Coordinated Reactive Power Management in Power Networks With Wind

Energy Conversion and Management 52 (2011) 2575–2586

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Energy Conversion and Management

journal homepage: www.elsevier .com/ locate /enconman

Coordinated reactive power management in power networks with windturbines and FACTS devices

Hortensia Amaris ⇑, Monica AlonsoUniversity Carlos III of Madrid, Electrical Engineering Department, 28911 Leganes, Madrid, Spain

a r t i c l e i n f o

Article history:Received 27 March 2010Received in revised form 1 February 2011Accepted 11 February 2011Available online 24 March 2011

Keywords:Variable speed wind turbineReactive power planningFACTS devicesOptimizationSystem operation

0196-8904/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.enconman.2011.02.012

⇑ Corresponding author.E-mail addresses: [email protected] (H. Am

(M. Alonso).

a b s t r a c t

Reactive power management is a critical issue when dealing with the planning and operation of powernetworks with high wind energy penetration. This paper is intended to introduce a coordinated ReactivePower Planning strategy among Doubly-Fed Induction Generator (DFIG) variable speed wind turbinesand Flexible AC Transmission Systems (FACTS) devices. According to this strategy, the reactive powercapability from DFIG wind turbines is obtained and the limitations on deliverable power are deducedfor each operation point. Furthermore, instead of using the reactive power limit as it is traditionally done,the reactive power injection from Static Var Compensator (SVC) is related to the existing physical limitsof the control variables. The optimization strategy is based on genetic algorithms and includes directly inits formulation both the reactive power capability from wind turbines and the reactive power injectionfrom SVC units. An existing 140-bus power system is used to validate the performance and effectivenessof the proposed method.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

Nowadays wind energy has proved to be one of the most com-petitive and efficient renewable energy sources and as a result, itsuse is indeed continuously increasing. Low wind penetration levelsare usually accommodated in present power networks consideringthat the network is passively controlled and operated. However,this assumption is no longer valid since as soon as the wind energypenetration starts increasing, a wide range of technical problemscould appear, such as: bi-directional power flow, voltage rise, in-creased power quality problems and altered voltage stability [1,2].

The further development of electricity production from renew-able energy sources in a safe and reliable system operation is push-ing transmission and distribution operators to use new operatingstrategies that are not currently available. One of the critical issuesof the challenging situation above defined is the reactive powermanagement which entails the requested operation and planningactions to be implemented in order to improve the voltage profileand the voltage stability in power networks [3]. An efficient reac-tive power planning could be obtained by choosing an optimumlocation of var sources during the planning stage, whereas efficientreactive power dispatch could be achieved by scheduling an opti-mum regulation of the voltage set point at the generators connec-tion point and at the VAR settings during the reactive power

ll rights reserved.

aris), [email protected]

dispatch [4]. Lately, and in the attempt to fill the existing gap,Flexible AC Transmission Systems (FACTS) devices have emergedas a feasible option to improve voltage stability by influencingpower flows and by improving voltage profiles [5]. The optimumusage of these devices implies the finding of the optimal locationwhere their influence would be more useful as well as to determi-nate their optimum sizing [6,7].

Wind farms are usually considered as PV or PQ nodes for loadflow or reactive power studies [3,8]. For instance, in [8] voltage sta-bility in power networks with wind generators is analyzed beingwind farms represented as PQ buses with cos phi = 0. However,there is not any aim to contribute to the voltage stability by usingthe DFIG as a reactive power generator.

At the same time [9], analyzes the reactive power capability ofthe wind induction machine to minimize the power line losses in adaily reactive power dispatch taking into account that the windfarms are already fixed at specific places in the network. However,the study does not expect to find out the optimal places for locat-ing wind farms in order to improve the voltage stability of the con-trol area either. Moreover the needed reactive power is supposedto be injected only by the induction electrical machine and notby the Grid side converter.

Ref. [10] includes the reactive power capability of the inductionmachine in the optimization problem in order to minimize powerlosses during normal operation. Reactive power injection from thegrid side converter has not been considered and consequently theresults do not reflect properly the reactive power capability fromDFIG wind turbines.

Nomenclature

NotationI, U complex phasors for current and voltageI, U RMS-values

SymbolsIS, IR stator and rotor current phasorImS magnetizing current representing stator fluxISd, ISq direct and quadrature components of stator currentIRd, IRq direct and quadrature components of rotor currentLS, LR stator and rotor inductancesL0 mutual inductancek loadalibility parameterNG number of generators connected at bus ithND number of loads connected at bus ithNsvc number of SVC units connected at bus ithNB number of busesRS, RR stator and rotor resistances

rS, rR stator and rotor leakage factorPmech mechanical powerPn nominal power wind turbinePS, PR stator and rotor active power DFIGPGi, QGi active and reactive power injection from generators

connected at bus ithPDi, QDi active and reactive power consumed by loads connected

at bus ithPLik, QLik active and reactive transmission power between bus ith

and kthQS, QR stator and rotor reactive power DFIGQsvc reactive power injected by SVC unit connected at bus ithSGSC grid side converter Apparent powerS slip velocityx rotor angular speedx1 synchronous angular speed

Fig. 1. DFIG wind turbine.

