dynamic modeling of high power static switching

8/3/2019 Dynamic Modeling of High Power Static Switching

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67 8 IEEE Transactions on Power Systems, Vol. 14, No. 2, Ma y 1999Dynamic Modeling of High Power Static Switching

Circuits in the dpframeBrian K. Perkins M.R. Iravani

Department of Electrical and Computer EngineeringUniversity of TorontoToronto, Ontario M5S 3G4

Abstract 1 Introduction

With the proliferation of static switching circuits inpower systems there is a need to understand not onlytheir steady-state behaviour but their dynamic inter-action with the network as well. Analysis is difficultdue to the nonlinear nature of switching processes asso-ciated with such devices. Linear, time-invariant (LTI)dynamic models are needed for standard analytical toolssuch as eigenanalysis for the study of inter-area as wellas higher frequency modes of the system. To this endLTI small-signal dynamic models are developed for highpower static switching circuits in a synchronous rotat-ing dqreference frame. Two conventional devices arechosen for study: a Thyristor Controlled Series Capaci-tor (TCSC) and a High Voltage direct current (HVDC)converter. The proposed models are shown to representthe t he dynamic response of the static switching circuitsto nominal cahnges in operating point. In particularthe damping characterizing the dynamic behaviour ofthe TCSC is shown to vary significantly with operatingpoint. This accounts for the observed responses arisingfrom field tests and simulations but not accounted for bythe quasi-static model. On the other hand, the dampingcharacterizing the dynamic behaviour of the HVDC con-verter is shown to vary little with operating point. Thelinearized models are validated by digital time-domainsimulation of the nonlinear static switching circuits.Keywords- FACTS,dynamic modeling, eigenanalysis,TCSC, HVDC converterPE-478-PWRS-0-06-1998 A paper recommended and approved bythe IE EE Power System Dynamic Performance Committee of the IEEEPower E ngineering Society for publication in the IEE E Transactions onPower Systems. Manusc ript submitted July 31 , 1996 ; made availablefor printing June 11, 1998.

As the application of static switching circuits in powersystems increases there exists a need to characterize notonly steady-state processes but dynamic behaviour aswell. Steady-state processes are generally well under-stood, however there is a need for the accurate char-acterization of dynamic behaviour for stability analy-sis. Conventional used quasi-static models do not ad-equately model the dynamic behaviour of the staticswitching circuit.

The objective of this paper is to develop linear, time-invariant (LTI) small-signal dynamic models for conven-tional high power static switching circuits in the syn-chronous rotating dq-reference frame. Such models findapplication in eigenanalysis of power system dynamics.Two thyristor-based devices are chosen for study:0 a Thyristor Controlled Series Capacitor (TCSC) as anexample of a static switching circuit with no commuta-tion process,0 a High Voltage direct current (HVDC) converter as anexample of a linecommutated stat ic switching circuit.

The TCSC is an emerging technology that providesthe means of regulating line reactance. HVDC convert-ers have long been applied to the efficient transmissionof bulk power over long distances.The underlying modeling approach is based on a lin-

earization of the advance map associated with the staticswitching circuit. The advance map projects the circuitsta te through a fraction of a cycle and thereby retainsthe time-domain nature of the equations describing theoperation of the static switching device and allows inher-ent symmetries to be exploited. The prerequisite math-ematical background can be found in [ l ] , 2]. This isin contrast to conventionally used quasi-static modelswhich are based on the linearization of the fundamentalsteady-state response of the sta tic switching circuit.The developed models are shown to capture the dy-namic response of the nonlinear sta tic switching circuitsto perturbations in operating point. The damping fac-tor associated with the small-signal dynamics is shownto exhibit different trends with respect to the operatingpoint of the device. In particular the damping associ-

