate boundedness of the closed-loop system. Thus, with an α outside this feasible region, the system does not necessarily become unbounded.

Simulation studies

The controller studied in the previous section is tested by computer simulations.

The simulated example concerns an MTDC grid of 5 AC areas, whose parameter values are presented in Table 6.1. The system is supposed to initially operate at the reference operating point. Then at time t = 2 s, the load demand of AC area 2 has a step increase by 30%.

Figs. 6.2 and 6.3 illustrate the frequency and the mechanical power response without any controller, i.e. α i = 0. In Fig. 6.2, the minimum value of f 2 is less than 49 Hz, which is beyond frequency safety range 50±1 Hz, and the nal value is 49.5615 Hz. The peak value of P m2 is nearly 106 MW and the nal value is nearly 97.6 MW. when the controller is implanted. Table 6.2 shows the values for α i calculated by LMI techniques. The proposed α i are in their feasible region that can make sure that the system is stable. Note that, the minimum value of f 2 is between 49.4515 Hz and 49.452 Hz which is in the safety range and the nal value f 2 is 49.7299 Hz. P m2 has a peak value of nearly 92.5 MW, and is stabilized at 91 MW. The above results show that our controller makes a signicant improvement for f 2 not only in the transient performance but also in the steady-state performance compared to the case without sharing the primary reserves (see Fig. 6.2).

Comparing the trajectories of P dc i in Figs. 6.4 and 6.8, when the frequency control is applied to counteract the sudden load change of AC area 2, the power injection of every AC area gives a corresponding response, which is sent to their respective converter stations.

Conclusions

In this chapter, we addressed the problem of developing a frequency controller that provides the active power reference values to the VSC and shares primary reserves between the AC areas connected by an MTDC system. In the case of a power imbalance, the controller reduces the burden of the aected AC area by making the other AC areas collectively react to that disturbance. In addition, the proposed control law is only based on local measurements without needing remote communication, thus avoiding the disadvantages caused by time-delay.

Conclusions

In the last decades, the development of VSC HVDC systems becomes a very important topic in scientic researches. Due to their exibility, controllability and eciency, VSC HVDC systems have been greatly promoted in today's electrical networks. In particular, there is a growing need to integrate large-scale oshore renewable energy sources via MTDC systems, whose generation outputs are usually intermittent. Nevertheless, the MTDC technology is not yet mature enough for wide commercial applications. There are still many issues that need to be solved. This dissertation is dedicated to the challenges in the control of MTDC systems. This thesis started by giving a brief introduction to HVDC technology where the limits of the classical AC transmission technology are shown and the rationale of HVDC systems is discussed. Besides, the comparisons between AC and DC as well as LCC and VSC are also listed. Then, we introduced several existing control structures of VSC HVDC systems according to their research groups in the literature review. Interestingly, we found that most of the control methods only focus on the operating results but neglect related theoretical analysis. As a consequence, extra time-consuming processes are usually needed to accompany these control approaches to achieve good performance. At the end of Chapter 1, we dened the objectives of this thesis: 1) developing new control methods for MTDC systems to improve the overall performance; 2) providing theoretical explanations for the system dynamic behaviors to give new insights into how and why the existing conventional (or empirical) control structures work; 3) fully developing the potential of MTDC systems for AC frequency regulation; 4) Developing a new control theory named Control Induced Time-Scale Separation. This general control theory was created aiming at the MTDC system, but indeed applies to a very broad class of nonlinear systems, and allows plug-and-play capabilities. In addition, it ts well in a Systems of Systems framework.

In Chapter 2, we rst gave an overview of the VSC technology to get a good understanding of the classic operation of VSC HVDC systems. Then, we introduced two dierent types of AC networks which are distinguished by their Short Circuit Ratio. In line with the control objectives and the type of the connected AC network, possible control modes of a single VSC terminal were discussed, which oer a basic knowledge for the control design of MTDC systems.

In the modeling, the time-averaged modeling method which is the most generally used modeling approach in the context of power systems was chosen and presented in Chapter 3. This modeling approach is much simpler and tractable than the detailed modeling one. In this thesis, the proposed MTDC system consists of N strong AC networks, M weak AC networks and a DC grid. One of the main contributions of this thesis is that a generic topology is considered for the structure of the DC grid, which can be mapped to a weakly connected directed graph. Thus, our research is not only suitable for the radial topology but also for the meshed topology. In addition, all connected AC networks are considered in balanced three-phase condition and hence, Park's transformation is applied to project all three-phase quantities onto a rotating two-axis reference frame. The overall modeling work is divided into three parts: 1) modeling of the strong AC network connected VSC terminals; 2) modeling of the weak AC network connected VSC terminals; 3) modeling of the DC grid. At the end of Chapter 3, an averaged state-space model of high order for the proposed MTDC system was established.

In Chapter 4, two of the most widely used conventional control methods for VSC HVDC systems, namely direct and vector control methods, were discussed. Both advantages and disadvantages of the aforementioned control methods were analyzed.

Based on the existing problems of the two control approaches, new improved control strategies were proposed in this dissertation.

In Chapter 5, new control structures by means of dierent nonlinear control design tools for VSC HVDC systems were developed. We started by applying feedback linearization technique to the development of a nonlinear controller, which can partly transform the nonlinear MTDC model to a linear one. A theoretical analysis of the nonlinear feedback linearization controller was carried out, which indicates that the stability of the equilibrium of the zero dynamics determines the stability of the overall system. Furthermore, a sucient condition on the system parameters was derived to ensure the stability of the zero dynamics. The performance comparisons between this nonlinear controller and the conventional vector controller were given, which show that the nonlinear control provides much better decoupling characteristics of the DC voltage and the AC q-axis current control or the AC voltages v sw j d and v sw j q control than the PI controller. Besides, during the simulation process, we 7.1. Conclusions 227 found that when a new operating point of the MTDC systems is to be achieved, it is usually needed to recalculate the proper PI control gains for the conventional vector controller to obtain an optimal performance. However, this is time-consuming and usually requires extra eorts. For the nonlinear feedback controller, we usually have no need to deduce new control gains for dierent operating conditions. In order to make the system more robust, a sliding mode control technique was applied to develop a novel controller where the uncertainties of the system parameters and other possible exogenous inputs were taken into account. During the design procedure of the feedback linearization controller, we naturally chose the controlled variables as the output. However, the theoretical analysis clearly shows that the MTDC system is not always minimum phase. There exists a restriction on the choice of the prescribed references and the system parameters (see Lemma 4.1.2). To overcome this problem, a passive output was deduced by means of passivity theory so that the MTDC system with the passive output is always minimum phase. Then, a new control strategy based on the input-output feedback linearizaton and the passivity theory was devised. Simulation studies were also carried out to evaluate the performance of the passivity-based state feedback controller.

In Chapter 6, the dynamic behaviors of the proposed MTDC system were analyzed. It is interesting to remark that these dynamics have never been extensively rigorously studied in the literature. The research done on this issue is one of the main contributions of this dissertation summarized as follow:

• Control induced time-scale separation design for a class of nonlinear systems

A control induced time-scale separation method was developed for a class of nonlinear systems, which enables the system to exhibit a multi-time-scale behavior characterized by the presence of fast and slow transients. Based on singular perturbation theory, a detailed theoretical analysis was performed, which shows that the time-scale decomposition of the system dynamics is qualied by the control gains of the driving subsystem. Besides, with the help of the proposed controller, the original system can be divided into two low-order subsystems and then, the behaviors of the system states can be approximated by the solutions of the two simpler subsystems. Furthermore, we gave an analytical expression to illustrate the trade-o between the system performance and the region of attraction.

• Analysis of the conventional control structures and the dynamics of the MTDC system For an MTDC system, the concept of the conventional vector control method is mainly based on the assertion that the dynamics of the system state variables can be divided into dierent time scales and hence, the two control loops can be designed independently. However, very few research has ever veried and explained this assertion. In this dissertation, we applied the theoretical results obtained from the study on the time-scale separation controller to

  diagram of the j th VSC connected wind farm. 35 3.6 DC circuit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.7 An example of the DC grid. . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 Block diagrams of direct control method. . . . . . . . . . . . . . . . . 41 3.9 Inner current loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 3.10 Outer current loop. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.1 A resistive load R L is connected to the i th SAC terminal. . . . . . . 51 4.2 Model of VSC HVDC link connected to a resistive load using Sim-PowerSystems toolbox. . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 The response of DC voltage for the change of u o cg i . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . 56 4.4 The response of i g i q for the change of u o cg i . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . . . 56 4.5 The response of i L for the change of u o cg i . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . DC voltage for the changes of R L and i o g i q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . .

  (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . 59 4.8 The response of i g i d for the changes of R L and i o g i q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . .

