Harmonic domain approach to STATCOM modelling

C.D. Collins, G.N. Bathurst, N.R. Watson and A.R. Wood

Abstract: Previous work conducted in the unified harmonic domain has been primarily focused onHVDC systems. The paper outlines the extension of this technique to a hard switched FACTSdevice. A unified harmonic domain model is developed for a STATCOM and solved using aNewton solution based on DC side current mismatches alone. This generic formulation maintainsmodularity, allowing the interaction between multiple STATCOMs to be studied. The model hasbeen verified against a time domain solution using PSCAD/EMTDC. The proposed techniqueprovides practically identical results, without the time domain’s inherent problems with respect tosteady-state simulation.

1 Introduction

The increasing prevalence of flexible AC transmissionsystem (FACTS) devices makes having accurate models ofthese devices essential. One attractive method for modellingthe steady-state performance of these devices is frequency(or harmonic) domain analysis [1]. This has been shown bynumerous authors to be a reliable modelling technique forswitching devices [2].

Both the frequency and harmonic domains use harmonicphasor representation of electrical quantities. Harmonicphasors contain the complex Fourier series coefficients [3],which, by definition, have both positive and negativefrequency terms to account for the phase dependenceinherent in the linearisation [4]. Acha [5] used this approachby having phasors which include the negative frequencyterms. In contrast, Smith et al. [6] showed that the problemsize could be reduced by using a tensor representation whichtakes advantage of the conjugated nature of the negativefrequency components. This leaves a positive frequencyspectrum representation which still fully accounts for thephase dependence.

Models using these harmonic phasor representations canbe broadly split into three categories: linearised models [3],which do not account for power frequency operating pointvariations; ‘sequential’ models [2], which have separateiterative loops for the operating point and harmonics; and‘simultaneous’ or ‘unified’ models [4, 7], which have a singleiterative loop, incorporating both operating point variationsand harmonic interactions. Unified techniques have beenshown to provide good convergence at the cost of havingcomplex heterogeneous Jacobians, making them lessmodular [8]. Bathurst [7] overcame this modularity problemby removing the switching variables from the main solutionand including their effect implicitly in the average firingangle.

This paper describes the derivation of a unified modellingtechnique for a distribution level STATCOM. Thealgorithm has been developed from the convolution basedtechnique used by Bathurst et al. [9] for HVDC systems.PWM STATCOMs differ significantly from HVDCsystems as they utilise hard switched modulation techniques.This makes it possible to represent the device in terms of theDC side mismatches alone, reducing the problem size, whilemaintaining modularity through the use of global controlvariables which internalize the solution for switching instantvariables.

2 STATCOM fundamentals

The STATCOM is a versatile shunt injection FACTSdevice, based on a voltage sourced inverter (VSI) [10]. Thisacts as a pseudo-sinusoidal voltage source of variable phaseand magnitude, the manipulation of which permits thecontrol of the real and reactive power flows [11]. The devicemodelled in this paper uses a single level PWM VSI and asingle proportional integral (PI) control loop, regulating thevoltage at the point of common coupling (VPCC) with phasecontrol (see Fig. 1). The phase control principle varies theDC voltage (by adjusting a) such that the reactive powerinjection into the PCC can be controlled, and hence aregulated voltage maintained. The modulation index isfixed in this simplistic control scheme, while the switchinginstants themselves are described using classic bipolar PWMtheory [12].

While the STATCOM time domain behaviour is lineartime variant, its harmonic transfers (between the AC andDC sides) are actually linear time invariant; this is whatmakes harmonic domain analysis so attractive. A Newtonsolution is required because the control system incorporatesfeedback from both fundamental and harmonic frequencies,making the transfers nonlinear time-invariant.

3 Unified harmonic domain model

The unified harmonic domain, proposed by Smith et al. [6]and developed further by Bathurst et al. [9], differs fromother frequency domain techniques in that both theharmonic interactions and the fundamental frequencypower flow are solved simultaneously. This multivariableunified Newton technique has been shown to have robust

C.D. Collins, N.R. Watson and A.R. Wood are with the Electrical andComputer Engineering, University of Canterbury, Private Bag 4800, Christ-church, New Zealand

G.N. Bathurst is with IPSA Power Ltd., 1 Echo Street, Manchester M1 7DP,UK

r IEE, 2005

IEE Proceedings online no. 20041227

doi:10.1049/ip-gtd:20041227

Paper first received 1st February and in revised form 22nd October 2004.Originally published online: 16th February 2005

194 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005

and rapid convergence [13], and has therefore been used forthe following model.