088.5-8950/99/$10.00 0 998 IEEE


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ated with the dynamics of the TCSC varies significantlywith operating point. This damping has been observedin the dynamic response obtained by field tests and intimedomain simulation but is not accounted for by thequasi-static model. At the Kayenta installation it isreported that the dynamic response varies significantlywith the operating point [3], [4]. his variation of thesystem dynamics with operating point is also reportedin [5] where the Kayenta system dynamics are studiedanalytically by linearization of the Poincark map. It isreported in [6] that the TCSC can have significant im-pact on the damping of subsynchronous resonance (SSR)modes of the turbinegenerator unit. Analysis of fieldtest data from BPAs Slatt TCSC installation revealedsignificant damping effectiveness with the TCSC oper-ating with very small feedback gains (virtually in openloop). This added damping of the SSR modes effectedby open loop operation of the TCSC is also reported in[7]. Eigenanalysis of the TCSC compensated IEEE firstbenchmark model for the study of SSR showed signifi-cant added damping of the mechanical torsional modeseffected by the TCSC operating in open loop. The eigen-analysis results were also validated by time-domain sim-ulation of the nonlinear machine-TCSC system equa-tions.

In contrast to the TCSC the damping associated withthe dynamics of the HVDC converter vary little with o perating point for typical parameters. In this respect theconventionally used quasi-static model is consistent withthe developed one. However, the proposed model resultsin a much simpler structure than the quasi-static modelwhich is obtained by linearizing about the fundamentalsteady-state response.The paper is organized as follows: Section 2 presentsan overview of the proposed modeling approach. Sec-tion 3 and 4 develop the TCSC and HVDC convertersmall-signal dynamic models in the synchronous rotat-ing dpreference frame. Section 5 discusses the variationof the dynamics with operating point. Section 6 com-pares the developed models with the conventionalTCSCand HVDC converter dynamic models. Finally, Section7 concludes the paper.

2 Overview of the modeling ap-proach

The dynamics of a periodic system can be studied bysampling the system states once per period. The systemdynamics can be described as the change in the statesfrom one sample to the next. This concept is formalizedas the Poincark map which advances the system statesforward in time from 8 to 8 + 27r. The Poincark mapwill henceforth be referred to as the 27r-advance map asa reminder of this property. For symmetrical systemsthe 27r-advance map can be expressed as the iterate ofsimpler maps.

The modeling approach is to obtain the lineariza-tion of the advance map about a steady-state operat-ing point with respect to perturbations in input. Thisapproach characterizes the dynamic behaviour as theswitching circuit is perturbed from one operating pointto the next. This is in contrast to linearizing about thesteady-state fundamental solution leading to a quasi-static model which effectively characterizes the changein operating point but does not model the dynamic pro-cess leading from one operating point to the next.

The advance map is a nonlinear function of the sys-tem states and inputs relating phase (abc) quantities a tthe sample instants. The nonlinearity arises due to thedependence of the switching process on the circuit state.Linearizing the advance map yields a linear system re-lating the perturbed phase quantities at the sample in-stants [5]. The linearized map is put into a more conve-nient form by transforming phase quantities (abc) to asynchronous rotating dq-reference frame. A continuous-time model is obtained from the linearized map on theassumption of zero-order hold with samplet ime com-mensurate with that of the advance map. For the sim-ple systems under study aliasing is not a concern in theabove transformation.

3 TCSC dynamic modelingFigure 1depicts the schematic diagram for a single phaseof a hree-phase TCSC and typical waveforms associatedwith phase a. Phase b and c waveforms are displacedby k l2O0 respectively for symmetrical operation. Evi-dently the waveforms exhibit odd half-wave symmetry.The 7r-advance map denoted

H : uabc(kT) -+uabc((k+ I (3.1)advances the three-phase TCSC capacitor voltagethrough half a cycle.