2576 H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586

Few papers give consideration to the reactive power capabilityfrom the DFIG power converters [11–13]. Ref. [11] deduces thereactive power capability of a doubly-fed induction generator bymeans of the use of performance capability curves in order to offerthe Low Voltage Ride Through (LVRT) requirements stated in thenational grid codes in case of abnormal situations such as faultsand voltage dips. However, those papers do not consider the imple-mentation of this extended reactive power capability to reactivepower planning analysis, such as optimal location of wind farms.In [13] a mathematical model for steady-state performance of DFIGis investigated in which the reactive power capability (from themachine and from the grid side converter) is already consideredinto the control of the DFIG system. The study focuses only onthe DFIG local connection point, whereas a coordinated voltagecontrol among different wind farms and FACTS devices has neitherbeen analyzed nor included into any coordinated reactive poweroptimization strategy.

In fact, in the current literature there is little work that gets intofocus the achievement of a coordinated voltage control or coordi-nated reactive power planning in networks with wind farms andFACTS devices. In [5] voltage stability in power networks withwind generators and FACTS units is analyzed by considering onlythe reactive power injected by a SVC. The objective of the proposedalgorithm in [5] is to obtain a stable voltage profile by injecting aSVC reactive power adjustable independently of the power injectedby the wind generator. The paper neither show the possibility ofperforming a coordinated voltage control among wind turbinesand FACTS devices nor consider the reactive power injection fromthe grid side converter of the DFIG.

Indeed, this paper aims to clarify the shortcomings identified inthe above-mentioned papers as well as to propose a new reactivepower management strategy for the coordinated handling of reac-tive power from FACTS devices and wind farms. The optimizationmethodology is based on genetic algorithms and includes, directlyin its formulation, the reactive power capabilities of the DFIG, bothfrom the electrical machine and from the grid side converter.

The proposed methodology will improve the existing knowl-edge of determining which are the optimal placements for locatingwind farms and FACTS devices in power networks and which is theamount of reactive power that should be injected in the network ina coordinated way to improve not only the voltage profile but alsothe voltage stability. The optimization formulation proposed in thepaper focuses on Static Var Compensator (SVC) and Doubly-FedInduction Generators (DFIG) variable speed wind turbines.

However, it should be emphasized that the developed methodcould also be applied to any other controllable FACTS devices suchas static synchronous compensator (STATCOM) or other variablespeed wind turbine such as a full power converter.

The paper is organized as follows: Section 2 is devoted to themathematical model of the DFIG variable speed wind turbine forobtaining its power capability curve and the relation with its lim-iting physical factors. In Section 3 the Reactive Power PlanningOptimization Strategy is described in which the available powercapability from wind turbines has been included. A description ofthe genetic algorithm is shown in Section 4. The methodology pro-posed in the paper is applied in an existing 140-bus power systemin which several wind farms and SVC devices are optimally located.The results of this methodology are shown in Section 5. Finally,Section 6 presents papers conclusions.

2. Reactive power capabilities from wind turbines

Currently, grid code requirements from wind farms demandwind turbines to offer voltage control and reactive power regula-tion capabilities at the connection point. Moreover, they must beable either to inject or absorb reactive power according to the sys-tem operator’s commands. Just to mention an example to clarifythese requirements, grid code [14] states that wind farms shouldoffer the capability of regulating their power factor from 0.95 lead-ing to 0.95 lagging. In the case of Danish grid code [15] wind farmsshould offer a reactive power regulation band between ±10%Pn.Although there is no standard grid code yet, all national grid codesagree about including the reactive power injection from wind tur-bines in both normal and faults situations. Fixed speed windturbines equipped with SVC or STATCOM at the wind farm substa-tion offer reactive power regulation capability. Variable speed

0 0.2 0.4 0.6 0.8 1

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

Mechanical Power Pmech [p.u.]

deepS rotoR

ω[p

.u.]

ωmin

ωmax

Fig. 2. Rotor speed evolution x ¼ KSPcmech , KS = 1.464, c ¼ 1

3, xmin = 0.7, xmax = 1.3.

Fig. 3. Steady state equivalent circuit of DFIG.

H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586 2577

wind turbines, such as DFIG, Fig. 1, or full power converter technol-ogy, are connected to the grid by electronic power converters and,consequently, they have the capability to provide voltage supportto the network as well as to fulfill the grid code requirements [16].