  (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . 60 4.10 The response of i L2 for the changes of R L and i o g i q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . 60 4.11 The response of i L3 for the changes of R L and i o g i q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . 61 4.12 A VSC HVDC link consists of one strong and one weak AC system. . 61 4.13 The response of v sw 1 d for the changes of v o sw 1 d and v o sw 1 q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . 67 4.14 The response of v sw 1 q for the changes of v o sw 1 d and v o sw 1 q . (a) PIcontroller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . 67 4.15 The response of v sw 1 ,abc for the changes of v o sw 1 d and v o sw 1 q using PI controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 4.16 Zoom of the response of v sw 1 ,abc for the changes of v o sw 1 d and v o sw 1 q using PI controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.17 The response of v sw 1 ,abc for the changes of v o sw 1 d and v o sw 1 q using feedback linearization controller. . . . . . . . . . . . . . . . . . . . . 69 4.18 Zoom of the response of v sw 1 ,abc for the changes of v o sw 1 d and v o sw 1 q using feedback linearization controller. . . . . . . . . . . . . . . . . . 70 4.19 f w 1 = 50 Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.20 The response of u cg 1 for the changes ofv o sw 1 d and v o sw 1 q . (a) PI con- troller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . 71 4.21 An MTDC system consists of two strong and two weak AC systems. 78 4.22 DC grid used to connect the four AC areas. . . . . . . . . . . . . . . 79 4.23 The response of u cg 1 . (a) PI controller. (b) Feedback nonlinear con- troller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.24 The response of i g 1 q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.25 The response of v sw 1 d . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 4.26 The response of v sw 1 q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.27 The response of v sw 1 ,abc using PI control technique. . . . . . . . . . . 82 4.28 The response of v sw 1 ,abc using feedback linearization technique. . . . 83 4.29 The response of v sw 2 d . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.30 The response of v sw 2 q . (a) PI controller. (b) Feedback nonlinear controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 4.31 The response of v sw 2 ,abc using PI control technique. . . . . . . . . . . 84 4.32 The response of v sw 2 ,abc using feedback linearization technique. . . . 85 4.33 Variations in the resistive load R. . . . . . . . . . . . . . . . . . . . . 89 4.34 Response of u cg i (curve (a)) and its reference u o cg i (curve (b)). . . . . 89 4.35 Response of i g i d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 4.36 Response of i g i q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 List of Figures xix 4.37 Response of m g i d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.38 Response of m g i q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.39 The response of u cg 1(a) Feedback nonlinear controller. (b) Passivity- based controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.40 The response of i g 1 q (a) Feedback nonlinear controller. (b) Passivitybased controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.41 The response of i g 2 d (a) Feedback nonlinear controller. (b) Passivitybased controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.42 The response of i g 2 q (a) Feedback nonlinear controller. (b) Passivitybased controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.43 The response of v sw 1 ,abc using feedback nonlinear controller. . . . . . 4.44 The response of v sw 1 ,abc using passivity-based controller. . . . . . . . 4.45 The response of u cg 1 (a) Feedback nonlinear controller. (b) Passivity- based controller. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.46 The response of v sw 1 d (a) Feedback nonlinear controller. (b) Passivity-based controller. . . . . . . . . . . . . . . . . . . . . . . . . 4.47 The response of v sw 1 q (a) Feedback nonlinear controller. (b) Passivity-based controller. . . . . . . . . . . . . . . . . . . . . . . . . 4.48 The response of v sw 2 d (a) Feedback nonlinear controller. (b) Passivity-based controller. . . . . . . . . . . . . . . . . . . . . . . . . 4.49 The response of v sw 2 q (a) Feedback nonlinear controller. (b) Passivity-based controller. . . . . . . . . . . . . . . . . . . . . . . . . 4.50 Variations in R f w 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.51 Variations in R f w 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.52 The response of v sw 1 d (a) Nominal passivity-based controller (b) Adaptive passivity-based controller (c) Reference values of v o sw 1 d = 40.82 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.53 The response of v sw 1 ,abc under the nominal passivity-based controller. 4.54 The response of v sw 1 ,abc under the adaptive passivity-based controller. 4.55 The response of v sw 2 d (a) Nominal passivity-based controller (b) Adaptive passivity-based controller (c) Reference values of v o sw 2 d = 40.82 V. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.56 The response of v sw 2 ,abc under the nominal passivity-based controller. 4.57 The response of v sw 2 ,abc under the adaptive passivity-based controller. 4.58 Comparison between (a) the actual parameter value ψ 3 1 and background . . . . . . . . . . . . . . . . . . . . . 1 1.2 Overview of HVDC technology . . . . . . . . . . . . . . . . . 3 1.2.1 Line-commutated current source converters . . . . . . . . . . 4 1.2.2 Voltage-sourced converters . . . . . . . . . . . . . . . . . . . . 4 1.2.3 LCC vs VSC . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.4 Advantages of HVDC over AC . . . . . . . . . . . . . . . . . 6 1.3 Literature review: related studies on control designs . . . . 8 1.3.1 Control of a single VSC terminal . . . . . . . . . . . . . . . . 8 1.3.2 Control of MTDC systems . . . . . . . . . . . . . . . . . . . . 9 1.4 Motivations and contributions of the thesis . . . . . . . . . . 11 1.4.1 Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.4.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . 14 1.6 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

Figure 1

 1 Figure 1.1: AC versus DC cost.

Figure 2

 2 Figure 2.1: A simplied representation of a VSC HVDC converter station. Usually, the harmonic lters, the phase reactors, the converters and the DC capacitors are located indoors while the transformers and the cooling system are outdoors.

Figure 2 . 2 :

 22 Figure 2.2: Three phase bidirectional VSC.

Figure

  Figure 2.3: Monopolar HVDC link with earth, water or metallic return path.

Figure 2

 2 Figure 2.4: Bipolar HVDC link with earth or metallic return path.

Figure 2

 2 Figure 2.5: Back-to-back HVDC link.

Figure 2

 2 Figure 2.6: Multi-terminal HVDC link with three terminals connected in parallel.

Figure 2

 2 Figure 2.7: A single line representation of the AC side of a VSC.

Figure 3

 3 Figure 3.1: A three-phase PLL system.

Figure 3

 3 Figure 3.2: A multi-terminal VSC HVDC system with strong and weak AC systems.

Figure 3

 3 Figure 3.3: A simplied schematic diagram of the i th VSC connected strong grid.

Figure 3

 3 Figure 3.4: A VSC connected to wind farm.

  high frequency AC lter represented by a simple capacitor C f w j is used to eliminate those unacceptable harmonics.

Figure 3

 3 Figure 3.5: A simplied schematic diagram of the j th VSC connected wind farm.

Figure 3

 3 Figure 3.6: DC circuit.

Figure 3 . 7 :

 37 Figure 3.7: An example of the DC grid.

  represent the node's total incoming and outgoing transmission branch currents respectively. U + (•) and U - (•) are the DC voltages of the two nodes connected by l k .

  to the N SAC converter nodes, the M WAC converter nodes and the P intermediate nodes, respectively. E = {e 1 , • • • , e L } is the set of the edges mapping to the L circuit branches. The incidence matrix of G = (V, E) is denoted by H ∈ R (N +M +P )×L and its element in the l th row and the k th column, i.e. H lk , satises current of e k ows into the node v l , -1 if the branch current of e k ows from the node v l , illustrated in Fig.3.7. The corresponding incidence matrix

Figure 3

 3 Figure 3.8: Block diagrams of direct control method.

  control system has a cascaded control structure consisting of two loops, the inner current and the outer control loops. The vector control structure involves the eld-oriented vector control technique, which is developed from the representation described by (3.20) of the AC quantities in the dq synchronous reference frame. The detailed control diagrams of the i th connection terminal are presented in Figs. (3.9) and (3.10).

Figure 3 . 9 :

 39 Figure 3.9: Inner current loop.

Figure 3 .

 3 Figure 3.10: Outer current loop.

Contents 4 . 1

 41 Feedback linearization control . . . . . . . . . . . . . . . . . . 48 4.1.1 Theoretical results . . . . . . . . . . . . . . . . . . . . . . . . 48 4.1.2 Application to a VSC HVDC link connected to a resistive load 51 4.1.3 Application to a VSC HVDC link consisting of a strong and a weak AC system . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.1.4 Application to an MTDC system using master-slave control conguration . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 4.2 Feedback linearization with sliding mode control . . . . . . 85 4.2.1 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.2.2 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . 88 4.3 Passivity-based control . . . . . . . . . . . . . . . . . . . . . . 92 4.3.1 Steady-state analysis . . . . . . . . . . . . . . . . . . . . . . . 93 4.3.2 Passive system . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.3.3 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.3.4 Control of zero dynamics . . . . . . . . . . . . . . . . . . . . 102 4.3.5 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . 116 4.4 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 125

Figure 4

 4 Figure 4.1: A resistive load R L is connected to the i th SAC terminal.

  Figure 4.2: Model of VSC HVDC link connected to a resistive load using SimPow-erSystems toolbox.

56

  Chapter 4. Control methods based on nonlinear control design tools

Figure 4 . 3 :FeedbackFigure 4 Figure 4 . 5 :

 43445 Figure 4.3: The response of DC voltage for the change of u o cg i . (a) PI controller. (b)

Figure 4

 4 Figure 4.6: The response of DC voltage for the changes of R L and i o g i q . (a) PI

Figure 4

 4 Figure4.7: The response of i g i q for the changes of R L and i o g i q . (a) PI controller.

Figure 4

 4 Figure 4.8: The response of i g i d for the changes of R L and i o g i q . (a) PI controller.

Figure 4 . 9 :

 49 Figure 4.9: The response of i L for the changes of R L and i o g i q . (a) PI controller. (b)

Figure 4 .

 4 Figure 4.10: The response of i L2 for the changes of R L and i o g i q . (a) PI controller.

Figure 4 .

 4 Figure 4.11: The response of i L3 for the changes of R L and i o g i q . (a) PI controller.

Figure 4 .

 4 Figure 4.12: A VSC HVDC link consists of one strong and one weak AC system.

Figure 4 .

 4 Figure 4.13: The response of v sw 1 d for the changes of v o sw 1 d and v o sw 1 q . (a) PI

Figure 4 .

