Power Flow and Transient Stability Models of FACTS Controllers for Voltage and Angle Stability StudiesClauclio A. CaiiizaresIJniversity of Waterloo Department of Electrical k Computer Engineering Waterloo, ON, Canada N2L 3G1 c .canizares@ece.uwaterloo. ca Abstract-This paper presents transient stability and power flow models of Thyristor Controlled Reactor ( T C R ) and Voltage Sourced Inverter (VSI) based Flexible AC Transniission System (FACTS) Controllers. Models of t h e Static VAr Coinpensator (SVC), the Thyristor Controlled Series Compensator (TCSC), t h e Static VAr Coinpensator (STATCOM). t h e Static Synclironous Source Series Compensator (SSSC), and t h e Unified Power Flow Controller ( U P F C ) appropriate for voltage and angle stability studies are disciissed in detail. Validation procedures obtained for a test system with a detailed as well as a simplified U P F C model are also presented and briefly discussed. Keywords: FACTS, SVC, TCSC, STATCOM, SSSC. U m n i u l a t i o n , rnodels. controls, transient stability. power flow. I. INTRODUCTION the system.

A . SI-CThe basic structure of an SVC operating under t,ypical bus voltage control is depicted in the block diagram of Pig. 1. Assuming balanced, fundamental frequency operation, an adequate transient stability model can he developed assuming sinusoidal voltages 4 . This model is depicted in Fig. 2 and can be representei Ly the following set, of p.u. equations:

0

The development and use of F.4CTS controllers for power transmission systems 11% led to the application of these controllers to improve the stability of power networks [I, 21. Many studies have been carried out an3 reported in the 1if.era.tureon the use of these controllers in a variety of voltagc and angle stability applirations, proposing diverse control schemes and location techniques for enhancing voltage and angle oscillation control [2]. Several distinct models have been proposed to represent FACTS in static and dynamic analysis [3]. This report describes in detail some of t,he inost appropriate models availahle for these types of studies with the following controllers: s\'(:, TC:SC:, ST.\TCOM, SSW and UPFC! represented in the system. These models allow the engineer t.o accurately arid reliably carry out power flow and transient stability studies of such system with its controllers. The latker is demonstrated in this paper by means of a comparat,ive study in a typical transient- stability problem on a test system using a detailed TTPFC! inodel aiicl the corresponding reduced model presented here. Section I1 describes i n detail the models for TCR-based coiitrollers, concentrating specifically on the W C and TCSC', and Section I11 discusses in detail t,he models for VSJ-based controllers, namely. the STATCORI, the SSSC atid the UPFC!. In Section IV, the test system used for ididating some of these modrls as well as the comparative results obtained for a detailed and t>hesimplified model of tlir IJPF(! are shown and discussed. Finally, Section V briefly s ~ ~ m m ; ~ r ithe smaterial presented in this paper as ze well as discussing t.he liniitations of the reducwl models.11.hlODEr,[NC:

I

I - \ < Be

where most variables are clearly defined on Fig. 2 , and x, and I(.) stand for the control system variables and equations. respectively. These equations represent liriiit,s not only on t,he firing angle a. but also on the crurrent 1. t,lie control volt.age I' and the capacitor voltage L.;. a s well as control variables other types of controllcrs suc:li as ii reactive power Q control scheme. The differential equations representmecl f ( ' j i t i ( 1 ) vary by with the t.ype of control system used. Fig. 3 d:piots a typical voltage control block diagram. which inclucles a droop to avoid continuous operation of the controller ancl to allow for proper coordination with other voltage cont>rollers in the network. It is important to highlight the fact that an admittance model is numerically more stable than the corresponding impedance model, i.e., using B, OII t,lir model averts numerical roblems when close to t,he controller's [ . ! resonant points 5 The bins no for this controller is determined by solving the equat.ions resulting from forcing B, = 0 in ( I ) , i.e., this value corresponds to t.lw resonant point of t.he SVC ( I = 0) and is obtained by solving the nonlinear equation%U,

- sin 2a,

- ~ ( ' - . l /.I-,-.) 2 Y. =0

TCR-BASED c!ONTROI,LERS

The steady stat,e V-I characteristics for this coiit,roller are depictmecl Fig. 4. and correspond t o tlie .cwll-knonn in control characteristics of a typical S\:C! [2]. . S I ' ( ' steady 4 sta.t.e model call be obtained by replacing t.lie cI ifferetitialequations in (1) with the corresponding eqiiat,ions repre-

