Aouini Raouia

Raouia Aouini
email: aaouinii@yahoo.com

Bogdan Marinescu
email: bogdan.marinescu@irccyn.ec-nantes.fr

Khadija Ben Kilani

Mohamed Elleuch

Keywords: Synchronverter, Synchronous generator/motor, SHVDC, transient stability, damping oscillatory modes

Synchronverter based HVDC Control in a Multi Machines AC Systems

The HVDC links are increasingly used not only two distinct power systems but are also embedded into a same meshed AC power system. Thanks to its speed and flexibility, the HVDC technology is able to provide transmission system advantages as transfer capacity enhancement and power flow control. In addition, studies have shown that the way of controlling the HVDC converters impacts the stability of the AC system. This can be particularly exploited to enhance the dynamic power system performances during transients. In this paper, the HVDC emulation by the synchronverter concept is investigated in a realistic power system. A specific tuning method for the parameters of the regulators based on the sensitivity of the poles of the neighbour zone of the HVDC with the respect to the latter parameters is used. As consequence, not only the local performances of the HVDC link, but also overall transient stability of the AC zone in which the HVDC is inserted are improved. Extensive tests are provided using Matlab/Simulink implementation of the IEEE 9 bus/3 machines test system.

Introduction

The high voltage direct current (HVDC) link is a mean of transmission of electric power based on high power electronics which offers an opportunity to enhance controllability, stability, and power transfer capability of AC transmission systems [START_REF] Bahrman | The ABCs of HVDC transmission technologies[END_REF]. Practical conversion of power between AC and DC became possible with the development of power electronics devices such as thyristors and voltage source converters (VSC). The VSC technology offers even more flexibility, new capabilities for dynamic voltage support, independent controls of active/reactive power and easier integration of wind farms [START_REF] Flourentzou | VSC-Based HVDC Power Transmission Systems: An Overview[END_REF].

It was first used in power systems to interconnect asynchronous AC systems as it is the case of the England-France interconnection [START_REF] Goodrich | The 2000 MW HVDC link between England and France[END_REF]. Nowadays, HVDC links are increasingly used into a same AC system in order to enhance the grid's transmission capability and flexibility of the power system. The project Spain-France interconnection [START_REF] Decoeur | France-Spain interconnections, first step for smart grids[END_REF], which will use the VSC technology scaled up to 2000MW, is such an example. In this context, the HVDC link co-exists with other AC system elements as for instance AC lines and generators.

It was shown that the strategy used to control the HVDC converters can impact the stability of the system in which the link is inserted [START_REF] Hauer | Robustness issues in stability control of large electric power systems[END_REF]- [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF].

Most control of a VSC-based HVDC system uses a nested-loop d-q vector control approach based on the linear PI technology [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF].

Recently in [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF], the synchronverter concept has been adapted to the converters of a VSC-HVDC link. The concept of the synchronverter control is to mimic the behavior of a synchronous generator (SG) along with its voltage and frequency regulations [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF].

More specifically, the sending-end rectifier emulates a synchronous motor (SM) and the receiving end inverter emulates a synchronous generator (SG). This resulting Synchronverter based HVDC was called SHVDC [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF].

In [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF], we have taken into account the new context of use of the HVDC during the synthesis of converters controllers. In fact, we have developed an analytic methodology which takes into account, in the tuning stage, the neighbor zone of the HVDC link and the dynamics that most impact transient stability. As a consequence, power system performances were enhanced in addition to local performances. However, the effectiveness of the tuning method of the SHVDC parameters had not been proved in the case of realistic power system.

The present paper shows that the strategy developed in [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF] is applicable to a realistic case which is the full model of the IEEE 9 bus system. In the tuning stage, this control technique allows one to take into account dynamic specifications and swing information of the neighbour AC zone in term to better dynamic performances and to improve the transient stability of the neighbor AC zone of the HVDC link; and this fact is proven on a realistic power system.

The rest of the paper is organized as follows: in Section II, the SHVDC structure is recalled. An analytic method to tune the parameters of the controllers of the SHVDC in order to meet the desired performances is given in Section III, while validation tests are presented in Section V.

Synchronverter based HVDC

In this Section, the converters of the HVDC line are designed to emulate synchronous machine using the synchronverter concept proposed in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. The latter is an inverter of which regulations are chosen such that the resulting closed-loop mimics the behavior of a conventional synchronverter generator (SG) [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. Therefore, to provide an HVDC structure, another synchronverter working as a synchronous motor (SM) based on the same mathematic derivation is necessary. As a result, the DC power is sent from the SM to the SG.

The resulting system in Fig. 1 is called Synchronverter High-Voltage Direct Current (SHVDC). On the figure, we depict: (i) the power part of the SG/SM consists of the inverter/rectifier plus LC filter;

(ii) the controls are assured by the electronic part. VSC converter technologies shown in Fig. 2 are used. The overall structure is shown to be equivalent to a SG/SM with capacitor bank connected in parallel with the stator terminal.

As shown in Fig. 2, controllers include the mathematical model of a three-phase roundrotor synchronous machine described by 1 ( )

g gm ge gp g g θ θ θ   =      . ( 12
)
The operator .,. < > denotes the conventional inner product in 3 � .

The phase terminal voltage of the SG and SM are

V g_abc = [V ga V gb V gc ] T , V m_abc = [V ma V gb V gc ] T respectively _ _ _ _ g abc g abc s g abc s g abc di V R i L e dt = - - + , ( 13
) _ _ _ _ m abc m abc s m abc s m abc di V R i L e dt = + + . ( 14
) _ m abc V , R L g g Bus1 Bus 2 _ m abc e _ m abc E ~ , m me P Q , g ge P Q PWM PWM 1 dc V 2 dc i 1 dc i , dc dc L R _ g abc e _ g abc V _ g abc E cc i ~ , Rs Ls , R L g g ,

Rs Ls

Fig. 2. Model of the synchronverter: power and electronic parts [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] The SHVDC given in Fig. 1 is connected to the grid via the impedance (L g , R g ) such that _ _ _123

( )

g abc g abc g f V i i C s = - , ( 15
) _123 _ _ 1 ( ) ( ) g g abc g abc g g i V E R L s = - + , ( 16
) _ _123 _ 1 ( ) m abc m m abc f V i i C s = - , ( 17
) _123 _ _ 1 ( ) ( ) m m abc m abc g g i E V R L s = - + , ( 18
)
where i g _ abc = [i ga i gb i gc ] T , i g _ abc = [i ga i gb i gc ] T , are respectively the stator phase currents of the SG and the SM, L s , R s are, respectively, the inductance and the resistance of the stator windings and e g_abc = [e ga e gb e gc ] T , e m_abc = [e ma e mb e mc ] T are respectively the back emf of the SG and the SM.

To mimic the droop of the SG, the following frequency droop control loop is proposed _ ( )

gm gm ref gp n g T T D s ω θ = + - , (19)
_ ( )

mm mm ref mp n m T T D s ω θ = + - . ( 20
)
T gm_ref is the mechanical torque applied to the rotor of the SG and it is generated by a PI controller as shown in Fig. 1 to regulate the real power output g P . In the SM case, T mm_ref is generated by a DC voltage control for power balance.

_ g ref V ~ _123 g i _ g ref P g M - ge T - n ω g θ  g θ 1 J *s g gp D (5) (7) (10) Eqn Eqn Eqn 1 k s g _ gm ref T (22) Eqn 1 s _ e g abc g V (27) Eqn _ g abc i _ g abc e Bus 2 , s s R L ~ , g ge P Q ge Q - cc i 2 dc i 2 dc V - f C _ i g abc g P gq D _ g ref Q dc C PWM , g g R L _ V g abc _ _ _ 1 _ ( )( ) i vdc mm ref p vdc dc ref dc k T k V V s = + - , (21)
_ _ _ _ ( )( ) g g i p gm ref p p g g ref k T k P P s = + - . ( 22
)
The reactive power Q gm (respectively Q mm ) is controlled by a voltage droop control loop using a voltage droop coefficient D gq (respectively D mq ), in order to regulate the field excitation M g (respectively M m ) , which is proportional to the voltage generated.