 4 Figure 4.14: The response of v sw 1 q for the changes of v o sw 1 d and v o sw 1 q . (a) PI

Figures 4 .Figure 4

 44 Figures 4.15-4.19 illustrate the responses of the AC voltage in abc frame. Figure4.19 shows that the AC voltage at the PCC are always kept at f w 1 = 50 Hz. By comparing Fig.4.16 and Fig.4.18, the feedback nonlinear controller gives a better transient performance for v sw 1 ,abc with a smaller overshoot than the PI controller when v o sw 1 q is set to 4.08 V at t = 1.5 s.

Figure 4 .Figure 4

 44 Figure 4.16: Zoom of the response of v sw 1 ,abc for the changes of v o sw 1 d and v o sw 1 q

Figure 4 .

 4 Figure 4.18: Zoom of the response of v sw 1 ,abc for the changes of v o sw 1 d and v o sw 1 q

Figure 4 .

 4 Figure 4.20: The response of u cg 1 for the changes of v o sw 1 d and v o sw 1 q . (a) PI con-

  Fig. 4.21 where the structure of the DC grid is illustrated by Fig. 4.22. Based on the arrangement of the VSC operation as stated at the beginning of this section, the control congurations of the four terminals are listed in Table 4.7. The system parameters of the four terminals are chosen according to Table 4.4.

Figure 4 .

 4 Figure 4.21: An MTDC system consists of two strong and two weak AC systems.

Figure 4 .

 4 Figure 4.22: DC grid used to connect the four AC areas.

  Figures 4.25

  and 4.26 ( or Figures 4.29 and 4.30) depict the responses of v sw 1 ,dq ( or v sw 2 ,dq ) and show that the feedback nonlinear controller provides faster damping and less overshoot than the PI controller. Besides, as plotted in Fig. 4.25 ( or Fig. 80 Chapter 4. Control methods based on nonlinear control design tools 4.25 ), the change of v osw 1 q ( or v o sw 2 q ) has a crippling eect on the performance of v sw 1 d ( or v sw 2 d ) under the PI controller while v sw 1 d ( or v sw 2 d ) under the feedback nonlinear controller is always kept at its reference value. It reveals that the feedback nonlinear controller also gives a better decoupling characteristics of v sw 1 d and v sw 1 q ( or v sw 2 d and v sw 2 q ) control. As shown inFigs. 4.27 and 4.28 (or Figs. 4.31 and 4.32), the feedback nonlinear controller also improves the performance of the AC voltage of WAC1 ( or WAC2) at the PCC in abc frame in comparison to the PI controller.

Figure 4 .

 4 Figure 4.23: The response of u cg 1 . (a) PI controller. (b) Feedback nonlinear controller.

Figure 4 .

 4 Figure 4.24: The response of i g 1 q . (a) PI controller. (b) Feedback nonlinear controller.

Figure 4 .

 4 Figure 4.25: The response of v sw 1 d . (a) PI controller. (b) Feedback nonlinear controller.

Figure 4 .Figure 4

 44 Figure 4.26: The response of v sw 1 q . (a) PI controller. (b) Feedback nonlinear controller.

Figure 4 .

 4 Figure 4.28: The response of v sw 1 ,abc using feedback linearization technique.

Figure 4

 4 Figure 4.29: The response of v sw 2 d . (a) PI controller. (b) Feedback nonlinear controller.

Figure 4 .Figure 4

 44 Figure 4.30: The response of v sw 2 q . (a) PI controller. (b) Feedback nonlinear controller.

Figure 4 .

 4 Figure 4.32: The response of v sw 2 ,abc using feedback linearization technique.

Figure 4 .Figure 4

 44 Figure 4.33: Variations in the resistive load R.

Figure 4 .

 4 Figure 4.35: Response of i g i d .

Figure 4

 4 Figure 4.36: Response of i g i q .

Figure 4

 4 Figure 4.37: Response of m g i d .

Figure 4

 4 Figure 4.38: Response of m g i q .

  on (4.90) and (4.91), a new model for the DC grid is established as ż = A p z + ϑ p (4.92) 4.3. Passivity-based control 93 where A p and ϑ p are given by

  Control methods based on nonlinear control design tools

  4.3.2 Passive systemLet (•) = (•) -(•) denote the error between the variable (•) and its steady-state value. By considering the steady-state condition (4.96), (•) satises the dierential equations

  the change of the variables i gρd = α gρd u cgρ and di gρd dt = α gρd du cgρ dt into the zero 4.3. Passivity-based control 103 dynamics (4.115) and then obtain the following transformed zero dynamics

g

  i d are considered equal to the steady-state values ᾱg i d = īg i d ūcg i

  Control methods based on nonlinear control design toolsLet us consider the reference model 

  rewrite the controller (4.141) in these new variables as

Figure 4

 4 Figure 4.39: The response of u cg 1 (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4 .

 4 Figure 4.40: The response of i g 1 q (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4 .

 4 Figure 4.41: The response of i g 2 d (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4 .Figure 4

 44 Figure 4.42: The response of i g 2 q (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4 .

 4 Figure 4.44: The response of v sw 1 ,abc using passivity-based controller.

Figure 4

 4 Figure 4.45: The response of u cg 1 (a) Feedback nonlinear controller. (b) Passivity- based controller.

Figure 4 .

 4 Figure 4.46: The response of v sw 1 d (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4 .

 4 Figure 4.47: The response of v sw 1 q (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4 .

 4 Figure 4.48: The response of v sw 2 d (a) Feedback nonlinear controller. (b) Passivitybased controller.

Figure 4

 4 Figure 4.49: The response of v sw 2 q (a) Feedback nonlinear controller. (b) Passivitybased controller.

  Figures 4.58 and 4.59 illustrate the actual system parameters and their estimated values. Interesting, we nd that the estimated values ( ψ3 1 and ψ3 2 ) are not exactly equal to their actual values (ψ 3 1 and ψ 3 2 ). This phenomenon does not surprise us

FigureFigure 4

 4 Figure 4.50: Variations in R f w 1 .

Figure 4 .

 4 Figure 4.52: The response of v sw 1 d (a) Nominal passivity-based controller (b) Adaptive passivity-based controller (c) Reference values of v o sw 1 d = 40.82 V.

Figure 4 .Figure 4

 44 Figure 4.53: The response of v sw 1 ,abc under the nominal passivity-based controller.

Figure 4 .Figure 4

 44 Figure 4.55: The response of v sw 2 d (a) Nominal passivity-based controller (b) Adaptive passivity-based controller (c) Reference values of v o sw 2 d = 40.82 V.

Figure 4 .

 4 Figure 4.57: The response of v sw 2 ,abc under the adaptive passivity-based controller.

Figure 4 ..

 4 Figure 4.58: Comparison between (a) the actual parameter value ψ 3 1 = 1 R f w 1 and

Figure 4 .

 4 Figure 4.59: Comparison between (a) the actual parameter value ψ 3 2 = 1 R f w 2 and

  Figure 5.1: A description of the proposed control scheme.

  Control induced time-scale separation for a class of nonlinear systems 143 Equation (5.28) means that, for any given L 1φ > 0, there exists a positive b 2 such that ||g φ (ỹ)|| ≤ L 1φ (5.29) for all ỹ ∈ By 2 = {ỹ ∈ Dỹ | ||ỹ|| ≤ b 2 }. It is worth noting that the values of L 1x and L 1ỹ depend on Ω 1x and Ω 1ỹ . In general, a larger Ω 1x (Ω 1ỹ ) may result in a larger L 1x (L 1ỹ ). The constants c i , i = 1, • • • , 4 are also strongly related to Ω 2ỹ .

  , and by using (5.25), (5.26), (5.29), 144 Chapter 5. Control induced time-scale separation the inequality (5.31) satises

  36) Two control objectives exist: Objective 1: making y = x 3 track the reference x o 3 ; Objective 2: making y = x 2 track the reference x o 2 .

  The response of x1. (b) The response of x2.(c) The response of x3 and its reference y o . (d) The response of u.

Figure 5 . 2 :

 52 Figure 5.2: Simulation results by using feedback linearization.

  42) At the same time, a sub-reduced model is generated as ẋ3 = x * 2 + x 3 (5.43) 148 Chapter 5. Control induced time-scale separation

5. 3 .

 3 Since both sets provide the same values of control gains to the reduced model, i.e. k 2 and k 3 , the quasi-steady-states of x 1 , x 2 and x 3 with k 1 = 4 are identical to those with k 1 = 20. As shown in Fig.5.3(a)-5.3(c), a larger k 1 leads to a smaller discrepancy between the values of the state variables and their corresponding quasisteady-state values, which exactly corresponds to our theoretical analysis.

5. 1 .

 1 Control induced time-scale separation for a class of nonlinear systems 149 (a) The response of x1. (b) The response of x2. (c) The response of x3 and its reference y o . (d) The response of u.

Figure 5 . 3 :

 53 Figure 5.3: Simulation results by using time-scale separation control: y = x 2 .

Figure 5 . 4 :

 54 Figure 5.4: Simulation results by using time-scale separation control: y = x 2 .

152Chapter 5 .

 5 Control induced time-scale separation Time-scale separation control: First, we design the fundamental module as in Example 1

  .77) is degenerated into the following reduced model ζ= A ζ ζ żr = A r zr + ϑ * r + A r zr + θic + θic (5.80) which consists the external and internal dynamics represented by ζ and zr , respec- tively. As ζ converges to zero, the dynamics of the reduced model (5.80) is governed by the internal dynamics żr = A r zr + ϑ * r | ζ=0 + A r zr + θic (5.81) Similar to Lemma 4.1.2 in Section 4.1.4, we have the following result:

  Control induced time-scale separation Table 5.6: Base quantities in the per-unit system. AC side S b = 3000 VA V b = 338.84 V I b ≈ 5.90 A DC side S dc,b = 3000 VA V dc,b = 700 V I dc,b ≈ 4.29 A 5.2.4.1 Scenario 1: Verication of time-scale separation Simulation results are shown in Figs. 5.6 and 5.7. Let us rst focus on the responses

1

  Figure 5.6: Simulation results with constant I w 1 ,dq (1).

