Thijs Peirelinck
email: thijspeirelinck@gmx.com

Antoneta Iuliana Bratcu
email: antoneta.bratcu@gipsa-lab.fr

Yvon Bésanger
email: yvon.besanger@g2elab.grenoble-inp.fr

Impact of IEC 61850 GOOSE Communication Quality on Decentralized Reactive Power Control in Smart Distribution Grids -a Co-simulation Study

Keywords: Co-simulation, Generic Object-Oriented Substation Event (GOOSE), IEC 61850, Reactive power control

Smart grids are expected to increase efficiency, reliability and sustainability of future energy usage. Employing stateof-the-art information and communication technologies within power grids, is defining for smart grids; therefore interaction between energy grids and communication networks requires thorough study. Combined simulation of the behaviour of the electrical and communication network would allow to analyse their influences on one each other.

In this paper, the results of a MATLAB R /Simulink R cosimulation of an electrical distribution grid containing two renewable sources in interaction with its communication network are presented. Communication between different nodes is based on the IEC 61850 GOOSE protocol, whose simulation model incorporates its different efficiency and reliability features. Communication network and electrical grid are modelled using SimEvents R blocks and SimPowerSystems R blocks, respectively. The effect of perturbations on the reliability of the data transfer links, while information to perform decentralised reactive power control is flowing on them, is examined.

I. INTRODUCTION

The energy sector faces many challenges, one of them is the need to allow integration of Distributed Energy Resources (DER) in the electrical grid. However, for this to be successful, grid operators will have to extensively use Information and Communication Technology (ICT) and, have to expand the communication network drastically and rapidly. Only the combination of electrical grids and communication networks will allow challenges to be overcome and the 'smart grid' to be truly smart [START_REF] Andrén | IEC 61850/61499 control of distributed energy resources: Concept, guidelines, and implementation[END_REF].

The merger of electrical grids and communication networks leads to an increasing reliability of the electrical grids stability on the communication networks performance. Since communication channels are not perfect, simulations will have to show these interdependencies [START_REF] Li | Cosimulation for smart grid communications[END_REF]. The nature of electrical dynamics This work was partially supported by the CNRS (Conseil National de la Recherche Scientifique, France) grant within the framework of CYBERGRID Exploratory Project. and communication system events is completely different, and they should thus be treated differently in simulations. Electrical dynamics are time-driven, whereas a communication system is event-based. Because of these contrasting features, they cannot simply be combined into one single piece of software. Experts have been trying to combine popular tools, including MATLAB R /Simulink R , with each other to ideally converge to a single platform able to simulate both networks [START_REF] Sharma | Co-simulation of a low-voltage utility grid controlled over IEC 61850 protocol[END_REF]. Advanced mechanisms were developed in order to ensure correct coordination between the two simulation tools, since they were not designed to cooperate [START_REF] Bian | Real-time co-simulation platform using OPAL-RT and OPNET for analyzing smart grid performance[END_REF].

Real-life examples of the usage of co-simulation and cyberphysical networks exist. The Wide Area Control System (WACS), using IEC 61850 GOOSE on a Wide-Area Ethernet network, to enable communication between three substations in Australia, is only one example [START_REF] Masoum | Power system management using wide area network digital control[END_REF].

The combination of an electrical grid and a communication network provides the ability for highly automated control. An example of an ancillary service, which can be provided if DER are able to communicate with each other, is distributed reactive power support for voltage control [START_REF] Calderaro | Coordinated local reactive power control in smart distribution grids for voltage regulation using sensitivity method to maximize active power[END_REF]. Two main approaches are seen in literature: the centralised [START_REF] Keane | Enhanced utilization of voltage control resources with distributed generation[END_REF] and decentralised [START_REF] Tanaka | Decentralised control of voltage in distribution systems by distributed generators[END_REF] approach, respectively. Using the communication grid, static voltage control based on load predictions, can easily be changed in a more dynamic automated control system.

Cyber-physical systems, with highly efficient and reliable communication protocols, allow for Advanced Distribution Automation (ADA) systems to react on real-time changes without human intervention. But, in order to understand the interdependencies, co-simulation of the event-based communication, and its effects on the time-driven electrical grids, is necessary. In this paper, a single software environment, which is MATLAB R /Simulink R , accommodates simulation models of both communication and electrical network, namely SimEvents R toolbox for the communication system and SimPowerSystem R toolbox for the electrical system. The effect of perturbations within the communication network, which is providing services for reactive power control, was implemented and assessed by simulation. Simulation models within the same software environment are expected to ease formal mathematical description.

The paper is organised as follows. Section II reviews the literature dedicated to IEC 61850 GOOSE. Section III reveals the adopted communication network configuration, whereas the electrical grid configuration is given in Section IV. The different scenarios simulated and their corresponding results are analysed in Section V. Finally, in Section VI, the conclusions are formulated and further work issues are presented.

II. IEC 61850 GOOSE FEATURES AND LITERATURE

REVIEW

GOOSE was intended to be a communication protocol used within the substation automation process. However, as it advanced in time, it became much wider and was used in other applications as well. The International Electrical Commission (IEC) proposed three main goals for the 61850 standard [START_REF] Kriger | A detailed analysis of the generic object-oriented substation event message structure in an IEC 61850 standard-based substation automation system[END_REF].

1) Intelligent Electronic Devices (IEDs) manufactured by different companies should be able to communicate with each other. 2) In order to allow further development a certain degree of freedom has to be available to the manufacturers. 3) Further development will lead to progress. The standard should always allow for this to happen and keep on track. GOOSE is one of the communication protocols within the IEC 61850 standard. The main focus of this paper is on modelling the GOOSE protocol most relevant features and functions. GOOSE main purpose is to enable lateral communication between IEDs. A visual representation of the position of GOOSE within a substation can be seen in Fig. 1. GOOSE is mainly responsible for fast real-time and timecritical services. Therefore it needs to be efficient and, with less importance, reliable. To achieve this, GOOSE is a multicast, publisher/subscriber, unconfirmed communication protocol which is directly operating on Ethernet. These different properties all have their advantages and allow the protocol to be very useful in different scenarios [START_REF] Fernandes | Testing of goose protocol of IEC61850 standard in protection IED[END_REF]. receive the same data at the same time. This allows for fast and non-redundant communication; • Publisher/Subscriber & Unconfirmed: enables less redundant communication, since establishing a connection between two communicators is unnecessary, which is beneficial for the speed; • Directly operating on Ethernet: ethernet is a widespread technology, prone to constant development. As Ethernet will evolve, this progress will also be reflected to GOOSE. Moreover, the advanced functionalities of Ethernet can easily be used. For example, the priority tagging functionality of Ethernet frames can be used efficiently within the GOOSE protocol.