Basic modcls for SVCk and TC!SC:s built around a TCR structure are described in this sec%ion. These inodels are Ixwed 011 representing tlie controllers R S variable impedances t.hal change wit,li the firing angle of [,he TC!R, wI~icItis iisecl l o coritrol voltage, current atid/or power iit0-7803-5935-6/00/$10.002000 IEEE (c)

silting the s t k d y state clia.ract,erisl.ics;~ t.Iius, flow" eyuatioiis of the SVC! in this case are 1," - \ r e / SSLI

- +

U

=

[

y(0.

r;

I ; , I , 9. B,)

1

1447

V

(-Jv

t

Crossing

-

IFig. 4. Typical steady state \'-I characteristics of a SVC.Q

Fig. 1 . Block diagram of a S\'C with voltage control.Yrj

< QminYrf

>C,

0 < Y C t < ,r ;

kf >*:tFig. 5. Handling of limits in the SVC' steady state nlodel.

Fig. 2. Transient stability model of a SVC:.

0 =

P

+ Vk\;,Be sin(& - d,,,) + li:V,,B,COS(^^ - d,) - Q,,

-~~B,+I.1~'",B,cos(Sk-6,) - - Q k

-\.,;;Be

iJ

Fig. 3 . Hasic SVC contrdler for voltage control.

[sc- r;: (7r0-7803-5935-6/00/$10.002000 IEEE (c) 1448

cos k,

(77

-N)

laFig. 6. Block diagram of a TCW operating in current control mode.

CiOSSIngA

Zero

Fig. 8. Block diagram of a STATCOM with P\Yhl voltage conlrol.

111.Fig. 7 . Transient stability moclel of a TCSC!.

VSI-BASED CONTROLLERS

In this section. the basic models of the most common \'SI-based FACTS controllers, namely, the STATC'ORI, SSSC and UPFC. are discussed. All the niodels presented here are based on the power balance equationpac

=

p d c $. Plms

k,

JX L "

It is important to mention that as the controller gets closer to its resonant point, the current deviates from its siiiusoidal condition, and hence the model presented should not be used to represent the controller under these conditions. .4 simple PI controller with limits can be used to control t,he current directly through the firing angle a ; in this case, the differential equat.ions f(.) in (3) can be replaced by the equations of the corresponding coiitrol system. Observe, however. that more sophisticated coiitrols such as impedance or power control can be readily implemented on this model. A dteady state model for this TCSC controller can be obtained by replacing t.he differential equat.ions on ( 3 ) with the corresponding steady state coiitrol equations. For example. for an impedance control model wit,li 110 droop, which yields the simplest set of steady state eyuatioiis froin the numerical point of view [SI, the "power flow." equations for the TCSC are

which barically represents the balance beteween the controller's ac power Pa, and dc power P d c under ba,laiiced operation at funda.menta1 frequency. For the niodels to bp accurate, it is important to represeiit all losses of the cons trollers ( PI,,,, ) especially i.hose related to the inverters. a discussed below. Although PWM control is currently not. practical in t.ypica1hi h-volta.ge applications of VSI-biLsed controllers, given theknitatioiis imposed by the high switching losses of GTOs, there have been some new recent developinerits on power electronic switches that will probably allow for the practical use of PWhl control techniques on these liintls of applirations in the near future [7]. The models discussed in this paper awuine P\;t'RI coiitrol techniques are assumed. These models are used to develop iiiore genera.1 models that can readily be adapted to represent phase angle control as well.~

A4. ST.4 TCOMThe basic strricture of a STATCOM wit,h PiYhI-based volt.age controls is depicted in Fig. 8. Eliniiiiating the dc voltage control loop on this figure woulcl yield llie basic block diagram of a controller with typical phase itngle cont,rols. Assuming balanced, fundamental frequency voltages, the controller can be a.ccurately represenkd in transient stability studies using the basic-. model slio~vii in Fig. 9 [& 9. lo]. The p.u. differential-algebraic equations ( DAE) crorresponding to this model are(5)

As previously indicated, iL is important to adequat,ely implenient. t,he controller 1iniit.s on the steady stmate model to accurately r~yr~si:nt operation [SI. its0-7803-5935-6/00/$10.00(c) 2000 IEEE

v,c

=

1 \.'. I cos(S - 0 ) - -\ -J C C' I ;Ic R(:: C'

R I' --C,I

1449

I

I

Magnitude

I

+ l

* , +pMagnitud Fig. 9. Transient stability model of a control.