(

)

g gm ge g M Q Q k s = - , ( 23
) g_ref _ ref ( ) gm gq g g Q Q D V V =+ - , (24) 1 (
)

m mm me m M Q Q k s - = - , ( 25
) m_ref _ ( ) mm mq m ref m Q Q D V V =+ - , ( 26
)
where V g (respectively V m ) , is the output voltage amplitude is computed by 2 3( )

g ga gb ga gc gb gc V V V V V V V = + + , (27) 2 3(
)

m ma mb ma mc mb mc V V V V V V V = + + . ( 28
)
The circuit equations of the DC line (Fig. 2) are 1 1

( )

dc dc cc dc V i i C s = - , (29) 2 2
1 ( )

dc cc dc dc V i i C s = - , (30) 1 2
1 ( )

cc dc dc dc cc dc i V V R i L s = - - . ( 31
)
The AC and DC circuits are coupled by the active power relation

1 1 1 dc dc dc P V i = , ( 32
) 2 2 2 dc dc dc P V i = , ( 33
) 1 dc m P P = , ( 34
) 2 dc g P P = .
(35) Fig. 3. IEEE 9 bus system

Control model description

The tuning methodology of the SHVDC parameters developed in [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF] is briefly recalled here and enriched such that the local performances are enhanced both for the HVDC as well as for the transient stability of the neighbour AC systems. The resulting control model must preserve the dynamics that most impact the stability of the region neighbouring the HVDC link. For this reason, it was necessary to start from a full model of a sufficient large zone arround the HVDC link. In this study, we considered the realistic of the IEEE 9 bus system.

Description of the test power system

The IEEE 9 bus system shown in Fig. 3 contains 3 generators, 5 lines, 3 loads and 3 two winding power transformers. The 100 km HVDC cable link has a rated power of 200 MW and a DC voltage rating of ±100 kV [START_REF]Simulink Model of IEEE 9 Bus System with load flow[END_REF]. The rating of each generator is 600 MVA and 20 kV. Each of the units is connected through transformers to the 100 kV transmission line. The detailed system data is given in the appendix. The loads L 1 , L 2 and L 3 are modeled as constant impedances. The simulations tool is in the Matlab/Simulink toolbox.

Specific framework for the tuning of the SHVDC parameters

HVDC Control Specifications

We first recall the usual full set of control specifications for a VSC-based HVDC. Setpoints for the transmitted active power, the reactive power and the voltage at the points of coupling have to be tracked with the following transient performances [START_REF] Li | Control of HVDC light system using conventional and direct current vector control approaches[END_REF][START_REF] Sangathan | Performance of high-voltage direct Current (HVDC) systems with Line-commutated converters[END_REF]: -the response time of the active/reactive power is normally in the range of 50 ms to 150 ms; -the response time for voltage is about 100 ms to 500 ms.

Control objectives

In our previous work [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF], we have developed a tuning methodology of the SHVDC parameters which takes directly into account; swing information at the synthesis stage in term of the less damped modes of the neighbor AC zone of the HVDC link. As a consequence, the stability limit in term of the CCT of the neighbor zone was improved in addition to local performances presented above. The local performances are ensured which are tracking of references for active power, reactive power [START_REF] Aouini | Synchronverter-based Emulation and Control of HVDC transmission[END_REF]. In this present work, the same tuning method mentioned above is used in the IEEE 9 bus system. More precisely, the synthesis of the SHVDC parameters is done such that the stability performances are ensured for several cases of fault along with the local ones.

Control Structure

In order to satisfy these control objectives, the open loop system shown in Fig. 3 is put into the feedback system structure presented in Fig. 4 where H(s) is the linear approximation of the system in Fig. 3

V V y=[ -sθ , V V , V V , , -sθ , s P P V V , P P , ] . s n n T ω ω - - - - - -

Regulators parameters and residues

Let ( ) H s be the transfer matrix of a linear approximation of ∑  and consider each closed-loop of the feedback system in Fig. 5 which corresponds only to input i u and output i y .More specifically, ( )

ii H s and ( ) ii K s are the (i,i) transfer functions of H(s), , respectively.

Proposition [START_REF] Rogers | Power System Oscillations[END_REF]: The sensitivity of a pole λ of the closed-loop in Fig. 2 with respect to a parameter q of the regulator ii K is ( ,

) ii K s q r q q λ λ ∂ ∂ = ∂ ∂ (37)
where r λ is the residue of ( )

ii H s at pole λ .

Note that, for our case. (36), ( , ) λ can be computed for each pole i λ starting from the control specifications given in Section 3.2.1. The connections between the dynamics of interest and the modes are established based on the participations factors given in Table 1. From the same column of Table 1; several poles have also significant participations to the same dynamic which led us to compute the gains K in a coordinated way. More specifically, Λ denotes the set indices j from 1 to 8 for which ( ) jj H s has i λ as pole, the contribution of each control gain in the shift of the pole is

1 ii s K s q q λ = ∂ = ∂ .
0 i i ij j j r K λ λ ∈Λ = + ∑ , ( 38
)
where 0 i λ is the initial (open-loop) location of the pole i λ and ij r is the residue of ( ) jj H s in i λ . Finally, the pole placement is the solution of the following optimization problem 2 * * { , 1...8} arg min

j j i i K i K j λ λ = = - ∑ , ( 39
)
where i λ is given by (38).

The optimal parameters in the appendix were obtained with (39) solved for the desired locations in Table 1. The specific tuning SHVDC parameters is tested on the IEEE 9 bus/3 machines benchmark [START_REF]Simulink Model of IEEE 9 Bus System with load flow[END_REF] shown in Fig. 3.

Simulations test

This section deals with performances and robustness of the proposed control. The tuned SHVDC parameters technique presented in the previous Section is tested and compared with the classic vector control. The latter has a cascade structure with a current inner loop more rapid than the outer one. The two controllers are tuned to satisfy the same performance specifications (the usual time setting for HVDC voltage and power control presented in Section 3.2.1). λ for each dynamic of interest. The optimal parameters K in the appendix was obtained with (39) solved for the desired locations in Table 1. The response of SHVDC for the test power system in Fig. 3 with these tuned parameters is given in Fig. 6 in solid lines in comparison with the ones in lines obtained with a classic vector control. Figs 6.a and 6.b show the responses of active and reactive powers to a +0.1 p.u step in P g_ref and to a -0.1 p.u step in Q g_ref.. A good tracking of the active power reference and satisfying control specifications for both responses is observed. It is noted that better dynamic responses are provided with the new coordinated.

Transient stability

Fig. 7.a, b and c, respectively show the responses of the angular speed of generator G1, G2 and G3, to a three phase short circuit fault of 100 ms duration, occurring near G2. Again better dynamic performances are obtained with the proposed controller: the transient oscillations with the SHVDC control are more damped.

The transient stability is assessed by the Critical Clearing Time (CCT), which is the maximum time duration that a short-circuit may act without losing the system capacity to recover to a steady-state (stable) operation [START_REF] Kundur | ): d-axis open circuit sub-transient[END_REF].

The obtained CCTs for a three-phase short-circuit occurring near each generator are presented in Table 2 for the classic vector control, and the proposed SHVDC control. We can see that the SHVDC control with tuned parameters improves the transient dynamics of the system and thus augments the transient stability margins of the neighbor network. This is due to the fact that the dynamics of the neighbour zone are taken into account at the synthesis stage via the oscillatory modes in Table 1. The gains of the controllers are computed to damp these modes and thus to diminish the general swing of the zone and not only for the local HVDC dynamics.

Dynamics of interest

0 i λ * i λ 0 i r λ Voltage V m -5

Robustness against variation of the operating point

As the synthesis of the SHVDC controller is based on the linear approximation as presented in Section III, robustness of performances against the variation of the operating point is required. Therefore, a new load flow setting is considered for the simulations. For example, the active power of load L1 is increased by 50 MW. The gains of the SHVDC controller are not recomputed and are thus the same as in the appendix. Fig. 8 gives the active power response to a +0.1 step in P g_ref in this case. The latter is comparable with the one obtained in Fig. 6.a. Indeed, Fig. 9 illustrates the response to a step of -0.5 p.u in P g_ref to test the performances of the controller to a sudden change in the direction of the transmitted power. Thus, the dynamic performance is better than the one obtained with the standard controller. These results confirm the good robustness of the performances of the proposed controller against variation of operating conditions.

Conclusion

A High Voltage Direct Current HVDC has been emulated using the synchronverter concept. A control strategy of the converters has been proposed based on a specific tuning method which uses, the sensitivity analysis of the system poles with respect to the control parameters. The control strategy has been validated on the realistic IEEE 9 bus benchmark. More specifically, better results than with the standard vector control were obtained for the following points: -Better dynamic performances (less overshoot and more damping) because this approach allows one to analytically take into account dynamic specifications at the tuning stage.

-Better stability margin of the neighbor zone for two reasons. First, swing information is directly taken into account at the synthesis stage in terms of the less damped modes of the neighbor zone and not only for the local HVDC dynamics as it is the case for the standard VSC control. It is noted that this kind of tuning is surely under optimal in the sense that the constraints to keep the structure of the classical generators and their regulations imposed by the synchronverter principle limits the performances of the .resulting closed-loop.

Fig. 4 .

 4 Fig. 4. Feedback system

Fig. 5 .

 5 Fig. 5. Single input/Single output feedback system

Fig. 6 .

 6 Fig.6. Responses of the IEEE 9 bus system (a)Response of Pg to a +0.1 step in Pg_ref (p.u) (a)Response of Qg to a -0.1 step in Qg_ref (p.u) (a)Response of Vg to a +0.1 step in Qg_ref (p.u)

Fig. 7 .