A. GOOSE Frame

Since GOOSE is operating directly on Ethernet, the frame structures are very similar. A visual representation of such a frame is shown in Fig. 2. The different components of this frame, will shortly be explained [START_REF] Fernandes | Testing of goose protocol of IEC61850 standard in protection IED[END_REF].

B. Retransmission Mechanism

The reliability of GOOSE communication is not ensured by acknowledgements, instead a retransmission mechanism is used. Every frame sent by an IED is repeatedly sent as long as the state of that device did not change. By increasing the amount of times a frame is sent, the chance of arriving will rise.

This retransmission mechanism can be explained more clearly based on Fig. 3. The system is in the first state when nothing is happening, and in the second state when an event took place. In the first state, the retransmission mechanism will make sure that every T max seconds, the same data will be sent. This way, some of the retransmissions can get lost, or be prone to errors while travelling, but the receiver will have the correct information in the end. In the second state the data changes and the new data will be sent. After a short amount of time (T min , with T min < T max ) the data will be sent again. This will happen a couple of times more, but every time with an increased retransmission interval. The interval will increase steadily until it reaches the same length as T max , after which the system is in the first state again. Next, the retransmissions happening in the second state will be referred to as 'fast retransmissions'. The length of these intervals can be chosen by the manufacturers [START_REF] Cochet-Muchy | IEC 61850 -advanced[END_REF].

III. IMPLEMENTATION OF IEC 61850 GOOSE

A simple communication network implementing the IEC 61850 GOOSE protocol was considered. Since GOOSE is a highly advanced communication protocol with a lot of functionalities outside the scope of this paper, a simplified version was implemented. Hence, the communication network consists of IEDs and a switch, whose main functionality is the routing of the data frames to the correct receivers. In what follows, the functionality of these devices and the protocol implemented by using blocks of the SimEvents R toolbox in MATLAB R /Simulink R , will be explained.

A. Physical devices

• IED: an IED can have different implementations according to its requirements. The requirements of the implemented IEDs are just sending and receiving data.

The sending part holds a register with the current dataset.

Connected to this register are the retransmission timers. These timers will create a trigger, which will initiate the creation of an event and an entity. The sending part of the IED also adds the necessary attributes: the data, the source address, the destination address (multicast address) and the VLAN Identifier (V-ID). The receiving part is slightly more complicated. The source MAC-address of the entity arriving from the switch will be read by the receiving IED. For each sending IED it wishes to retrieve data from, it will have a receiving register. Based on the source address, the receiving IED will decide in which register the data have to be stored. Before the entity arrives at the register though, it will pass through a FIFO buffer. This will allow for multiple entities to be buffered. When a read signal is given, the data attribute of the first entity in the FIFO will be read and will be put into the register;

IED 1 IED 2 IED 3 IED 1 IED 2 IED 3
Fig. 4. Internal connections of the switch.

• Switch: the switch tasks consist of routing the packets to the correct IEDs, buffering arriving packets and detecting collisions of packets. In this network, three IEDs are communicating with each other, therefore, the switch will have three ports. These ports are internally connected with each other as in Fig. 4. Based on the subscription table and the source address the switch is able to decide to which ports the arriving entity needs to be forwarded. The first row of the subscription table consists of the subscriber IEDs, whose subscription is given on the corresponding column in the second row. This means that for every subscriber-subscription-pair there is one column in the table. In a real-life implementation of GOOSE this table is dynamic and IEDs can add and/or delete subscriptions in real-time. However, this functionality is not really needed in this case, since it is assumed that the electrical network configuration will never change. Thus, the subscription table is implemented as a static table.

The switch also performs collision detection. All the arriving packets will come together in a priority queue, in which the packets will be sorted by their corresponding priority (based on the V-ID).

Based on [START_REF]Latency on a switched ethernet network[END_REF] the delay inside the switch is equal to 5.2 µs, existing due to finding the subscription in the lookup table and priority queue buffering. Any packet can be subject to an additional delay based on the time it needs to spend inside the buffer.

B. Protocol

The protocol makes sure there is a standard way of communication between the IEDs, and that the switch can find the necessary information within every frame. Its operating features can be divided into three main parts.

1) Multicast: all the IEDs are able to communicate with all the other IEDs. An IED just sends the data once, not knowing who will receive it, and the switch duplicates the frame while making sure only the IEDs that actually want the data get them; 2) Event-based: if the data change, the communication chain is started; 3) Retransmission: the two most important implementation characteristics of this mechanism are the two retransmission timers held by the IEDs. As soon as a dataset is communicated for the first time, these will be reset.

The first timer will make sure the packet gets sent in a predetermined interval, called T max in the literature.

The second timers responsibility is the so-called 'fast retransmissions' mechanism. This makes sure that after new data are sent it will send the same data quickly again.

When the chance of non-reception is reduced because of the fast retransmissions, this will stop and the other timer will only send the data once in a while (interval T max ). The values of these parameters used in the simulations are 10 ms and 200 ms, for T min and T max respectively.

IV. ELECTRICAL GRID

The distribution grid consists of a Low Voltage (LV) feeder with three load buses. Every bus consists of a 3-phase load, and two buses have a distributed generating source connected to it, one a Photovoltaic (PV) system and the other one a small wind turbine. Details about these energy production units will be provided later. Both loads at the energy nodes are constant at 12 kW, with a power factor of 0.95. The main load on the third bus has a more industrial nature, with a power factor of 0.8. Its profile can be seen in Fig. 5, the voltage at this point is called V load (Fig. 6). This bus will be referred to as 'the load'. The bus with the PV system attached will just be referred to as the 'PV system' and similar for the 'wind turbine'.

The distribution grid is connected to a Medium Voltage (MV) grid with a 400 V / 20 kV 3-phase Y / ∆ transformer (160 kVA). The MV grid is modelled by a perfect 50 Hz, three-phase voltage source. Every branch has a set of mutual inductances and line resistances.