-4 -&

STATCOM with PWM voltage

P - I - I cos(8 - e )Q - L7 I sin(6 - e)

Fig. 10. Basic STATC'OhI PWM voltage control.

o =

P - I-? G

+k

L ' d C

'L B sin(6 - a)

+ k I'dc

I*,.c: cos(6 - a)

Q + 1 B - XI \Idc V B COS(S - a ) " V G' sin(6 - a) +kf

g ( n , IC. CLS. ,'L

I , e. P, Q )

the biases m, and a, [13]. The steady state model can be readily obtained froiii (5) by replacing the differential equations with the steady state equations of the dc voltage and the voltage control characteristics of the STATCONI (see Fig. 11 r2]). Notice that t.he controller droop is directly represented on the \'-I characteristic curve, with the controller limits being defined by its ac current limits. Hence. the steady state equations for the PtYhI controller are

where most of the variables are explained on Fig. 9. The admittanre G' + j B = ( R + j-S)-l is used to represent the transformer impedance, any ac series filters, and the "switching inertia" of the inverter due to its high frequency in, is .directly proswitching. The conshnt k = portional to the pulse width modulation index m and ., r represents the interna.1 control system variables. .4 siniple PWM voltage controller is shown in Fig. 10 Ill, 121. which basically defines the differeiit,ial equations represented by f (.) in (5). Observe that the ac bus voltage magnitude is controlled t.hrough the niodulation index m since this has a direct effect on the ac side \:SI voltage magnitude. Whereas the phase angle. C L , which basically determines the active power P flowing into the cont,roller is used to directly control the dc voltage magnitude since the power flowing into the cont.roller charges and discharges t,he capacitor. The controller limits are defined in terms of the coiitroller current limits, which are directly related to the switching device current limits, as these are the basic limiting factor in VS1-based controllers. In simulations. these limits can bo directly defined in terms of the iii:\simum and minimum converter current.s I,,, a! In,in, id rcnpectiwly, i.e.?the ;ntcgrat,or blocks are "stopped tvhenever the converter current I reaches a limit which would allow t.0 closely duplicate the steady state V-I clia.racteristics of the controller shown i n Fig 11. Another option is to compute 1.11eselimits by solving the steady state equations of t,Ilz converter: these erlitations are also used to compute0-7803-5935-6/00/$10.00 (c) 2000 IEEE

1 I - - Vr.f +

SS'LI

1

L !/(n,k.\z73 1idc.6,1,8.P,Q)1A phase control technique can be readily inocleled by simply replacing the dc voltage control equation in (6) with an equation for k , i.e., for a 12-pulse \:SI, replace 0 = [Vd, - VdC,.,] with 0 = [k - 0.91 in the above set of equations. In this case the dc voltage changes as fi changes, thus charging and discharging the capacitor to control the inverter volt age inagnit ude. Tho limits on the current I , as well a s any other limits on the st.eady state model variables. such as the tlr voltage I h the niodulat~oiiratio represent.ed by A. or the volt,age , phase angle CL. can be directsly introduced in this niodel. It is important to properly represent the switching of coutrol Inodes when these liinits are readied, a.s this is a significant fact,or for properly modeling FACTS cnntrollers i n steady state studies [SI. Thus. the mode switchiiig logic depicted in Fig. 5 for t,he S\'C! can be readily inotlified to represent the steady st,a.te conl.rol inotle switdiing for the STATCOM, by simply replacing the firing angle limits wit.11 current limits.

1450

XSL

Crossing

210

pLL

SwitchingI

Magnitud

B

nt(PWM)

v,I,

+Fig. 11. Typical steady state V-I characteristics of a STATCOM.

n 0

Magnitud

%,U,

Fig. 12. Block diagram of a SSSC with PWM current control

B. SSSC The basic controller structure for the SSSC operating on current control mode is depicted in Fig. 12. The corresponding transient stability model is shown in Fig. 13 [9], and ca.n be represented by the following p.u. equations:

'UPWM

RijX

-P kPm

1 Cos(6k - e)

Qk - V I sin(& - 8) k

o =

+V I m Qm + V I sin(6, m P - P + Pm kCOS(&

- 8) - e)

Fig. 13. Transient stability model of a SSSC

Q-Qk+Qm

P - V' (2 k Vdc V G C O S - p) (~ +k vdc V Bsin(6 - p)

+

L +k

Q + V 2 B - k VdeV B cos(6 - p) vdc V Gsin(6 - P)

the phase angle ,L? and the capacitor voltage \Idcq i.e., the current is controlled by direct control of the series voltage VL6. A more sophisticated dq controller to control the active and reactive powers on the line is discussed in the next section for the series branch of a UPFC, which is basically a SSSC. The steady state model equations, for a PWM controller with no droops, are then

where most variables are defined on Fig. 13, k = 772, a.nd t, a.nd f(.) s h i d for the dyna.mic variables and equations of the control syst,eiii, respectively. The basic VSI model follows from t,he niodel developed for the STAT-

o =

COM.