2.1. Slip wind turbine operational point

The available wind power increases with the wind speed [17]according to:

Pmech ¼12qCpðkv ;bÞAv3 ð1Þ

Where q is air density, kv = kv(x,v) is tip speed ratio, x is rotationalrotor speed, v is wind speed, A is rotor swept area, and Cp is a powercoefficient value depending on kv and pitch blade angle b.

In the existing literature there are several optimization strate-gies that could be used to find the working point as a function ofwind speed. In this way, (1) can be considered as a dependent func-tion on the rotor speed:

Pmech ¼ PmechðxÞ ð2Þ

and, consequently, its inverse:

x ¼ xðPmechÞ ð3Þ

The generic relationship of (3) could be refined in order to includeany further restriction, such as: minimum and maximum mechan-ical speed, security limits as well as other design aspects. A conve-nient way to use (3) is by parameterizing it as an exponentialrelation, as shown in Fig. 2:

x ¼ KSPcmech; xmin 6 x 6 xmax ð4Þ

where KS and c are constants depending on wind turbine machinedesign parameters, and xmin, xmax define the limit values for xvariation.

Considering x1 as the grid angular frequency, the slip S shall bedefined as:

S ¼ x1 �xx1

ð5Þ

Therefore, the rotor slip S shall be regarded as a dependent functionon Pmech:

S ¼ SðPmechÞ ð6Þ

2.2. Wind farm models for reactive power management studies

Traditionally, wind farms have been represented as PV or PQmodels in power flow studies [9,18]. The methodology imple-mented till now seems to be quite simple, however it presentsthe main drawback that the available reactive power range is lim-ited either to a maximum cos(/) or to a fixed regulation band[19,10,20]. Additionally, this representation is not completelyaccurate and therefore does not allow to take full advantage ofthe reactive power injection from the wind turbine.

In this paper, a better wind turbine model is proposed that doestake into account the actual available reactive power capability foreach working operation point. The proposed formulation could beincluded thus in any modified power flow analysis for optimumreactive power management. At the same time, this methodologywill enable to regulate the reactive power injection either locally,at the wind farm, or globally, in the whole network. As a resultof the optimum and coordinated reactive power planning, the volt-age stability in the power network will be significantly improvedand enhanced. This consideration is not tacking into account inthe actual literature.

2.2.1. Steady state model of the DFIGThe steady-state stator and rotor equations, considering a sym-

metrical three-phase system could be written following the com-plex phasor notation [21]:

US ¼ ðRS þ jx1LSÞIS þ jx1L0IR ð7ÞUR

S¼ RR

Sþ jx1LR

� �IR þ jx1L0IS ð8Þ

Introducing the stator and rotor leakage factors, the stator and rotorself inductances would be defined as:

LS ¼ ð1þ rSÞLo ð9ÞLR ¼ ð1þ rRÞLo ð10Þ

And so that the mathematical model becomes:

US ¼ ðRS þ jx1rSLoÞIS þ jx1Lo IS þ IRð Þ ð11ÞUR

S¼ RR

Sþ jx1rRLo

� �IR þ jx1Lo IS þ IRð Þ ð12Þ

And it is graphically represented in Fig. 3. Neglecting stator and ro-tor resistances results in:

US ¼ jx1rSLoIS þ jx1LoðIS þ IRÞ ð13ÞUR

S¼ jx1rRLoIR þ jx1LoðIS þ IRÞ ð14Þ

Defining the stator magnetizing current ImS as an extended magne-tizing current phasor responsible for the stator flux including statorleakage [21]:

ImS ¼ ð1þ rSÞIS þ IR ð15Þ

Phasor frames are widely used in Field Oriented Control in order tosimplify the equations. Using the ImS phasor as the d � q reference,

Fig. 4. Phasor ImS used as d � q frame reference.

2578 H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586

as shown in Fig. 4, the stator voltage Eq. (13) may be written in theform:

US ¼ jx1LoImS ð16Þ

It should be noted that ImS is equal to its module ImS:

ImS ¼ jImSj ¼ ImS ð17Þ

Hence, the stator voltage phasor is purely imaginary and alignedwith the q-axis

US ¼ jx1LoImS ð18Þ

2.2.2. Active power delivery to the gridActive power delivered by the DFIG stator could be deduced

from the real component of the apparent power at stator terminals.