A. Power Generating Units

The PV system is implemented by means of a detailed model [START_REF]Grid-connected PV array -file exchange -MATLAB central[END_REF], to which a reactive power control loop was added. This PV system consists of a PV array, a MPPT regulated boost-converter (average model) and an average model of a Voltage Source Converter (VSC) (converting 500V DC to 260 V AC, represented by an equivalent voltage source). The VSC has a sample time of 0.1 ms. A capacitor bank filters out the harmonics produced by the VSC. The PV array consists of 7 parallel strings of 5 series-connected PV modules, having a total maximum power of 10.7 kW at the reference irradiance value, 1000 W/m 2 . The irradiance curve used as an input for the PV array was obtained by quarter hourly measurements from a real production unit in France. However, for simulation reasons, the 24-hour period of the measurements was mapped to a span of 3 seconds, which corresponds to very fast irradiance variations.

The wind turbine was modelled by a dynamic load. Based on real measurements of a wind turbine in France, the power curve that this source is generating was determined. The measurements of one day were also scaled to only take 3 seconds. The obtained generated power by the wind turbine has the same order of magnitude as the PV system. The reactive power of the load is externally controllable.

B. Reactive Power Compensation Algorithm

A simple algorithm was developed and implemented, to illustrate the principle of reactive power control. A flowchart of this algorithm is shown in Fig. 7, with S in and S out the incoming and outgoing signal respectively and Q thresh , arbitrary chosen to equal 9.5 kVAR.

V. RESULTS

Since both the communication network model and electrical grid model are implemented in MATLAB R /Simulink R , it is possible to simulate them at the same time and inspect their interactions. Three test cases will be investigated. First, a scenario in which all the communications work, without disturbances. Secondly, a situation where the link used to send the power reference is highly disturbed. A third scenario, simulating high disturbances on the whole communication network and their effects on the electrical grid, is analysed.

A. No communication disturbances

According to the reactive power control algorithm, the PVsystem will generate 4/3th of the reactive power reference received from the load, noted by Q ref (Fig. 5). Its reactive power generation curve can be seen in Fig. 8a. However, between second 0.5 and 1 the power reference received is higher than Q tresh and the PV system will thus ask for help of the wind turbine. In Fig. 8b it can seen that the wind turbine successfully starts generating 2/3th of the power reference received. The PV system will, during this interval, reduce its generated power to 2/3th of Q ref as well. This results in V load as shown in Fig. 9.

Q gen = 4/3 Q ref Q ref > Q tresh S out = 1 S in == 1 Q gen = ⅔ Q ref

B. Disturbances on the reactive power reference link

If the channel used to communicate the reactive power reference, is disturbed, it will affect the performance of the electrical grid. Due to the retransmissions mechanism in GOOSE, a considerable effect is obtained, only at high rates of packet loss. In Fig. 10 the value of Q ref received by the PV system and the wind turbine, at a packet loss probability of 0.9, is shown. By comparing this figure with the reactive power part of Fig. 5, which is the power reference sent, it is clear that the received Q ref is not correct between second 1 and 2. So, reactive power generation profiles will be different, which results in a slightly different voltage profile (Fig. 11).

C. Heavily loaded grid

In the previous simulations, the properties of the electrical grid were based on real-life urban distribution grid configurations. As a result, lines were fairly short and oversized. These assets are advantageous, if the voltage drop is considered as reference for a well functioning grid. In order to consider a less favourable case, in the following simulation a distribution grid, whose line lengths are doubled compared to the previous cases, is considered.

To show the importance of communication, V load is measured, both with and without the communication network functioning. The result of these simulations is shown in Fig. 12. This figure shows the voltage profile in both cases. The French grid codes related to LV admissible variations are ±0.1 p.u. [START_REF] Erdf | Principes d'étude et de développement du réseau pour le raccordement des clients consommateurs et producteurs BT[END_REF]. As seen in the figure, the inferior admissible boundary was crossed in the case of a non-functioning communication network. These results show that the effect of communication failure on the electrical network can be severe, since crossing of these boundaries should be avoided.

In order to identify the probability of packet loss (P loss ) at which the voltage variation boundary is crossed, the simulations were run repeatedly with an increasing value of P loss . By plotting the maximum voltage drop at the load, in function of this packet loss chance, Fig. 13 is obtained. By linear interpolation the value of P loss , where the boundary is crossed, is determined to be 0.9580. Increasing chances of packet loss can sometimes lead to lower voltage drops, since non-received packets might lead to reactive power over-compensation.

VI. CONCLUSION AND FURTHER WORK

A smart grid consisting of two grids, one that allows electrical energy to flow and the other one that allows data to be transmitted, has been considered in this paper. In a highly automated electrical grid both of them are important and one cannot work without the other. MATLAB R /Simulink R cosimulation results obtained here showed a direct influence of the performance of the communication network on the electrical grid. Packet loss on different communication links have various effects, which cannot always be treated lightly; especially when the grid is heavily loaded, and has a rather rural than urban nature, the consequences can be as severe as crossing the grid codes adopted for safe operation of a distribution grid. In less critical cases, the behaviour of electrical dynamics in response to malicious events in the communication network was less destructive.

The co-simulation of an electrical grid together with a communication network is required to understand all their mutual interdependencies. The main goal of the work in this paper was to provide a starting point for designing a simulation combining both above mentioned systems. To this end, some simplifications were made in the progress. One of them concerns the reactive power control algorithm. One can expect that some more complex algorithms have a better performance. Other simplifications concern the GOOSE com-munication protocol, whose model implemented here is based on the most relevant of its numerous features. Other functions, like dynamic subscription of IEDs, could be implemented. In real-life situations there is also vertical communication. In the IEC 61850 standard, this data would run over TCP/IP in the form of MMS packets, which may also be prone to errors and may affect the functioning of the electrical grid.

In order to fully describe the consequences of disturbances, the work must be continued with expanded simulation models, allowing thorough mathematical modelling.

Fig. 1 .