 7 Fig.7. Responses of the angular speed to a 100 ms short circuit near to G2 (a)Angular speed of G2 (p.u) (b)Angular speed of G1 (p.u) (c)Angular speed of G3 (p.u)

  .

	All control parameters are grouped in the following diagonal matrix:
	_ p Vdc K s q diag D D K K ( , ) ( , , , _ , , , gp mp i Vdc gq mq D D K K _ , _ ) p p i p g g =	(36)
	where gp D , mp D are, respectively, the static frequency droop coefficients of the SG and
	the SM; gq D , mq D are, respectively, the voltage droop coefficient of the SG and of the
	SM;	K	_ dc p V	, _ dc i V K	are the DC voltage PI control parameters; and	K	_ g p P	,	_ g i P K are the
	active power PI control parameters.
	Note that all elements of the matrix	( , ) K s q are tuned via the pole placement presented
	in Section IV.B to meet HVDC performance specifications given in Section IV.A.
	The inputs u and the outputs y are
	mm u=[T ,Q ,T _ mm mm	ref_kp	mm ,T _	ref_ki	gm , T ,Q ,T _ gm gm	ref_kp	,
	gm T _	ref_ki	T ]
										dc_ref	dc
				m		m_ref	m	dc_ref	dc	g
								g_ref	g
	g_ref		g	g_ref	g

Table 1 :

 1 Desired modes meeting the HVDC specifications

	i e	+	i u	'	( ) H s ii	y	i
			i u		( , ) K s q ii

Table 1 (

 1 column 2) presents the desired location of modes

* i

Table 2 :

 2 Critical Clearing Times with both control strategies Fig.8. Response of Pg to a +0.1 step in Pg_ref (p.u) with a new operating point Fig.9. Response of Pg to a -0.5 step in Pg_ref (p.u)


Raouia Aouini
email: aaouinii@yahoo.com

Khadija Ben Kilani
email: khadijakilani@yahoo.fr

Bogdan Marinescu
email: bogdanmarinescu@hotmail.fr

Mohamed Elleuch
email: melleuch2005@yahoo.fr

Virtual Synchronous Generators dynamic performances

Keywords: synchronverter, synchronous generator, frequency/voltage drooping. I

In this paper, dynamic performances of virtual synchronous generator are investigated. The virtual synchronous generator is based on the synchronverter (VSG) concept which is an inverter that mimics a synchronous generator (SG). The real and reactive power delivered by VSGs connected in parallel and operated as generators can be automatically shared using the well-known frequency and voltage drooping mechanisms. The VSG method is tested in comparison with the standard SG of the same parameters and capacity on a two bus test network. A comparative assessment is presented in terms of local performances and the fault critical clearing time (CCT). The results indicate good local performances and the contribution of the synchronverter to enhance the stability margin of the neighbouring AC zone.

INTRODUCTION

OR economic, technical and environmental reasons, more renewable energy sources are being connected to power systems, generally via power electronic converters. The International Energy Agency (IEA) predicts that gross renewable electricity output will continue to pick up speed in the medium term to 2018, rising by approximately 40% and forecasts installed renewable capacity increasing from 1 580 GW in 2012 to 2 350 GW in 2018 [START_REF]Worldwide electricity production from renewable energy[END_REF]. Consequently, the AC/DC/AC converted power injected into AC networks has been increased. This means that more and more inverters will be connected to the grid and will eventually dominate power generation. When focusing eyes on system stability analysis, compared with traditional transmission network; one of the most features for renewable energy sources is the randomness output active and reactive power supplied to AC grids. Thus, as inverters are used to interface with grid, inverters dynamic behavior has important influences on system performance. Since inverter's corresponding part in transmission networks, synchronous generator SG is also an energy transforming devices. Thus it's natural to mimic SG character when designing inverter controller.

Generally, inverter controller can be divided into three types. The first one is called constant-PQ controller which controls inverter output active and reactive power as given values.

The controller is based on a current-controlled voltage source converter (CC-VSC), to inject the required real and reactive powers via current control [START_REF] Bueno | Current Control of Voltage Source Converters connected to the grid through an LCL-filter[END_REF]. The second is to mimic SG external characteristic and called droop controller. When system frequency or node voltage amplitude is out of its normal range, this type of inverter will automatically adjust its output active or reactive power, and try to control these electrical qualities back into normal ranges [START_REF] Zhong | Robust droop controller for accurate proportional load sharing among inverters operated in parallel[END_REF]. The last type is to mimic SG characteristics directly and called synchronverter [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF], or virtual synchronous generator [START_REF] Gao | A control strategy for a distributed generation unit in grid-connected and autonomous modes of operation[END_REF][START_REF] Chen | Improving the grid power quality using virtual synchronous machines[END_REF]. Since the operation of AC systems via SGs voltage/frequency regulation is rather well known [START_REF] Kundur | Power system stability and control[END_REF], the synchronverter concept led to new applications. For example, in [START_REF] Nguyen | Synchronverter-based Operation of STATCOM to Mimic Synchronous Condensers[END_REF], a STATCOM controller is synthesized from the mathematical model of synchronous generators operated in a compensator mode.

In this paper, we investigate the transient stability of a two-bus power system, considering two configurations of coupling system: coupling of AC-AC system and hybrid DC-AC system. For each configuration, local performances are studied and transient stability of the systems is assessed in terms of the CCT. The VSG parameters are adapted to the conventional SG parameters. Static and dynamic responses of the VSG and the SG are compared under different operating conditions. The VSG benefits its operation frequency and voltage drooping mechanisms for load sharing. It is shown that local performances obtained by VSG are better than the ones of SG. The transient stability margin is improved with the VSG, compared to the SG. Moreover, it is not necessary to have a large inertia as for a physical SG, where a larger inertia means that more energy is stored mechanically. A small inertia is good for overall grid stability. The rest of the paper is organized as follows: in Section II, the synchronverter concept is briefly overviewed. Test power systems are presented in Section III, where different tests validations are presented in Section IV.

II. OVERVIEW OF THE SYNCHRONVERTER CONCEPT

A synchronverter VSG is an inverter of which regulations are chosen such that the resulting closed-loop mimics the behavior of a conventional SG. The VSG concept consists of a power part, as shown in Fig. 1, and an electronic part, i.e., the controller, as shown in Fig. 2. It is assumed that the DC bus of the synchronverter is constant.

F

An inverter plus an LC filter is the power part of the VSG as shown in Fig. 1. VSC inverter technologies shown in Fig. 1 are used. The overall structure is shown to be equivalent to a SG with capacitor bank connected in parallel with the stator terminal [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. The phase terminal voltage

V abc = [V a V b V c ] T is abc abc s abc s abc di V R i L e dt     , (1)
where i abc = [i a i b i c ] T is the vector of the stator phase currents, L s , R s are, respectively, the inductance and the resistance of the stator windings and e abc = [e a e b e c ] T is the back emf. The controller includes the mathematical model of a three-phase round-rotor synchronous machine described by

2 m e p H T T D      , (2)
,

e abc T M i sin    , (3)
sin

abc e Ms  , (4)
,

s abc P Ms i sin      , (5)
,cos

s abc Q Ms i      . ( 6
)
where T m is the mechanical torque applied to the rotor, T e is the electromagnetic torque, e are the three-phase generated voltage,  is the rotor angle, s=d/dt is the derivation operator, P and Q are the active and the reactive power, respectively, H is the inertia constant of all the parts rotating with the rotor, M is the field excitation,  is the angular speed of the machine and also the frequency of the control signal e abc sent to the pulse width modulation (PWM) generator, i abc is the stator current vector flowing out of the machine, The equations of sin and cos are defined as 22 sin sin sin( -) sin( + ) 33

T          , 22 cos cos cos( - ) cos( + ) 33 T          .
The operator .,.

 denotes the conventional inner product in 3 . The electronic part controls the switches in the power part. These two parts interact via the signals e abc and i abc . The VSG given in Fig. 1 is connected to the grid via the impedance (L g , R g ) such that 123 1 ( )

abc abc f V i i Cs   , (7) 1 ( ) (
)

abc gg i V V R L s    . ( 8
)
To mimic the droop of the SG and to operate the VSG in a useful way, a controller that generates T m and M is needed such that system stability is maintained and the desired values of real and reactive power are followed. Thus, the VSG operations are discussed.

A. Frequency drooping and regulation of real power

The static frequency droop control loop is proposed where _ ( )

m m ref p n T T D s     . (9)
Thus, VSG shares load with the other generators of the AC grid to which its is connected in proportion, with the static droop coefficient D p . Typical values for the frequency droop are a 100% increase in power for a frequency decrease between 3% and 5% (from nominal values). In (10), T m_ref is the mechanical torque applied to the rotor and obtained from the set point (or reference value) of the real power P ref by dividing it with the nominal mechanical speed n  , as shown in Fig. 2.