Different kinds of controls can he implemented for v x i -

troller t,hal clirecily operates on the phase angle P. The PWM controller represented on the SSSC figures in this report., indirectly cont,rols t,he current, I by opera.ting on0-7803-5935-6/00/$10.00 (c) 2000 IEEE

oils controller variables. The simplest is a. PI current con-

For a. phase controller, the dc voltage eclua.tion is replaced by a n equation defining the variable k. Once again, it is iinportant to properly model talle controller limits i n orcler to have an adequat,e stea.dy statme rnoclel of t,he SSSC!.

1451

I

I

i----eiUPFC CONTROLLER

UPFC CONTROLLER

Fig. 14. Block diagram of a UPFC. Fig. 15. Transimt stability model of a IJPFC.

C' [TPFC ..As shown in Fig. 14, the UPFC! can be viewed as a ST.4TCORI and a SSSC with a shared dc bus. The corresponding t,ransient stabi1it.y model reflects this fact, as shown in Fig. 15. Thus, the model equations then can he defined as a combination of the STATCOM and SSSC ecluatioiis (5) and (7), respectively, as discussed in detail in [I2, 13, 14. The shunt controller is basically the same as the one described for the ST.STC!Ohl above. A control syst,em diagram for the UPFC:'s series branch is depicted in Fig. 16. This controller, originally proposed in [l5]?is a PQ coiitroller based on a d y-axis decornposition to decouple the active and reactive powers of the inverter [ I I , 12, 14; this PQ controller performs better than other PQ controls proposed in the literature [12 . However, a current control strategy for the SSSC! coul he also used in this case. The steady state model can be obtained from tlie transient stability model equations and the corresponding controls, ils previously done for all other models. Once more. it is important to properly niodel the controller liiiiits to ohta.in reliable results in steady state studies.plied at Bus 6 at 4 s. This triggers the circuit breaker of the Bus 4-Bus 5 line at -1.15 s (9 cycles later), removing the fault as well as the load a t Bus '7. The generator at Bus 3 recovers successfully, keeping its terminal voltage at about 1 p.u.. as shown in Fig. 18. The UPFC also recovers, maintaining its power and terminal vokages at the desired levels. Observe liow close the results for both the IJPFC! detailed model and the simplified model are. The most significant differences are in the internal ITPFC' variable (e.g.?capacitor voltages), as expected, but the effect. of the UPFC on the system is fairly accurat,ely captured by the lllodel.

d

IT. CONCLUSIONSThe transient stability and power flow models presented here are based on models that. have beer1 proposed on the current literature. and can be considered to be simple. adequate models for voltage and angle stability studies of networks with these kinds of F.4CTS controllers. These models are all based on the assumption that voltages and curreiits are sinusoidal, balanced. and operate near fundaniental frequency, which are the typical assuniptions in transient stability and power flow studies. Hence. they have several limitations, especially when studying large system changes occurring close to these FACTS controllers:

IV.

VALIDATION STUDIES

The test system of Fig. 17 was used in [12, 14 t o validate the simplified niodel discussed here. The who e system is tnodeled in detail in the EMTP, i.e.. 3-phase generators, transmission lines, etc. The detailed l.rPFC! model of Fig. 14) wit,h all its switches, was modeled n well as the cors responding simplified inodel of Fig. 15, are represented in detail. The generator is assumed to have an .4VR coilt,rollirig its terminal voltage, and the UPPC is designed to control the power through the line a well as the voltages at s Bus 4 and Bus 8 , using a simplified PiVhl power controller proposed in [I'L]. . ba~lniiced %phase fault through an inipcdatice is apI0-7803-5935-6/00/$10.00 (c) 2000 IEEE 1452

1

1. These models cannot be reliably used to represent unbalaiiced system condit.ions, as they are all based on halancccl voltage and current, conditions. 2. Large disturbances that yield voltages and/or currents w i t h high harmonic conlent. which is usually the case when large faults occur neilr poww elect.ronics-based controllers, cannot. he accurately studiecl with t.hesc:

Generator Phase Angle

-0 7;

3.5

tenerator Terminal Volta&

5.5

6

i

I3.5 LoadPowerDemand

5.5

6I

KI+ Kl/S

8

I 105.130

1801

-

2

U )

80 03

3.5 Sending End Voltage 5.5

6

1 5r .

i3.5 1.51Receiving End Voltage 5.56

' 1

0' 3 023

I3.5 4 4.5 Senes Inserted Voltage 5 5.56

n o 083.5

voltage

5.5

6

Fig. 16. Basic series branch dq control of UPFC w i t h respect to the bus voltagr. KL61. All variables are in P.u., and w g stands for the fundamental frequency of the system in rad/s.