The input current is expressed in the dq-frame coor-dinates

i a b c ( 6 ) = w e ) . d q ( 3 4for Parks transformation defined in Appendix A SO thatthe q-axis is oriented with the phase a line current. The7r-advance map can be expressed in the form

u a b c ( ( k + 1)T) = DH .uabc(kr) k ( -1)kh(4, d ,) (3 .3 )where expressions for DH and h ( 4 , i d q ) are discussedin Appendix A. The (-l)k term arises due to the oddhalf-wave symmetry. Linearizing (3.3) with respect toperturbations about the operating point yields

Gabc((k+ 1 ) ~ )D H . 6,bC(kr)+ (-1)Dh(3 .4 )


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*--(a)Figure 1: (a) Single phase of the TCSC circuit, (b)steady-state waveforms associated with phase a.

4= 272v).-X2 26.50 1 2 3 4 5 6

(c) time (cycles)Figure 2: TCSC d-axis capacitor voltage response to a- lo step change in firing angle for the operating points(a) u = 15O, (b) o = 30" and (c) u = 45" where (-)denotes nonlinear model response and (- - ) denotesLTI model response.

damping factor0.0250.108

-0.0248-0.1082

-1 -0.1082

) 1 0.286-0.2983-1 -0.2983450 I75O -1293.9-601.4-5556.2-4244- 4227

0.337 12.539912.8126 1.37961.4238 12.745314.6337 3.98563.3813 13.3618

I . . , . . ITable 1: TCSC dynamic model parameters obtained

where &bc, phi and l d q denote perturbations in TCSCvoltage, firing instant and input current respectively.Note that Park's transformation at the sample instantsbecomes

V ( k 7 r )= (-1)"(0) W(k7r)= (-1)"(0) ( 3 . 5 )which applied to (3.4) yieldsedq((k -k 1 ) T ) = - V ( O ) D H W ( O ). fidq(k7r) - v(0) Dh

(3.6)Transforming to a continuous-time system whose re-sponse matches that of the above at the sampleinstan tsyields

AGdq 4- Bp$ + Bi&q (3 .7)ddq-=dt3.1 ExampleThe parameters are chosen as in [9]:

L = 15mH C = 212pF.The power system is modeled as a current source with2000 A peak line current. A 60 Hz fundamental fre-quency is assumed.

Table 1 depicts the model parameters of (3.7) ob-tained for the operating points c = 15', 30' and 45' cor-responding to the steady-state conduction angle. Theparameters in the table are obtained by applying themethod outlined in Section 3. The damping factor as-sociated with the eigenvalues of A depicted in Table 1increases significantly as the conduction interval is in-creased. This variation of damping with operating pointaccounts for the observations described in Section 1.

Figure 2 compares the TCSC capacitor voltage re-sponse of the LTI dynamic model with th at of the non-linear model for a -1" step change in the firing instants


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3

eFigure 3: (a) HVDC rectifier with RL-load (b) asso-ciated generic commutation circuit (c) typical steady-sta te waveforms associated with (b)

associated with the operating points enumerated in Ta-ble l . The TCSC is operating in steady-state for onecycle before the step change is applied. The nonlin-ear model response (solid line) exhibits ripple associatedwith the voltage harmonics generated by TCSC opera-tion. The LTI model (dashed line) captures the dynamicresponse of the TCSC associated with the transient pro-cess leading from one cyclic mode to the next. Further-more, the variation in damping with operating point isevident in both the nonlinear and LTI model responses.

4 HVDC dynamic modelingFigure 3(a) depicts the schematic diagram of a six-pulserectifier with ohmic-inductive load. Under the assumption of mode I operation (overlap angle less than SO0 )the circuit evolves through a succession of two intervals:0 commutation intervals in which all three phases areinvolved,

full conduction intervals in which current flows in onlytwo phases.The dc current is symmetric with respect to the 60'

interval, thus the n/3-advance map is the simplest map.