_ ref m ref n P T   . ( 10
)
The transfer function of the frequency regulation is:

1/ 1 2 1 p m e p f D T T Hs D s         . ( 11
)
The frequency loop time constant is:

2* f p H D   . ( 12
)
The choice of τ f determines the dynamic response of the loop.

It is proportional to the moment of inertia H. A large τ f is equivalent to having a large H, which makes the control less sensitive to the variations in the grid frequency and the system more stable. However, the response is slow. A small τ f is equivalent to having a small H, which leads to fast frequency tracking.

Eqn

Eqn Eqn

1 k*s _ m ref T (11) Eqn 1 s m V q D (16) Eqn abc e ref Q s Q    n  V abc

B. Voltage drooping and regulation of reactive power

The reactive power Q s is controlled by a voltage droop control loop using a voltage droop coefficient D q , in order to regulate the field as 1 ( )

m s M Q Q ks   , ( 13
) ref ref () m q m Q Q D V V    , ( 14
)
where V m is the output voltage amplitude computed by using a phase locked loop. Another elementary method is as follows: Assume that

a b b c c a m a b b c c a V V V V V V V            ² (cos( ) cos( ) cos( )) 2 m a b b c c a V             (15) ² (cos( ) cos( ) cos( )) 2 m a b b c c a V             .
When the terminal voltages are balanced, i.e, when 22 33

c a b        
, then the last term above is zero and we

obtain 2 3( m a b a c b c V V V V V V V    . ( 16
)
The time constant of the voltage loop is proportional to

* v nq k D    . ( 17
)
Hence the choice of τ v affects the dynamic response of the amplitude tracking.

III. THE TEST POWER SYSTEM

The proposed test systems consist of two finite control areas are designed as Area 1, Area 2 feeding a common area load via AC lines, as illustrated in Fig. 3. Two cases are considered:

-Case 1: Area 1 is a SG equipped with a standard power frequency and voltage reactive power controls.

-Case 2: Area 1 is a VSG emulating a SG, modeled by (1)-( 16). The AC systems are chosen to have identical AC generator, with nominal voltage, short-circuit powers and frequency and given by V = 100 kV, S cc = 200 MVA, f = 50 Hz. For the VSG parameters, they are firstly adapted to match standard SG parameters [START_REF] Kundur | Power system stability and control[END_REF], [START_REF]IEEE Recommended Practice for Excitation System Models for Power System Stability Studies[END_REF]: a 5% governor droop, an inertia constant H=3.2 s and a voltage regulation gain of 3%.

IV. SIMULATION RESULTS

The main advantage of the VSGs is to put into evidence the structure of classic SGs for which the controls are well known. The VSG operations are verified in a test power shown in Fig. 3.b. Dynamic and transient performances of VSG are compared with SG of same capacity for different faults.

A. Response of VSG vs SG A.1 Load share of the VSG

An important mechanism for VSGs and SGs for evenly sharing load (in proportion to their nominal load) is to vary the real power delivered to the grid according to the grid frequency. The VSG and the SG coupled via an AC lines as shown in Fig. 3.b are set in a parallel arrangement and are initially at nominal frequency f 0 =1.007 p.u with power outputs, respectively, P s =0.36 p.u and P m =0.12 p.u given in Table I. When an active load increase 0.13 . L P p u  , at t=10 s, causes the units to slow down, the governors increase output until they reach a new common operating frequency f'. The system responses are shown in Fig. 4 and5. The frequency of the SG followed the VSG frequency, and the real and reactive power tracked their set points. The amount of load picked up by each unit depends on the droop characteristic such that 0 '

L sm P f f f DD       . ( 18
)
The VSG and the SG increased their real power output by 0.064 p.u that is 6.5% of the power rating '* s s s s

P P P f D      , ( 19
) '* m m m m P P P f D      . ( 20
)
amount of load increase, correspond to a 0.32 % drop of the frequency. The load share of the VSG and the SG are summarized in Table . I. Hence,

ss mm PD PD   
, where D s and D m are , respectively, the static droop of VSG and SG, which are equal to 1/5% . When a sudden change in load occurred, at t= 10 s, as shown in Fig. 5, the reactive power dropped by about 0.042 p.u because the local terminal voltage V abc was about 0.1 % higher than the nominal value, which depends on the droop characteristic 1 1.0012 0.04 . 0.03

q V Q p u D       . ( 21

A.2 Local performances of VSG and equivalent SG

To test local performances, a +0.15 p.u step is applied to the reference of the active power and a +0.1 p.u step is applied to the reference of the voltage of the VSG at t=10 s. Fig. 6 shows the responses of active powers and speeds responses of the VSG and the SG. It can be noted that the VSG speed attaint the steady state faster than the SG speed. Fig. 7 shows the responses of reactive powers and voltages responses of the VSG and the SG. It can be observed that better dynamic responses with the VSG concept are obtained, e.g. more damped dynamics oscillations and fast responses are attained with VSG. We note that the overshoot in the voltage response is of limited magnitude.

A.3 Transient stability

The robustness of the VSG is evaluated by assessing the transient stability of the whole test power system, following severe fault. The latter is quantified by the Critical Clearing Time (CCT), which is the maximum time duration that a short-circuit may act without losing the system capacity to recover to a steady-state stable operation. To evaluate the CCT, a three phase short-circuit fault was simulated near the load. According to this study, the fault critical clearing time for case 1 (AC-AC) estimated at CCT AC-AC = 230 ms is less than for a mixed system case 2 (DC-AC) with a CCT DC -AC = 350 ms. As a result, the VSG concept improved also the transient stability of the neighboring AC system. Furthermore, a severe short-circuit of 200 ms duration is applied. Fig. 8 shows the responses of active power and speed in case 1 and 2. It can be observed that the transient oscillations obtained with the VSG are more damped than the ones obtained with the SG. The active power response of SG is more oscillatory than the VSG; oscillations are attributed to electromechanical dynamics of the rotor to reach steady state, for the case of VSG, power control is electronic.

B.1 Effect of inertia values

Fig. 9 shows the responses of the VSG to a +0.13 p.u load change and with different inertia constant. We note that local performances with a small inertia are better.

H=3.2s H=1s

In addition to the previous simulations and in order to better investigate the transient stability, the critical clearing times are obtained with different inertia constant. The results are resumed in Table II and confirm the improvement of transient stability with a small inertia. As a result, it is not necessary to have a large inertia as for a physical SG, where a larger inertia means that more energy is stored mechanically. We can observe that a small inertia is good for overall grid stability. 10 shows that local performances of VSG with the same parameters of SG are improved with PID governor.

V. CONCLUSION

This paper presented a dynamic performances study of the virtual synchronous generator. The latter is based on the synchronverter control strategy. The study is based on a comparative analysis of a SG/VSG two-area system, feeding a common load. For each configuration, dynamic performances to different system fault are studied and the transient stability of the system is assessed in terms of the fault critical clearing time. Due to the controllable features, the virtual synchronous generator of the same capacity and parameters with a standard SG can be used to improve the dynamic system oscillations, the response time and the transient stability of interconnected AC systems in terms of increasing the fault critical clearing time.

Fig. 1 .

 1 Fig. 1. Power part of a VSG

Fig. 3 .

 3 Fig. 3.a. Coupling of AC-AC system Area 1: SG

Fig. 4 . 2 Fig. 5 .

 425 Fig. 4. Responses of the SG in case 2

Fig. 8 .Fig. 9 .

 89 Fig. 8. Comparison of case 1 & 2 for a 200 ms short-circuit

Table II :

 II CCTs with different constant inertiaB.2 Frequency regulation by PID governorStandard parameters of SG presented above are applied to the VSG. For performances improvement, the static droop control loop is replaced by the PID governor. Results presented in Fig.