 1 Fig. 1. GOOSE within the IEC 61850 family.

•

  Dest.-& Source Address: these bytes represent the destination and the source MAC-address. Since in GOOSE a message is always sent in a multicast fashion, the destination will be the multicast address; • Tag: part of this tag is the priority Virtual Local Area Network (VLAN) ID, GOOSE frames can skip the queues that arise at switches, or they can get ahead of other GOOSE frames within a queue, using this ID; • Ethertype: The Ethertype 2-byte part of an Ethernet frame is used to show which communication protocol is encapsulated within this frame; • Protocol Data Unit (PDU): This part of the frame contains the actual data along with some useful metadata; • Frame Check Sequence (FCS): This part of the frame is a value calculated based on the data and allows for the receiving IED to check if the data was prone to errors during transmission.

Fig. 3 .

 3 Fig. 3. Retransmission mechanism.

Fig. 5 .

 5 Fig. 5. Electrical load.

Fig. 6 .

 6 Fig. 6. Electrical grid.

Fig. 7 .Fig. 8 .

 78 Fig. 7. Graphical representation of the reactive power compensation algorithm. (a) PV-side. (b) Wind-side.

Fig. 9 .

 9 Fig. 9. V load without communication disturbances.

Fig. 10 .

 10 Fig. 10. Q ref received by the generation units.

Fig. 11 .

 11 Fig. 11. V load with communication disturbances.

Fig. 13 .

 13 Fig. 13. V load drop in function of packet loss probability (P loss ).

  Fig. 12. V load with and without communication network.

			260
						With Q compensation
			250			Without compensation of Q
						0.9 and 1.1 pu
	Voltage [V]	220 230 240
			210
			200
			0	0.5	1	1.5	2	2.5	3
					Time [s]
			0.11
		0.105
	Voltage Drop [p.u.]	0.09 0.095 0.1
		0.085
			0	0.2	0.4	0.6		0.8	1
						P loss


Quang Linh Lam
email: quang-linh.lam@g2elab.grenoble-inp.fr

Antoneta Iuliana Bratcu
email: antoneta.bratcu@gipsa-lab.grenoble-inp.fr

Delphine Riu
email: delphine.riu@g2elab.grenoble-inp.fr

Systematic Multi-variable H ∞ Control Design for Primary Frequency Regulation in Stand-Alone Microgrids with High Penetration of Photovoltaic Energy Sources

Keywords: Microgrids, primary frequency control, robust control, multi-variable control. 1

In this paper, a systematic design of a robust multi-variable control structure for primary frequency regulation in microgrids with high rate of renewable source penetration is proposed. The considered microgrid represents a diesel-photovoltaic-supercapacitor hybrid power generation system operating in stand-alone mode. The proposed control structure relies on a two-level architecture: classical PI-based current tracking controllers are placed on the low control level and receive references from an H∞-control-based upper level. A comprehensive methodology that casts the specific engineering demands of microgrid operation into H∞ control formalism is detailed. It is also shown how closed-loop dynamical performance requirements must at their turn be taken into account in the initial microgrid setup and sizing, namely in choosing and rating the energy storage system. Numerical simulations carried out in MATLAB R /Simulink R show the effectiveness of the proposed H∞ robust control strategy on a MVA-rated microgrid.

I. INTRODUCTION

The concept of microgrid appears first in the technical literature in [START_REF] Lasseter | Microgrids [distributed power generation[END_REF] and [START_REF] Lasseter | Microgrids[END_REF] as a promising solution to overcome the challenges of integrating distributed energy resources, including renewable energy sources, into power systems. Microgrid technology allows global reliability to be enhanced, reduction of carbon footprint along with more diversified energy sources, all with smaller investment costs [START_REF]The smart grid: An introduction[END_REF].

Reliable operation of microgrids -in either grid-connected or stand-alone mode -claims design of special protection schemes and control systems. From the main grid viewpoint, a microgrid can be seen as a single element responding to appropriate control signals. Such perception goes towards defining microgrids as "energy clusters" that contain loads, Distributed Generation (DG) units and Energy Storage Systems (ESSs) [START_REF] Olivares | Trends in microgrid control[END_REF].

Stability issues and problems related to low inertia and uncertainties due to presence of renewable energy sources are among the main control challenges [START_REF] Olivares | Trends in microgrid control[END_REF].

In stand-alone mode it is required to confine the frequency and voltage deviations within a small range to satisfy operating requirements. Presence of high-speed storage sys-tems, such as batteries or supercapacitors, has thus become necessary, leading to new grid configurations that do not longer operate well with conventional control laws. More complex robust control structures are needed to achieve better performance despite uncontrollable disturbances and model uncertainties.

A plethora of control methods have been proposed to handle this kind of problem. Proportional-integral-derivative (PID) control is a widely known and used technique [START_REF] Dong | Control strategies of DC-bus voltage in islanded operation of microgrid[END_REF], [START_REF] Sundaram | Load frequency control using PID tuned ANN controller in power system[END_REF]. Despite its intrinsic robustness, this method has limited possibility to ensure satisfactory trade-off among dynamic performances when the operating conditions vary significantly. Fuzzy logic control is proposed in [START_REF] Dhanalakshmi | Application of multi stage fuzzy logic control for load frequency control of an isolated wind diesel hybrid power system[END_REF]- [START_REF] Sakamoto | Output power leveling of wind turbine generators using pitch angle control for all regions in wind farm[END_REF], but it is difficult to obtain a model serving for a systematic control design. H ∞ control is considered in [START_REF] Goya | Frequency control in isolated island by using parallel operated battery systems applying H∞ control theory based on droop characteristics[END_REF]- [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF]; it appears as a powerful tool able to handle multiple requirements in a systematic manner such as to guarantee stability and overall robust performance.

In this paper, the frequency stability problem of microgrids with a high rate of decentralized, renewable and intermittent production is approached. In [START_REF] Han | Robust control of microgrid frequency with attached storage system[END_REF] it is shown that relatively small storage units can significantly reduce frequency deviation, provided that saturation conditions are avoided by dynamically coordinating storage with other generation sources. In this paper a systematic design procedure for computing a multi-variable H ∞ robust controller for primary frequency regulation is proposed. It is also shown how closed-loop operation demands must at their turn be taken into account in the initial microgrid setup and sizing, namely in appropriately choosing and rating the ESS.

The remainder of this paper is organized as follows. In Section II, microgrid description, coordinated strategy and choice of energy storage technology are presented. Modeling for the H ∞ control is described in Section III. H ∞ control design is detailed in Section IV. Numerical simulation results of the proposed control approach is analyzed in Section V. Section VI is dedicated to conclusion and future work.