The nlbadvance map is obtained by considering thegeneric commutation circuit represented in Figure 3(b)whose inputs and outputs are related to the ac sidephase quantities by the switching matrices S and T pre-sented in Appendix B. The n/3-advance map denoted

advances the dc current one sixth of .a cycle. The inputac voltage is expressed in the dq-frame coordinates

for Park's transformation defined in Appendix B. Theadvance map can be expressed in the form

where expressions for DG nd g($, udq ) are presented inAppendix B. Linearizing (4.3) with respect to perturba-tions about the operating point yields

(4.4)where ;d, 4 nd iidq denote perturbations in dc-link cur-rent, firing instant and input ac voltage, respectively.Transforming to a continuous-time system whose re-sponse matches that of the above at the sample instantsyields

(4.5)The three-phase ac side current a t the sample instants

is given by

where S k is the switching matrix associated with theswitch states in the interval 8 E (kx/3, (k+ 1)r/3) andQ a is the injection matrix (both are presented in Ap-pendix B). In the perturbed dq-frame the readout func-tion becomes

Due to symmetries in V ( k n / 3 ) nd s k

which results in a linear, time-invariant readout func-tion. The readout function relates the perturbed accurrent in the dq-frame directly to the perturbed dc cur-rent.


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682U A B6 Bu10' 1 --0.0877 1 -286.4 I (4.566 -0.776) x 1

(4.456 -1.154 j x 10-3(4.312 -1.522) x I 2.2r1 1 h

Table 2: HVDC converter dynamic model parameters g 2 . I8obtained 5 2 ,0 1 2 4 5 6 7 8 9 IOI.9lg 2 . 54.1 ExampleThe parameters,are chosen to match as closely as pos-

sible the CIGRE benchmark model [lo]: E 2X7 10 I 2 3 4 5 6 7 8 9 IOL, 0.072068 H Ld = 0.82888 H. + 1.5 '

R d = 5 f l ud = 495kVThe power system is modeled as a three-phase 426.9114kV (peak L-L) voltage source. A 60 Hz fundamentalfrequency is assumed. Table 2 depicts the model pa-rameters of (4.5) obtained for firing angles q5 = loo , 15Oand 20' corresponding to overlap angles p = 29', 23Oand 16' respectively. Note that A which characterizesthe damping associated with the dynamics shows littlevariation with operating point. Figure 4 compares thecurrent response of the HVDC converter LTI dynamicmodel with that of the nonlinear model for a -2O stepchange in the firing angle from q5 = 15O (the responsesassociated with the other operating points are similar).The HVDC converter is operating in steady-state forone cycle before the step change is applied.

- O rb -210 1 2 3 4 5 6 7 8 9 IOtime (cycles)Figure 4: HVDC converter current response to a -2'step change in firing angle from the operating point q5 =15O where (-) denotes nonlinear model response and(- - ) denotes LTI model response.

5 Variation of dynamicsIt can be inferred from Tables 1 and 2 that the twostatic switching circuits under consideration exhibit dif-signal dynamics with operating point. In particular, theTCSC dynamics are strongly dependent on the operat-ing point whereas the HVDC converter dynamics are

100-90 -80

ferent trends with respect to the variation of the small--

3 70-B 6 0 -.s 5 0 -6virtually independent of the operating point. This is nhighlighted in Figure 5 which depicts the dynamics (ex-

pressed as percent damping per cycle) as a function ofthe operating point for both the TCSC and the HVDCconverter with parameters as in the preceeding exam-ples. The operating point of the TCSC is characterizedby the conduction angle whereas the operating point ofgle. The variation of the dynamics with the operatingulation results [3], [ 5 ] . The damping is not accountedfor by the quasi-static model as will be demonstratedin Section 6. The damping associated with the dynam-ics of the HVDC converter is well accounted for by thequasi-static model for typical parameters.

the HVDC converter is characterized by the overlap an-point of the TCSC is consistent with field test and sim-

conductionloverlapangle (degrees)

Figure 5: Variation of dynamics with operating point


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time (cycles)ime (cycles)Figure 6: TCSC small-signal d-axis voltage response toa -1 step change in firing angle of the proposed (-)and quasi-static (- - ) dynamic model.

Figure 7: HVDC converter small-signal current responseto a - 2 O step change in firing angle of the proposed (--) and quasi-static (- - ) dynamic model.