.01, Xq (p.u): q-axis synchronous reactance = 0.053, Xq (p.u): qaxis synchronous reactance = 0.474, X''q (p.u): q-axis sub transient reactance =0.243, T''q0 (s): q-axis open circuit sub transient time constant=0.1, Coupling Delta/ Yg Primary resistance (p.u) =0.002, Primary inductance (p.u) =0.12 Secondary resistance (p.u) =0.002, Secondary inductance (p.u) =0.12 M =2H (kWs/kVA): Mechanical starting time = 6.4. Governor control system Dm(%): permanent droop =5, servo-motor: ka = 10/3, ta (s) = 0.07, regulation PID: kp = 1.163, ki= 0.105, kd= 0 Excitation control system Amplifier gain: ka = 200, amplifier time constant: Ta (s) = 0.001, damping filter gain kf = 0.001, time constant te (s) = 0.1 Generator transformers Rated 400 MVA, 20/ 100 kV Load: PL=200 MW AC transmission lines: Resistance per phase (Ω/km) =0.03, Inductance per phase (mH/km) =0.32 , Capacitance per phase (nF/km) =11.5 Pi line length= 100 km AC filter: reactive power=40 MVAR DC system: voltage= ±100 kV, rated DC power=200 MW


Raouia Aouini
email: aaouinii@yahoo.com

Bogdan Marinescu
email: bogdanmarinescu@yahoo.com

Khadija Ben Kilani
email: khadijakilani@yahoo.fr

Mohamed Elleuch
email: melleuch2005@yahoo.fr

Improvement of transient stability in an AC/DC system with synchronverter based HVDC

Keywords: synchronverter, Synchronous generator/motor, SHVDC, transient stability, damping oscillatory modes

The HVDC emulation by the synchronverter concept is investigated in a realistic power system. A specific tuning method for the parameters of the regulators based on the sensitivity of the poles of the neighbor zone of the HVDC with the respect to the latter parameters is used. As consequence, not only the local performances of the HVDC link, but also overall transient stability of the AC zone in which the HVDC is inserted are improved. Extensive tests are provided using Matlab/Simulink implementation of the IEEE 9 bus/3 machines test system.

I. INTRODUCTION

Due to environmental, technical and economical reasons, the installation of HVDC lines is favored in order to maximize the electric power transmission efficiency. As a result, the past five decades witnessed significant development in HVDC transmission systems [START_REF] Velasco | Power transmission in direct current: Future expectations for Colombia. Renewable and Sustainable Energy Reviews[END_REF]. Recently, rapid advancement was achieved in the field of power electronic devices such as Insulated Gate Bipolar Transistor (IGBT). That opens opportunities for the power industry via the utilization of HVDC based on Voltage Sourced Converters (VSC) with IGBT technology which is commercially known as HVDC Light TM by ABB [START_REF] Hingorani | High-voltage DC transmission: a power electronics workhorse[END_REF].

VSC-HVDC have enabled the interconnection of asynchronous networks, the connection of offshore wind farms, and the control of power flow especially for long distances such as the NorNed interconnection: the longest submarine DC cable between Norway and Netherlands. Nonetheless, other equally important HVDC applications concern complex AC interconnected system in order to enhance the power transmission capacity and meet the growing demand. These emerging applications gave rise to the co-existence of parallel HVDC/HVAC, and consequently, an increased level of AC/DC/AC converted power injected into AC networks.

Lately, the problem of VSC-HVDC transmission system's stabilization has attracted renewed attention in both power systems and control communities. Therefore several studies have shown that the methods of controlling HVDC converters have an impact on stability of the system in which the link is inserted [START_REF] Hammad | Improving the dynamic performance of a complex AC/DC system by HVDC control modifications[END_REF]. Most control of a VSC-based HVDC system uses a nested-loop d-q vector control approach based on the linear PI technology. The controller design is processed in the same manner for different operating conditions. In case of parameter uncertainties; changed operating conditions and faults; undesirable poorly damped or even unstable oscillations may result. More clearly, controller parameters yielding satisfactory damping and enhancing system's dynamic behavior for one operating condition may no longer provide sufficient damping for others [START_REF] Ruihua | VSCs based HVDC and its control strategy[END_REF]. Although, conventional controls are lack robustness and become difficult to implement especially for cases in which: different controllers are to be coordinated for a variety of operating conditions, and certain performance specifications have to be satisfied. Consequently, to overcome the undesired problems probably encountered by PI conventionally tuned controllers under operating conditions variations and to guarantee system stability, different robust controllers design [START_REF] Latorre | Improvement of power system stability by using a VSC-HVDC[END_REF] and adaptive control structures have been proposed [START_REF] Rostamkolai | An adaptative optimal control strategy for dynamic stability enhacement of AC/DC power systems[END_REF].

Recently, in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]- [START_REF] Nguyen | Synchronverter-based Operation of STATCOM to Mimic Synchronous Condensers[END_REF], the authors have proposed a different control method for which an inverter can be operated to mimic the behavior of a synchronous generator (SG) and the resulting closed-loop has been called a synchronverter [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF]. Since the operation of AC systems via SGs voltage/frequency regulation is rather well known [START_REF] Kundur | Power system stability and control[END_REF], the synchronverter concept led to new applications. For example, in [START_REF] Nguyen | Synchronverter-based Operation of STATCOM to Mimic Synchronous Condensers[END_REF], a STATCOM controller was synthesized from the mathematical model of synchronous generators operated in a compensator mode. In [START_REF] Aouini | Synchronverterbased Emulation and Control of HVDC transmission[END_REF], the synchronverter concept was adapted to converters of an HVDC transmission. The idea is a conceptual control strategy of the DC line, where the sending-end rectifier controls emulate a synchronous motor (SM), and the receiving end inverter emulates a SG, both along with their controls. This resulting Synchronverter based HVDC was called SHVDC. In this paper, the SHVDC system is tested in the IEEE 9 bus system. In the tuning stage, this control technique allows one to take into account dynamic specifications and swing information of the neighbour AC zone in term to better dynamic performances and to improve the transient stability of the neighbor AC zone of the HVDC link; and this fact is proven on a realistic power system. The rest of the paper is organized as follows: in Section II, the SHVDC structure is recalled. An analytic method to tune the parameters of the controllers of the SHVDC in order to meet the desired performances is given in Section III, while validation tests are presented in Section V. . Fig. 1. Model of the SHVDC power and electronic parts [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] II. SYNCHRONVERTER BASED HVDC In this section, the converters of the HVDC line are designed to emulate synchronous machine. The synchronverter proposed in [START_REF] Zhong | Synchronverters: Inverters that mimic synchronous generators[END_REF] is an inverter of which regulations are chosen such that the resulting closed-loop mimics the behavior of a conventional synchronverter generator (SG). For a complete DC system, a rectifier and an inverter are connected by a DC link as shown in Fig. 1. Therefore, to provide an HVDC structure, another synchronverter working as a synchronous motor (SM) based on the same mathematic derivation is necessary. As a result, the DC power is sent from the SM to the SG. The resulting system in Fig. 1 is called Synchronverter High-Voltage Direct Current (SHVDC). On the figure, we depict: (i) the power part of the SG/SM consists of the inverter/rectifier plus LC filter; (ii) the controls are assured by the electronic part. VSC converter technologies shown in Fig. 1 are used. The overall structure is shown to be equivalent to a SG/SM with capacitor bank connected in parallel with the stator The phase terminal voltage of the SG and SM are

V g_abc = [V ga V gb V gc ] T , V m_abc = [V ma V gb V gc ] T respectively _ _ _ _ g abc g abc s g abc s g abc di V R i L e dt     , (1)
_ _ _ _ m abc m abc s m abc s m abc di V R i L e dt    . ( 2
)
where i g _ abc = [i ga i gb i gc ] T , i g _ abc = [i ga i gb i gc ] T , are respectively the stator phase currents of the SG and the SM, L s , R s are, respectively, the inductance and the resistance of the stator windings and e g_abc = [e ga e gb e gc ] T , e m_abc = [e ma e mb e mc ] T are respectively the back emf of the SG and the SM.

As shown in Fig . 1, controllers include the mathematical model of a three-phase round-rotor synchronous machine described by 1 ()

g gm ge gp g g T T D s J     , (3) 1 (
)

m me mm mp m m T T D J      , (4) _
, ge g g abc g T M i sin    , (5)
_ ,

me m m abc m T M i sin    , (6)
_ , g g g g abc g P M s i sin      , (8)
_ ,cos

ge g g g abc g Q M s i       , (10)
_ ,cos

me m m m abc m Q M s i      , (11)
_ ,

m m m m abc m P M s i sin      . ( 12
)
where T gm and T mm are, respectively, the mechanical torque applied to the rotor of the SG and the SM. T ge and T me are, respectively, the electromagnetic torque applied to the rotor of the SG and the SM. θ is the rotor angle, J g and J m are the combined moment of inertia of generator and turbine. s=d/dt is the derivation operator. P g (respectively P m ) and Q g (respectively Q m ) are the active and the reactive power, respectively, of the SG (respectively of the SM). M g and M m are, respectively, the field excitation of the SG and the SM. The equations of sin and cos are

_ m ref V m  , s s RL , g g RL Bus1 _123 m V , m me P Q PWM 1 dc V 1 dc i cc i , s s RL , g g RL Bus2 2 dc i 2 dc V dc C PWM _ g ref P _123 g V _ m abc e _ g abc e _ m abc V _ g abc V , g ge P Q g  _ g ref V n  g P g M  ge T  g  1 J *s g gp D (3) (5) (7) Eqn Eqn Eqn 1 ks g _ gm ref T (19) Eqn 1 s _ e g abc g V (27) Eqn ge Q   _ i g abc gq D _ g ref Q _123 g V  _ i g abc m  n  dc V m M  me T  1 J *s m mp D (4) (6) (8) Eqn Eqn Eqn 1 ks m _ mm ref T (20) Eqn 1 s _ e m abc m V (28) Eqn me Q   _ i m abc mq D _ m ref Q _123 m V  _ dc ref V _ i m abc 22 sin sin sin( - ) sin( + ) 33 T          , ( 13
) 22 cos cos cos( - ) cos( + ) 33 T          . ( 14
)
The operator .,.