II. MICROGRID DESCRIPTION, COORDINATED STRATEGY AND CHOICE OF ENERGY STORAGE TECHNOLOGY A. Microgrid Description

The block diagram of the considered microgrid operating in stand-alone mode is shown in Fig. 1. This microgrid contains a diesel generator as a classical source, a photovoltaic (PV) unit as a renewable source and an energy storage unit. Power sources are connected in parallel to a point of common Fig. 2. Performance specification on the microgrid frequency variation in response to a load step of ± 5% of the rated load power (± 60 kW) in the time domain [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF]. coupling (PCC) and supply a common load. The bank of PV panels is connected to the PCC through a one-quadrant chopper -inverter which only allows unidirectional power flow, whereas the energy storage system can be operated in charging or discharging mode.

B. Coordinated Strategy

In stand-alone microgrids with high penetration of PV energy sources, the frequency is risky to be unstable or its variations can be considerable and unacceptable [START_REF] Delille | Dynamic frequency control support by energy storage to reduce the impact of wind and solar generation on isolated power systems inertia[END_REF]. Energy storage systems can then be used to maintain the stability and/or improve the transient response of the frequency in case of disturbances. In the studied microgrid, the proposed coordinated strategy is participation of both the storage system and the diesel generator in primary frequency control. The role of the storage system is to guarantee faster recovery of the microgrid frequency (f grid ) consequent to a load power variation, i.e., to improve dynamic performances (overshoot, response time, steady-state error). A template for the frequency variation in response to a step load disturbance of ± 5% of the rated load power (± 60 kW) is shown in Fig. 2; it is defined based on its time-domain response in the case where only the diesel generator participates in primary control [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF].

C. Choice of Energy Storage Technology

Energy storage devices can be separated into two different groups with respect to their power supply ability [START_REF] Thounthong | Energy management of fuel cell/solar cell/supercapacitor hybrid power source[END_REF]:

• Sources with high power density which are able to supply high power for a short period of time with fast dynamic characteristics, supercapacitors (or ultracapaci-tors, or electrochemical double-layer capacitors) belong to this class of sources; • Sources with high energy density which are able to provide power over long periods of time with slow dynamic characteristics, fuel cells and long-time batteries are typical examples of such type of sources. This classification is illustrated by the Ragone plot [START_REF] Kuperman | Battery-ultracapacitor hybrids for pulsed current loads: A review[END_REF]. The choice of energy storage technology depends on the desired service. The own frequency f p (Hz) of each technology of a storage device is determined by f p = ρ p /ρ e , where ρ p (W/kg) and ρ e (J/kg) are the power density and the energy density of the storage device respectively.

The choice of storage devices for primary frequency control participation is based on analyzing the transfer function between the frequency deviation and the power variation in the frequency domain. The most critical situation of the frequency variation is first studied, then a suitable interval around the critical frequency will be defined for participation of the storage device.

The power variation in the frequency domain ∆P diesel (s), simplified dynamic modeling and frequency control of the diesel generator are expressed by

∆P diesel (s) = - 1 T diesel s + 1 Simplified dynamic modeling K p + K i s PI controller for frequency control ∆f grid (s) , (1)
where the underline denotes per-unitized values. The proportional gain of the PI controller corresponds to primary control, with K p = 1/s diesel , where s diesel is the droop value. The integral gain, K i , corresponds to secondary control and is the inverse of the response time of secondary control. T diesel denotes the time constant of the diesel generator.

Without storage device, based on results in [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF], [START_REF] Delille | Dynamic frequency control support by energy storage to reduce the impact of wind and solar generation on isolated power systems inertia[END_REF], [START_REF] Delille | Contribution du stockage à la gestion avancée des systèmes électriques, approches organisationnelles et technicoéconomiques dans les réseaux de distribution[END_REF], [START_REF] Kundur | Power System Stability and Control[END_REF], the microgrid frequency deviation in the frequency 3) with only primary control (Kp = 0, K i = 0), both primary and secondary control (Kp = 0, K i = 0) without the dynamic part of the diesel generator (T diesel = 0), both primary and secondary control with the dynamic part (T diesel = 0).

v s R c L c i s v c H i c i m v dc C dc R dc i dc A B C R f L f i r1 i r2 i r3 v m1 v m2 v m3 v r1 v r2 v r3
Storage Choke Chopper DC bus Three-phase inverter Output filter AC bus

u c u c u 1 u 1 u 2 u 2 u 3 u 3 (a) Sp regu Pro (mo Actuator (valve) Supply (fuel) Y diesel_reg (P diesel_reg ) Y diesel ( Superc. R sc C sc v sc dc C i dc R i p sc R Fig. 4.
The full electrical scheme of the energy storage system [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF].

domain ∆f grid (s) is given as follows

∆f grid (s) = ∆P diesel (s) + ∆P dif f (s) ∆P P V (s) -∆P load (s) 2Hs + D load , (2)
where H is the microgrid equivalent inertia constant and D load is the load damping constant. Substituting (1) into (2) yields

∆f grid (s) ∆P dif f (s) = (T diesel s + 1) s (2HT diesel ) s 3 + 2H + T diesel D load s 2 + D load + K p s + K i . (3)
The Bode diagram of the transfer function in (3) is shown in Fig. 3. The system with only primary control can be assumed as a low-pass filter. High-frequency power fluctuations are then filtered by the microgrid equivalent inertia. The DC gain of the transfer function is constant and equal to the droop value s diesel . Low-frequency power variations are damped by secondary control. The critical frequency value (or resonant frequency) is given in Fig. 3. Power fluctuations around the critical frequency can cause very large frequency variations. Use of a storage device to limit microgrid frequency variations is thus suitable.

