model order reduction of wind farms: linear...

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Model Order Reduction of Wind Farms:Linear Approach

Husni Rois Ali, Linash P. Kunjumuhammed, Senior Member IEEE, Bikash C. Pal, Fellow, IEEE,Andrzej G Adamczyk Member, IEEE, and Konstantin Vershinin Member, IEEE

Abstract—This paper presents three types of linear modelorder reduction (MOR) technique, namely singular value de-composition (SVD)-based, Krylov-based, and modal truncation-based type applied to large-scale wind farm models. The firsttype includes a Balanced Truncation (BT) and AlternatingDirection Implicit (ADI)-based BT method, while the secondtype encompasses a Rational Krylov (RK), and Iterative RationalKrylov Algorithm (IRKA) method. In the third type, a SubspaceAccelerated MIMO Dominant Pole Algorithm (SAMDP) methodis used. The effectiveness of these methods are tested on practical-sized wind farms with 90, 120 and 210 doubly-fed inductiongenerators (DFIGs). Merits and demerits of each method arediscussed in detail. The reduced order model (ROM) of windfarm is validated against the full order model (FOM) in term offrequency domain indices and waveform agreement at the pointof common coupling (PCC).

Index Terms—wind farm model order reduction, BalancedTruncation, Alternating Direction Implicit-based BT, RationalKrylov, Iterative Rational Krylov Algorithm, Subspace Acceler-ated MIMO Dominant Pole Algorithm (SAMDP).

I. INTRODUCTION

OVER the past twenty years, there has been phenomenalgrowth in installed capacity of wind power worldwide[1]. Such growth has not been possible without meetingvarious challenges. Most often these problems are related togrid interconnection, so analysis of wind farm connected toexternal grids are required to understand such problems andseek appropriate solutions.

A detailed dynamic model of a typical wind turbine genera-tor (WTG) may be described using about 20 differential equa-tions. Inevitably, representation of a medium size wind farmconsisting of about 100 WTGs in power system study requiresthousands of differential equations. With the inclusion ofcollector system component dynamics such as cable sectionsand transformers, the overall model order becomes extremelylarge to handle in power system simulation studies. However,such detailed representation poses computational burden fortypical power system simulation software, preventing the studyof behaviour of wind farms connected to a power systemvirtually infeasible. Hence, a reduced model of wind farm,which can capture important dynamics of the high ordersystem, is of importance for power system study. In addition tocomputational issue, for control engineers, reducing dynamic

Husni Rois Ali, L. P. Kunjumuhammed and B. C. Pal are with the De-partment of Electrical and Electronic Engineering, Imperial College, LondonSW7 2AZ, U.K. (e-mail: {h.ali15;linash.p.k; b.pal}@imperial.ac.uk)

Andrzej Adamczyk and Konstantin Vershinin are with the GE′s GridSolutions, St Leonards Building Harry Kerr Drive Stafford ST16 1WT (e-mail: {andrzej.adamczyk;konstantin.vershinin}@ge.com).

model of complex systems has become an integral part ofcontroller design [2]. A complex model leads to not onlya complex controller design but also a high cost and alsopoor reliability. With a reduced order wind farm model, asimple controller design can be easily pursued. For developersand WTG manufacturers, this reduced model of WF is alsouseful to demonstrate and validate stability of their system andequipment while safe guarding their intellectual properties.

In order to obtain the reduced order model of wind farms,the aggregation-based method, which was originally used insynchronous machines model, is by far the most popularmethod [3], [4], [5]. The method can be further divided intothe single-machine and multiple-machine aggregation. In theformer method [3], it is assumed that the whole wind farmoperates at uniformly in wind speed and power output terms;thus an aggregated WTG is sufficient to represent the dynamicsof wind farms. Since wind farms are normally situated acrossa very large area, this assumption can be hardly fulfilled dueto the wake effects, size of each WTG, etc. Consequently,the single-machine aggregation cannot accurately representthe dynamics of a wind farm. To overcome this problem,the multi-machine representation is used. In this method, theWTGs are grouped according to the similarity of operatingcondition, known as a coherent group, and the generators witha similar operating condition are represented by an equivalentmodel [4], [5]. Despite its popularity in the literature, multi-machine aggregation has several weaknesses: 1) The operatingcondition of wind farm is continuously changing, as a result acoherent group of WTGs may contain different set of gener-ators, 2) Unlike synchronous generators where the concept ofcoherent group is well defined by the rotor angle, in WTGs,this is not easy due to the weak connection between rotorangle and WTG’s output power.

In this paper, the application of linear reduced order model(ROM) techniques, which do not rely on identification ofcoherent group of WTGs, is demonstrated on practical-sizedwind farm models [6]. The methods include Balanced Trun-cation (BT) [7], Alternating Direction Implicit (ADI)-basedBT [8], [9], [10], Rational Krylov (RK) [8], [11], IterativeRational Krylov Algorithm (IRKA) [12], [13] and SubspaceAccelerated MIMO Dominant Pole Algorithm (SAMDP) [14].To the knowledge of the authors, the last four methods arethe first of their kind in wind farm MOR. Applicability andeffectiveness of these five methods are compared and con-trasted. The reduced order models are validated by comparingeigenvalues, bode plots and dynamic response waveform withthose from the the full order model (FOM).