Figs. 5 Figure 5

 55 Figure5.9: Simulation results with variations in the wind power production (1).

5. 2 .

 2 Figure 5.10: Simulation results with variations in the wind power production (2).

  j d e i w j d -εk 2w j d e Ii w j d -ε di * w j d dt de Iiw j q dt

  Control induced time-scale separation Proof. It is obvious that the origin of the boundary-layer model (5.105) is globally exponentially stable. Besides, the reduced model (5.106) has a unique solution.

196Chapter 5 .

 5 Control induced time-scale separation of the fast transient during the initial interval corresponds to the solution of the boundary-layer model. However, such two-time-scale behavior is not signicant in the DC voltage and the AC voltage, which are considered to have slow dynamics. As depicted in Figs. 5.15(a)-5.15(c), there is no apparent fast transient that can be found between the trajectories of u cg 1 and u re cg 1 (v sw 1 d and v re sw 1 d ).It can be observed that u cg 1 and v sw 1 d are well approximated by the solution of the reduced model, i.e. u re cg 1 and v re sw 1 d . Moreover, v sw 1 d asymptotically converges to its setpoint v o sw 1 d together with v re sw 1 d . Responses of i g 1 d and i * g 1 d . Zoom of i g 1 d and i * g 1 d . Zoom of the error between i g 1 d and i * g 1 d .

  5.18(d) -5.18(f ), the DC voltages start to drop and remain around 0.985 p.u. as shown in Figs. 5.19(e) and 5.19(f ).

Figure 5 .

 5 Figure 5.20: Responses of active power during the interval [0, 8] s.

Figure 5 .

 5 Figure 5.21: Comparison between Set 1 and Set 2.

First, in

 in Section 5.1, a general control design problem was studied where a timescale separation based control structure consisting of two loops, the fast inner and the slow outer loop, was established for a class of nonlinear systems. Further theoretical studies have been carried out, showing that the proposed control algorithm can divide the dynamics of the system's state variables into dierent time scales and the original closed-loop system can be divided into two models, the boundary-layer and the reduced model. Interestingly, by analyzing the two models, we nd that the control gains in both control loops have dierent eects on the system behavior. The state variables with fast dynamics exhibit a signicant two time-scale dynamical behavior by presenting a fast and a slow transient. Moreover, the fast transient exactly corresponds to the solution of the boundary-layer model, whose dynamics are dominated by the control gains of the fast inner loop.Then, we have applied the theoretical results to the control of MTDC system where two mostly used control congurations, master-slave and droop control conguration, have been investigated in this chapter. According to the dierent VSC operations of the two control congurations, two dierent time-scale separation based controllers are presented. As stated in Lemma 5.2.2 of Section 5.2, we point out that in master-slave control conguration, the developed controller can make the system exhibit dierent time-scale dynamics when the system's physical structure and the prescribed (or scheduled) references satisfy certain conditions. In addition, Response of ucw 1 during the initial interval.

Figure

  Figure 5.22: Comparison between Set 1 and Set 3 (1).

Figure 5 .

 5 Figure 5.23: Comparison between Set 1 and Set 3 (2).

208Chapter 5 .

 5 Control induced time-scale separationwe nd that in this control conguration, with the proposed controller, there exist some uncontrolled internal dynamics (for example, the state variables of the DC grid) whose behavior strongly depends on the system inherent characteristics.Subsequently, we have applied the theoretical results of control induced timescale separation to droop control conguration. As stated in Lemma 5.3.3 of Section 5.3.3.3, the proposed time-scale separation controller has no specic requirements on the system parameters, topology or those prescribed references. Unlike master-slave control conguration, the dynamics of the DC grid are determined by the droop gains and hence, there is no uncontrolled internal dynamics. The theoretical studies carried out in this chapter have been veried with simulation results where various simulated scenarios have been considered.

  .e. P l i = P o l i

Figure 6

 6 Figure 6.2: Frequencies of 5 AC areas without any controller.

Figure 6

 6 Figure 6.4: Active power injections P dc i of 5 AC areas without any controller.

Figure 6

 6 Figure 6.5: DC voltage V dc i of 5 AC areas without any controller.

Figure 6

 6 Figure 6.6: Frequencies of 5 AC areas with controller.

Figure 6

 6 Figure 6.8: Active power injections P dc i of 5 AC areas with controller.

Figure 6

 6 Figure 6.9: DC voltage V dc i of 5 AC areas with controller.

Figs. 6

 6 Figs. 6.6 and 6.7 illustrate the frequency and the mechanical power response

  Small scalar to qualify the time scales L g i i th SAC connected VSC terminal's phase reactor inductance R g i i th SAC connected VSC terminal's phase reactor resistance f g i

	t	Time
	τ	Time variable for fast dynamics
	ε

  s loss C g i i th SAC converter node's capacitance C w j j th WAC converter node's capacitance C t h h th intermediate node's capacitance i c k k th transmission line's branch current R c k k th transmission line's resistance L c k k th transmission line's inductance i c Transmission branch circuit current vector:i c = [i c 1 • • • i c L ] T u cg i DC voltageof the i th SAC converter node u cw j DC voltage of the j th WAC converter node u ct h DC voltage of the h th intermediate node u cg DC voltage vector:

  harmonic lters, the phase reactors, the converters and the DC capacitors are located indoors while the transformers and the cooling system are outdoors. . . . . . . . . . . . . . . . . . . . . . . . 18 2.2 Three phase bidirectional VSC. . . . . . . . . . . . . . . . . . . . . . Back-to-back HVDC link. . . . . . . . . . . . . . . . . . . . . . . . .

	1.1	AC versus DC cost. . . . . . . . . . . . . . . . . . . . . . . . . . . . .	7
	2.1	A simplied representation of a VSC HVDC converter station. Usu-
		ally, the 20
	2.3	Monopolar HVDC link with earth, water or metallic return path. . .	20
	2.4	Bipolar HVDC link with earth or metallic return path. . . . . . . . .	21
	2.5		22
	2.6	Multi-terminal HVDC link with three terminals connected in parallel. 22
	2.7

A single line representation of the AC side of a VSC. . . . . . . . . . 24 3.1 A three-phase PLL system. . . . . . . . . . . . . . . . . . . . . . . . 31 3.2 A multi-terminal VSC HVDC system with strong and weak AC systems. 32

  Key components . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Operating principles of VSC . . . . . . . . . . . . . . . . . . . 19 2.3 Congurations of HVDC systems . . . . . . . . . . . . . . . . 19 2.4 AC network and control modes of VSC terminals . . . . . . 23

	1.6. Publications
	Chapter 2
	VSC HVDC technology
	Contents
	2.1

• "Control Induced Explicit Time-Scale Separation to Attain DC Voltage Stability for a VSC-HVDC Terminal", Yijing Chen, Gilney Damm, Abdelkrim Ben-2.4.1 AC network connected to VSC terminal . . . . . . . . . . . . 23 2.4.2 Control modes of VSC terminals . . . . . . . . . . . . . . . . 23 2.5 Chapter conclusions . . . . . . . . . . . . . . . . . . . . . . . . 25

In order to get a good understanding of VSC HVDC technology, this chapter rst introduces the main components of a VSC HVDC system in Section 2.1, each performing a dierent function. Subsequently, the operating principles of VSC are described in Section 2.2. Next, several congurations of HVDC systems are presented in Section 2.3. Finally, various control modes of VSC terminals are discussed in Section 2.4.

  As depicted in Fig.3.6, the i th SAC converter node, the j th WAC converter node and the h th intermediate node are characterized by their corresponding DC voltages, i.e. u cg i , u cw j and u ct h , and DC capacitors, C g i and C w j and C t h . The k th branch transmission line l k is modeled by a lumped π-equivalent circuit[START_REF] Beerten | [END_REF]] composed of the aggregated resistance R c k and inductance L c k . The branch current of l k is denoted as i c k . Every branch circuit is used to

	i cw j =	P w j u cw j	(3.32)
	3.4.3 DC network
	A generic DC network topology is formed by N SAC converter nodes, M
	WAC converter nodes, P intermediate nodes and L transmission branches
	[Prieto-Araujo 2011]. connect two adjacent nodes and every node can be connected to a number of trans-
	mission lines.

Table 4 .

 4 1: Parameters of the VSC HVDC link.

	Parameters	Value
	Power System Rating	4 KVA
	Source Voltage Amplitude (V rms phase-to-phase)	127 V
	Source Voltage Frequency	50 Hz
	Switching Frequency for PWM	1500 Hz
	Transformer Primary Voltage	127 V
	Transformer Secondary Voltage	50 V
	Inductance

s, it is set back to 150 V. As depicted in Fig. 4.3, both PI and feedback nonlinear 4.1. Feedback linearization control 55

  illustrate that the feedback nonlinear controller gives a faster response (a faster rising time) as well as a smaller undershoot to the DC current than the conventional

	PI controller.
		154
									(a)
									(b)
		152
	DC voltage (V)	148 150
		146
		0.8 144	1	1.2	1.4	1.6 Time (s) 1.8	2	2.2	2.4

  60 Chapter 4. Control methods based on nonlinear control design tools

		3							(a) (b)
		2.8
	L (A)	2.6
	i
		2.4
		2.2
		2
		0.8	1	1.2	1.4	1.6 Time (s) 1.8	2	2.2	2.4

  46) 64 Chapter 4. Control methods based on nonlinear control design tools Linearizing (4.46) around the equilibrium denoted by īc 1 ūcw 1

	T	yields the follow-
	ing Jacobian matrix

  sw 1 ,dq can always converge to their new references under both control strategies. But for the PI controller, v sw 1 q is readily inuenced by the change of v o sw 1 d and v sw 1 d is also disturbed remarkably when v o Chapter 4. Control methods based on nonlinear control design tools Table 4.4: Parameters of the VSC HVDC link with two terminals.