PS ¼ �Re USI�S� �

ð19Þ

And so that by substituting US by (18) and IS by its d � q compo-nents: IS = ISd + jISq, the stator active power results in:

PS ¼ �x1LoImSISq ¼ �USISq ð20Þ

In the same way, active power delivered by the rotor may be ob-tained from the real component of the apparent power at rotorterminals:

PR ¼ �Re URI�R� �

ð21Þ

And so that by substituting UR by (14) and IR by (15), the expressionfor the active power delivered by the rotor becomes:

PR ¼ Sx1LoImSISq ¼ SUSISq ð22Þ

Note that rotor power PR is a fraction of the stator power PS depend-ing on the slip S:

PR ¼ �SPS ð23Þ

In this way, the total active power delivery to the grid is expressedthus:

P ¼ PS þ PR ¼ USISqðS� 1Þ ð24Þ

2.2.3. Reactive power delivery to the gridReactive power delivered by the stator could be obtained from

the imaginary part of the apparent power as shown in:

Q S ¼ �Im USI�S� �

¼ �USISd ð25Þ

Reactive power delivered by the rotor will depend on the DFIG con-trol strategy defined in the grid side converter. It is usually designedto maintain the DC voltage between the two converters at a con-stant operational level as well as to flow the rotor active power tothe grid PGSC = PR. Because of the fact that the reactive power refer-ence is usually kept to zero, the only suitable way to regulate thetotal power factor of the machine at the connection point PCC (Pointof Common Coupling) would be by controlling the stator reactivepower by (25).

However the grid side converter could be used to control thereactive power and to improve the total reactive power capabil-ity of the wind turbine [22,11] too. This potential usage is a veryimportant aspect since it may be quite useful to system opera-tors in order to perform a coordinated reactive power manage-ment in the whole power network. The proposed methodologycould be applied in the available converter designs not beingnecessary to perform any physical modification to the currentDFIG commercial converters. And therefore the grid side con-verter could be treated as a reactive power source dynamicallycontrolled.

Total reactive power injected to the grid will be composed bythe superposition of both, the reactive power injected by theinduction machine (stator) and the reactive power injection fromthe Grid Side Converter (GSC).

Q ¼ Qsþ Q GSC ð26Þ

2.3. DFIG limitations on deliverable power

The maximum power capability of the DFIG wind turbine is de-fined by the following constraints:

� Stator side: Both maximum voltage and current are bounded bystator machine rating.

IS < IS;max ð27ÞUS < US;max ð28Þ

Voltage at the PCC, US is allowed to range between 10% UPCC

according to the voltage grid regulations.� Rotor side converter: Both maximum voltage and current are

restricted by rotor machine rating and by Rotor Side Converter(RSC) rating:

IR < IR;max ð29ÞUR < Ur;max ð30Þ

� Grid side converter: Apparent GSC power should not exceed itsrated or nominal value.

SGSC < SGSC;nominal ð31Þ

� Slip speed range may oscillate between the maximum and min-imum values {Smin,Smax} that are specified by the manufacturer.Slip operation point of the DFIG is obtained from (3).

Keeping in mind the main purpose of getting a better under-standing of the implications related to each constraint, a PQ dia-gram will be obtained and analyzed individually for each ofthem in following sections.

2.3.1. Rotor current limitThe rotor current phasor could be characterized by the maxi-

mum complex modulus where the phase, u, is swept over theoperating range.

IR ¼ IR;maxeju ¼ IR;maxðcos uþ j sinuÞ ð32Þ

For each working condition the steady-state of the DFIG machine isobtained and a PQ locus for active and reactive powers can bedrawn.

Stator current could be expressed in the d � q reference frame,according to:

IS ¼ ISd þ jISq ð33Þ

0 2 4 6 8 10Active Power P [p.u.]

S=−0.3S=−0.2S=0.2S=0.3

Rea

ctiv

e Po

wer

Q [p

.u.]

−14

−12

−10

−8

−6

−4

−2

0

2

4

−14

−12

−10

−8

−6

−4

−2

0

2

4

Fig. 6. PQ locus for rotor voltage UR,max.

H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586 2579

Using (15), the d � q components are:

ISd ¼Imsd � IR;max cos uð1þ rSÞ

ð34Þ

ISq ¼�IR;max

ð1þ rSÞsinu ð35Þ

Substituting (34) and (35) in (24) and (25), the resulting expressionsfor active and stator reactive power will be formulated as follows:

P ¼ �USIR;max

1þ rSðS� 1Þ sin u ð36Þ

Q S ¼USIR;max

1þ rScos u� U2

S

x1Loð1þ rSÞð37Þ

Resulting in the power locus PQ diagram

P

SIro ðS� 1Þ

" #2

þ Q S � Q Iro

SIro

" #2

¼ 1 ð38Þ

where SIro , QIr

o are defined as:

SIro ¼

USLo

LSIR;max ð39Þ

Q Iro ¼

U2S

x1LSð40Þ

The resulting PQ diagram considering the rotor current constraint isan ellipse centered in ½0;QIr

o � with semi-axes ½SIro ðS� 1Þ; SIr

o �, asshown in Fig. 5. Moreover, when slip is zero (synchronous speed),the PQ diagram becomes a circumference with center in ½0;QIr

o �and radius SIr

o .