 denotes the conventional inner product in 3 . The SHVDC given in Fig. 1 is connected to the grid via the impedance (L g , R g ) such that

_ _ _123 1 ( ) g abc g abc g f V i i Cs   , ( 15
) _123 _ _123 1 ( ) ( ) g g abc g g g i V V R L s    , ( 16
) _ _123 _ 1 () m abc m m abc f V i i Cs  , ( 17
) _123 _123 _ 1 ( ) (
)

m m m abc gg i V V R L s    . ( 18
)
To mimic the droop of the SG, the following frequency droop control loop is proposed _ ( )

gm gm ref gp n g T T D s     , (19)
_ ( )

mm mm ref mp n m T T D s     . ( 20
)
T gm_ref is the mechanical torque applied to the rotor of the SG and it is generated by a PI controller as shown in Fig. 1 to regulate the real power output g P . In the SM case, T mm_ref is generated by a DC voltage control for power balance.

_ _ _ 1 _ ( )( ) i vdc mm ref p vdc dc ref dc k T k V V s    , (21)
_ _ _ _ ( )( ) g g ip gm ref p p g g ref k T k P P s    . ( 22
)
The reactive power Q gm (respectively Q mm ) is controlled by a voltage droop control loop using a voltage droop coefficient D gq (respectively D mq ), in order to regulate the field excitation M g (respectively M m ) , which is proportional to the voltage generated.

1 ()

g gm ge g M Q Q ks  , ( 23
) g_ref _ ref ( ) gm gq g g Q Q D V V    , (24) 1 (
)

m mm me m M Q Q ks   , ( 25
) m_ref _ ( ) mm mq m ref m Q Q D V V    . ( 26
)
where V g (respectively V m ) , is the output voltage amplitude is computed by: 2 3(

g ga gb ga gc gb gc V V V V V V V   , ( 27
) 2 3( m ma mb ma mc mb mc V V V V V V V   . ( 28
)
The circuit equations of the DC line (Fig. 1) are 11 1 ()

dc dc cc dc V i i Cs  , (29) 22 1 (
)

dc cc dc dc V i i Cs  , (30) 1 2 1 (
)

cc dc dc dc cc dc i V V R i Ls    . ( 31
)
The AC and DC circuits are coupled by the active power relation

1 1 1 dc dc dc P V i  , ( 32
) 2 2 2 dc dc dc P V i  , ( 32
) 1 dc m PP  , ( 34
) 2 dc g PP  . ( 35
)
III. TUNING OF THE SHVDC PARAMETERS The SHVDC parameters should be tuned simultaneously and in a coordinated way such that the global stability performances of HVDC are ensured i.e: -satisfy HVDC local dynamics. The SHVDC parameters are calculated to reach desired modes starting from HVDC specifications control. The connections between the dynamics of interest and the modes are based on the participations factors [16].

-enhance the transient stability of the neighboring AC system. This is done such that the dynamics of the neighbor zone are taken into account at the synthesis stage via the oscillatory modes. The gains of the SHVDC controllers are computed to damp these modes and thus to reduce the swing of the zone. For this tuning, the SHVDC shown in is put into the feedback system structure presented in Fig. 2 where H(s) is the linear approximation of the system modeled by equations ( 1) to (34) with the exception of equations ( 19), ( 20), ( 21), ( 22), ( 24) and (26). All control parameters are grouped in the following diagonal matrix and tuned based on the following result. Proposition [16]: The sensitivity of a pole  of the closed- loop in Fig. 2 with respect to a parameter q of the regulator

ii K is ( , ) ii K s q r q q        (37)
where r  is the residue of ()

ii Hs at pole  .

Note that, for our case. (36), ( , )

1 ii s K s q q      .
The contribution of each control gain in the shift of the pole is

0 i i ij j j rK       , ( 38
j i i K i Kj      , ( 39
)
where i  is given by (38) [15].

IV. IEEE 9 BUS TEST SYSTEM The IEEE 9-bus system shown in Fig. 3 contains 3 generators, 5 lines, 3 loads and 3 two winding power transformers. The 100 km HVDC cable link has a rated power of 200 MW and a DC voltage rating of ±100 kV. The rating of each generator is 600 MVA and 20 kV. Each of the units is connected through transformers to the 100 kV transmission line. The detailed system data is given in the appendix. The loads L 1 , L 2 and L 3 are modeled as constant impedances.

V. SIMULATIONS TEST This section deals with performances and robustness of the proposed control. Simulation tests are performed using MATLAB SIMULINK software. The tuned SHVDC parameters technique presented in Section III is tested and compared with the classic vector control which has a cascade structure with a current inner loop more rapid than the outer one. The two controllers are tuned to satisfy the same performance specifications.

A. Local performances

Table I (column 3) presents the desired location of modes * i  for each dynamic of interest. The optimal parameters K in the appendix were obtained with (39) solved for the desired locations in Table I. The response of SHVDC for the link in Fig. 3 with these tuned parameters is given in Fig. 4 in solid lines in comparison with the ones in dotted lines obtained with a classic vector control. Figs 4.a and 4.b show the responses of active and reactive powers to a +0.1 p.u step in P g_ref and to a -0.1 p.u step in Q g_ref.. A good tracking of the active power reference and satisfying control specifications for both responses is observed. It is noted that better dynamic responses are provided with the coordinated. Indeed, Fig. 5 illustrates the response to a step of -0.5 p.u in P g_ref to test the performances of the controller to a sudden change in the direction of the transmitted power.

B. Transient stability

Fig. 6.a, b and c, respectively show the responses of the angular speed of generator G1, G2 and G3, to a three phase short circuit fault of 100 ms duration, occurring near G2. Again better dynamic performances are obtained with the proposed controller: the transient oscillations with the SHVDC control are more damped. The transient stability is assessed by the Critical Clearing Time (CCT), which is the maximum time duration that a short-circuit may act without losing the system capacity to recover to a steady-state (stable) operation. The obtained CCTs for a three-phase short-circuit occurring near each generator are presented in Table II for the classic vector control, and the proposed SHVDC control. We can see that the SHVDC control with tuned parameters improves the transient dynamics of the system and thus augments the transient stability margins of the neighbor network. This is due to the fact that the dynamics of the neighbour zone are taken into account at the synthesis stage via the oscillatory modes in Table I. The gains of the controllers are computed to damp these modes and thus to diminish the general swing of the zone and not only for the local HVDC dynamics.

C. Robustness against variation of the operating point

As the synthesis of the SHVDC controller is based on the linear approximation as presented in Section III, robustness of performances against the variation of the operating point is required. Therefore, a new load flow setting is considered for the simulations. For example, the active power of load L1 is increased by 50 MW. The gains of the SHVDC controller are not recomputed and are thus the same as in the appendix. Fig. 7 gives the active power response to a +0.1 step in P g_ref in this case. The latter is comparable with the one obtained in Fig. 4.a. In addition, the dynamic performance is better than the one obtained with the standard controller. This confirms the good robustness of the performances of the proposed controller against variation of operating conditions. A High Voltage Direct Current HVDC has been emulated using the synchronverter concept. A control strategy of the converters has been proposed based on a specific tuning method which uses, the sensitivity analysis of the system poles with respect to the control parameters. The control strategy has been validated on a realistic IEEE benchmark. More specifically, better results than with the standard vector control were obtained for the following points: -Better dynamic performances (less overshoot and more damping) because this approach allows one to analytically take into account dynamic specifications at the tuning stage.

-Better stability margin of the neighbor zone for two reasons.

First, swing information is directly taken into account at the synthesis stage in terms of the less damped modes of the neighbor zone and not only for the local HVDC dynamics as it is the case for the standard VSC control.



  is the initial (open-loop) location of the pole i

Fig. 3 .

 3 Fig. 3. Test power system

Fig. 4 .Fig. 5 .Fig. 6 .

 456 Fig.5. Response of Pg to a -0.5 step in Pg_ref (p.u)

Fig. 7 .