	(a)
	(b)
	Simulation results sw 1 ,dq are subjected to a step As shown in Figs. 4.13 and 4.14, every time v o are plotted in Figs. 4.13-4.20. SAC terminal WAC terminal Power System Rating 4 KVA 4 KVA Switching Frequency for PWM 1500 Hz 1500 Hz Transformer Primary Voltage 127 V Transformer Secondary Voltage 50 V Inductance L g i 2.1 mH 3.3 mH Resistance R g i 0.142 Ω 0.165 Ω Table 4.5: Values of v o sw 1 ,dq . v o sw 1 d (V) v o sw 1 q (V) t = 0 s 40.82 0 t = 1 s 61.23 0 t = 1.5 s 61.23 20.41 Table 4.6: Two dierence control strategies by comparison. SAC terminal WAC terminal (a) Feedback linearization control PI control (m g 1 ,dq in (4.52)) (b) Feedback linearization control Feedback linearization control (m g 1 ,dq in (4.52)) (m w 1 ,dq in (4.52)) 0.8 1 1.2 1.4 1.6 1.8 Time (s) v sw 1 change, v Parameters Source Voltage Amplitude (V rms phase-to-phase) 127 V Source Voltage Frequency 50 Hz 50 Hz d (V)

sw 1 q is set to another value. t reveals that the feedback nonlinear controller gives better decoupling characteristics of d -q voltage control than the PI controller. 66

  76 Chapter 4. Control methods based on nonlinear control design tools Lemma 4.1.3. Consider the MTDC system described by (3.20), (3.28), (3.29) and (3.36). If P o gρ , ρ = 2, • • • , N and P o w j , j = 1, • • • , M are non-negative, the MTDC system is minimum phase.

  For any complex matrix Ψ ∈ C n×n , it can be expressed as Ψ =

	H(Ψ) + S(Ψ) where H(Ψ)	1 2	(Ψ + Ψ H ) is a hermitian matrix and S(Ψ)	1 2	(Ψ -
	Ψ

H ) is an anti-hermitian matrix. Lemma 4.1.6. Matrix Ψ ∈ C n×n is positive denite if and only if its hermitian part H(Ψ) is positive denite. Lemma 4.1.7. If matrix Γ is invertible and Ψ is positive denite, then Γ H ΨΓ is also positive denite. Lemma 4.1.8. If Ψ 1 is positive denite and Ψ 2 is semi-positive denite, we have Ψ 1 + Ψ 2 ∈ P + C . Lemma 4.1.9. If Ψ ∈ P + C , then Ψ is invertible and Ψ -1 ∈ P + C . Based on the above statements, we can now demonstrate Lemma 4.1.3 as follows.

Table 4 .

 4 7: Control conguration for the four-terminal VSC HVDC system.

	SAC 1 (master terminal)	SAC 2
	Constant DC voltage control	Active power control
	Reactive power control	Reactive power control
	WAC 1	WAC 2
	AC voltage control	AC voltage control
	(constant frequency and magnitude)	(constant frequency and magnitude)
	4.1.4.2 Simulation studies

To test the proposed nonlinear controller, an MTDC system with two SAC and two WAC connected VSC terminals is simulated in

Table 4 .

 4 8: Initial values of the system state variables.

	SAC 1	SAC 2

  For the prescribed referencesu o cg 1 , Q o g 1 , P o gρ , Q o gρ , v o sw j ,dq, for every state variable, only one steady-state value exists in their respective domain of interest dened in Section 3.4.4.

	Assumption 4.3.2. Consider the MTDC system described by (3.20), (3.28), (4.94)
	and (4.92).

  101) 96 Chapter 4. Control methods based on nonlinear control design tools

where θ p is expressed as

  Control methods based on nonlinear control design tools 4.3.4 Control of zero dynamics Substituting (4.109) and the control algorithm (4.124) into (4.92) and (4.94), the dynamics of the DC grid and the AC voltages v sw j ,dq will be governed by the zero dynamics (4.115) if y g i ,dq and y w i ,dq exactly converge to zero (or if i g i ,dq

  Iucg 1 ėIi gρd = ĩgρd α * gρd = -k P i gρd ĩgρd -k Ii gρd e Ii gρd

	(4.156) Of course, there are dierent approaches to choosing the PI control gains. As with the integrated errors and positive control gains k I(•) . presented in [Lee 2003], the absolute tracking results for Lur'e plants is applied to get a criterion for tuning the control gain. It can also be used to deduce the Table 4.9: Initial values of the system state variables. constraints for k 116Chapter 4. Control methods based on nonlinear control design tools SAC 1 SAC 2

P (•) and k I(•) via linearization. In this thesis, we are not tangled on this issue. Finally, the overall control structure are built, which is composed of (4.122), (4.156) and the adaptive law (4.146).

  developed. When the output perfectly approaches to zero, the dynamics of u cg 1 , i gρd and v sw j ,dq are eventually governed by the zero dynamics (4.115). Regulating i g i ,dq u cg i , indirectly drives u cg 1 , i gρd and v sw j ,dq to the prescribed values. However, due to the uncertainties of the system, it is usually dicult to get the accurate Control induced time-scale separation for a class of nonlinear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

	Chapter 5
	and (4.96)values of i g j ,dq u cw j īg i ,dq to the desired Control induced time-scale īg i ,dq ūcg i and īg j ,dq ūcw j , obtained from the steady-state condition ūcg i and īg j ,dq ūcw j . Moreover, the convergence rate of the uncontrolled states separation
	of the zero dynamics totally depends on the system parameters. Thus, to get a
	better tracking performance and improve the dynamics of those uncontrolled state variables, the control law (4.156) for α * Contents g i ,dq and α w j ,dq is derived based on the zero dynamics. 5.1

5.1.1 Introduction and motivation . . . . . . . . . . . . . . . . . . 132 5.1.2 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . 134 5.1.3 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . 136 5.1.4 Theoretical study . . . . . . . . . . . . . . . . . . . . . . . . . 139 5.1.5 Study cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.2 Control induced time-scale separation for MTDC systems using master-slave control conguration . . . . . . . . . . . . 152 5.2.1 Control design . . . . . . . . . . . . . . . . . . . . . . . . . . 153 5.2.2 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . 156 5.2.3 Plug-and-play operations . . . . . . . . . . . . . . . . . . . . 164 5.2.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . 165 5.3 Control induced time-scale separation for MTDC systems using droop control conguration . . . . . . . . . . . . . . . . 174 5.3.1 VSC operation . . . . . . . . . . . . . . . . . . . . . . . . . . 174 5.3.2 Droop control structure . . . . . . . . . . . . . . . . . . . . . 175 5.3.3 Theoretical analysis . . . . . . . . . . . . . . . . . . . . . . . 178 5.3.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . 193 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205

Table 5 .

 5 1: The parameter values.

Table 5 .

 5 2: The values of control gain (1).

		k 1 k 2	k 3
	Set 1	4	2	0.125
	Set 2	20	2	0.125
	Table 5.3: The values of control gain (2).

Table 5

 5

	.4: The VSC terminal parameters.
		SAC 1	SAC2	WAC 1
	R g,w 9.9 mΩ 9.4 mΩ 8.4 mΩ
	L g,w 3.2 mH 3.1 mH 2.7 mH
	f g,w	50 Hz	50 Hz	60 Hz
	C g,w 20.4 µF 20.4 µF 27.2 µF
	Table 5.5: The transmission branch parameters.
		Branch 1	Branch 2
	R c	0.27 Ω		0.33 Ω
	L c	0.045 H		0.055 H

  1g i d e i g i d -k 2g i d e Ii g i d de Iig i q

	5.3. Control induced time-scale separation for MTDC systems using droop control conguration 179
	loop system
	de Ii g i d dt di g i d dt	= e i g i d = -k
	Before going further, we give a brief analysis of the equilibruim point. Substituting
	the controller composed by (5.90), (5.94) ,(5.96) and (5.97) into the plant model de-
	scribed by (3.20), (3.28), (3.29) and (3.36) leads to the following augmented closed-

  • • • e i g N ,dq e Ii g N ,dq T e i w,dq e i w 1 ,dq e Ii w 1 ,dq

  • • • e i w M ,dq e Ii w M ,dq T e v w,dq e v sw 1 ,dq e Iv sw 1 ,dq

  • • • e v sw M ,dq e Iv sw M ,dq T Consider a Lyapunov function W = dW 1 + (1 -d)W 2 where d ∈ (0, 1). W 1 and W 2 are designed as gd F i gd e i gd + e T igq F igq e igq + e T i wd F i wd e i wd + e T iwq F iwq e iwq ) wd F v wd e v wd + e T vwq F vwq e vwq + e T z F ez e z )

	W 1 = i and 1 2 (e T
	W 2 =	1 2	(e T v

  Control induced time-scale separation for MTDC systems using droop control conguration 191 and the other expressions are given by

  dq j Since the dq current reference trajectories only depend on the state variables with || → 0 as ||e i gd || → 0 and ||e igq || → 0 ||ϑ w 4 || → 0 as ||e i wd || → 0 and ||e iwq || → 0 ||ϑ w 2 || → 0, ||ϑ w 3 || → 0 as ||e v wd || → 0 and ||e vwq || → 0 and then there exist positive β k , k = 1, • • • , 8 and convex regions B e i g,dq , B e i w,dq , B ev w,dq such that ||ϑ g 3 || ≤ β 1 ||e i gd || + β 2 ||e igq || ||ϑ w 4 || ≤ β 3 ||e i wd || + β 4 ||e iwq || ||ϑ w 2 || ≤ β 5 ||e v wd || + β 6 ||e vwq || ||ϑ w 3 || ≤ β 7 ||e v wd || + β 8 ||e vwq || for all e i g,dq ∈ B e i g,dq , e i w,dq ∈ B e i w,dq and e v w,dq ∈ B ev w,dq . Note that, in general, a larger size of B (•) leads to larger values of β k . || + α 0 ||e z || + β 1 ||e i gd || + β 2 ||e igq || +β 3 ||e i wd || + β 4 ||e iwq || + β 5 ||e v wd || + β 6 ||e vwq || +β 7 ||e v wd || + β 8 ||e vwq || Control induced time-scale separation holds for all e z ∈ B ez , e i g,dq ∈ B e i g,dq , e i w,dq ∈ B e i w,dq and e v w,dq ∈ B ev w,dq .