The basic structure of any system being solved using theNewton method is well established. A system of mismatchequations are formed, MðXiÞ, which are a function of thesolution variables. These variables are then updated using amatrix of partial derivatives (a Jacobian, J) such that themismatches tend to zero. Figure 2 outlines this methodol-ogy, while (1) is Newton’s classic equation. Note how thesolution variables are initialised with a power-flow, and howthe full Jacobian is calculated after this initialisation andthen fixed:

MðXiÞ ¼ JiDXi

Xiþ1 ¼ Xi � DXi

ð1Þ

3.1 Fundamental frequency representationThe fundamental frequency backbone of the system(including the STATCOM) has been represented with apositive sequence power-flow, permitting the use oftraditional constant power loads. A positive sequencepower-flow has been used in this model because of itssimplicity and hence computational speed. Obviously theuse of a positive sequence power-flow assumes that thesystem is balanced at the fundamental frequency. Themodel does not, however, assume that the system isbalanced at harmonic frequencies.

3.1.1 System power-flow: The power-flow repre-sentation of the linear system at the ith busbar contributesthe well known P and Q mismatches, which are included in

Vrms

+−

PI

system representation

VSet

VPCC

P + jQ

Rt + jXt

VConvVdc

C Rc

I ∠�

∠�

Conv

load at PCC

Idc

�

Fig. 1 Single line diagram of STATCOM representation

form Y matrix

positive sequence power-flow

formulate DC currentmismatches

calculate PQ andSTATCOM mismatches

calculate Jacobian(numerically)

combine mismatches

converged

calculate updates

harmonic mismatchesfundamental mismatches

no

calculate AC sidevariables, Vpcc etc.

yes

calculate switchinginstants

Fig. 2 Flow chart outlining unified Newton solution adopted for this model

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 195

(2), the derivation of which can be found in [14]:

DPi

DQi

" #¼

P schi �

Pnj¼1jVijjVjjjYijj cosðyij � di þ djÞ

Qschi þ

Pnj¼1jVijjVjjjYijj sinðyij � di þ djÞ

26664

37775 ð2Þ

3.1.2 STATCOM power-flow: The STATCOMitself is represented at the fundamental frequency by thepower-flow proposed by Canizares [15]. This modelcontributes seven additional mismatch equations (3). Thefirst four mismatch equations consider the real and reactivepower being transferred into or out of the STATCOM,while the fifth and sixth describe the steady-state controlscheme, and finally the seventh mismatch accounts for theSTATCOM losses.

The power-flow implemented in this model differs fromthat proposed by Canizares in that the control variable is nolonger the fundamental frequency voltage magnitude butthe true RMS voltage, at the PCC. This true RMS accountsfor the impact of voltage distortion at the PCC and isdefined by (4). Variables used in (3) and (4) are defined inFig. 1, with the exception of the admittances G and B,which account for the transformer impedance, and M themagnitude of the sinusoidal modulating function.

0 ¼

P � VPCCIConv cosðd� yÞ

Q� VPCCIConv sinðd� yÞ

P � V 2PCCGþ VConvVPCCGcosðd� aÞ þ VConvVPCCBsinðd� aÞ

Qþ V 2PCCB� VConvVPCCBcosðd� aÞ þ VConvVPCCGsinðd� aÞ

VPCCRMS � Vset

M �Mset

P � V 2dc

Rc� RtI2Conv

266666666666666666664

377777777777777777775ð3Þ

where VRMS is defined as

VRMS ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXnh

h¼1

jVhjffiffiffi2p

� �2

vuut ð4Þ

and Vi is defined as a peak valued phasor, nh being thehighest frequency of interest.

3.2 Harmonic representation

3.2.1 Harmonic spectra: From a computationalperspective, the only difference between a 6-pulse waveformand a PWM waveform is the number of pulses which needto be summed to produce the complete spectrum. Theharmonic spectrum of the PWM switching function cantherefore be found by extending the positive frequencyrepresentation used by Smith et al. [6] for a 6-pulseconverter.

The spectrum is derived by first solving for the switchinginstants, a nonlinear problem itself. The switching instantsare defined by the intersection of the triangular carrier, Vtri,and the sinusoidal modulating function, m. This vector ofswitching instants c, which contains an ON and OFF anglefor each of the Np conduction periods, is found using asingle variable Newton solution. These switching angles arethen used in (5), the exponential form of that used by Smithet al. [6]. This produces the positive frequency complex

conjugate switching spectrum S; a time domain example ofS is included in Fig. 3.