 7 Fig.7. Response of Pg to a +0.1 step in Pg_ref (p.u) with a new operating point VI. CONCLUSION


Raouia Aouini
email: aaouinii@yahoo.com

Khadija Ben Kilani
email: khadija.kilani@ept.rnu.tn

Bogdan Marinescu
email: bogdanmarinescu@hotmail.fr

Mohamed Elleuch
email: melleuch2005@yahoo.fr

IMPROVEMENT OF FAULT CRITICAL TIME BY HVDC TRANSMISSION

Keywords: HVDC transmission, transient stability, fault critical clearing time

This paper investigates the impact of High Voltage Direct Current (HVDC) transmission on the transient stability of a two-machine power system, considering three transmission line configurations: parallel HVAC-HVAC, parallel HVDC-HVDC, and a hybrid HVAC-HVDC operation. The faults are balanced three-phase shortcircuits in AC lines, and single phase faults on DC lines, applied in the mid-point of the interconnection. For each configuration, transient stability of the AC systems is assessed in terms of the fault critical clearing time (CCT), and for different DC power levels. The results indicate the contribution of HVDC transmission in increasing the critical clearing time; and therefore enhancing the systems stability margin and operational security.

INTRODUCTION

Fault critical clearing time (CCT) is commonly used as the conventional transient stability measure of power system robustness to withstand a given disturbance [START_REF] Kundur | Power system stability and control[END_REF]. This parameter corresponds to the maximum time duration that a disturbance may last without losing the system capacity to recover to a steady-state stable operation. The CCT depends on both the initial operating state of the system, the location ant the severity of the disturbance.

Transient instability in a power system is exhibited in the form of swings in machine angle and grid power. In an interconnected network, power swings in one area may affect the connected areas and cause the instability of the entire network. The instability of frequency or voltage may cause loss of load in an area, transmission line tripping, equipment damage and degradation of power system performance [START_REF] Bevrani | Robust Power System Frequency Control[END_REF].

Several control schemes may be used to prevent this event, for example using power electronic equipment devices, in the form of Flexible Alternative Current Transmission Systems (FACTS) devices, or employing HVDC systems. In a purely AC system these control devices consist of voltage controls and turbine governor controls. Transmission in Direct Current (DC) offers attractive features such as fast controllability of power through converter control, bulk power transmission, ability to enhance transient stability problems associated with HVAC lines, asynchronous interconnections, environmental and economical advantages [START_REF] Meah | Comparative Evaluation of HVDC and HVAC Transmission Systems[END_REF].

Controllability in a DC link is assured by hierarchically organized controls: the master control, the pole control and the converter control [START_REF] Kundur | Power system stability and control[END_REF]. The pole control system is responsible for the firing of the thyristor. On the rectifier side, a proportional integrator (PI) regulator maintains the DC current, whereas on the inverter side, a (PI) regulator controls DC voltage so that minimum losses are archived.

Several studies have emphasized the performance improvements of power systems brought on by HVDC controllability functions [START_REF] Wang; Redfern | A ''the advantages and disadvantages of using HVDC to interconnect AC networks[END_REF]. However, in earlier HVDC works, voltage and angle rotor stability have not thoroughly been addressed. Only in recent years, have papers [5 -9] addressed the issue of voltage stability in AC/DC interconnection. It has been recognized that weak AC/DC interconnection points can limit the power transfer capability of DC links during dynamic operation on account of the transient voltage stability phenomenon [START_REF] Pilotto | Transient AC Voltage Related Phenomena for HVDC Schemes Connected to Weak AC Syslems[END_REF].

Transient stability has been investigated for a parallel AC and hybrid AC-DC configurations in [START_REF] Vovos | Transient stability of ac-dc system[END_REF]. Both configurations were subjected to a fault near inverter bus. It has been shown that the presence of a DC link suitably controlled improves the transient stability of the system.

HVDC schemes in parallel operation with AC transmission are prone to both angle and voltage instabilities even for relatively "strong" AC systems [START_REF] Hammad | Stability and Control of HVDC and AC Transmission in Parallel[END_REF]. In [START_REF] Hammad | Stability and Control of HVDC and AC Transmission in Parallel[END_REF] it has been shown that a HVDC with constant DC power control does not contribute to system synchronizing torque even with supplementary damping signal.

Fundamentally, transient stability is assessed based on synchronizing and damping torques. Lack of sufficient synchronizing torque results in an aperiodic instability, and lack of damping torque results in an oscillatory instability [START_REF] Kundur | Power system stability and control[END_REF].

An important measure for system security is the maximum time interval by which a fault must be cleared in order for the system to preserve its stability this maximum duration is the fault critical clearing time (CCT)

In this paper, we investigate the transient stability of a two-machine power system, considering three transmission line configurations: parallel HVAC, parallel HVDC, and a hybrid HVAC-HVDC. The faults are applied in the mid-point of interconnections. For each configuration, transient stability of the systems is assessed in terms of the CCT. The results indicate the importance of installing HVDC to grid power system not only on increasing the stability margin and the tie-line power control but also increasing the power system critical clearing time.

SYSTEM MODELING

HVDC system

HVDC transmission systems transport very large amounts of electric power which can only be accomplished under tightly controlled conditions. Direct current and voltage are precisely controlled to affect the desired power transfer P d . It is necessary therefore to continuously and precisely measure system quantities which include at each converter bridge, the DC current I d , its DC side voltage, the delay angle  , and for an inverter, its extinction angle  . The principles of HVDC control can be well described by the two-terminal monopolar HVDC link schematized in Figure 1, where the rectifier and the inverter are 12-pulse converters using two 6-pulse thyristor bridges connected in series. The corresponding equivalent circuit is shown in Figure 2. The strength of the two AC systems connected by HVDC transmission has a significant impact on the AC/DC system interactions. This impact could be measured by the short-circuit ratio (K cc ) when AC filter ratings are excluded [START_REF] Kundur | Power system stability and control[END_REF]:

rating (MW) converter DC system AC of (MVA) it shortcircu K cc  (1)
Based on this parameter, the AC system is considered strong or weak. If the K cc > 3.0 the AC system is considered strong and control systems stability and robustness is not a problem. If K cc < 3.0, the AC system is considered weak. For such a system, converter controls are difficult to adjust and appropriate tuning is needed [START_REF] Lefebvre | Adaptive control for HVDC power transmission systems[END_REF]. A minimum K cc level of 2.5 has often been used as a lower limit for acceptable DC operation [START_REF] Meah | A self-coordinating adaptive control scheme for HVDC transmission systems[END_REF]. The basic converter equations, for both rectifier and inverter operations, describing the relationship between the AC and DC variables can be written as follows [START_REF] Kundur | Power system stability and control[END_REF]:

0 2( cos ) dr dr cr d V V R I    (2) 3 cr cr X R   (3) 0 2( cos ) di di ci d V V R I    (4)
0 V dr  cos 0 V di  Figure 2. Equivalent HVDC transmission system 3 ci ci X R   ( 5
)
dr di L d V V R I   (6) ac d d d P P V I   (7) ac d Q P tg  (8) 0 0 cos cos - 2 dr cr d r dr dr V R I V V     (9)
where V d0r and V d0i represent the converter transformer noload DC voltage of the rectifier and inverter respectively. The equivalent commutation resistances of the rectifier and inverter are respectively R cr and R ci . The commutation reactance of the converter transformer at the rectifier and inverter are respectively X cr and X ci . The DC line resistance is denotes by R L . The firing angle for rectifier is  , the extinction angle for inverter is  ; the current of DC line is I d , the DC terminal voltage of the rectifier and the inverter ends are defined by V dr and V di respectively. Although several control strategies are suggested for DC link operation, most DC transmission systems use the control concept of constant extinction angle at the inverter with constant current control at the rectifier end. The shift logic of these controllers is implemented by the current in DC line I d which could be derived from the equivalent circuit:

0 0 cos cos dr di d cr ci L V V I R R R       (10)
Figure 3 illustrates the basic control scheme of an HVDC link. An HVDC system can be divided into several levels. The master control layer determines the reference current I d_ref , which indeed decides the active power to be transmitted. The actual reference current I d_ref_lim used by the controllers is limited by the Voltage Dependent Current Order Limiter (VDCOL), which is implemented to help the fault recovery. This control automatically reduces the reference current I d_ref set point when V d_mes decreases (as, for example, during a DC line fault or a severe AC fault).

R L R ci R cr V di V dr I d
AC System Filtrer AC System Filtrer Three-phase AC Three-phase AC

DC Line

Reducing the I d_ref currents also reduces the reactive power demand on the AC system, helping to recover from the fault. The difference between both settings is the current margin I margin and its value is normally fixed in the range of 10%-15% of the system rated current.

The current error I margin provides a transition between the current control and voltage control to facilitate control stabilization. The Pole control is the core of HVDC control and activates the appropriate controller of the rectifier and inverter station according to the state of AC/DC systems. Then it produces the firing angle ord  for both rectifier and inverter stations. The constant current mode at the rectifier is achieved using a typical PI regulator, which acts according to the error obtained from the measured and the desired current to keep the direct current constant. Pole control of the inverter station includes a constant current (CC) controller and a PI a constant extinction angle (CEA) controllers in order to maintain a constant voltage at the inverter end. Finally, the bridge or converter unit control determines the firing instants of the valves within a bridge. This has the fastest response within the control hierarchy.