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Wind Farm 1

Wind Farm 2

Area 1 Area 2

Area 1Area 2

WF2 TR2

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WF1 TR2

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WTG TR

0.6/33 kV

WTG 091

cVpccV

ref

pccV

VSC controller

Fig. 1: Wind farm connected to grid

II. APPROACH FOR WIND FARM MODEL ORDERREDUCTION

The basic layout of a 120×5 MW wind farm used in thisresearch is shown in Fig. 1. This layout will be modified inSection V to investigate the reliability of MOR techniquesfor different sizes of test systems. In Fig. 1, triangles indicateDFIG type WTG. The model is divided into two wind farms,namely Wind Farm 1 and 2, where each has several stringsof WTGs connected to a wind farm transformer (WFT). TheWFTs are connected to Voltage Source Converter (VSC)-HVDC using three high voltage cables (HVCs). The DC-link of the HVDC that connects wind farm to main grid isnot considered in this work. The dynamic model of a DFIGuses a 22nd order model. The converter controllers use avector control approach to allow independent control of torqueand reactive power [15]. Each collector system component isrepresented using a fourth order model [6].

The system is modelled using Matlab/Simulink R© and lin-ear representation is obtained using the command linearise.The order of system is 3874 with 1673 conjugate pairs ofeigenvalue. The frequency of the modes ranges from 1.7 Hz to3.5×104 Hz. The lower frequency modes are related to DFIG’smechanical system, while the higher frequency modes are dueto a capacitor-inductor combination in the collector system.The model contains four critical modes listed in Table I. Usingthe q−axis of PCC voltage and d−axis of converter voltage asthe input and output, respectively, these modes are evident assignificant peaks in the Bode plot magnitude given in Fig. 2.Further, a nonlinear time domain simulation is performed byapplying a 1% step change in the PCC reference voltage att = 0.3 s. Fig 3 shows the responses of PCC active power andvoltage, which are dominated by the critical modes.

To perform MOR, the wind farm system is partitionedinto two areas [16], namely, study-area and external-area as

100 102

Frequency (Hz)

-40

-30

-20

-10

0

10

Mag

nitu

de (

dB)

Fig. 2: Bode magnitude plot for full order model of the wind farm

0.4 0.6 0.8 1Time (s)

2.86

2.88

2.9

2.92

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p.u.

)

49.36 Hz mode

(a) Active power at PCC

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tage

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10 Hz mode

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1.138

1.14

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tage

(p.

u.)

219 &315 Hzmode

(b) Voltage at PCCFig. 3: Response due to a fault at PCC voltage reference

illustrated in Fig. 4. The wind farm with the collector systemsis considered as the external-area, and the study-area includesonly the VSC controller and impedance. The external-areais linearised and subsequently reduced using the methodspresented in Section III. The ROM is reconnected to studyarea, and its response is compared to the response of windfarm FOM using the following three criteria:

1) The critical modes of both systems are compared.2) The Bode plot of both systems is compared specifically

around the frequency of critical modes.3) The PCC power and voltage of both system are com-

pared using Mean Square Error (MSE),

MSE =( N∑n=1

T∑t=1

(yn,t − ŷn,t)2/NT)1/2

(1)

where y and ŷ denote output of the FOM and ROM,respectively. N is the number of outputs to be compared,and T is total number of time steps.

III. MODEL ORDER REDUCTION METHODSTo obtain a linear model, the nonlinear model in Fig. 1 is

linearised around an operating condition x(0), y(0) and u(0),ẋ(t) = Ax(t) +Bu(t), y(t) = Cx(t) +Du(t) (2)

where A ∈ Rn×n, B ∈ Rn×m, C ∈ Rk×n, and D ∈ Rk×mare the state, input, output, and feedforward matrices, respec-tively, while x(t) ∈ Rn, y(t) ∈ Rk, and u(t) ∈ Rm denotestate, output, and input vectors, respectively. In the wind farmsystem studied, matrix D is zero. The transfer function of (2)is expressed as,

G(s) =

[A BC D

]= C(sI −A)−1B +D (3)

G(s) is controllable if for an initial states x(−∞), there isinput u(t) that brings the initial states x(−∞) to x(0). This is

TABLE ICRITICAL MODES OF FULL ORDER SYSTEM

Mode Frequency(Hz)

Damping(%)

Dominant states

M1 315 2.9 VPCC, IVSC, VHVC, IWFTM2 220 5.1 VPCC, IVSC, VHVC, IWFTM3 49.36 2.6 IVSCM4 10.33 4.2 IVSC, IWFT, VPCC, VSC controller states

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External area

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Re( )

Im( )

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PCC

V

Vu

Converter

impedancecV

pccV

ref

pccVVSC controller

Re( )

Im( )

PCC

PCC

I

Iy

Fig. 4: Model order reduction approach

equivalent to controllability Gramian P defined by (4) beingpositive definite,

P =

∫ ∞0

e−AtBBT e−AT tdt (4)

P is the solution to the Lyapunov equation,AP +ATP +BBT = 0 (5)

Further by assuming that x(−∞) = 0, P is related to theenergy to control states,

‖u(t)‖22 = x(0)TP−1x(0) (6)

If the SVD is performed on P , small singular values corre-spond to the states which are difficult to control.