6 Comparison with quasi-static than the proposed one.modelsQuasi-static models are obtained by linearizing theabout the fundamental steady-state response of thestatic switching circuit. Although such quasi-staticmodels can accurately account for perturbations in thesteady-state response, they do not model the dynamicprocess associated with such perturbations.

Figure 6 depicts the TCSC small-signal voltage re-sponse to a -lo step change for both the proposedand the conventional models. The same three operatingpoints as in Section 3 are considered. The conventionalmodel (dashed line) exhibits an undamped response in-dependent of the operating point. Furthermore, the con-ventional model understates the change in the TCSC d-axis voltage - an error which is more pronounced as theconduction angle increases. This is due to the neglectof the loop current and its associated distortion of thecapacitor voltage [9],

Figure 7 depicts the HVDC converter small-signalcurrent response to a - 2 O step change in firing angleof the proposed and the quasi-static models for the o perating point corresponding to a firing angle of 15. Thedynamic responses show good agreement in spite of thefact that both models are derived by different meth-ods. The reason is that for typical parameter values theresistance that models the steady-state voltage regula-tion associated with the commutation reactance coinci-dentally models the damping associated with the small-signal dynamics quite accurately. However, the quasi-static model has a more complicated readout function

7 ConclusionsA dynamic modeling approach based on the lineariza-tion of the advance map associated with the staticswitching circuit is proposed. This is in contrast to con-ventional approaches based on the linearization of thefundamental steady-state response yielding quasi-staticmodels. The developed linear, timeinvariant (LTI)models are shown to represent the dynamic responseof the nonlinear static switching circuits. The dampingassociated with the dynamics of each of the two devicesunder study is shown to exhibit different trends. Inparticular, the damping associated with the dynamicsof the TCSC varies significantly with operating pointwhereas the damping associated with the dynamics ofthe HVDC converter does not. Comparisons are madewith the quasi-static models which do not adequatelyrepresent the dynamics of the TCSC, or in the case ofthe HVDC converter, yield cumbersome expressions as-sociated with the readout function.

The proposed models find application in the standardtools for the linearized analysis of power system dynam-ics, such as eigenanalysis and can be easily integratedwith other power system components as outlined in [111.

References[I] 1. Dobson, S. Jalali, and R. Rajar aman, Damping and reso-nance in high power switching circuits, in System and Con-trol Theory for Power Systems (eds. J.H. Chow, P.V. oko-


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tovic an d R.J. Thomas) IMA Volume 64 in Mathematics andits Applications, Springer-Verlag, 1994.A. Ghosh and G. Ledwich, Modeling and control ofthyristor-controlled series compensators, IEE Proceedingson Generation, %nsmission and Distribution, vol. 142,pp. 297-304, May 1995.N. Christl, R. Hedin, K. Sadek, P. Liitzelberger, P. Krause,S. McKenna, A. Montoya, and D. Torgerson, Ad-vanced series compensation (ASC) with thyristor controlledimpedance, CIGRE 14/37/38-05, August 1992.R. Johnson, P. Krause, A. Montoya,N. Christl, a ndR . Hedin,Power system studies and modeling of th e Kayenta 230kvsubstation advance series compensation, in IEE Fif th In-ternational Conference on A C and D C Power Tmnsmission,(London, UK) , September 1991.S. Jalali, R. Lasseter, and I. Dobson, Dynamic response ofa thyristor controlled switched capacitor, IEEE T-P WRD,D. Trudnowski, M. Donnely, and J. Hauer, Estimatingdampin g effectiveness of BPAs thyristor controlled series caipacitor by applying time and frequency domain methods tomeasured response, IEEE T-PWRS, vol. 11,pp. 761-767,May 1996.B. Perkins and M. Iravani, Dynamic modeling of a TCSCwith appiication to SSR analysis, Presented at the 1997IEEE PES Winter Meeting, New York.S. Helbing and G. Karady, Investigations of an advancedform of series compensation, IEEE T-PWRD, vol. 9,pp. 939-947, April 1994.M. Szechtman, T. Wess, and C. Thio, First benchmarkmodel for HVDC control studies , Electra, vol. 135, pp. 54-73, April 1991.P. Anderson, B. Agrawal, and J. van Ness, SubsynchronousResonance in Power Systems. IEE E Press, 1990.