Definition of cut-off frequencies to separate frequency intervals would be useful to define participation bandwidth of the storage device, which is then applied for choosing appropriate storage technology. In order to calculate these cut-off frequencies, the dynamic part of the diesel generator is firstly neglected (i.e., T diesel = 0) to simplify the expression of the transfer function given in (3) as follows

∆f grid (s) ∆P dif f (s) = s K i 2H Ki s 2 + D load +Kp Ki s + 1 , (4)
which corresponds to a second-order system with a zero at the origin, the corner frequency ω n = K i 2H and the damping ratio ζ = D load + K p 2 2HK i . The magnitude of this transfer function is illustrated in Fig. 3. For the considered microgrid, ζ is equal to 4.17, therefore, two cut-off frequencies denoted ω c1,2 can be calculated as

ω c1,2 = 1 4H -D load + K p ± D load + K p 2 -8HK i . (5)
Substituting microgrid parameter values into (5) gives the low cut-off frequency ω c1 = 0.12 rad/s (or f c1 = 0.02 Hz) and the high cut-off frequency ω c2 = 8.21 rad/s (or f c2 = 1.31 Hz). Moreover, as analyzed in Fig. 3, the dynamic part affects the medium-frequency region which corresponds to primary control and it hardly affects cut-off frequencies. Hence, the frequency interval f c ∈ [0.02, 1.31] Hz is chosen for primary control participation of the storage device. By comparing this frequency interval with the own frequency of each storage technology deduced from the Ragone plot in [START_REF] Kuperman | Battery-ultracapacitor hybrids for pulsed current loads: A review[END_REF], one can conclude that supercapacitor storage technology with its own frequency f p ∈ [0.00278, 27.78] Hz is the most appropriate one and consequently selected for primary control participation.

The supercapacitor power source is connected to the PCC and controlled by means of a two-quadrant chopper -inverter which allows bidirectional power flow. The full electrical scheme of the storage system is presented in Fig. 4 [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF]. Another control objective which must be taken into account is to regulate the DC-bus voltage (v dc ) at the desired value of 1000 V. This performance specification (Fig. 5), relied on the one proposed recently in [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF]. By imposing suitable small-signal performances, large-signal standards can be guaranteed.

III. MODELING FOR H ∞ CONTROL

In this paper, the energy storage system is modeled by using the averaged modeling. The non-linear averaged model is obtained from its topological model [START_REF] Maksimovíc | Modeling and simulation of power electronic converters[END_REF]. By linearizing the non-linear averaged model around an equilibrium point supposing that the PCC voltages in d-axis and q-axis respectively (v rd , v rq ) remain constant (i.e., ∆v rd = 0, ∆v rq = 0), the linear averaged model is obtained in [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF].

In some cases, for instance, H ∞ control design, it is often more convenient to express a power-electronic converter system in a per-unit term. For the considered storage system, the base values for AC-side quantities are given in Appendix. The DC-side base values are determined based on those of the AC side [START_REF] Yazdani | Voltage-Sourced Converters in Power Systems: Modeling, Control, and Applications[END_REF] per-unitized value by the underline, then the linear averaged model of the storage system is described by the following per-unitized set of equations

1 ω b d∆v sc dt = - 1 C sc ∆i s - 1 R scp C sc ∆v sc , (6)
1 ω b d∆v dc dt = 1 C dc α ce ∆i s + 1 C dc i se ∆α c - 1 C dc β de ∆i rd + i rde ∆β d + β qe ∆i rq + i rqe ∆β q - 1 R dc C dc ∆v dc , (7)
1 ω b d∆i s dt = 1 L c ∆v sc - R sc + R c L c ∆i s - 1 L c α ce ∆v dc - 1 L c v dce ∆α c , (8)
1 ω b d∆i rd dt = 1 L f β de ∆v dc + 1 L f v dce ∆β d - R f L f ∆i rd + ω gride ∆i rq + i rqe ∆ω grid , (9)
1 ω b d∆i rq dt = 1 L f β qe ∆v dc + 1 L f v dce ∆β q - R f L f ∆i rq -ω gride ∆i rd -i rde ∆ω grid , (10)
where the state variables are: the supercapacitor voltage (∆v sc ), the DC-bus voltage (∆v dc ), the storage device current (∆i s ) and the inverter output currents in d-axis and q-axis respectively (∆i rd , ∆i rq ). α c is the average value of the chopper switching function, β d and β q are the average values of the inverter switching functions in d-axis and qaxis respectively, ω grid is the microgrid pulsation.

The power variation of diesel generation in the time domain (∆P diesel ) for only primary control participation (i.e., K p = 0, K i = 0) is obtained by applying the inverse Laplace transform to (1)

d∆P diesel dt = - 1 T diesel ∆P diesel - 1 T diesel s diesel ∆f grid . (11)
Based on results in [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF], [START_REF] Delille | Dynamic frequency control support by energy storage to reduce the impact of wind and solar generation on isolated power systems inertia[END_REF], [START_REF] Delille | Contribution du stockage à la gestion avancée des systèmes électriques, approches organisationnelles et technicoéconomiques dans les réseaux de distribution[END_REF], [START_REF] Kundur | Power System Stability and Control[END_REF] and supposing that PV source power is constant (i.e., ∆P P V = 0 ), the per-unitized equation expressing the microgrid frequency deviation in the time domain ∆f grid is written as

d∆f grid dt = 1 2H i se ∆v sc + v sce -2R sc i se ∆i s + ∆P diesel -∆P load - D load 2H ∆f grid . ( 12
)
Subscript e indicates the steady-state equilibrium point. Steady-state real-unit values are given in Table I. This section details the control approach used to achieve the control objectives previously mentioned in Section II. Its basic idea is to consider the current variations ∆i s and ∆i rd as control inputs for the linearized system (6), ( 7), [START_REF] Singh | Robust H-infinity load frequency control in hybrid distributed generation system[END_REF] and [START_REF] Lam | Multivariable H-infinity robust control applied to primary frequency regulation in microgrids with large integration of photovoltaic energy source[END_REF], which should result from the disturbance rejection requirement ∆P load . The global control structure is illustrated in Fig. 6, where control loops are separated into two cascaded levels.

A. Current Control Level

The current variations ∆i s , ∆i rd and ∆i rq must be controlled and prevented from exceeding admissible limits. All current control loops have fast closed-loop dynamics compared to the H ∞ control loop, therefore, they are grouped together.