Sh ¼XNp

p¼1

j2p

cOFFp� cONp

� �; h ¼ 0

Sh ¼XNp

p¼1

ejhcONp � ejhcOFFp

� ��hp

; h 6¼ 0 ð5Þ

The proposed model represents the switching function by aspectrum of bandwidth twice the highest frequency ofinterest, nh, fulfilling the Nyquist rate. This switchingspectrum is then convolved with the harmonic spectrum ofVdc to produce the converter terminal voltage, VConv.To account for the conjugated negative frequency termsin the spectrum, the convolution defined by (6) is used(Smith et al. [6]):

F � Sð Þk

¼

j2�2F0S0 þ

Pnh

m¼0FmS�m

� �; k ¼ 0

j2

Pnh

m¼0FmS�mþkð Þ

� ���Pkm¼0

FmS k�mð Þ þPnh

m¼kFmS�m�kð Þ

� �;

k40

8>>>>>>>><>>>>>>>>:

ð6Þ

3.2.2 Harmonic interaction: Accounting forthe interaction between the harmonics produced bymultiple nonlinear devices through the linear powersystem is of critical importance. To account for theseinteractions the AC side converter currents, IConv, are foundby applying the harmonic voltages present at each converterbusbar, VConv, to an admittance matrix which representsthe linear time invariant system, in a similar fashion to

0 1 2 3 4 5 6

−1.0

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

angle, rad

cont

rol s

igna

ls

modulating signaltriangular carrier signal

convolved switching function PWM switching−represented by convolution

Fig. 3 PWM switching function found using convolution,ftri¼ 9� fsys

196 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005

Bathurst et al. [9]. First consider the partitioned harmonicadmittance,

IVsourcehi

IBushi

IConvhi

2664

3775 ¼

A B C

D E F

G H J

2664

3775�

VVsourcehi

VBushi

VConvhi

2664

3775 ð7Þ

where IV source and VV source refer to the harmonic currentsand voltages present at ideal voltage source buses. LikewiseIBus and VBus refer to the currents and voltages at theremaining busbars (e.g. PQ load busbars).

By assuming that VConv is the only source of harmonicdistortion and that ideal voltage sources are used, thissystem of equations can be simplified down to (8):

0

IConvhi

" #¼

E F

H J

" #�

VBushi

VConvhi

" #ð8Þ

Using matrix algebra it is possible to reduce this systemto (9), which accounts for the effect that multiple convertershave on IConv and VPCC:

IConvhi¼ ½J � HE�1F �V Convhi

VBushi¼ �½E�1F �V Convhi

ð9Þ

3.2.3 Harmonic mismatch equations: The ma-jor difference between the proposed model and otherharmonic domain models is that it characterises the devicein terms of the DC side only. This is possible because of thehard switched nature of GTOs and IGBTs, leading to atransfer function which is only dependent on the sinusoidalmodulating function m. The terminal voltage is thereforeonly dependent on the DC voltage, Vdc, and the transferfunction or spectrum, S. This significantly reduces theproblem size, by eliminating the AC side mismatches usedby Smith and Bathurst to model HVDC systems.

To form the DC current mismatch at each harmonic theDC current is calculated from both the AC and DC sides ofthe converter. The difference between these two currentsforms the mismatch. Given the present estimate of the DCside voltage, Vdc, the DC side current, Idc, can be found bymultiplying the voltage by the DC side admittance, Ydc (10).The second method transfers the AC side current, IConv,to the DC side, summing the contribution from eachphase (11).

Idc ¼ Ydc½ �Vdc ð10Þ

Idc ¼X3ph¼1

IConvph � Sph ð11Þ

Assuming that all distortion is coming from any number ofSTATCOMs, the AC side currents at harmonic frequenciesare also defined by the DC bus voltages. If other linearharmonic injections are present the model still holds;however, (9) will contain extra terms.

To calculate the AC side currents from the DC voltagesfirst the converter terminal voltages VConv are foundusing (12):

V Convi ¼ V dci � Si ð12ÞThese terminal voltages can now be applied to (9), the resultbeing the AC side currents, IConv (13). It is important to notethat (12) and (13) are replicated to account for all three

phases on the AC side of each STATCOM:

IConvhi¼ ½J � HE�1F �ðV dci � SiÞ ð13Þ

The basic harmonic mismatch is defined by (14) where(10) and (11) have been equated and IConv substituted for(13). By using this mismatch on the DC side of eachconverter it is possible to characterise the harmonicperformance of the system.