2.3.2. Rotor voltage limitRearranging the rotor voltage phasor (14) using IS from (15) an-

dUS from (18) results in:

UR ¼ jS ð1þ rRÞUS þ ISa½ � ð41Þ

where a is defined as:

a ¼ x1Lo 1� ð1þ rSÞð1þ rRÞ½ � ð42Þ

By considering the rotor voltage phasor UR as,

UR ¼ UR;maxeju ¼ UR;maxðcos uþ j sin uÞ ð43Þ

0 0.2 0.4 0.6 0.8 1 1.2 1.4−1.5

−1

−0.5

0

0.5

1S= −0.3S= −0.1S= 0.0S= 0.1S= 0.3

Active Power P [p.u.]

Rea

ctiv

e Po

wer

Q [p

.u.]

Fig. 5. PQ locus for rotor current IR,max.

and using (41), the d � q components are:

URd ¼ �aSISq ð44ÞURq ¼ S aISd þ ð1þ rRÞUSð Þ ð45Þ

Active power and stator reactive power could be rewritten thus:

P ¼ US 1� Sð ÞSa

UR;max cos u ð46Þ

Qs ¼1þ rR

aU2

s �US

SaUR;max sinu ð47Þ

Resulting in the power locus PQ diagram:

P

SUro ðS� 1Þ

" #2

þ Qs � QUro

SUro

" #2

¼ 1 ð48Þ

where SUro , QUr

o are defined:

SUro ¼

US

SaUR;max ð49Þ

QUro ¼

1þ rR

aU2

S ð50Þ

The PQ diagram is an ellipse centered in ½0;QUro � with semi-axes

½SUro ðS� 1Þ; SUr

o � as shown in Fig. 6. When slip is zero, the PQ diagram

becomes a circumference with center in ½0;QUro � and radius SUr

o .

2.3.3. Stator current limitActive power and stator reactive power could be rewritten as:

P ¼ USð1� SÞIS;max sinu ð51ÞQS ¼ �USIS;max cos u ð52Þ

Resulting in the power locus PQ diagram:

P

SISo ðS� 1Þ

" #2

þ QS

SISo

" #2

¼ 1 ð53Þ

where SISo is defined as:

SISo ¼ USIS;max ð54Þ

The PQ diagram is an ellipse centered in the origin [0,0] and semi-axes ½SIS

o ðS� 1Þ; SISo � as shown in Fig. 7. When slip is zero, the PQ

diagram becomes a circumference with radius SISo .

0 0.5 1 1.5

Active Power P [p.u.]

Rea

ctiv

e Po

wer

Q [p

.u.]

S= −0.3S= −0.2S= 0.0S= 0.2S= 0.3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Fig. 7. PQ locus for stator current IS,max.

Fig. 8. PQ locus for grid side converter.

Fig. 9. PQ locus capability.

2580 H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586

2.3.4. Grid side converter limitThe grid side converter rating SGSC will define the limits of the

reactive power injected to the grid according to the followingexpression graphically shown in Fig. 8:

Q GSC ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS2

GSC � P2R

qð55Þ

Fig. 10. Voltage loadability.

2.4. Maximum DFIG capability

Taking into account all previous constraints, the resulting capa-bility curve is shown in Fig. 9. It should be highlighted that whenthe machine is delivering reactive power to the grid, rotor currentis the critical constrain; whereas when the machine is consumingreactive power, stator current works as the limit factor.

In the same Figure it is to be noted two regions: the inner areacorresponds to the PQ diagram without considering the reactivepower injection from the GSC, and the outer area considers thereactive power capability from the GSC (extended region in Fig. 9).

This extended capability could be very useful to system opera-tors not only by performing voltage stability or contingency anal-ysis, but also in every situation where power reactive reservesare critical factors for keeping stable the network.

3. Reactive power planning

Reactive Power Planning (RPP) could be formulated as an optimi-zation process [23] in which the objective function to minimize is:

minx2Rn

f ðx;uÞ

subject togðx;uÞ ¼ 0hðx;uÞ 6 0

� ð56Þ

where u are the control variables, x�Rn stands for all the systemsteady-state variables, f(x,u) objective function to minimize, g(x,u)involves the equality constrains, h(x,u) corresponds to the inequal-ity constrains.

3.1. Objective function

The target of the optimization algorithm is to allocate a specifiednumber of wind farms and SVC’s units in a power network in order toimprove the voltage stability and the voltage profile of the powernetwork. The main factor that tends to cause voltage instability isthe inability of a power system to maintain an adequate reactivepower management in the network and voltage level [24].