AC and DC transmission line model

The DC transmission line is represented using the distributed parameter line model with lumped losses. For an AC transmission line, the resistance, inductance, and capacitance are uniformly distributed along the line. Unlike the distributed parameters line block, which has an infinite number of states, the  sections linear model has a finite number of states that permits to compute a linear state-space model.

TRANSIENT STABILITY CRITERIA

Transient stability is the ability of the power system to maintain synchronism when subjected to a severe transient disturbance such as a short-circuit on a transmission line. The resulting system response involves large excursions of generator rotor angles and is influenced by the nonlinear power-angle relationship [START_REF] Kundur | Power system stability and control[END_REF]. The conventional transient stability measure of power system robustness to withstand a given disturbance (e.g., its stability degree) is named Critical Clearing Time (CCT), which is the maximum time duration that the disturbance may act without losing its capacity to recover to a steady-state (stable) operation. The CCT depends on both the initial operating state of the system, the location and the severity of the disturbance [START_REF] Kundur | Power system stability and control[END_REF].

TEST POWER SYSTEM

The proposed test power systems consist of two finite control areas interconnected via different HVDC and HVAC parallel line configurations. Each control area is represented by a turbine generator set equipped with power frequency and voltage reactive power control systems, an area load (P load ), and a system interconnection bus. An overview of the test power systems is shown in Figure 1.

Figure 3. Basic HVDC control scheme

For each AC system, the synchronous generator is represented by a sixth order dynamic model; the excitation system is represented by a third order dynamic model, the model parameters are given in the Appendix. The AC systems have identical nominal voltages, frequency, and short-circuit powers (U = 500 kV, f = 50 Hz; S cc = 5000 MVA). They are considered as strong systems since the corresponding short-circuit ratio (K cc ) is equal to 5). The model presented in Figure 4 includes two types of buses according to Table 1: Table 1: Model load flow data

The DC system is a bipolar, 12 pulse, rated 1000 MW (1000 A, 500 kV). The rectifier (sending end) and the inverter (receiving end) are separated by a 100 km transmission line. The HVDC transmission line has the configuration with two ends (point to point). The converters stations use thyristor valves. The simulations are performed with the Toolbox SimPower Systems of MATLAB. The no-load DC voltage is computed by: 0 3 2 3 2 0.96 * 200 288.1 0.9

dr c V V kV      (11)
The equivalent commutation resistance of the rectifier is computed from equation ( 3 Where V c is the line-to-line RMS commutating voltage which depends on the AC system voltage and the transformer ratio. The terminal DC voltage V dr at the rectifier end is computed from its expression in (2): 0 2( cos ) 514.7

dr dr cr d V V R I kV     (13)
Referring to equations ( 8) and ( 9), the power factor and the reactive power consumption are about:

0 0 cos cos 0.893 2 dr cr d r dr dr V R I V V       (14) 503 ac d Q P tg MVAR    (15)

SIMULATION RESULTS

The time domain simulation method is chosen to assess the transient stability of a power system because it is the most accurate method compared to direct methods. The differential equations to be solved are nonlinear ordinary equations with known initial values. In this work, the trapezoidal technique is used considering the fact that it is widely used for solving electro-mechanical differential algebraic equations. When a three-phase fault occurs at any line in the system, a breaker will operate and the respective line will be disconnected at the Fault Clearing Time (FCT) which is set by the user. If the relative rotor angles with respect to the slack generator remain stable after the fault is cleared, it implies that FCT < CCT and the power system is stable. However, if the relative angles go out of step after the fault is cleared, FCT > CCT and the system is unstable.

Methodologically, the type of contingencies considered are three-phase balanced faults created at the mid-point if the AC tie-lines and single phase faults applied on the DC tie-lines. For comparison, the study considers three different tie-line configurations:

Comparative Analysis

The simulation tests were carried out to compare system dynamic responses of the above three configurations for four scenarios are tested:

Case 1: Configuration 1-Parallel HVAC-HVAC

A three phase short-circuit fault occurs at the mid-point of the AC transmission line. A critical time noted CCT AC//AC is determined (Table 2). The simulation results depicted in Figure 5 shows the speed machine 2 for the stable and the unstable cases.

Case 2a: Configuration 2a -Hybrid HVAC-HVDC

A three phase short-circuit fault occurs at the middle of AC transmission line. A critical time noted CCT AC//DC-a is determined (Table 2). Figure 6 shows the angular speed of machine 2 for the last stable simulation (FCT<CCT AC//DC-a ) and for the unstable case (FCT > CCT AC//DC-a ).

Case 2b: Configuration 2b-Hybrid HVAC-HVDC

A single phase short-circuit fault occurs at the mid-point of the DC transmission line. The critical clearing time noted CCT AC//DC-b is determined (Table 2) and the results are shown in Figure 7.

Case 3: Configuration 3-Parallel HVDC-HVDC

A single phase short-circuit fault occurs at the mid-point of the DC transmission line. The critical clearing time noted CCT DC//DC is determined (Table 2, Figure 8).

In all the above cases, the CCT is obtained by increasing the fault duration gradually until the system loses its stability. This result further confirms the improvement of system transient stability by adding the DC transmission.

Influence of AC-DC interconnection ratio

In the following, the influence of the relative capacity of the HVDC and the parallel AC transmission is investigated. The ratio between the active power transmitted and the continuous power to HVDC line is defined by a ratio R:

ac d P R P  (16)
where P ac is the active power transmitted by the AC line (MW) and P d is the continuous power line DC (MW).

According to [START_REF] Hammad | Stability and Control of HVDC and AC Transmission in Parallel[END_REF], the strength of an AC line is classified as high or low depending on the value of R:  R ≥ 1.5: the AC line is strong  R ≤ 0.7: the AC line is weak

The reference power of the DC line noted P d_ref DC is varied (P d_ref_DC = 800 MW, 1000 MW, and 1300 MW) keeping constant the total transmitted power: Two different cases were considered: the case when the capacity of the parallel AC transmission P ac is much less than the DC power P d with R=0.63; and the case when the capacity of the AC transmitted power P ac is higher than the DC power P d with R=1.66. The results presented in Table 3 show the positive impact of the increased ratio on the CCT.

P T = P d + P ac = constant (16)
The increase of the DC current significantly reduces the momentary AC power transfer capability; improving therefore the CCT, and hence increasing the stability margin of the network when the AC line becomes weaker. In fact, in a DC transmission, DC transmitted power is independent from the AC transmission angle δ. This possibility is often used to improve the performance and efficiency of the connected AC networks. The controllability of the HVDC power is often used to improve the operating conditions of the AC networks where the converter stations are located.

CONCLUSION

In this paper, the impact of HVDC interconnections on transient stability is investigated. Three transmission line configurations are considered: parallel HVAC, parallel HVDC, and a hybrid HVAC-HVDC transmission line. For each configuration, transient stability of the system is assessed in terms of the fault critical clearing time. Due to the rapid and controllable features, HVDC systems can be used to improve the transient stability of interconnected AC systems in terms of increasing the fault critical clearing time.

For the hybrid HVAC/HVDC transmission system, the rapid and controllable features of HVDC system can also be used to control the power flow in AC systems, so as to increase AC lines transmission capacity and transient stability margin. The protection systems in HVDC transmission are ignored in this work.

Figure 1 .

 1 Figure 1. Single pole 12-pulse HVDC transmission system

Figure 4 (

 4 Figure 4(a). Parallel HVAC transmission line

Figure 4 (

 4 Figure 4(c). Parallel HVDC transmission line

Figure 5 . 2 Figure 6 .

 526 Figure 5. Angular speed of machine 2

Figure 7 . 2 Figure 8 .

 728 Figure 7. Angular speed of machine 2

Table 2 :

 2 Critical clearing time for different casesAccording to Table 2, the fault critical clearing time for case 1 estimated at CCT AC // AC = 424 ms is less than for the mixed system of case 2a where the CCT was estimated at CCT AC//DC-a = 519 ms. As a result, the addition of parallel DC lines to AC transmission circuit improved

	CCT AC//AC	CCT AC//DC-a	CCT AC//DC-b	CCT DC//DC
	(ms)	(ms)	(ms)	(ms)
	424	519	100	180

Table 3 .

 3 Critical clearing time with DC power levels

	Direct Power P d (MW)	AC power P ac (MW)	R		ac d P P	CCT (ms)
	800	1330		1.66	450
	1000	1130		1.13	519
	1300	830		0.63	526

Acknowledgment This work has been supported by the Tunisian-French project CNRS/DGRDRT project Code: 09/R11-09 entitled "Stability of interconnected power systems integrating decentralized production. Case of the south Mediterranean countries". "This work was supported by the Tunisian Ministry of High Education, Research and Technology"

Appendix

The parameters of the system are: Generators: Rated 5000 MVA, 13.