V O ~ .9, pp. 1609-1615, July 1994.

Appendix to Section 3Parks transformation associated with TCSC modeling is definedin term s of )Wa (0) cose -sinew e ) = (wb(@)) = ( os(0- 2n/3) -sin(@- 2n/3)WC(@) cos(8+ 2 ~ / 3 ) - in(B+ 2 ~ / 3 )(A.1)with V(e) = $W(S) t .Now consider the TC SC circuit depicted inFigure 1 associated with phase a. When either thyristor conductsthe circuit has sta te vector z = ( v a it, ) t and system dynamicsdescribed byx = Ax + Bi,, A=(L!l ;I), B = ( i l ) . (A.2)

In nonconduction mode it, = 0 and the system dynamics aredescribed byua = PAQv, + PBi,

where P = ( 1 0 ) and Q = Pt are t he projection and injectionmatrices associated with t he switching process [l].The n-advancemap associated with phase a Ha va(kn) + va((k + 1)n) can beexpressed in th e form

(-4.3)

va((k + 1)T)= D H a .va(kn)+ (-1lkha(4, idq) (A.4)where

D H a = PeA(r-@)Q (A.5)ha(4,idq) = PeA(r-@)Q

Similar expressions are derived for phases b and c with 4 and rdisplaced accordingly. Consequently, th e three-phase n-advancemap is expressed as in Section 3 with

D H = diag( D H , D H b D H , ) ( -4 . 6 )h ( 6 dq) = ( h a ( ) hb ( ) h e ( ) t . (A.7)Dh of (3.4) is obtained by differentiating (A.7) with respect t o 4

and idq.

B Appendix to Section 4Parks transformation associated with HVdc conv erter modelingis defined in terms of

(B.1))sin(0+~ / 6 )w(e)= -case sin 0cos(! +n/6)( in(8+ 5n/6) cos($ + 5x/6)with V( e )= 4 W ( q t. Consider the six-pulse rectifier configura-tion depicted in Figure 3(a). The generic commutation circuitdepicted in Figure 3(b) describes the commutation process inde-pendent of th e particular sta te of th e rectifier. In a commuta-tion interval both thyristors CY and f l conduct and th e circuit hasstate ide = (i d i, ) t . The switching matrices SI, an d Tk relatethe phase quantities (abc) associated with the ac side to thoseassociated with the generic commutation circuit in the interval0 E (kn/3, (k+ l)n/3) as follows

iabc = S k i d e ude = Tkuabc (B.2)whereTh = S i . For example in th e interval 6 f (0 , / 3 )hyristorsQl-Q5-Q6 conduct and

so=(:l -1 ) . (B.3)Figure 3(c) suggests that th e generic circuit cycles through thre eintervals of operation. During commutation when both t hyri stor sconduct th e system dynamics are described by

(B .4)de = Aide + BTkU,bc 4- dUd, i abc = S k i d e

In th e interval th at preceeds commutation, thyristor CY is open andth e circuit has dynamics described byi d = PaA&aid+Po,BTkUabc+PaBdUd, i a b c = SkQaid (B.5)

are t he associated projection and injection matrices [l]. n the in-terval th at follows commutation th e circuit dynamics are similarlydescribed with associated projection and injection matrices

DG nd g(4, udq) of (4.3) are expressed in term s of th e aboveDG = e 4 ~ / 3 - ~ ) pe A ( ~ - @ ) ~ a e A 4 (B .8 )

where 2 = PaAQa = PpAQp and U ( $ )= BT0W(@)21dq+BdUd.Dg of (4.4) is obtained by differentiating (B.9) with respect to 4and Udq.

dynamic modeling of high power static switching

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