To overcome any eventual inefficient performing, the algorithmmanages to find the optimum location and the optimum reactivepower injection among wind farms and SVC’s in a coordinated way.

In most of the cases the driving force of voltage instability is theload and that is why the algorithm tries to maximize the loadalibil-ity factor taking into account the minimum voltage allowable limit(normally Umin = ±10%UN) according to the utilities grid voltageregulations.

In this situation the load change scenarios, PD and QD could bemodified as:

PD ¼ PD0ð1þ kÞ ð57ÞQD ¼ Q D0ð1þ kÞ ð58Þwhere PD0, QD0 original load, base case and k loadalibility parameter.

H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586 2581

Voltage stability is usually studied by a P–V diagram [24] asshown in Fig. 10. Loads in all buses are increased proportional to theirinitial load levels and the generators outputs are also increasedproportional to their initial generations. The point where the loadparameter becomes tangent to the network characteristic definesthe Point of Collapse (PoC). At this point k = kcritical. In the sameway, if it takes place a load increase beyond this critical value, anunstable equilibrium will be produced, and consequently the systemwould be unable to operate any longer. For all the reason abovementioned, in this paper the objective function tries to maximizethe loadalibility parameter considering the minimum allowedvoltage value according to the utility regulations (Umin,klimit).

3.2. Constraints

3.2.1. Equality constraintsBasic equality constrains correspond to the power flow equa-

tions in every buses:

DPi ¼ Pgi � Pdi � Pi ð59ÞDQ i ¼ Qgi � Qdi � Q i ð60Þ

where Pgi and Qgi are real and reactive powers of generator at bus i,respectively; Pdi and Qdi the real and reactive load powers, respec-tively; Pi and Qi the power injections at the node

� Active and reactive transmission power between buses ith andkthBus ith is linked to bus kth through a transmission line whoseadmittance is Yik = Gik + jBik. Voltage buses at buses ith and kthare: Ui = Ui\hi and Uk = Uk\hkwhere:

hik ¼ hi � hk ð61Þ

PLik ¼ Ui

XN

k¼1

UkðGikcoshik þ BiksinhikÞ ð62Þ

Q Lik ¼ Ui

XN

k¼1

UkðGiksinhik � BikcoshikÞ ð63Þ

� Power consumed by loads connected at bus ithLoads models are a key point for voltage stability studies. In thispaper, loads are characterized by a static exponential model inwhich active and reactive power consumed by the load isexpressed as:

PDi ¼ PD0UU0

� �a

ð64Þ

Q Di ¼ Q D0UU0

� �b

ð65Þ

Where U0, PD0 and QD0 are the reference voltage, active and reac-tive power at bus.

Fig. 11. Static var compensator diagram.

In this paper a power constant model (a = b = 0) has been se-lected. It should be highlighted that the methodology proposedis not affected by the load parameter values. Different valuesof a and b parameters enable to represent different models, suchas an impedance constant model (a = b = 2), a current constantmodel (a = b = 1) or a mixed model.� Reactive power injected by SVC units connected at bus ith

A Static Var Compensator (SVC) would be defined as a devicewhose output is adjusted to exchange capacitive or inductivecurrent to maintain or control specific parameters of the electri-cal power system. In this paper, the SVC considered correspondsto a TCR (Thyristor Controlled Reactor) as shown in Fig. 11.In this situation steady-state current injected is expressed thus:

I ¼UXL

cosasvc � cosxtð Þ; asvc 6 xt < asvc þ r0; asvc þ r 6 xt < asvc þ p

(ð66Þ

where U voltage at SVC connection point, it is the voltage that itis being controlled, XL total inductance, Xc capacitor, asvc is thefiring delay angle and r is defined as the SVC conduction angleaccording to:

r ¼ 2ðp� asvcÞ ð67Þ

Following Fourier [25], the variable susceptance Bsvc could be ex-pressed as:

BsvcðasvcÞ ¼2p� asvc þ sin 2asvc

pXLð68Þ

Many research studies represent the SVC as a variable reactancetaking into account the reactive power and the voltage limits atthe connection point. However this representation it is not fullycorrect, because SVC limits are really defined by the maximumphysical limits of the firing delay angle. Under low voltage situ-ations, if Q limits were to be considered instead of the physicalfiring angle limits, the necessary Q would be unreachable dueto the existing physical ceilings of the asvc angle.In [6] asvc angle is a state variable whose value is updated in eachiteration and the reactive power is computed according to it. Inthis paper, this approach is followed and the reactive power in-jected by the SVC is:

Q svcðasvcÞ ¼U2

Xc� U2BsvcðasvcÞ ð69Þ

� Steady state operation point of DFIGThe wind power production for each operating point is consid-ered to be known, and consequently the slip speed can beobtained from Eq. (3) and it will be used together with (13)and (14) in order to obtain the steady-state working operationpoint of the machine.