Similar to controllability, G(s) is observable if given thevalues of u(t) and y(t) for any t ≥ 0, it is possible todetermine a unique initial state x(0) [17]. This can be checkedby using observability Gramian defined as,

Q =

∫ ∞0

eAT tCTCAAtdt (7)

If Q is positive-definite, then the system is observable. Q isthe solution to the Lyapunov equation,

ATQ+QA+ CTC = 0 (8)It is related to the energy observation of states. Suppose that

an autonomous system is released from initial states x(0), theenergy of y(t) can be computed using,

‖y(t)‖22 = x(0)TQx(0) (9)

If SVD is performed on Q, small singular values are relatedto the states which are difficult to observe.

To obtain ROM of G(s), the state variables x(t) ∈ Rn isprojected into a new set of state variables x̂(t) ∈ Rr withr

4

Using (21) with the proper shift parameters αi, P can befound efficiently. Further, realizing the fact that in BT, theaim is to calculate a low-rank factorization of P i.e. R as in15, it is possible to modify (21) to directly calculate R [19].Assuming that R0 = 0 and RiRTi = Pi, then (21) can berewritten as,R1 =

√−Re(α1)(A+ α1In)−1B

Ri =[√−Re(α1)(A+ α1In)−1B, (A+ αiIn)−1(A− ᾱiIn)Ri−1

](22)

The shift parameters used in (37)-(21) are obtained fromthe solution of a min-max problem,

αi = minαi∈C

5

0

20

40

10-1

100

102

-90

0

90

Full

6th

18th

33rd

Frequency (Hz)

M

ag (

dB

); P

hase (

deg)

Fig. 6: Bode plot of external-area from Vqrefpcc to Idpcc: FOM vs ROM by BT

-40

-20

0

20

10-1 100 102

-900

90

Fullreduced

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)

(a) Bode plot from Vdrefpcc to Vqpcc

-80 -60 -40 -20 0Real axis

0

100

200

300

Imag

inar

y ax

is (

Hz)

FullReduced

(b) Eigenvalue plotFig. 7: Frequency domain comparison of complete system: FOM vs ROM byBT

Let G(s) be expressed in terms of pole-residue representa-tion,

G(s) =

n∑i=1

cibTi

s− λi=

Ris− λi

(30)

where AXv = Xvdiag(λ1, ..., λn), ci = CXvei, andbi = (e

Ti Y

vB)T with Y v = Xv−1. Columns of Xv andY v are constructed from the right and left eigenvectors of λ,respectively. A triplet {λi, xvi , yvi } is commonly referred as aneiganpair. Ri is known as the residue of λi. The IRKA triesto find Ĝ(s),

Ĝ(s) =

r∑i=1

ĉib̂Ti

s− λ̂i=

R̂i

s− λ̂i(31)

which satisfies (29). The first order optimality conditions forthis problem is proposed in [24]. If Ĝ(s) is the solution of(29), then the following conditions hold,

ĉTi G(−λ̂i) = ĉTi Ĝ(−λ̂i), G(−λ̂i)b̂i = Ĝ(−λ̂i)b̂iĉTi G

′(−λ̂i)b̂i = ĉTi Ĝ

′(−λ̂i)b̂i, ∀i = 1, . . . , r

(32)

where λ̂i are poles of Ĝ(s) with associated residues ĉi, b̂i.Therefore, equation (32) suggests that the interpolations pointsshould be selected from mirror image the eigenvalues of ROM,while the corresponding tangential directions are selected fromthe associated left and right eigenvector.

E. Modal Truncation

In the IRKA, one tries to find R̂i and λ̂i in (31), so that(29) is minimised. On the other hand, the modal truncationconstructs a ROM so that (31) preserves r most dominanteigenvalues of (30). In this way, system stability is alwaysguaranteed to be preserved. One way to define dominantmodes is through the corresponding residue Ri of λi. A modeλi is called dominant if it has a large ratio of ‖Ri‖2 /|Re(λi)|;thus, the modal truncation selects the r largest ratio.

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FullReduced


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tage

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FullReduced

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tage

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(b) Voltage at PCCFig. 8: Time domain comparison of complete system: FOM vs ROM by BT

When the modal truncation is applied to a large-scalesystem, the calculation of all eigenpairs {λi, xvi , yvi } andthen selection of dominant modes based on the ratio of‖Ri‖2 /|Re(λi)| become extremely expensive. To overcomethis, the eigenpairs {λi, xvi , yvi } should be calculated one-at-a-time until r eigenpairs are found. In [14], a new method knownas a Subspace Accelerated MIMO Dominant Pole Algorithm(SAMDP) has been successfully used to extract dominantmodes of large power system models. This method startsfrom an initial value of a pole and performs iterations until apole with largest ratio of ‖Ri‖2 /|Re(λi)| is found. Duringthe iteration, the corresponding right and left eigenvectorsare also calculated. Once this pole has been found, it isdeflated to prevent the iterations to converge to the same pole.This process is repeated until r eigenpairs are found. Havingcompleted this process, the right and left eigenvectors are putinto columns of V and W , respectively, as follows,

W =

yv1∣∣∣∣∣ · · ·

∣∣∣∣∣ yvr , V =

xv1∣∣∣∣∣ · · ·

∣∣∣∣∣ xvr (33)

The ROM is obtained by applying V and W to the FOMas in (11).

IV. WIND FARM MODEL ORDER REDUCTION

A. Reduced order model using BT

Prior to reducing the system using the BT, the external-areaneeds to be linearised. This produces a linear system with anorder of 3868. Furthermore, it has two inputs (d and q-axisPCC voltage) and also two outputs (d and q-axis PCC current).The study-area, by contrast, is retained using nonlinear model.

The order of reduced model is easily determined throughinspection of HSV plot shown in Fig. 5. There are significantdrops in the magnitudes after the 6th, 18th and 33rd HSV, andthey indicate as appropriate orders of reduced system. TheBode plots from Vqrefpcc to Idpcc for these three ROMs arecompared with that of the FOM of external-area in Fig.6.It is evident that the 6th order system is able to capture thepeak at frequency 47.9 Hz, whereas it performs very poorlyfor other frequency range. Increasing order to 18th and furtherto 33rd can significantly enhance the accuracy of ROM. Thisis consistent with the theory of the twice sum of the tail ruleof BT in (19).

It is important to pause here and discuss stability preserva-tion of BT. As previously discussed, the ROM obtained fromBT preserves stability of the FOM if ΣH1 and ΣH2 in (16)have no common terms or entries. In other words, the stability

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TABLE IIPERFORMANCE OF ALL METHODS

Model Order Elapsedtime (s)

Extractiontime(s)

MSE PPCC MSE VPCC

Full nonlinear 3874 78.18 - - -Full linear 3874 68.54 - - -BT 39 0.27 433 3.77×10−4 2.96×10−5

ADI 39 0.30 19.18 4.80×10−5 2.05×10−6

RK 42 0.44 0.32 4.3×10−3 9.9×10−4

IRKA 42 0.23 7.41 1.45×10−5 1.52×10−5

SAMDP 307 5.10 31.17 2.42×10−4 1.61×10−5

is always preserved when the HSVs are distinct. It can beobserved that in case of wind farm model used in this study,the HSVs are always unique. Although Fig. 5 shows that someHSVs appear to be the same, in fact they slightly differ. Hence,this makes it possible to always obtain a stable ROM of windfarm using BT. This is also supported by a large number ofexperiments. For further analysis, the 33rd order is selected.

In order to study the complete system, the 33rd order isreconnected to the study-area and the combined system isthen compared to the FOM in term of Bode plot, eigenvalueplot, and waveform agreement. The Bode plots from Vdrefpccto Vqpcc are given in Fig. 7a and it is clear that they matchvery accurately. The same information is also conveyed by theeigenvalue plot in Fig. 7b in which the four critical modes canbe accurately retained in the ROM. Further, multiple modesat the frequency close to 50 Hz of the FOM are replacedonly by just two modes in the ROM. To investigate waveformagreement at the PCC, the study-area is perturbed with a1% step change in the PCC reference voltage at t = 0.3 s.The responses of power and voltage waveform for both fulland reduced order are depicted in Fig. 8, which indicates theROM is able to mimic the oscillations of FOM very accurately.The corresponding MSE of power and voltage given in TableII demonstrates accuracy of the ROM. Further, Table II alsoshows comparison of the elapsed time in order to perform aone second simulation. As anticipated, for the same simulationduration, the 33rd order system only requires 0.27s, while thenonlinear FOM and linear FOM require 78.18s and 68.54s,respectively.

However, to solve the Lyapunov equations in the BT provescomputationally prohibitive for large-scale systems. For theapplication in wind farm MOR exercise, this computationalcost is even more expensive than simulating the full ordersystem for one second as illustrated in Table II. This showsthat a special method is required to calculate the solution ofLyapunov equation more efficiently, enabling extension of BTto the high order model of wind farm.

0 20 40State

0

50

100

HS

V 6th

order

0 20 40State

0

0.05

0.1

HS

V

18th

order

33rd

order

Fig. 9: HSV plot of external area using ADI

0

20

40

10-1 100 102

-900

90

Full

6th

18th

33rd

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)

Fig. 10: Bode plot of external-area from Vqrefpcc to Idpcc: FOM vs ROM byADI.

B. Reduced order model using ADI-based BT

The bottleneck of BT is solving the Lyapunov equations.To overcome this, an Alternating Direction Implicit (ADI)iteration is used to solve these equations. A set of shift pa-rameters need to be carefully selected to ensure a satisfactoryperformance of the ADI method. In this paper, the heuristicapproach proposed in [20] is adopted due to its simplicityand Matlab implementation is given in [25]. Since calculatingthe spectrum of A is computationally prohibitive for largesystems, the method estimates these values using the Arnoldiiteration. The spectrum with large magnitudes is determined byperforming Arnoldi for 100 iterations, while the spectrum withsmall magnitudes is calculated by running the same algorithmfor 50 iterations and then calculating inverse of the result.The final spectrum, λ, is union of the results from these twoprocesses, but choosing only the values with negative realpart. The shift parameters, αi, is then selected from this finalestimation by solving (23) heuristically. The number of shiftparameters is set to be 40.

Having obtained the shift parameters, the ADI algorithm isrun for 500 iteration. Since there are fewer number of shiftparameters than the number of iterations, the shift parametersare used in a cycle. The estimated HSV of the external-area isgiven in Fig. 9. Although the magnitude is different from theexact HSV in Fig. 5, the same orders as in the case of BT, 6,18, and 33, are selected for the sake of fair comparison. Thecorresponding HSV is given in Fig. 9. The Bode plots fromVqrefpcc to Idpcc of these three ROMs are compared to that of theFOM of external-area in Fig. 10. The 6th order system is onlyable to capture the peak of full order system at frequency of47.9 Hz. By increasing the order to 18 and 33, more accurate

-40

-20

0

20

10-1 100 102

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Fullreduced

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)


-80 -60 -40 -20 0Real axis

0

100

200

300

Imag

inar

y ax

is (

Hz) Full

Reduced

(b) Eigenvalue plotFig. 11: Frequency domain comparison of complete system: FOM vs ROMby ADI

7

0.4 0.6 0.8 1Time (s)

2.86

2.88

2.9

2.92

2.94

2.96

2.98R

eal p

ower

(p.

u.)

FullReduced


0.4 0.6 0.8 1Time (s)

1.134

1.136

1.138

1.14

1.142

Vol

tage

(p.

u.)

FullReduced

0.28 0.3 0.32 0.34 0.36Time (s)

1.134

1.136

1.138

1.14

1.142

Vol

tage

(p.

u.)

FullReduced

(b) Voltage at PCCFig. 12: Time domain comparison of complete system: FOM vs ROM by ADI

result can be attained. Further analysis is carried out only to the33rd reduced order system. The Bode and eigenvalue plots ofthe 33rd ROM when connected to the study-area are validatedagainst the FOM and the results are given in Fig. 11. It isobvious that the critical modes are preserved in the ROM.This result is also consistent with the time domain simulationwhen the study area is perturbed by the same fault as in thecase of BT. Fig. 12 shows that the ROM behaves very similarto the FOM. The MSE of power and voltage at the PCC asgiven in Table II.

With regard to stability preservation, although to the knowl-edge of the authors, there is no guarantee of stability preserva-tion using the ADI method, authors have never found a case inthis research where the ROM of wind farm is unstable. Even ifa small number of Arnoldi iterations and shift parameters areused, a stable ROM has always been obtained. For instance,when the number of Arnoldi iterations are set to be 10 and5, while the number of shift parameters are set to be 5, thereduced models with order 6, 18, and 33 are all stable.

The performance comparison above clearly indicates thatthe ADI based BT is able to attain comparable accuracy to theresults of BT. However, the main advantage of ADI iterationis the ability to solve Lyapunov equations efficiently for largescale systems; thus giving rise to a significant reduction inextraction time as given in Table II.

C. Reduced order model using RK

In the RK, it is necessary to provide a set of interpolationpoints where the moments of FOMs are preserved. It has beensuggested in [13] that one can choose the interpolation pointsfrom the mirror image of poles of G(s), specially those withthe highest residue. For the wind farm model used in thispaper, those poles are given in Table. I.

Three scenarios of moment matching shown in Table. III areconsidered to examine the influence of interpolation points onRK accuracy. The first scenario, RK1, aims to match momentsat two critical modes, namely, M3 and M4. In the RK2, morepoints are added to match the moments of high order system atall critical modes M1-M4. Finally in RK3, a similar scenarioto RK2 is considered with addition of some points, which arenot the dominant modes, to enhance accuracy of the ROM

TABLE IIIINTERPOLATION POINT SCENARIO FOR RK

Scenario Order Interpolation pointsRK1 8 8.17 ± 310.17i (P1), 2.72 ± 64.94i (P2)RK2 20 P1, P2, 58.50 ± 1979.16i (P3), 70.83 ± 1379.46i (P4)RK3 36 0,±1i, ±10i, ±40i, P1, P2, P3, P4

0204060

10-1 100 102

-900

90

FullRK1RK2RK3

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)

Fig. 13: Bode plot of external-area from Vqrefpcc to Idpcc: FOM vs ROM byRK

-40

-20

0

20

10-1 100 102

-900

90

Fullreduced

Frequency (Hz)

Mag

(dB

); P

hase

(de

g)(a) Bode plot from Vdrefpcc to Vqpcc

-80 -60 -40 -20 0Real axis

0

100

200

300

Imag

inar

y ax

is (

Hz) Full

Reduced

(b) Eigenvalue plotFig. 14: Frequency domain comparison of complete system: FOM vs ROMby RK

at low frequency. These scenarios are carefully selected topreserve the stability of the external-area as it is well-knownthat the RK does not preserve stability. Although it is found inthis paper that selecting interpolation points from a subset ofthe negative poles of G(s) often produces a stable ROM, thisapproach sometimes produces an unstable reduced system. Forexample, if the interpolation point are 2.72±64.94i (M2) and70.83± 1379.46i (M4), the ROM has two unstable modes.

The Bode plots from Vqrefpcc to Idpcc of the RK1-RK3are compared to that of the FOM of external-area in Fig.13. Although the both RK1 and RK2 are able to produceaccurate Bode plot response at the preselected points, theyare erroneous at low frequency range. With the incorporationof interpolation points at low frequency, RK3 produces moreaccurate results. This shows that the choice of interpolationpoints significantly affects accuracy of the ROM. For furtheranalysis, RK3 is selected.

The Bode and eigenvalue plots when the RK3 is reconnectedto the study-area are shown in Fig. 14. Both plots indicatethat the ROM is generally able to capture the critical modes ofFOM. However, it can be seen that there are small errors in theBode plot. Further, the time domain simulation is performedby using the same scenario of fault as in the previous sectionsand the responses of power and voltage at PCC are given inFig. 15. It is likely that the errors previously shown in the Bodeplot might influence accuracy of time-domain simulation. TheMSE for both power and voltage at PCC is slightly largerthan in the previous cases as indicated in Table. II. Finally, itremains to be seen that the main advantage of RK as shown inTable II is a significant reduction in the extraction time, whichaccounts only for 0.32 s.

8

0.4 0.6 0.8 1

Time (s)

2.86

2.88

2.9

2.92

2.94

2.96

2.98R

ea

l p

ow

er

(p.u

.)

Full

Reduced


0.4 0.6 0.8 1

Time (s)

1.134

1.136

1.138

1.14

1.142

Voltage (

p.u

.)

Full

Reduced

0.28 0.3 0.32 0.34 0.36

Time (s)

1.134

1.136

1.138

1.14

1.142

Voltage (

p.u

.)

Full

Reduced

(b) Voltage at PCCFig. 15: Time domain comparison of complete system: FOM vs ROM by RK

0

20

40

10-1 100 102

-900

90

FullIRKA1IRKA2IRKA3

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)

Fig. 16: Bode plot of external-area from Vqrefpcc to Idpcc: FOM vs ROM byIRKA

D. Reduced order model using IRKA

The performance of RK methods is determined by selectionof the interpolation points, which is often difficult to do. Thisis exacerbated by the fact that the operating condition ofwind farms continuously changes, which in turn affects thedominant modes. With the aim to overcome these difficulties,the iterative rational Krylov algorithm (IRKA) is proposed toreduce the model of wind farm. In this way, a set of initialinterpolation points with corresponding tangential directionsare updated until an optimal point, which solves (29), is found.

To demonstrate the effectiveness of IRKA, three ROMs,namely, IRKA1, IRKA2, and IRKA3 are considered. Thesemodels have the same order as RK1, RK2 and RK3, respec-tively. The interpolation points with the corresponding rightand left tangential directions for these models are selectedrandomly. Using these random parameters, the IRKA is able toconverge within 16, 81, and 23 iterations and requires elapsedtime of 2.51s, 19.41s, and 7.41s for IRKA1, IRKA2, andIRKA3, respectively. There are two stopping criteria adoptedin this study, namely, the H2-norm for interpolation pointsand reduced system. If any of these criteria falls below the

-40

-20

0

20

10-1 100 102

-900

90

FullReduced

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)


-80 -60 -40 -20 0Real axis

0

100

200

300

Imag

inar

y ax

is (

Hz)

FullReduced

(b) Eigenvalue plotFig. 17: Frequency domain comparison of complete system: FOM vs ROMby IRKA

tolerance, then the IRKA iteration stops. The final error for theinterpolation points and system are 2.0×10−4 and 2.1×10−3for IRKA1, 8.8 × 10−3 and 8.5 × 10−4 for IRKA2, and 2.6and 8.7×10−4 for IRKA3. The Bode plots from Vqrefpcc to Idpccfor these ROMs are compared to that of the FOM of external-area in Fig. 16, which in general shows superior performanceof the IRKA to the RK. It is also obvious that the IRKA3produces more accurate results than IRKA1 and IRKA2. Forfurther analysis, only the IRKA3 is considered.

The response of complete ROM with IRKA3 is comparedto the responses of FOM. The Bode and eigenvalue plots aregiven in Fig. 17 and it shows generally better performancesof IRKA than RK. This result is also compatible with thetime-domain simulation when the system is subjected by thesame disturbance as in the previous sections. The power andvoltage responses at the PCC are given in Fig. 18 with thecorresponding MSE equal to 1.45×10−5 and 1.52×10−5,respectively, as shown in Table II. It is to be noted that interm of stability preservation, to the knowledge of authors, theIRKA essentially does not preserve stability of the FOM. Thisis observed from a large number of experiments that unstableROMs are sometimes generated from a stable FOM.

E. Reduced order model using Modal TruncationIn this section, application of the SAMDP method to obtain

ROM of the wind farm is investigated. Different from the pre-vious methods, it is found that the SAMDP cannot be applieddirectly to the original linearised model of the external-area.It is because there is a problem with eigenvalue condition-ing. This can be studied through the condition number of theeigenvector matrix, i.e.

cond(Xv) = ‖Xv‖2∥∥∥Xv−1∥∥∥

2= ‖Xv‖2 ‖Y

v‖2 (34)

The original matrix has this number around 1250, which in-dicates that the eigenvalues are very sensitive to any changesin elements of A, for instance due to rounding error. As theSAMDP calculates explicitly the eigenpair {λi, xi, yi}, thismethod cannot converge unless cond(Xv) is reduced. To dothis, a similarity transform is performed,

Asim = T−1AT (35)

where Asim has equal row and column norm whenever pos-sible. This is carried out by using a Matlab’s command bal-ance. Performing this step, cond(Xv) decreases significantlyto 803.80, which in turn enables the SAMDP to converge.

Three scenarios are considered to test the performance ofSAMDP, namely SAMDP1 (18 dominant modes), SAMDP2

0.4 0.6 0.8 1Time (s)

2.86

2.88

2.9

2.92

2.94

2.96

2.98

Rea

l pow

er (

p.u.

)

FullReduced


0.4 0.6 0.8 1Time (s)

1.134

1.136

1.138

1.14

1.142

Vol

tage

(p.

u.)

FullReduced

0.28 0.3 0.32 0.34 0.36Time (s)

1.134

1.136

1.138

1.14

1.142

Vol

tage

(p.

u.)

FullReduced

(b) Voltage at PCCFig. 18: Time domain comparison of complete system: FOM vs ROM byIRKA

9

0

20

40

10-1 100 102

-900

90

FullSAMDP1SAMDP2SAMDP3

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)

Fig. 19: Bode plot of external-area from Vqrefpcc to Idpcc: FOM vs ROM bySAMDP

-40

-20

0

20

10-1 100 102

-900

90

FullReduced

Frequency (Hz)

M

ag (

dB);

Pha

se (

deg)


-80 -60 -40 -20 0Real axis

0

100

200

300

Imag

inar

y ax

is (

Hz) Full

Reduced

(b) Eigenvalue plotFig. 20: Frequency domain comparison of complete system: FOM vs ROMby SAMDP

(40 dominant modes), and SAMDP3 (180 dominant modes).The orders of these models are 36, 74, 301 for SAMDP1-SAMDP3, respectively and an initial value of the pole is setto be 1i for all cases. The Bode plot from Vqrefpcc to Idpcc forthese models are compared with that of the FOM of external-area in Fig. 19. It is obvious that to improve accuracy ofSAMDP, higher number of poles need to be incorporated.Notice that SAMDP requires significantly higher order thanthe other methods in order to obtain comparable accuracy. Itis due to the fact that the SAMDP gives a high priority to themodes whose residues are high. However, it is true that thesemodes might not be always important for system dynamics;hence selecting these modes does not always improve accuracyof the ROM. Another reason is that the SAMDP extracts allpoles which are close each other. The SAMDP3 is selected forfurther analysis. Fig 20 shows the performance of SAMDPin frequency domain. It is obvious that both the Bode andeigenvalue plot are able to capture the responses of FOM.This translates into time domain performance as illustrated inFig. 21, where very small errors are observed in the waveformof power and voltage at the PCC. The values of MSE for theseplots are given in Table II. Further increase the order shouldminimise the error further.

There are two important technical details to be discussed.Firstly, the SAMDP is able to efficiently extract 180 modeswithin only 31.17 s, see Table II. Secondly, due to the factthat the SAMDP extracts the poles of FOM one-at-a-time,stability of ROM is always guaranteed whenever the FOM isstable, which is confirmed by experiments.

V. ROM OF DIFFERENT ORDER OF WIND FARM MODELS

This section presents performance of the MOR methods ondifferent wind farm models. Two new wind farm models are

0.4 0.6 0.8 1Time (s)

2.86

2.88

2.9

2.92

2.94

2.96

2.98

Rea

l pow

er (

p.u.

)

FullReduced


0.4 0.6 0.8 1Time (s)

1.134

1.136

1.138

1.14

1.142

Vol

tage

(p.

u.)

FullReduced

0.28 0.3 0.32 0.34 0.36Time (s)

1.134

1.136

1.138

1.14

1.142

Vol

tage

(p.

u.)

FullReduced

(b) Voltage at PCCFig. 21: Time domain comparison of complete system: FOM vs ROM bySAMDP

TABLE IVMOR ON DIFFERENT WIND FARM MODELS

Method Performance System 1 (2902) System 2 (3874) System 3 (6770)

BT

Order 39 39 51Extraction 186.42 s 433.00 s 2168.55 sMSE PPCC 8.35×10−6 3.77×10−4 2.33×10−4MSE VPCC 1.39×10−6 2.96×10−4 1.14×10−5MSE Bode 5.29×10−4 8.10×10−3 5.20×10−3

ADI


RK


IRKA


SAMDP


obtained from the modification of the wind farm shown in Fig.1. Firstly, the Wind Farm 2 of Fig. 1 is removed, which yieldsa new wind farm with an order of 2902. This wind farm hasa total of 90 DFIGs and is referred as System 1. Secondly,the Wind Farm 1 of Fig. 1 is duplicated, resulting in a newsystem with a total of 210 DFIGs and an order of 6770. Thissystem is termed as System 3. The original wind farm modelconsisting of 90 DFIGs is referred as System 2.

Table IV shows performances of the five methods on theSystem 1, System 2, and System 3. Note that the results forSystem 2 are presented again here for the sake of clarity. It isobvious that the BT consistently has the highest computationtime for all systems. The increase in computational time ofBT is at a rate of around O(n3), which stems from solvingthe Lyapunov equations. This practically limits the use ofBT for wind farm MOR. The introduction of ADI methodto solve these equations reduces the computational time toapproximately O(n) as evident from Table IV. The accuracyof ADI in all cases is practically similar to that of BT. It isalso confirmed that unstable ROMs calculated from ADI arenever found in all wind farm models.

Table IV also indicates that the IRKA outperforms the RK inboth time and frequency domain for all test systems, but theIRKA requires significantly higher computational time thanthe RK. This observation is more pronounced as the system

10

size increases. The computational time of RK in all casesis less than 0.4 s. But, it is to be noted that the RK oftenproduced unstable ROMs and stability preservation becomesmore difficult for larger systems. In particular for System 3,the ROM are unstable for many cases and only the reducedmodel with an order of 28 is stable. For the IRKA, stabilityfor System 3 is essentially also not always preserved. Finally,in order to achieve comparable accuracy to other methods, theSAMDP constantly requires higher order that other methodsand always produces stable ROMs.

VI. CONCLUSIONS

This paper describes linear MOR techniques for wind farmsystems, namely BT, ADI-based BT, RK, IRKA, and SAMDP.The models are practically-sized wind farm systems with 90,120, and 210 DFIGs. Due to solving Lyapunov equations, theapplication of BT to the wind farm model is computationallyexpensive. With the aim of reducing the computational time,the ADI-based BT solves Lyapunov iteratively. This methodcan attain accuracy of BT at much lower computational cost.In RK, the moments of full order system are retained in thereduced order system at a set of pre-specified interpolationpoints. These points determine the accuracy of the reducedorder model and therefore should be chosen carefully at mirrorimage of the dominant modes. As the operating conditionof wind farms changes, the interpolation points need to bechanged accordingly, making this a very difficult task to do.To overcome these problem, the IRKA iteratively calculatesa set of initial interpolation points to minimise error betweenFOM and ROM. Finally, the SAMDP is able to extract thepoles of FOM one-at-a-time to construct the ROM, but itstrongly depends on the eigenvalue conditioning on systemsmatrices. Although the comparison between full and reducedorder model in term of Bode plot, eigenvalue plot, and andwaveform at the PCC show that all methods can provide anacceptable reduced order model, it seems that IRKA is themost suitable method for large-scale wind farm applications.This is due to the fact that IRKA is able to provide a veryaccurate model within reasonable extraction time and order.Furthermore, IRKA is also less sensitive to initial parameterset up.

APPENDIX ASOLVING LINEAR EQUATIONS USING ADI ITERATION

let’s define a linear system of equations [19],Ax = b (36)

with A being a symmetric positive definite matrix. Matrix Acan be factorized as a sum of two positive definite matrices, Land R. As a result, (36) can be rewritten as Lx+Rx = b. Thisequation can be solved using ADI iteration by performing thefollowing routines,

(L+ αiIn)xi− 12

= (αiIn −R)xi−1 + b(R+ αiIn)xi = (αiIn −R)xi− 1

2+ b

(37)

where αi are ADI shift parameters. A careful selection ofthese parameter is critical for ADI iteration to converge to thesolution x.

ACKNOWLEDGMENT

The first author fully acknowledges the Indonesia Endow-ment Fund for Education (LPDP) scholarship, Ministry ofFinance, The Republic of Indonesia for funding his PhDresearch at Imperial College London. The authors wouldalso like to acknowledge Dr. Nelson Martins of CEPEL forproviding technical discussion on the SAMDP method.

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IntroductionApproach for wind farm model order reductionModel order reduction methodsBalanced truncations (BT)Alternating Directions Implicit (ADI)-based BTRational Krylov (RK)Iterative Rational Krylov Algorithm (IRKA)Modal Truncation

Wind farm model order reductionReduced order model using BTReduced order model using ADI-based BTReduced order model using RKReduced order model using IRKAReduced order model using Modal Truncation

ROM of different order of wind farm modelsConclusionsAppendix A: Solving linear equations using ADI iterationReferences

model order reduction of wind farms: linear...

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