	192 Chapter 5. Taking (5.112), (5.126) and (5.127) into account, it can be veried that there
	exist positive parameters
	slow dynamics, we then have
		dη i gd dt dη i wd dt dη iwq dt	= = =	dη i gd de z dη i wd dη iwq de v wd de v wd	de z dt de v wd , dη igq dt dt + dη iwq dη igq = de z de vwq de vwq de z dt dt + de v wd dt + dη i wd de vwq de vwq dt +	de z dη iwq dη i wd de z	dt de z de z dt	(5.126)
	From the aforementioned sections, we know that
	||ϑ g 3 Applying the above inequalities and (5.124) to (5.123), we then get that
	||	de z dt	||≤ ||Φ|| • ||e z (5.127)

  5.128)for all e z ∈ B ez ⊂ B ez , e i g,dq ∈ B e i g,dq ⊂ B e i g,dq , e i w,dq ∈ B e i w,dq ⊂ B e i w,dq and e v w,dq ∈ B ev w,dq ⊂ B ev w,dq . These positive coecients (b i k and c j l ) are determined by the size of the domains B (•) , the droop gains k d i , k 1,2d j , k 1,2q j , the system parameters and the prescribed setpoints while they are independent of k 1g i ,dq , k 2g i ,dq , k 1w j ,dq and k 2w j ,dq .

	Using Young's inequality
	||x|| • ||y|| ≤	1 2µ	||x|| 2 +	µ 2	||y|| 2
	for the cross terms in (5.128) where µ is a positive constant which can be chosen
	arbitrarily, we obtain
	D 2 ≤ Γ 3 + Γ 4		(5.129)
	with

  Responses of v sw 1 d and v re sw 1 d . Zoom of v sw 1 d and v re sw 1 d . Evaluation of the system performance in case of disturbance At t = 2 s, I w 1 d is changed to 0.6 p.u. because of the increase in the active power generated by the wind farms. Consequently, both SAC connected VSC terminals should share the duty of eliminating the power unbalance caused by this increase of the power production. The simulation results are presented in Figs. 5.16-5.17. Since more power needs to be transmitted via the DC grid, in order to make the MTDC system operate normally, the two SAC terminals should absorb more power from the DC grid. As shown in Figs. 5.16(a) and 5.16(b), i g 1 d attains a new steady value around -0.5 p.u. from -0.4 p.u. and īg 2 d is also changed from -0.5 p.u. to -0.6 p.u.. According to (5.101), if more power is absorbed by the SAC terminals, this forces the DC voltages to rise and reach new steady levels. This phenomenon is clearly presented in Figs. 5.17(a) -5.17(c). The new DC voltage transmission

	5.3. Control induced time-scale separation for MTDC systems using droop control conguration 197 198 Chapter 5. Control induced time-scale separation
	5.3.4.2
	1.04			u cg	1	(p.u.)	1.04			u cg	1	(p.u.)
	1.03			u cg re (p.u.) 1	1.03			1 re (p.u.) u cg
	1.02			u cg o	1		1.02			u cg o	1
	1.01							1.01
	1							1
	0.99							0.99
	0.98							0.98
	0.97							0.97
	0.96							0.96
	0	0.2	0.4 Time (s) 0.6	0.8	1	0	0.05	0.1 Time (s)	0.15	0.2
	(a) Responses of ucg 1 and u re cg 1 .			(b) Zoom of ucg 1 and u re cg 1 .
	1.1							1.1
	1							1
	0.9			v sw 1	d (p.u.)	0.9			v sw 1	d (p.u.)
	0.8			v sw 1 re v sw 1 o	d d	(p.u.) (p.u.)	0.8			v sw 1 re v sw 1 o	d d	(p.u.) (p.u.)
	0.7							0.7
	0.6							0.6
	0 0.5 (c) 0 0.02 0.04 0.06 0.08 0.1 Time (s) 0.5 (d) Figure 5.15: Simulation results with the control gains in Set 1 (2). 0.2 0.4 0.6 0.8 Time (s)	1

  Responses of i g 1 d and i * g 1 d . Responses of i g 2 d and i * g 2 d . Responses of i w 1 d and i * w 1 d . Zoom of i w 1 d and i * w 1 d . Responses of ucg 1 and u re cg 1 . Responses of ucg 2 and u re cg 2 . Responses of v sw 1 d . Responses of i g 1 d and i * g 1 d . Responses of i g 2 d and i * g 2 d . Responses of ucw 2 and u re cw 2 . Responses of v sw 1 d and v re sw 1 d . Responses of i w 1 d and i * w 1 d . Responses of ucg 2 and u re cg 2 .

	5.3. Control induced time-scale separation for MTDC systems using droop control conguration 199 2 2.1 2.2 2.3 2.4 2.5 -0.5 -0.48 -0.46 -0.44 -0.42 -0.4 Time (s) i g 1 d (p.u.) i g 1 d * (p.u.) (a) 2 2.1 2.2 2.3 2.4 2.5 -0.6 -0.58 -0.56 -0.54 -0.52 -0.5 Time (s) i g 2 d (p.u.) i g 2 d * (p.u.) (b) 2 2.1 2.2 2.3 2.4 2.5 0.4 0.45 0.5 0.55 0.6 Time (s) i w 1 d (p.u.) i w 1 d * (p.u.) (c) 1.99 2 2.01 2.02 0.4 0.45 0.5 0.55 0.6 Time (s) i w 1 d (p.u.) i w 1 d * (p.u.) (d) Figure 5.16: Simulation results with the variation in I w 1 d . 200 Chapter 5. Control induced time-scale separation 2 2.1 2.2 2.3 2.4 2.5 0.99 1 1.01 1.02 1.03 1.04 1.05 Time (s) u cg 1 (p.u.) u cg 1 re (p.u.) u cg 1 o (a) 2 2.1 2.2 2.3 2.4 2.5 0.99 1 1.01 1.02 1.03 1.04 1.05 Time (s) u cg 2 (p.u.) u cg 2 re (p.u.) u cg 2 o (b) 2 2.1 2.2 2.3 2.4 2.5 0.99 1 1.01 1.02 1.03 1.04 1.05 Time (s) u cw 1 (p.u.) u cw 1 re (p.u.) (c) Responses of ucw 1 and u re cw 1 . 2 2.1 2.2 2.3 2.4 2.5 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 Time (s) v sw 1 d (p.u.) v sw 1 d o (p.u.) (d) Figure 5.17: Simulation results with the variation in I w 1 d . 5.3. Control induced time-scale separation for MTDC systems using droop control conguration 201 4 4.1 4.2 4.3 4.4 4.5 -0.5 -0.48 -0.46 -0.44 -0.42 -0.4 -0.38 -0.36 Time (s) i g 1 d (p.u.) i g 1 d * (p.u.) (a) 4 4.1 4.2 4.3 4.4 4.5 -0.6 -0.58 -0.56 -0.54 -0.52 -0.5 -0.48 Time (s) i g 2 d (p.u.) i g 2 d * (p.u.) (b) 4 4.1 4.2 4.3 4.4 4.5 0.3 0.35 0.4 0.45 0.5 Time (s) i w 2 d (p.u.) i w 2 d * (p.u.) (c) Responses of i w 2 d and i * w 2 d . 4 4.1 4.2 4.3 4.4 4.5 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 Time (s) u cg 1 (p.u.) u cg 1 re (p.u.) u cg 1 o (d) Responses of ucg 1 and u re cg 1 . 4 4.1 4.2 4.3 4.4 4.5 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 Time (s) u cg 2 (p.u.) u cg 2 re (p.u.) u cg 2 o (e) Responses of ucg 2 and u re cg 2 . 4 4.1 4.2 4.3 4.4 4.5 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 Time (s) u cw 2 (p.u.) u cw 2 re (p.u.) (f) 202 Chapter 5. Control induced time-scale separation 6 6.2 6.4 6.6 6.8 7 0.88 0.9 0.92 0.94 0.96 0.98 1 Time (s) v sw 1 d (p.u.) v sw 1 d re (p.u.) v sw 1 d o (p.u.) (a) 6 6.2 6.4 6.6 6.8 7 0.598 0.599 0.6 0.601 0.602 0.603 0.604 0.605 0.606 Time (s) i w 1 d (p.u.) i w 1 d * (p.u.) (b) 6 6.2 6.4 6.6 6.8 7 -0.4 -0.395 -0.39 -0.385 -0.38 -0.375 -0.37 -0.365 Time (s) i g 1 d (p.u.) i g 1 d * (p.u.) (c) Responses of i g 1 d and i * g 1 d . 6 6.2 6.4 6.6 6.8 7 -0.5 -0.495 -0.49 -0.485 -0.48 -0.475 -0.47 -0.465 Time (s) i g 2 d (p.u.) i g 2 d * (p.u.) (d) Responses of i g 2 d and i * g 2 d . 6 6.2 6.4 6.6 6.8 7 0.982 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 Time (s) u cg 1 (p.u.) u cg 1 re (p.u.) u cg 1 o (e) Responses of ucg 1 and u re cg 1 . 6 6.2 6.4 6.6 6.8 7 0.982 0.984 0.986 0.988 0.99 0.992 0.994 0.996 0.998 1 Time (s) u cg 2 (p.u.) u cg 2 re (p.u.) u cg 2 o droop control conguration 203 the power injection reduces, both SAC terminals decrease their power absorption accordingly. 0 1 2 3 4 5 6 7 8 -0.6 -0.4 0 0.2 0.4 0.6 Time (s) P g 1 (p.u.) P g 2 (p.u.) P w 1 (p.u.) (f) 5.3. Control induced time-scale separation for MTDC systems using -0.2 2 P w (p.u.)