∆i rq (or ∆i rq ) in ( 9) is considered a high-frequency perturbation, which is further significantly reduced by the d -q decoupling structure. The same applies for ∆i rd (or ∆i rd ) in [START_REF] Goya | Frequency control in isolated island by using parallel operated battery systems applying H∞ control theory based on droop characteristics[END_REF]. ∆v dc (or ∆v dc ) and ∆ω grid (or ∆ω grid ) in (8), ( 9), ( 10) are regarded as low-frequency perturbations that are rejected by the upper-level control. Hence, the transfer functions relating the current variations with the duty ratio variations (inner plants) are computed straightforwardly from (8), ( 9) and (10)

G is-αc (s) = ∆i s (s) ∆α c (s) = K s T s s + 1 , (13)
G i rdq -β dq (s) = ∆i rd (s) ∆β d (s) = ∆i rq (s) ∆β q (s) = K rdq T rdq s + 1 , (14)
where Small-signal state-space model G(s) Let the inner closed-loop dynamical performance be described by the bandwidth 1/T 0 and the damping coefficient ζ 0 . In our case study, we impose the closed-loop dynamics ten times faster than the original current plants, T 0s = T s /10 = 11.4 ms, T 0 rdq = T rdq /10 = 9.6 ms and the damping ratios ζ 0s = ζ 0 rdq = 0.8.

K s = -v dce /(R sc + R c ), T s = L c /(R sc + R c ), K rdq = v dce /R f and T rdq = L f /R f . _ + _ + _ + H ∞ Controller K (
mes rd i  mes rq i  ref s i  ref rd i  ref rq i  ref s i  ref rd i  0 ref rq i   0 ref sc v   0 ref dc v   0 ref grid f   mes sc v  mes dc v  mes sc v 
_ + P _ + H ∞ Controller K(s) K _ + ∆w ∆u ∆z ∆v load P  ref s i  ref rd i  sc v  dc v  grid f  0 ref grid f   0 ref dc v   0 ref sc v   1 e  2 e  3 e  4 e  1 ( ) u W s  2 ( ) u W s  1 ( ) perf W s 2 ( ) perf W s

B. H ∞ Control Level

The current variations ∆i s and ∆i rd are controlled by the very fast-dynamic loops which satisfy tracking of all current reference variations generated by the H ∞ controller, therefore, the outer H ∞ plant "sees" ∆i s ≡ ∆i ref s and ∆i rd ≡ ∆i ref rd . The linearized system (6), ( 7), ( 11) and ( 12) is represented in the state-space form as follows (with the terms 1 C dc i se ∆α c , 1 C dc i rde ∆β d and 1 C dc i rqe ∆β q in (7) being neglected in the outer H ∞ control loop design, ∆i rq = ∆i ref rq = 0)

∆ ẋ = A∆x + B 1 ∆u + B 2 ∆w ∆y = C∆x + D 1 ∆u + D 2 ∆w , ( 15
)
where ∆x = ∆v sc ∆v dc ∆P diesel ∆f grid The multi-variable H ∞ control formulation proposed in Fig. 7 is afforded by the general control configuration in [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. In our case study, focus is on the disturbance rejection problem corresponding to the load power variation, therefore, the S/KS mixed-sensitivity optimization must typically be solved [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]. This design method requires definition of frequency weights W perf (s) to formalize the disturbance rejection objectives (performance specifications) and frequency weights W u (s) to translate the practical implementation constraints of the control inputs.

The generalized plant P has three inputs, namely, the load power variation ∆P load acting as a disturbance input ∆w and the current reference variations ∆i ref s , ∆i ref rd which are the control inputs ∆u. The output vector, ∆y, is composed of the supercapacitor voltage variation ∆v sc , the DC-bus voltage variation ∆v dc and the frequency variation ∆f grid .

Weighting functions selection is the key to cope with the performance requirements. The DC-bus voltage variation and the frequency variation are bounded by first-order weighting functions W perf (s) of the following form [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]

1 W perf (s) = s + ω b A ε s/M s + ω b . (16)
The function 1/W perf (s) can be representative of timedomain response specifications, where the high-frequency gain M s has an influence on the system overshoot, the cutoff frequency ω b tunes the desired response time and the low-frequency gain A ε allows limiting the steady-state error.

The first weighting function W perf1 (s) is designed for ∆v dc to impose a bandwidth forty times smaller than the ∆i rd inner closed-loop one, ω b1 = 1 40T 0 rdq = 2.61 rad/s (or a response time of approximately t r1 ≈ 1.2 s). M s1 = 0.01v dce 0.05P loade = 0.833 is chosen to limit a maximum overshoot of 1% of the rated DCbus voltage (10 V) in response to a load step of 5% of the rated load power. Similarly, one imposes A ε1 = 0.005v dce 0.05P loade = 0.417 to ensure a desired steady-state error being less than or equal to 0.5% of the rated DC-bus voltage (5 V).

The parameters of the second weighting function W perf2 (s) designed for ∆f grid are chosen to improve the transient response of the frequency compared to its performance specification in Fig. 2. ω b2 = 2ω b1 = 5.22 rad/s is chosen to ensure a response time of approximately t r2 ≈ 0.6 s. Imposing M s2 = [(50.12 -50)/f b ] 0.05P loade = 0.08 allows limiting a maximum frequency overshoot of 0.12 Hz in response to a load step of 5% of the rated load power. A ε2 = [(50.06 -50)/f b ] 0.05P loade = 0.04 is chosen to ensure a desired steady-state error being less than or equal to 0.06 Hz.

As analyzed in Section II, the frequency interval f c ∈ [0.02, 1.31] Hz (or ω c ∈ [0.12, 8.21] rad/s) was selected for primary control participation of the storage device. Moreover, the active power injection or absorption of the storage system is controlled via i s . Thus, the storage device current reference variation ∆i ref s is bounded by a band-pass weighting function W u1 (s) written as follows

1 W u1 (s) = A u1 Mu 1 ω bc 11 s + 1 Au 1 ω bc 12 s + 1 Au 1 ω bc 11 s + 1 Mu 1 ω bc 12 s + 1 , ω bc11 A u1 < ω bc12 M u1 . (17)
M u1 = i smax -i se 0.05P loade = 9.62 is chosen to prevent the storage device current reference from exceeding its admissible limit i smax consequent to a load step of 5% of the rated load power. A small value of A u1 = 0.1M u1 = 0.962 is imposed. In our case study, the own frequency of the selected supercapacitor technology is f p = 0.48 Hz (ω p = 3.04 rad/s), therefore, one chooses ω bc11 /A u1 = 3.04 rad/s resulting in ω bc11 = 2.93 rad/s. Additionally, ω bc12 /M u1 = ω cmax = 8.21 rad/s is chosen, which leads to ω bc12 = 78.98 rad/s.