U"

22 '

I1

3w33.5 Angle Alpha 5.5

6

I 5 -70-

4 -

Iw

0 3 11F.ull

-1.21 3 12 .1

I3.5 Series Modulation Index 5.56

Fig 17. Test systwm designed f o r validation studies of UPFC controller iiiodcls [12, 1.11.

-1.21 3

I

3.5

4

4.5

5

5.5

6

Fig. 18. Test system results for a 3-phase fault at h i s 6 [12). Tile cont.inuous line was obtained with t,he siniplifircl IJPF'C! i n c d c l . w h e r e a s t,he dashed line w a s obtaincc1 wit.!) a detailerl IJPFC: i n i d e l .

0-7803-5935-6/00/$10.00(c) 2000 IEEE

1453

models, as they are all based on the assumptions of having sinusoidal signals.

[13] C. A. C'ariizares, "Modeling of T C R and VSI Based FACTS Controllers," internal report, ENEL a n d Politecnico tli Milano, October 1999. available a t www.power.uwaterloo.ca. [14]

3. The above also applies for caSes where voltage and current signals undergo large frequency deviations.4. Internal faults as well as some of the internal variables of the controller cannot be reliably represented with

E. Uzunovic, C'. A. Cariizares, and J. Reeve, "Fundamental Frequency Model of Unified Power Flow Controller," Proc. iVAPS, ('leveland, Ohio. October 1998. pp. 294-299.

these models.For these cases, detailed EMTP types of studies are required t,o obtain reliable results. Observe that these limitations also apply to most models typically used to represent other devices in transient stability and power flow studies.

[15] I. Papic, P. Zunko, and D. Povh, "Basic Control of Unified Power vol. 12. no. 4. Flow C!ontroller," IEEE Trrzns. Power S ~ s t e m s , November 1997. pp. 1734-1739.

1'1.

.ACKNOWLEDC:E~IENTS

The author would like to thank the National Science and Engineering Research Council (NSERC:) of Canada for its direct support of the research discussed in this paper, as well as his. Edvina 11zunovic for providing some of the information, gra.phs and results presented here.

Claudio A. Caiiizares received in April 1984 the Electrical Engineer diploma from the Escuela PolitCcnica Nacional (EPN), QuitoEcuador, where he held different teaching a n d adininistrative positions from 1983 t o 1993. His MS (1988) and P h D (1391) degrees in Electrical Engineering are from t h e University of WisconsinhIadison. Dr. Cafiizares is currently a n Associate Professor a t the University of Waterloo, E W E Department, and his research activities are mostly concentrated in studying stability, modeling a n d computational issues in ac/dc/FACTS systems.

REF EREN c ESN. G. Hitigorani. "Flexible AC! Transmission Systems," IEEE Spectruni, April 1993. pp. 40-15.

"FACTS Applications," technical report 96TP116-0. [EEE P E S ,1996.

Convener Terond, " hlodeling of Power Electronics Equipment (FACTS) in Load Flow and Stability Programs: A Representation Guide for Power System Planning ancl Analysis," technical report T F 38-01-08. CIGRE. September 1998.

N . Christl. R. Heiden. R. Johnson, P. Krause. a n d A. Montoya, "Power System Studies and hfocleling for t h e Kayenta 230 K V Subsfatioi Advanced Series Compensation." A C and DC P o ii'e r Trans m i ss 1 on IEEE C!o nfe re n ce Publrcut r on S I n t e rn a : lronrrl Conjerence on .A C rrnrl DC' Power Trarisniissron, Sep!tember 1991, pp. 33-37.

C. A. Catiizares and Z. T. Faur, "Analysis of SVC and TCSC: Controllers in Voltage Collapse," IEEE Truns. P o w e r Systems, vol. 14. no. 1 , February 1999, pp. 15.9-165.S. G.Jalali, R. A. Heclin. M . Pereira. and K. Sadek, "A Stability hlndcl for the Advanced Series Compensator (ASC!)," IEEE Truna. Potiter Delivery, VOi. 11. no. 2. April 1996. pp. 1128-1137.

P. I