Figure 5.18: Simulation results with the variation in I w 2 d .

Figure 5.19: Simulation results with AC voltage regulation.

  Introduction and motivation . . . . . . . . . . . . . . . . . . . 209 6.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 6.2.1 DC grid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2.2 AC areas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211 6.2.3 Reference operating point . . . . . . . . . . . . . . . . . . . . 212 6.3 Control strategy . . . . . . . . . . . . . . . . . . . . . . . . . . 212 6.3.1 Control law . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 6.3.2 Choice of control gains . . . . . . . . . . . . . . . . . . . . . . 213 6.3.3 Algorithm for the denition and verication of control gains and the associated region of attraction . . . . . . . . . . . . . 218

	Chapter 6
	Frequency control using MTDC
	systems
	Contents
	6.1

6.4 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . 219 6.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224

  in Section 6.3.2.1. Then, the derivative of Lyapunov function is

	6.3. Control strategy					217
	is chosen as
	function candidate
	N		N
	W =	a 4i	f 2 i +	a 2i	P 2 mi	(6.23)
	i=1		i=1

Table 6 .

 6 1: Parameter values of each AC area (`-'means dimensionless).

	Parameter	1	2	Area 3	4	5	Unit
	f nom	50	50	50	50	50	Hz
	P s m	50	80	50	30	80	MW
	P nom	50	80	50	30	80	MW
	J	2026	6485	6078	2432	4863	kg m 2
	D g	48.4	146.3	140.0	54.9	95.1	W• s 2
	δ	0.02	0.04	0.06	0.04	0.03	-
	T sm	1.5	2.0	2.5	2	1.8	s
	P l	100	60	40	50	40	MW

Table 6 .

 6 2: Values of α i calculated by LMI.

	α 1	α 2	α 3	α 4	α 5
	1700	397	185	548	868

  Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 7.2 Perspectives for future work . . . . . . . . . . . . . . . . . . . 228 7.2.1 Further research on the droop control conguration . . . . . 228 7.2.2 Further research on the connection of other types of weak AC systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 7.2.3 Further research on the operation of MTDC systems . . . . . 229 7.2.4 Further research on system modeling . . . . . . . . . . . . . . 229 7.2.5 Implementation on a real MTDC system . . . . . . . . . . . . 229 7.2.6 Control induced time-scale separation for the MMC . . . . . 229

	Chapter 7
	Conclusions
	Contents
	7.1

See the notations.

Unless otherwise stated, the subscripts i, j, t and k mean∀i ∈ N = {1, • • • , N }, ∀j ∈ M = {1, • • • , M }, ∀h ∈ P = {1, • • • , P } and ∀k ∈ L = {1, • • • , L}.

• The values of all physical parameters, i.e. the resistances, the inductances and the the capacitors, are positive. Moreover, in this chapter, we have also presented two most investigated conventional control strategies, direct and vector control strategies, and introduced their control principles. In addition, we have also discussed their advantages and disadvantages.

Unless otherwise stated, the subscript ρ mean ∀ρ ∈ N-1 = {2, • • • , N }.

K 1[1] = K 1[2] = K 2[1] = 1972 • I 2×2 , Γ 1[1] = Γ 1[2] = Γ 2[1] = 1.86 × 10 4 • I 2×2 , k g 1 d = k w 11 d = k w 11 q = 40, k g 2 d = k w 21 d = k w 22 q =400 and hence ε =

5.071 × 10 -4 . 4 Recall the denitions of the control gains in (5.60).

Recall that the notation (•) denotes the steady-state value of the variable (•).

Acknowledgments

My deepest gratitude is to my three thesis supervisors, Dr. Françoise LAMNABHI-

Chapter 5. Control induced time-scale separation

The derivative of V 1 along the trajectory of (5.10) is

(5.12) Hence, Lemma (5.1.6) is proved.

The values of γ 1 , a 1 and b 1 are strongly dependent on each other. For example, considering the following system ẋ = -ax + x 3 -x we then have x 0 = 0, a 1 = a and r 1 = x 3 -x. If γ 1 = 3, there exists b 1 = 2 such that (5.11) is satised. In this case, a 1 should be chosen larger than 3 + c 1 . If a larger γ 1 is given, i.e. γ 1 = 8, a new b 1 = 3 is then deduced. We obtain a new condition for a 1 , i.e. a 1 > 9 + c 1 . It is clear that a larger γ 1 results in a smaller feasible region for a 1 .

When γ 1 is xed, the linear term -A 1 (x -x 0 ) dominates the behavior of the autonomous system (5.10) in B x 1 . Consequently, the dynamics of (5.10) can be regulated by A 1 .

Step 5.1.7. Supposing that there exists a reference x * ∈ D x and with the acknowledge of (5.10), we then develop a subcontroller u = h 1 (x, y, x * ) as u = g -1 1 (x, y)[-A 1 (x -x * ) + r 1 (x, x * ) -f 1 (x, y)]

(5.13) such that the closed-loop system of the driving subsystem becomes

The only information on x * is that it has a much slower dynamic than x. As seen in (5.13), the derivative of x * is neglected in the subcontroller.

Design of the driven subsystem

Step 5.1.8. The main task in this step is to develop the unknown reference x * that must be tracked by the driving subsystem. Since we suppose that x * has a very slow dynamic, a large a 1 can then be chosen such that the state variable of the driving subsystem x quickly enters x * . Then, the driven subsystem becomes

which is called the reduced model. Because x 1 is required to track its reference trajectory, x * 1 can be directly given by:

x * 1 = x o 1 (5.15) 5.1. Control induced time-scale separation for a class of nonlinear systems 139

Thus, the reduced model (5.14) becomes

According to Assumption 5.1.4, there exists

(5.17)

such that the reduced model (5.14) is exponentially stable.

Until now, the entire control algorithm is developed, consisting of two subcon-

] T , which are derived from two lower order subsystems, i.e. the driving and the driven subsystems. As mentioned in the previous section, the driving subsystem (5.2) with its subcontroller u = h 1 composes the fundamental control module and the reduced model of the driven subsystem (5.14) with its subcontroller x * constitutes the outer control module. Furthermore, the proposed control structure has plug and play capabilities.

When a new objective is required, we only need to design a new corresponding outer control module that provides a new x * to the fundamental control module without abandoning the whole existing control structure. Dierent outer control modules share the same fundamental control module. x * is thus considered as the tie that binds the fundamental and the outer control modules together. In addition, the new outer control module is designed based on the reduced model where the dynamics of x are neglected. This greatly simplies the problem.

Theoretical study

During the design procedure in Section 5.1.3, we assume that the dynamics of the driving and the driven subsystems can be divided into dierent time scales with the aid of the proposed control methodology. In this section, we rst verify this assumption by means of singular perturbation theory. Subsequently, we carry out a theoretical stability analysis, from which sucient conditions for the construction of the fundamental and the outer control modules are derived.

Multi-time-scale dynamics

In this section, we present how the system (5.2)-( 5.3) exhibits a multi-time-scale behavior under the proposed controller. Substituting the control algorithm (5.13),

(5.15) and (5.17) into the original system (5.1), we then obtain the following closed-

For the subsystem (5.76), we choose

as a Lyapunov function candidate where

Then, the derivative of V b is deduced as 3

Now, we propose a Lyapunov function V based on V r and V b for (5.73)

where d is yet to be designed. The derivative of V along (5.76) and (5.77) becomes

We have that [ θζ θr ] T vanishes at ||x|| = 0 and moreover, θζ and θr are continuously dierentiable functions. Hence, there exists a Lipschitz parameter L 2 such that:

where

3 Recall the denition of x in (5.67).

Control induced time-scale separation for MTDC systems using master-slave control conguration 163

Since x * is a function of (ζ z r ), we then have:

Taking the above mentioned inequalities into consideration, V in (5.85) becomes

where β 2 is a positive constant given by

The quadratic expression (5.86) is negative denite as

In addition, we can obtain an estimation of the region of attraction

where the sets D (•) ∈ R (•) are bounded and connected containing the origin and C is given by

For all initial points starting from the attraction region R, when the inequality (5.87) is satised, the solution of the problem (5.73) converges to the origin. Consequently, when the control gains of the control algorithm (5.54), (5.58) and (5.61) are chosen such that (5.87) is satised, the equilibrium point P1 of the MTDC system (5.50),

(5.51) and (3.36) is locally asymptotically stable.