The DC-bus voltage is regulated via i rd . The inverter output current reference variation in d-axis ∆i ref rd is bounded by the following first-order weighting function [START_REF] Skogestad | Multivariable Feedback Control: Analysis and Design[END_REF]

1 W u2 (s) = A u2 s + ω bc2 s + ω bc2 /M u2 . (18)
M u2 = i rdmax -i rde 0.05P loade = 8.48 is imposed to limit the current reference below its maximum value i rdmax consequent to a load step of 5% of the rated load power. A small value of A u2 = 0.1M u2 = 0.848 is chosen. ω bc2 = 1/(100T si ) = 40 rad/s, where T si = 1/f si , is chosen to impose a bandwidth one hundred times smaller than the inverter switching frequency f si .

According to the system modeling and the selected weighting functions, a full-order H ∞ controller is designed using the MATLAB R software environment. The obtained result corresponds to the minimization of the norm

W perf S W u KS ∞ < γ, (19)
where the sensitivity functions are defined as follows

S 1 = ∆v dc ∆P load , S 2 = ∆f grid ∆P load , (20)
KS 1 = ∆i ref s ∆P load , KS 2 = ∆i ref rd ∆P load . ( 21
)
Design procedure may yield unstable controllers. In this case, the value γ min must be slightly increased to reduce the constraint on the control and result in stable controllers. The full-order H ∞ controller found has 9 states with a conditioning value of γ = 169.83. In summary, the systematic procedure for H ∞ control design can be carried out as follows control and choosing a storage technology which is the most appropriate to this frequency interval;

V. NUMERICAL SIMULATION RESULTS

Numerical simulations under MATLAB R /Simulink R are performed to show the effectiveness of the proposed control approach. The non-linear averaged model is used for the time-domain simulations. The load profile is varied in step of + 5% of the rated load power (60 kW) at t = 1 s, -5% at t = 9 s, -5% at t = 17 s and + 5% at t = 25 s. Fig. 8 presents the time-domain response of the microgrid frequency. One can see that the desired time-domain performance corresponding to the parameter choice of the weighting function W perf2 (s) in Section IV is well-respected. Primary frequency control participation of the storage device has allowed improving significantly the dynamic performances (i.e., lower overshoot, faster response time, smaller steady-state error) compared with the case where only the diesel generator participates in primary control. Fig. 9 shows that the DC-bus voltage control objective is satisfied with respect to the parameter selection of the weighting function W perf1 (s) in Section IV (the desired time-domain performance is met). Therefore, the proposed H ∞ controller ensures the desired performance specifications. The timedomain responses of the energy storage device current and the inverter output current in d-axis are given in Fig. 10.

VI. CONCLUSION AND FUTURE WORK

This paper has presented a systematic design procedure for computing a multi-variable H ∞ robust controller for primary frequency regulation in stand-alone microgrids highly penetrated by renewable energy sources. Effectiveness of the proposed H ∞ robust control strategy has been validated via numerical simulation results. Future work can concern robustness analysis of the synthesized H ∞ controller taking into account power output variation of the PV source. Use of Linear Parameter Varying (LPV) control will also be envisaged to ensure good performance if the synthesized H ∞ controller is not robust to significant variations of the operating point (storage system capacity, PV power output,...). The next main prospect of this paper is to design a robust control strategy for PCC voltage regulation. Practical implementation of the proposed control algorithms on a realtime test bench must also be envisaged.

APPENDIX

Matrices in the small-signal state-space model for the H ∞ control level

A =       -ω b Rsc p Csc 0 0 0 0 -ω b R dc C dc 0 0 0 0 -1 T diesel - 1 T diesel s diesel 1 2H i se 0 1 2H - D load 2H       B 1 =       -ω b

Fig. 1 .

 1 Photovoltaic panels

Fig. 3 .

 3 Fig.3. Bode diagram of the system in (3) with only primary control (Kp = 0, K i = 0), both primary and secondary control (Kp = 0, K i = 0) without the dynamic part of the diesel generator (T diesel = 0), both primary and secondary control with the dynamic part (T diesel = 0).

  currents Moderate-dynamic control loop used to generate current reference variations

Fig. 6 .

 6 Fig. 6. Block diagram of the proposed global control structure.

Fig. 7 .

 7 Fig. 7. Control configuration in the so-called P -K form, where P denotes the plant together with the weighting functions and K denotes the H∞ controller.

  input vector, ∆w = ∆P load is the load power variation, which represents the disturbance input, and ∆y = ∆v sc ∆v dc ∆f grid T is the measured output vector. Matrices A, B 1 , B 2 , C, D 1 and D 2 are given in Appendix.

Fig. 8 .

 8 Fig.8. Time-domain response of the microgrid frequency f grid under small step load disturbances of ± 5% of the rated load power (± 60 kW).

  Fig.10. Time-domain response of the energy storage device current is and the inverter output current in d-axis i rd under small step load disturbances of ± 5% of the rated load power (± 60 kW).

C dc α ce -ω b C dc β de 0 0 1 2H

 1 v sce -2R sc i se 0

  and summarized in Appendix. Let us note a

			Load step
		dc (V)	1100 V (110%)
		v DC-bus voltage	1000 V (100%)	1050 V (105%) 950 V (95%)
			900 V (90%)
			t	t + 0.6 s	t + 1.2 s	Time (s)
	Fig. 5.	Performance specification on the DC-bus voltage variation in
	response to a load step in the time domain.

TABLE I STEADY

 I -STATE REAL-UNIT VALUES OF THE LINEAR SYSTEM

	Variable	vsc e	v dce	v rde	vrq e	is e
	Value	780 V	1000 V	400 V	0 V	-3.4 A
	Variable	i rde	irq e	αc e	β de	βq e
	Value	-21.6 A	0 A	0.78	0.4	-0.003
	Variable	P diesele	P P Ve	P loade	f gride	ω gride
	Value	1.0026	0.2	1.2	50	100π
		MW	MW	MW	Hz	rad/s

IV. H ∞ CONTROL DESIGN

  System modeling and casting the specific engineering demands of microgrid operation into H ∞ control formalism; • Step 3: Selecting the weighting functions according to control objectives and control input constraints; • Step 4: Synthesizing an H ∞ controller and performing numerical simulations to verify if the control objectives are satisfied or not. If not, go to Step 3.

• Step 2: