1

Comprehensive Phase-Domain Model of a WindPower Generator and a STATCOM for Reactive

Power SupportL. Contreras-Aguilar, T. Venegas, J. Arroyo, R. J. Betancourt, J. H. Aguirre-Ruiz and R. Rodríguez-Flores

Abstract—This paper presents a comprehensive phase-domainmodel of a wind-power generator and a STATCOM for reactivepower support. The model includes the implementation of awind and turbine models, while the generator model is studiedin the natural coordinate reference frame (abc). In order togive reactive power support a STATCOM system was embeddedto the wind-power generator and implemented in the sameframe. Transformers and lines models for coupling purposes arealso included. Besides, a per-unit formulation is used in orderto ensure a high efficiency during simulation, this due to thestiffness problems associated to discontinuities of the VSC in theSTATCOM system. The scheme implemented was validated withSimulink of MATLAB and good agreement is observed with thewaveforms of the selected signals. Finally, a 60 seconds simulationwith large variability of wind speed is implemented to test case.

Index Terms—Wind farm, STATCOM system, phase-domainmodelling, time domain.

I. INTRODUCTION

Nowadays renewable energy technologies are having in-creasing presence in electric power systems around the world.In this context, wind power generation is the technologywhich has experienced the fastest growing among all types ofrenewable technologies currently being investigated [1]. Suchthat, the integration of large wind parks in power systems willaffect considerably the dynamic behavior of the system, sincewind based generation systems and conventional systems withsynchronous generators present inherently different dynamiccharacteristics [2].

The first practical experiences of the integration of windpower comes from regions like West Denmark, the northof Germany and Galicia in Spain [3], but is still ratherlimited in most of countries. Researches in wind power havebeen addressed in many directions, for example, in differentaspects related to the design and capabilities of the inductionmachines that are commonly used. Early wind parks employedInduction Generators of Squirrel Cage (SCIG), whereas theuse of a Doubly-Fed Induction Generator (DFIG) and powerelectronics in wind parks is already a fact. A comparativestudy of these technologies is presented in [4]. Anotherimportant research direction of wind power has been focusedon controlling the optimum operation of the wind turbines. A

L. Contreras-Aguilar, T. Venegas, J. Arroyo and R. J. Betancourt are withthe Facultad de Ingeniería Mecánica y Eléctrica, Universidad de Colima,28400 Coquimatlán, Colima, México. (Phone: +52 (312) 3161000).J. H. Aguirre-Ruiz and R. Rodríguez-Flores are students at the Facultad deIngeniería Mecánica y Eléctrica, Universidad de Colima, 28400 Coquimatlán,Colima, México.

control strategy which enables the DFIG-based wind farms toparticipate in voltage control and network damping followinga disturbance is presented in [5]. The implementation of PowerSystem Stabilizer (PSS) for a DFIG is also presented. Usuallythere is not a maximum accepted level of wind penetration.The limit for a particular grid will depend on the existinggenerating plants, pricing mechanisms, capacity for storage ordemand management and other factors [1].

On the other hand, the demand of reactive power in windfarms is a topic of interest, mainly because the main technol-ogy used for wind generator is the induction machine. There-fore, capacitor banks and the FACTS devices, such as STAT-COM (Static Synchronous Compensator) and SVC (StaticVAR Compensator), play an important role in the operationof the facilities of modern wind farms. Recent contributions[6-10] have been reported advantages and disadvantages ofthese two FACTS devices on wind farm installations in whichthe dynamic performance based on STATCOM has prevailedover the SVC device. For this reason, this paper has opted toinclude the STATCOM as reactive power support.

The modeling of wind parks in dynamic and steady statehave been a topic of interest by many research groups aroundthe world [11], [12]. Thanks to the help of different tools,it is now possible to develop models and to simulate withhighly accurately the proposed approach. In such a way,comprehensive model of wind power system including aSTATCOM system is implemented. The paper focuses onlyin the modeling of components in the natural reference frameor phase domain. So that, the controls systems, such aspitch angle control in wind power system and voltage controlin STATCOM system, are not included for this work. Asimulation of 60 seconds is developed in order to observe theeffect of stochastic wind speed within of wind power system.

II. COMPREHENSIVE WIND POWER GENERATOR MODELINCLUDING A STATCOM

Figure 1 shows the general configuration of the comprehen-sive model including the wind power generator and STATCOMsystem, which is implemented in this paper. This configurationconsists a set of independent models. e.g. wind model, turbinemodel, generator model, coupling transformer model, STAT-COM model and transmission lines model. The generator ismodelled with a squirrel cage induction generator of full-order,where the state variables are the stator and rotor currents,speed and angle of rotor position. The STATCOM system is

2

Infinite

bus

Transformer

model

Wind

model

Turbine

model

G

STATCOM

model

Generator

model

Transmission

line model

DC link

PCC

Figure 1. Scheme of the wind generator and STATCOM system.

implemented by using a wye-delta connection transformer inorder to coupling the voltage source converter (VSC) to thepoint of common coupling (PCC).

A. Wind model

The wind speed model used in this paper is reported in [1],and includes the sum of four components,

vw(t) = vwa(t) + vwr(t) + vwg(t) + vwt(t) (1)

where vwa(t) is the wind speed average, vwr(t) is the rampcomponent, vwg(t) is the gust component and vwt(t) is theturbulence component. Detail of equations for implementingthese components are shown in [1], [13], [14].

B. Turbine model

The well-known relation between mechanical power ex-tracted from the wind speed is [1],

Pm = cp(λ, β)ρAv3w (2)

where Pm is the mechanic output power from the turbine,cp is referred to as the coefficient of efficiency of power, ρairis the air density, A is the turbine swept area and vw is thewind speed.

When is used the per unit value, the equation (2) can benormalized as,

Pm−pu = kpcp−puv3w−pu (3)

where the pu values are computed by using base quantitiesand kp is a power gain with value kp ≤ 1, in this paperkp = 0.73.

The coefficient of efficiency of power cp(λ, β), which isrelated with the tip speed ratio λ and its blade pitch angle βis,

cp(λ, β) = c1(c2λi

− c3β − c4)e−c5λi + c6λ (4)

with

1

λi=

1

λ− 0.08β− 0.035

β3 + 1(5)

vs

a

vs

b

vs

c

wr

vr

av

r

b

vr

c

is

a

is

c

is

b

Figure 2. Equivalent circuit for three-phase, wye connected squirrel cageinduction machine.

where c1−6 are constant values chosen fromSimulink/MATLAB®. From cp − λ characteristic withβ = 0° and λ = 8.1 the nominal value of cp−nom is equal to0.48 from Simulink/MATLAB®, Then

cp−pu =cp(λ, β)

cp−nom(6)

The mechanical torque of the wind turbine is defined as[15],

Tm−pu =Pm−pu

ωr−pu(7)

On the other hand, for wind power generator the mostcommon topology is the horizontal-axis wind turbine. It isa lift based wind turbine with very good performance, thus itis a popular for wind power applications. In this configura-tion, the mechanical coupling system turbine-generator can bedescribed by means of one-mass equivalent model [16],

1

Htotal=

1

Htur+

1

Hgen(8)

where Htur is the inertia of wind turbine, Hgen is the inertiaof generator and Htotal is the total inertia used to compute thespeed of each wind power generator.

C. Generator model

Figure 2 shows the three-phase squirrel cage inductionmachine implemented in this paper. Voltage equations forphases a, b, c can be expressed as a per unit quantity [15],

vabcs,r = rs,riabcs,r +

1

ωb

d

dtψabcs,r (9)

where the superscript abc denotes the phases of the system,the subscript s, r indicates the stator and rotor variables, ψabcs,rare the stator and rotor flux linkage, iabcs,r are the stator androtor currents, rs,r are the stator and rotor resistances, vabcs,r

are the stator and rotor voltages and ωb is the base angularvelocity. The flux linkages ψ can be related to the currents bymeans of the inductance matrix L as follows,

ψabcs,r = L·iabcs,r (10)

with

L =

[Ls LsrLTsr Lr

](11)

3

and

Ls =

Lls + Lm − 12Lm − 1

2Lm− 1

2Lm Lls + Lm − 12Lm

− 12Lm − 1

2Lm Lls + Lm

(12)

Lr =

Llr + Lm − 12Lm − 1

2Lm− 1

2Lm Llr + Lm − 12Lm

− 12Lm − 1

2Lm Llr + Lm

(13)

Lsr =

cosθr cos(θr +2π3 ) cos(θr − 2π

3 )cos(θr − 2π

3 ) cosθr cos(θr +2π3 )

cos(θr +2π3 ) cos(θr − 2π

3 ) cosθr

(14)

where the inductances Lls, Llr and Lm are the stator leakage,rotor leakage inductances and magnetizing inductance, respec-tively [15]. By combining (9) and (10) the next formulationin terms of current is obtained,

d

dtiabcs,r = ωb

[L−1(vabcs,r − (rs,r +Gs,r) · iabcs,r )

](15)

with the matrix

Gs,r = ωr−pu

[d

dθrL

](16)

where rs,r is the diagonal matrix of the stator and rotorresistances L is the matrix of inductances, ωr−pu is the rotorspeed in pu value and θr is the rotor position.

The electromagnetic torque, rotor speed and rotor positionassociated to the mechanical equations are defined as follows[15],

Te−pu =

(2

3

)iabcTs

d

dθr[Lsr]i

abcr (17)

d

dt

[ωr−pu

]=

(Te−pu − Tm−pu)

2Htotal(18)

d

dtθm =

[ωr−pu

]ωb (19)

where Te−pu is the electromagnetic torque in pu and Tm−pu

is the mechanic torque in pu supplied by wind turbine.

D. STATCOM model

A typical STATCOM, which consists of a voltage sourceconverter (VSC) and coupling transformer connected in shuntwith the AC system is shown in Figure 3. The power trans-fomer is connected wye-delta with the power flow directionto VSC switches.

Sa

Sb

Sc

vDC

S´

aS

´

bS

´

c

VSC

va

iDC

vb

vc

vSa

vSb

vScPower

transformer

ia

p

ic

p

ib

p

Figure 3. Equivalent circuit for three-phase, wye connected squirrel cageinduction machine.

1) transformer model: The wye-delta transformer is mod-eled as a three-phase transformer bank built with three single-phase linear transformers. The impedance is placed in theprimary side and an ideal transformer with ratio a : 1 allowsthe wye-delta connection. Applying Kirchhoff voltage andcurrent laws to the linear transformer the following three basicequations can be obtained,

diapdt

=ωbxp

[va −Rpi

ap − a(vSa − vSb)

](20)

dibpdt

=ωbxp

[vb −Rpi

bp − a(vSb − vSc)

](21)

dicpdt

=ωbxp

[vc −Rpi

cp − a(vSc − vSa)

](22)

where a transformer ratio, Rp is the resistance at the primaryside, xp is the reactance at the primary side, vabc are thevoltage at the point of coupling of the STATCOM and vS−abc

is the voltage at AC side of the switches in the VSC.2) VSC model: Three-phase six pulse bidirectional switches

is used to build the well-known two-level VSC. Each switch isan arrangement of an GTO, IGBT or MOSFET in antiparallelwith a diode. Besides, the losses in the semiconductors areneglected, so that the ideal switch model is used in this work.Then, discontinuities in the bidirectional switching functionsis identified by S and S’ for each phase, which can be onor off, 1 or 0 respectively. Also, S and S’ in each phase arecomplementary, thus S+S’=1. The voltage vS−abc to groundat the AC side is,

vSa = SavDC − (Sa + Sb + Sc)

3vDC (23)

vSb = SbvDC − (Sa + Sb + Sc)

3vDC (24)

vSc = ScvDC − (Sa + Sb + Sc)

3vDC (25)

where the vDC is the voltage at the DC side and Sabc are 1or 0 according to control with Pulse-Width Modulation (PWM)techniques.

4

3) DC capacitor link: The dynamic behavior of the DCcapacitor link is defined as,

d

dtvDC =

ωbiDCBDC

(26)

with

iDC = [ a(iap − icp) a(ibp − iap) a(icp − ibp) ]×

SaSbSc

(27)

E. Transmission line model

Appropriateness of line model depends on the line lengthand the highest frequency to be simulated. In this paper ashort-transmission lines is considered, thus a simple lumpedmodel is used,

d

dtiL =

ωb(vk −RLiL − vm)

xL(28)

where vk is the voltage at the k side of the transmissionline, where vm is the voltage at the m side of the transmissionline, R is the resistance of the transmission line and xL is thereactance of transmission line.

III. TEST CASE

The test case presented in this section consists of a STAT-COM system embedded into wind power system shown inthe Figure 1. A per unit three-phase time cosine waveformstakes place in the infinite busbar. All simulations the initialconditions are selected in zero. The whole system is describedwith a set of 27 ODEs, which are solved in the time domainusing the explicit Runge-Kutta 4th order algorithm and a 10µs time step. The wind power system and STATCOM systemparameters used in this work are reported in the Appendix.

A. Validation of scheme implemented

In order to validate the scheme implemented, the block setin Simulink of MATLAB is used to compare the transientbehavior during the startup of the wind power system. Themodels including for this simulation are: turbine block con-sidering a pitch angle of 0°, a wind speed average is taken in12 m/s and the actual rotor speed of the induction machine.An asynchronous machine block where it is selected thesquirrel cage mode. A three-phase transformer block operatedas a linear model. An universal bridge block in combinationwith the PWM generator, a base frequency of 60 Hz, acarrier triangular signal of 900 Hz, a modulation index of 0.8and a phase of output voltage of 60°. The main simulationparameters in Simulink are: the solver used is ode23tb, themaximum step size of 1e-6 and relative tolerance of 1e-6. The signals selected for comparing are rotor speed andelectromagnetic torque, as well as capacitor voltage for phase(a) in the PCC and capacitor voltage in DC link at STATCOMsystem.

Figure 4 shows the rotor speed and electromagnetic torquefor wind power system during startup using the comprehensive

0 0.5 1 1.5 2 2.5 30

0.5

1

1.5

time, s

roto

r sp

eed,

pu

abc modelsimulink

0 0.5 1 1.5 2 2.5 3−2

−1

0

1

time, s

Te,

pu

abc modelsimulink

Figure 4. Start-up of a wind power system a).- electromagnetic torque andb).-rotor speed.

2.97 2.975 2.98 2.985 2.99 2.995 3−2

−1

0

1

2

time, s

volta

ge a

t the

PC

C, p

u

abc modelsimulink

2.97 2.975 2.98 2.985 2.99 2.995 32.6

2.65

2.7

2.75

time, s

DC

vol

tage

, pu

abc modelsimulink

Figure 5. Toward the steady-state solution after startup a).- voltage at thePCC and b).- DC voltage in the STATCOM.

model implemented in this work and Simulink/MATLABsoftware. On the other hand, Figure 5 shows the voltage atthe PCC for phase a and the DC voltage in the DC link onthe STATCOM. Excellent agreement can be observed duringthe transient behavior and following the steady-state solutions(see Figure 5). The small differences can be associated mainlyby the different integration methods.

B. Random wind speed

This section provides a simulation of 60 seconds, wherethe wind model provides a stochastic behavior. Figure 6 (a)shows the wind speed, (b) active power and (c) reactive power.The first two seconds shows the startup of the wind powersystem. Here it can be seen a high consumption of reactivepower by the generator (see Figure 6 (c)). The active powergeneration is highly variable and also an oscillatory pattern

5

0 10 20 30 40 50 600

5

10

15

20

25

(a)

win

d sp

eed,

m/s

0 10 20 30 40 50 60−1

0

1

2

3

4

(b)

activ

e po

wer

, pu

0 10 20 30 40 50 600

1

2

3

4

(c)time, s

reac

tive

pow

er, p

u

Figure 6. Transient behavior for (a) wind speed, (b) active power and (c)reactive power.

0 10 20 30 40 50 600.95

1

1.05

1.1

(a)

win

d sp

eed,

pu

0 10 20 30 40 50 60−2

−1

0

1

(b)time, s

elec

trom

agne

tic to

rque

, pu

Figure 7. Transient behavior for (a) wind speed and (b) electromagnetictorque.

can be observed, this is due mainly due to the presence ofSTATCOM. Furthermore, in about 17.4 seconds a transientresponse is observed. This behavior is due to the protectionof over speed which is set in 20 m/s. The behavior of thewind speed and electromagnetic torque for this case is shownin Figure 7.

17.3 17.4 17.5 17.6 17.7 17.8 17.9 18

−1

−0.5

0

0.5

1

(a)

volta

ge a

t the

PC

C, p

u

17.3 17.4 17.5 17.6 17.7 17.8 17.9 18

2

2.5

3

(b)time, s

DC

vol

tage

, pu

Figure 8. Transient behavior for (a) voltage at the PCC for phase a and (b)DC voltage at the STATCOM.

29.95 29.96 29.97 29.98 29.99 30

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

time, s

curr

ents

, pu

phase aphase bphase c

Figure 9. Transient behavior in currents for phase a,b,c in the stator machine.

A close-up of the transient is shown in Figure 8. The statevariables that are shown are voltage at the PCC for phase aand DC voltage at the STATCOM. The Figure 8 (a) showsa flicker behavior for this condition while DC voltage showsa transient oscillation during disconnection and reconnection(see Figure 8 (b)). Figure 9 shows the currents in the statormachine for any given time, where it can see distortion in thewaveforms under this condition.

It is also significant to note that this test case was notincluded any control for both wind turbine and the STATCOM,only the model response is obtained under specified conditions.

IV. CONCLUSION

In this paper shows the detailed implementation of acomprehensive phase-domain model of a wind-power gener-ator. The scheme implemented was validated by using the

6

Simulink/MATLAB software. Excellent agreement can beobserved during the transient and steady-state behavior of thedeveloped model. The small discrepancies are attributed tothe different integration algorithms used to solve the ODEs.Moreover, a dynamic solution including stochastic wind speedwill allow assessing the impact of wind power into the powersystem. Also, the STATCOM implemented allows showing theassociated problems to power quality. Since the controls arenot included, the variable wind simulation is unrealistic. Butthe main focus is based on the modeling of components in thephase coordinates. Further future work can be addressed forcontrol schemes and the use of DFIG.

APPENDIX AThe parameters of the wind-power generator and STAT-

COM used to develop this research are shown in Table I.

ACKNOWLEDGMENT

The authors wishes to thank the Programa para el DesarrolloProfesional Docente (PRODEP) for financial support to carryout this research, under PRODEP project No. DSA-103.5-15-6897.

REFERENCES

[1] T. Ackermann, Wind Power in Power Systems, England: John Wiley &Sons, 2005.

[2] J. G. Slootweg, Wind Power: Modelling and Impact on Power SystemDynamics, PhD thesis, Technische Universiteit Delft, Dec. 2003.

[3] H. Holttinen, “Estimating the Impacts of Wind Power on PowerSystems–Summary of the IEA Wind Collaboration”, EnvironmentalResearch Letters, No. 3, pp. 1-6, April 2008.

[4] L. Holdsworth, X.Wu, J. B. Ekanayake, and N. Jenkins, “Comparisonof fixed speed and doubly-fed induction wind turbines during powersystem disturbances,” Proc. Inst. Elect. Eng., Gener., Transm., Distrib.,vol. 150, no. 3, pp. 343–352, May 2003.

[5] F. M. Hughes, O. Anaya-Lara, N. Jenkins, and G. Strbac, “Controlof DFIG-based wind generation for power network support,” IEEETransactions on Power Systems, Vol. 20, No. 4, pp. 1958-1966, Nov.2005.

[6] L. Xu, L. Yao y C. Sasse, “Comparison of Using SVC and STATCOMfor Wind Farm Integration”, International Conference on Power SystemTechnology, Reino Unido, 2006, págs.1-7.

[7] N. Hasan , S. Farooq, “Dynamic Performance Analysis of DFIGbased Wind Farm with STATCOM and SVC” International Journal ofEmerging Technology and Advanced Engineering, Volume 2, Issue 7,July 2012.

[8] R. Fadaeinedjad, G. Moschopoulos y M. Moallem, “Using STATCOMto Mitigate Voltage Fluctuations Due to Aerodynamic Aspects of WindTurbines”, School of Engineering Science, Simon Fraser University,Surrey, Canadá, 2008.

[9] L. Qi, J. Langston y M. Steurer, “Applying a STATCOM for StabilityImprovement to an Existing Wind Farm with Fixed-Speed InductionGenerators”, U.S. Department of Energy, EUA, 2008, págs. 1-6.

[10] Z. Saad-Saoud, M.L. Lisboa, J.B. Ekanayake, N. Jenkis y G. Strbac,“Aplication of a STATCOMs to Winds Farms”, IEE, Proc. Gener. Trans.Distrib , England,Septiembre 1998, Vol. 145, No. 5, págs. 511-516.

[11] Z. Saad-Saoud y N. Jenkins, “Simple Wind Farm Dynamic Model,” IEEProc. Gener. Trans. Distrib., Vol. 142, No. 5, pp. 545-548, Sep. 1995.

[12] J. B. Ekanayake, L. Holdsworth, X. Wu and N. Jenkins, “DynamicModeling of Doubly Fed Induction Generator Wind Turbines,” IEEETransactions on Power Systems, Vol. 18 No. 2, May 2003.

[13] P.M. Anderson, A. Bose, “Stability simulation of wind turbine systems”,IEEE Transactions on Power Apparatus and Systems, v.102, n.12,Diciembre 1983, pp.3791 3795.

[14] O. Wasynczuk, D.T. Man, J.P. Sullivan, “Dynamic behavior of a classof wind turbine generators during random wind fluctuations”, IEEETransactions on Power Apparatus and Systems, v.100, n.6, June 1981,pp.2837-2845.

Table IWIND PARK PARAMETERS BASED ON 1 MW RATING, STATCOM SYSTEM

AND ADDITIONAL PARAMETERS.

wind generator wind turbine STATCOMnp 6 R 27 m α 60°Xls 0.1248 pu ρair 1.225 Kg

m3 m 0.8Xlr 0.0884 pu c1 0.5176 mf 15Xm 1.8365 pu c2 116 a 1√

3rr 0.0073 pu c3 0.4 Rp 0.05 purs 0.0076 pu c4 5 xp 0.1 pu

Hgen 0.250 pu c5 21 BDC 0.5 puHtur 2.410 pu c6 0.0068

Additional parametersRL 0.01 pu xL 0.1 pu Bc 0.2 pu

[15] P. C. Krause, O. Wasynczuk and S. D. Sudhoff, Analysis of ElectricMachinery, New York, McGraw-Hill, 1994.

[16] Christian Pérez, “Análisis de la Respuesta de Estado Estable de ParquesEólicos de Generación Usando el Método de Diferencias Finitas”, Tesisde maestría, UMSNH, Mayo 2011.

BIOGRAPHIES

L. Contreras-Aguilar received the mechanical-electrical engineering de-gree from the University of Colima, Colima, México, in 2002. He got his MScdegree and the PhD degree from the UMSNH, Morelia, México, in 2005 and2011, respectively. At present, he holds a permanent position as a researchprofessor at the University of Colima. His area of research is dynamic andsteady-state analysis of power systems.

T. Venegas received his BEng degree from University of Colima, México,in 1996, his MSc degree (first class) from Instituto Tecnológico de Morelia,México, in 2000. He is at present an Associated Professor at the Universityof Colima. His main research interests lie on the steady-state and dynamicanalysis of FACTS, CUSTOM POWER, Efficiency and Renewable Energyand Real-Time Modelling and Analysis.

J. Arroyo received his BEng degree from Instituto Tecnologico deMorelia, México, in 1999, his MSc and hisD.Sc. degree from CINVESTV-IPN Guadalajra, México, in 2002 and 2007 respectively. He is at present anAssociated Professor at the University of Colima. His main research interestslie on the steady-state and dynamic analysis of the electric power system.

R. J. Betancourt received his BEng degree from Instituto Tecnológicode Tepic, México, in 1994, his MSc and hisD.Sc. degree from CINVESTV-IPN Guadalajara, México, in 1999 and 2007 respectively. He is at present anAssociated Professor at the University of Colima. His main research interestslie on the analysis of electric power systems with nonlinear methods.

J. H. Aguirre-Ruiz and R. Rodríguez-Flores are currently towards hisBEng, at the University of Colima, México.