Theorem 5.2.3. Consider the MTDC system (5.50), (5.51) and (3.36).

are non-negative, then, the control algorithm (5.54), (5.58) and (5.61) exists such that the equilibrium point P1 of the MTDC system (5.50), (5.51) and (3.36) is locally asymptotically stable, with a region of attraction given by (5.88).

Chapter 5. Control induced time-scale separation

Again, we dene

with the notations

Due to (5.116), we assume that k dq > k dq,min , q ∈ N , where

Now, Φ 2 can be rewritten as

where Λ q 3 is a diagonal matrix whose q th element on the main diagonal is σ q and other elements are zero. Λ q 3 and H f are given by Λ q 3 = Λ 3 -Λ q 3 and H f = HΛ 1 2

.

Let us dene Φ 4 Λ q 3 +H f H T f and then the determinant of Φ 4 can be calculated as

5 is a full rank matrix, i.e. rank(Λ

invertible and positive denite. Since k dq satises (5.121), we get σ q > 0 and then, det(Φ 4 ) = σ q det(R(H f ) q R(H f ) T q ) > 0. On the other hand, Φ 4 and Λ q 3 are semi- positive denite because of (5.115) and (5.116) and hence, Φ 4 is positive denite.

Recalling that Φ 3 = Φ 4 + Λ q 3 , it turns out that, Φ 3 is also positive denite. As presented in (5.122), we know that H(Φ 2 ) = Φ 3 and then, according to Lemma 4.1.6 5.3. Control induced time-scale separation for MTDC systems using droop control conguration 189

and Lemma 4.1.9, Φ 2 is positive denite and invertible. Therefore, the following result is obtained:

for α ≥ 0. This leads to a contradiction to (5.118). Thus, all eigenvalues of the Jacobian matrix Φ must have negative real part, i.e. α < 0.

, k 1,2d j and k 1,2q j such that the equilibrium S of the closed loop system (5.98) is locally asymptotically stable. Thereby, the proposed control strategy can stabilize the MTDC system.

Proof. As previously mentioned, to study the stability of the origin of the error system (5.110) is equivalent to investigating the stability of the equilibrium S of the closed-loop system (5.98). So we still focus on the stability property of the error system (5.110).

We rewrite the last equation in (5.110) in the following special form

(5.123)

According to Taylor's theorem, f zero can be expressed by for all e z ∈ B ez . In addition, since Φ is a Hurwitz matrix, for any positive denite matrix G ez , there exists a positive denite matrix F ez such that

(5.125) d < k 1w j q and hence, Γ 2 + Γ 3 < 0. Consequently, there exist k 1g i d , k 2g i q , k 1w i d , k 2w i q , k 1,2d j and k 1,2q j such that the derivative of W is non-negative. In addition, Ẇ = 0 contains no trajectory of the system except the trivial trajectory. All solutions starting from R att will converge to the origin. According to LaSalle theorem, the origin of the error system described by (5.110) is locally asymptotically stable. Thereby, the equilibruim S of the original closed-loop system (5.102) is also locally asymptotically stable. Finally, we can say that the control strategy can ensure the asymptotic stability of the MTDC system.

There is a trade-o between the size of R att and the performance of system. If we want to get a large region of attraction, then we will have large values of k 1g i ,dq and k 1w j ,dq . As seen in (5.90), large k 1g i ,dq and k 1w j ,dq maybe lead to large peak values of the control variables.

Simulation studies

In this section, the MTDC system as depicted in Fig. 5.13 consisting of two WAC and two SAC connected VSC terminals is simulated. As described in the previous section, the wind farm is modeled as a controlled current source and hence, I w 1 ,dq and I w 2 ,dq represent the power productions of the wind farms. The setpoints and the initial values of I w 1,2 d are given by Table 5.10. I w 1 q and I w 2 q are set to zero. Furthermore, some system variables' initial values are provided by Table 5.11. For all the simulations in this section, the integral parts of the fast control gains are set to zero, i.e. k 2g i ,dq = k 2w j ,dq = 0. Three dierent sets of control gains as presented in Table 5.12 are chosen to verify the theoretical analysis.

Verication of two-time-scale behavior

The control gains in Set 1 are considered for the converters in this part. The simulation results are displayed in Figs. 5.14-5.15. The trajectory of i g 1 d in Fig. 5.14(c) illustrates the error between i g 1 d and i * g 1 d . At the initial instant, |i g 1 d -i * g 1 d | is nearly 0.29 p.u.. After t = 0.06 s, the discrepancy between the two trajectories is less than 0.015 p.u.. It turns out that during the initial interval [0, 0.06] s, the trajectory of i g 1 d approaches that of i * g 1 d . It is seen that the exponential decay Table 5.9: Base quantities used in the per-unit system. AC side S ac,base = 4.5 kVA V ac,base = 415 2/3 V DC side S dc,base = 3 kVA V dc,base = 700 V

Control induced time-scale separation for MTDC systems using droop control conguration 195

Table 5.10: Setpoints and initial values of the current source.

V ac,base 0 0.4I ac,base WAC 2

V ac,base 0 0.5I ac,base

bridge the gap between the practice and the theory where the master-slave and the droop control congurations were studied. From the theoretical analysis of the two control congurations, we identied several points: 1) how and why the system exhibits a multi-time-scale behavior; 2) the rules of tuning the control gains such that parts of those gains regulate the fast transient and the remaining determines the behavior of the slow transient; 3) the existence of uncontrollable state variables when the master-slave control conguration is used. Moreover, we pointed out that the master-slave control conguration based on vector control method is not always applicable for the MTDC system.

The stability of the MTDC system with the master-slave control conguration can be ensured only when the system parameters and the scheduled references satisfy certain conditions while for the droop control conguration, there is no special limitation on them.

In Chapter 7, a potential application of MTDC systems was introduced where an AC frequency support strategy via the MTDC system was proposed. This strategy enables the asynchronous AC systems to share their primary reserves via an MTDC system. The control method used in this application, namely the DC-voltage-based controller, is based on local measurements without needing any remote communication, and avoids possible problems caused by time-delay. Simulation studies showed that the proposed control strategy achieves the control objective with satisfactory performances.

Perspectives for future work

This thesis have met some challenges in the area of MTDC systems. However, there still exist many issues to be investigated.

Further research on the droop control conguration

In Section 5.3, we discussed the dynamic behavior of the MTDC system under the droop control conguration. We found that the choice of the droop gains has great impacts on: 1) the steady-state values of the system variables; 2) the distribution of power sharing between the interconnected AC networks in case of power imbalance;

3) the dynamics of the DC grid. In this thesis, the droop gains are considered constant. It would be interesting to consider the droop gains not as constants but as variables, which could vary according to dierent operating conditions. For example, each converter terminal has a dierent capability against the power imbalance. The droop gain of each converter terminal could vary depending on its existing available headroom of the capacity. adaptive control technique could be applied to improve the system performance.

Further research on the operation of MTDC systems

In Section 4.1.4.1, we presented that the MTDC system has plug-and-play capabilities. This ability is very important for the extension of the existing MTDC systems.

In this thesis, we provided a qualitative approach to determine whether an AC system could be added or removed. One of possible further work is that a quantitative approach need to be developed from a security point of view.

Further research on system modeling

In this thesis, the ideal three-phase AC source was used for every strong AC network and the controlled three-phase current source was used for every weak AC network.

Therefore, we can take the dynamics of the generator and the wind turbine into consideration when modeling an MTDC system. This would help us study the system behaviors and the interactions between AC and DC.

Implementation on a real MTDC system

In this thesis, the developed nonlinear controllers were tested by using SimPowerSystems toolbox of MATLAB/Simulink. From the simulation results, both advantages and disadvantages of each control method are clearly presented. However, due to the limit of the condition, these control methods are not yet implemented on a real MTDC system. It will be interesting to test these proposed controllers on a real MTDC system. For a matrix A ∈ R n×m , A ij is the element of A in the i th row and the j th column.

A(:, i) and A(j, :) denote the i th column and the j th row of A, respectively. A

, and the other elements of A are zero. The transpose of A is denoted by A T . The inverse of A is denoted by A -1 . The notation rank(A) means the rank of A. 0 n×m ∈ R n×m represents a zero matrix, with all its elements equal zero. I n×n ∈ R n×m represents the identity matrix. 0 n ∈ R n represents a zero vector, with all its elements equal zero. For

x ∈ C, the real part of x is denoted as Re(x) while its imaginary part is denoted as Im(x). For A ∈ R n×n , ||A|| = λ max (A H A) where A H is the conjugate transpose of A and λ max (•) represents the maximum eigenvalue of matrix (•). The notation D x ⊂ R n denotes the safe operating domain of variable x ∈ R n , which is convex. The notation x represents the value of x in steady-state condition and the initial value of x is denoted by x(t 0 ). x o is the prescribed (or reference value) of the variable x. x * means the reference trajectory of the variable x. x is the estimate of x. Any complex matrix A can be expressed as A = H(A) + S(A) where H(A) = 1/2(A + A H ) is a hermitian matrix and S(A) = 1/2(A -A H ) is an anti-hermitian matrix. For any matrix A, R(A) l is a reduced matrix by deleting the l th row of A. Ψ is invertible. Moreover, because H(T ) is positive denite, according to Lemma 4.1.6, T is also positive denite. Finally, we get the result that Ψ -1 is also positive denite.