0 ¼X3ph¼1

½J � HE�1F �ðV dci � SphiÞ�

� Sphi � Ydci½ �V dci

ð14Þ

3.3 Numerical formulation of the JacobianThe final component of the model is the Jacobian, which isused to update the DC voltage harmonics and the ACfundamental frequency system voltages. Analytical deriva-tions for HVDC systems [7] have been shown to offersignificant computational advantages and provide excellentresults. However, the analytical Jacobian has the distinctdisadvantage of not being as versatile as its numericalequivalent. For example, changes in the switching fre-quency, fs, and modulation index, M, are easily accom-modated within the numerical framework at the cost of atime penalty. This time penalty is not particularly criticalsince the full Jacobian is only calculated once.

In contrast, it is anticipated that gaining such versatilitywith an analytic derivation would be difficult. Overmodula-tion (when M exceeds jVtrij) could in the case of ananalytical Jacobian cause da=dM to become undefined, assome switching angles would no longer exist. This wouldlimit the region over which the model is valid.

As an example, the Jacobian for a single STATCOMsystem is shown in Fig. 4. The device being simulated used aswitching frequency of 600Hz (12th harmonic), while thehighest frequency of interest was 2500Hz (50th harmonic).It clearly conforms to the expectation that the AC harmonicspectrum should fit (15) [12]. For example the first pair ofoff-diagonal terms are spaced 500 and 700Hz from thefundamental, i.e.

fh ¼ ðjmf � kÞf1; j; k ¼ 1; 2; 3; ::: ð15Þ

4 Model validation

4.1 Test system specificationThe proposed model was tested against a time domainPSCAD/EMTDC simulation which was run until a steady-state operating point had been reached. The heavilydistorted test system includes a Thevenin equivalentimpedance representing the system, three constant powerloads, and two STATCOMs providing reactive powersupport to the loads.

The system parameters and basic layout are included inTable 1 and Fig. 5. Both STATCOMs are of the formoutlined in Fig. 1, where the star-delta linking transformerhas been assumed to be ideal, and the DC side resistancehas been sized such that the DC losses match the converterlosses measured in PSCAD/EMTDC. A synchronousPWM controller has been used, such that the triangularcarrier wave is synchronised with the ‘a’ phase modulatingsignal, as shown in Fig. 3. The system is considered to haveconverged when the 1-Norm of the mismatch vector iso10�6.

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 197

The system was simulated in PSCAD/EMTDC using atime step of 5ms and a snapshot startup. The data includedwere recorded 1 s after the steady-state operating point wasreached. Where spectra have been included, these have beenfound by taking the FFT of the time domain data providedby PSCAD/EMTDC.

4.2 Simulation resultsThe two models were found to produce practically the sameresults at both fundamental and harmonic frequencies.Consider, for example, the DC side voltage waveformsshown in Fig. 6. The two results are almost identical, with

the exception of extremely high dV =dt regions, these errorsresulting from the truncation of the harmonic spectrum.

The fundamental frequency component and controlvariables are also well matched as indicated by thefundamental frequency results (Table 2). The small differ-ences which exist result primarily from the contrasting loadrepresentation used by the proposed harmonic domain(HDA) model (constant power) and PSCAD/EMTDC(constant impedance).

20

40

60

80

100 1020

3040

5060

7080

90100

00.51.0

numerically derived Jacobian for a PWM STATCOM

Fig. 4 Numerically derived Jacobian (perturbation¼ 10�6)Note that large terms which would obscure the view have been set to 1

Table 1: Test system parameters

System parameters Setting STATCOM parameters 1 2

Power base 20MVA Switching frequency 450Hz 750Hz

Voltage base 11kV DC capacitor Xc �j1.052pu �j1.052pu

System voltage Vinf 1.0pu DC resistor Rc 25pu 12pu

System frequency 50Hz VRMS Set-point 1.0pu 1.0pu

Convergence tolerance 10�6 Transformer Xt 0.1pu 0.1pu

nh 50 Modulation index 1.0 1.0

0.0125 + j0.025Vinf

0.0125 + j0.025

0.5 + j0.4

1

0.05 + j0.1

2

0.5 + j0.4

0.0125 + j0.025

0.5 + j0.4

Vpcc1 Vpcc2

Fig. 5 Single line diagram of multiple STATCOM test system

0.880.900.920.940.960.981.001.02

DC

vol

tage

ST

AT

CO

M1,

pu

time domain comparison of PSCAD and HDA models

0 1 2 3 4 5 6 70.90

0.95

1.00

1.05

1.10

1.15

angle, rad

0 1 2 3 4 5 6 7angle, rad

DC

vol

tage

ST

AT

CO

M2,

pu

HDAPSCAD

Fig. 6 Time domain comparison of DC side voltages

198 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005

The heavily distorted voltage at each of the STATCOMbuses, VPCC, also shows an extremely strong correlation (seeFig. 7).