3.2.2. Inequality constraintsInequality constraints constitute the physical limits of the com-

ponents or operational constrains in the system.

� Voltage limits at busesVoltage level at buses are not allowed to be outside the maxi-mum and minimum values according to grid voltageregulations.

Ui;min 6 Ui 6 Ui;max ð70Þ

� Limits of the loadalibility factorLambda is the loading factor by which the load is increased atall buses and k P 0.� Limits on DFIG generators

The maximum power injected by wind farms is constrained bythe maximum capability of each wind turbine.

Table 1Chromosome structure.

Stability Wind farm SVC

2582 H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586

Active power injected by the wind turbine will depend on thewind availability for each wind turbine. Active power outputis restricted by lower and upper limits:

WF1 . . . WFN SVC1 . . . SVCM

where Pgi,min, Pgi,max are the minimum and maximum operatingpower.The capability constraints are related to current and voltage lim-its in the stator, rotor and grid side converter as expressed in(27,29,30 and 31).� Limits on SVC units

SVC reactive power generation limit is constrained by the phys-ical range of the firing angle. It is considered that the angle isallowed to be set between a maximum and a minimum value.

aSVCi;min6 aSVCi

6 aSVCi;maxð72Þ

� Physical constrains in the wind farm connection pointThe potential connection point from the wind farm to the grid islimited to the geographical area in which the available windresource is higher.

BusLocWFi ;min 6 BusLocWFi6 BusLocWFi ;max ð73Þ

4. Genetic algorithm

Genetic Algorithms (GA) belong to the evolutionary optimiza-tion algorithms and they were introduced by Holland in 1975[26]. Starting with an initial population, the algorithm evolves toa new generation of individuals by executing reproduction, muta-tion and crossover operations among the individuals of this popu-lation. The main advantages offered by GA over conventionaloptimization algorithms are:

� GAs do not need initial information about the system to beginthe searching process since they work only with the coding(chromosomes) which will be optimized according to the objec-tive functions and the proper constraints.� The algorithms would be able to explore simultaneously various

regions in the search space by using multiple points of the pop-ulation and iterative characteristics. This represents one of themost important distinctions from the traditional optimizationalgorithms where only one direction in the search space couldbe followed.� Best individuals are selected among parents and offspring gen-

eration making the process more likely to converge to a globalminimum.

4.1. Encoding

The genetic information of each individual is encoded in itschromosome which is a string of real numbers that corresponds to:

� Bus location,� Reactive power injected,� Loadalability factor.

If there are n wind farms and m SVC’s units to be located, thechromosome length will have(see Table 1):

� 2n genes: Bus location and reactive power injected for eachwind farm,� 2m genes: Bus location and firing delay angle for each SVC,� 1 gene: Optimal loadalibility factor.

Active power injected by each farm is not included in the chro-mosome since it is assumed that active power from wind farm is

merely a piece of information previously known according to thewind power production forecasts.

4.2. Evaluation

The Fitness Function (FF) assigns a goodness value at each indi-vidual of the population and it is employed to drive the evolutionprocess. Each chromosome is assigned a fitness value which deter-mines not only the individual adaptation capacity to the environ-ment but also its estimated survival probability in the followinggeneration. In this study the Fitness Function is assigned to bethe loadalability parameter k.

4.3. Selection

The individuals that scored the largest fitness values are se-lected as possible parents because of the fact that they have higherlikelihood of survival. So that, the larger the chromosomes fitness,the higher the survival probability in the next generation. There-fore, a roulette wheel selection is used to distinguish which arethe best individuals (parents) to be reproduced.

4.4. Crossover

The method employs a one-point crossover operation. Thus,after arbitrarily selecting the two chromosome parents the methodcan arbitrarily choose one crossover point for creating new off-spring chromosomes.

4.5. Mutation

The ultimate aim pursued by the mutation operator is to intro-duce variety in the population. So that, this operator selects arbi-trarily individuals from the population and alters some of itscharacteristics. In this case the mutation operation is achievedwith a small probability after crossover and it creates new individ-uals whose information was not included in previous generationsyet.

The complete optimization process is shown in Fig. 12.

4.6. Example

Firstly, a GA approach is applied to a modified IEEE-14 buspower system (as shown in Fig. 13), in which the original synchro-nous generator in bus 8 has been replaced by a wind farm with arated power of 30 MW. Network data are to be found at [7].

The main objective is to find out the optimal reactive powerinjection from the wind farm in order to maximize the voltage loa-dability keeping at the same time a voltage profile within ±10% ofthe nominal value in all buses. This type of optimization problemcould be solved by several optimization algorithms, such as theclassical Optimal Power Flow algorithm (OPF) [3,8,27] that hasbeen thoroughly applied in reactive power management studies.