The strong correlation between the PSCAD/EMTDCand HDA results is further illustrated in Fig. 8, whichshows the harmonic spectrum present at the PCC, Vpcc1.Table 3 gives a quantitative comparison of the two models,by comparing the absolute magnitude of the harmonicvoltages present at the PCC for STATCOM 1. Allharmonics of magnitude 40.001pu have been included,those harmonics smaller than this show larger inconsisten-

cies (up to 10% for nh¼ 50), yet because of their tinymagnitude they have little impact on the result.

High frequency inconsistencies can be reduced byincreasing the highest harmonic of interest, nh; however,this has a considerable impact on the solution time. Forexample, when considering nh ¼ 31, the harmonic domainmodel takes B40 seconds to converge (using a 3GHz PC).If the highest harmonic of interest is doubled, then thesolution time increases to 3min. In comparison in, thePSCAD/EMTDC simulation used for comparison tookB100 s to reach a steady state.

The harmonic domain models computational inefficiencyresults primarily from the use of a numerical Jacobian andthe use of less than optimal code being run in theMATLAB environment, making the model slower thanan equivalent harmonic domain model of an HVDCsystem [16].

5 Conclusions

A unified harmonic domain model capable of representinggeneric systems with multiple distributed converters hasbeen developed and verified. The model has been shown toprovide almost identical results to PSCAD/EMTDC, whilestill having the inherent speed advantages of frequencydomain modelling.

Table 2: Fundamental frequency results

Parameters PSCAD HDA model

Converter 1 Converter 2 Converter 1 Converter 2

jVpcc j 0.9827pu 0.9707pu 0.9828pu 0.9691pu

ffVpcc �3.3261 �10.841 �3.3241 �10.671

Firing angle �33.541 �41.351 �33.531 �41.391

Vdcav0.9628pu 1.0176pu 0.9627pu 1.0146pu

time domain comparison of PSCAD and HDA models

0 1 2 3 4 5 6 7−1.0

−0.5

0

0.5

1.0

angle, rad

0 1 2 3 4 5 6 7angle, rad

volta

ge V

pcc2

, pu

L-G

−1.0

−0.5

0

0.5

1.0

volta

ge V

pcc2

, pu

L-G HDA

PSCAD

Fig. 7 Time domain comparison of AC side voltages

0

0.02

0.04

0.06

0.08

0.10harmonic voltage spectrum at Vpcc

volta

ge m

agni

tude

, V

pu L

-G

0 10 20 30 40 50 60−3

−2

−1

0

1

2

3

harmonic order

0 10 20 30 40 50 60harmonic order

volta

ge a

ngle

, rad

PSCADHDA

Fig. 8 Harmonic domain comparison of AC voltage at PCC1

Table 3: Harmonic frequency results

VPCC

pu

Harmonicorder

PSCAD HDA Error as a% of fun-damental

5 0.00509 0.00516 0.00707

7 0.08608 0.08610 0.00279

11 0.08464 0.08493 0.02910

13 0.02007 0.02031 0.02465

17 0.05602 0.05663 0.06127

19 0.04439 0.04451 0.01238

23 0.04448 0.04471 0.02421

25 0.01555 0.01567 0.01195

29 0.00587 0.00609 0.02296

31 0.01053 0.01095 0.02222

35 0.02023 0.02038 0.01502

37 0.00383 0.00385 0.00168

41 0.02331 0.02376 0.04543

43 0.00214 0.00209 0.00585

47 0.02005 0.02031 0.02697

49 0.00386 0.00404 0.01847

IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005 199

The implementation in MATLAB has resulted in aversatile modular model at the cost of some computationalefficiency, making it slower than equivalent harmonicdomain models for HVDC systems.

By accounting for the nonlinear impact that characteristicharmonics have on the operating point, the model fullyrepresents a basic control scheme, yet lacks the ability tomodel a fundamental frequency system impedance imbal-ance. Possible improvements therefore stem from the needto improve computational efficiency (analytical derivationof the Jacobian and adaptive frequency selection) and fullymodel unbalanced situations (inclusion of a 3-phase powerflow).

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200 IEE Proc.-Gener. Transm. Distrib., Vol. 152, No. 2, March 2005