Wind farm location is fixed at bus 8 and not included in theoptimization process. Constraints and models of the wind farmsare the same in both formulations obtaining the results that couldbe seen at Table 2.

Fig. 12. Optimization process.

Fig. 13. Modified IEEE 14-bus test system.

Table 2Genetic algorithm.

Algorithm Reactive power injection kcritical Criticalfrom the wind farm (Mvar) (p.u.) bus

GA 45 0.667 14OPF 45 0.666 14

Fig. 14. 140 Bus Spanish test system.

Table 3Load distribution in the 140-bus system.

Voltage level (kV) Power load (% Ptotal load)

132 2045 2215 190,380 39

H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586 2583

Results show how both formulations achieve the same optimalreactive power injection at the wind farm and the maximumloadalability as expected. However, genetic algorithm offers theadvantages of being more flexible, as well as it has the ability ofperforming multiobjective optimization searches and could copewith more complex systems converging to the global solution.All these characteristics make GA very convenient for the optimi-zation problem defined in this paper.

5. A 140 bus Spanish test system

5.1. Test system description and problem formulation

In the next section the proposed RPP methodology is appliedupon a real power system. The system is made up of 140 nodes

with five voltage levels from 380 kV, 132 kV, 45 kV, 15 kV to lowvoltage at 380 V. The structure of the 45 kV network is a ring asit can be seen in Fig. 14 where 15 kV and 380 V grids have been re-moved to clarify the description. Although most loads are con-nected at low voltage level, there are indeed some located atmedium voltage level and even a few of them at high voltage levelas is shown in Table 3.

Bus #2 is chosen as the slack bus because it is connected to therest of the Spanish network through the 380 kV network.

According to Spanish national legislation every utility is obligedto maintain the voltage supply within an admissible margin of 7%around nominal voltage supply. For low voltage networks SpanishNational Standard UNE 21301 recommends the voltage to be keptwithin the range of ±10%Un. Initial voltage profile, in per unit, atevery node is shown in Fig. 15.

0 20 40 60 80 100 120 140

Volta

ge (p

.u.)

# Buses

0.92

0.94

0.96

0.98

1

1.02

1.04

Fig. 15. Initial voltage profile.

2584 H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586

The target is to allocate three wind farms in three areas with thehigher wind resource availability.

� Area 1: It is planned to install a fixed speed wind farm in theconfined zone between power network buses 8-9-10-11. Inorder to offer reactive power capability it would be necessaryto add a SVC at the wind farm power station.� Area 2 (DFIG2): In this region the wind resource availability is

high and thus it allows the installation of a 5 MW DFIG variablespeed wind turbine. The potential connection points to the net-work are limited to buses 23-24-25-26-27-28-29-30-31 or 32.� Area 3 (DFIG1): This area has a very high wind availability

resource and therefore it is planned to install a 5 MW DFIG var-iable speed wind turbine. The possible connection points to thepower network are restricted to bus numbers 12-13-14-15-16-17-18-19-21-21-22.

In order to improve the voltage profile, two additional SVC’s areto be installed in the whole network. Both, location points andreactive power injection (from the three wind farms and fromthe two SVC’s) will be optimally obtained by the RPP optimizationstrategy which has been largely explained in previous sections andshown in Fig. 12.

The validation approach of the optimization formulation is car-ried out applying the GA optimization algorithm to different prob-lems increasing gradually the complexity in each stage:

� Stage 1: The objective is to optimally locate several SVC’s and tofind out the optimal reactive power injection from the SVC’s inorder to improve voltage stability in the whole network. In thiscase, there is no wind farm connected to the grid. This is theclassical way to perform a RPP study as done in [7] and willbe the first problem to be analyzed.� Stage 2: The objectives are to optimally locate several SVC’s and

their optimal reactive power injection and to optimally locatewind farms in order to improve voltage stability in the wholenetwork. In this stage, DFIG’s are modeled as PQ node with afix power factor that implies that the reactive power injectionfrom the DFIG is fixed and consequently it is not a variable inthe RPP algorithm. This approach is the one followed by[5,28,29].� Stage 3: The objectives are to optimally locate several SVC’s and

their optimal reactive power injection and to optimally locatewind farms and their optimal reactive power injection in orderto improve voltage stability in the whole network. In this stage,the electrical machine is considered to be able to inject reactivepower to the grid and consequently the reactive power injection

capability from the DFIG electrical machine is included in theoptimization strategy too. This formulation has been consideredin[9,10,30].� Stage 4: The objectives are to optimally locate several SVC’s and

their optimal reactive power injection and to optimally locatewind farms and their optimal reactive power injection in orderto improve voltage stability in the whole network. In this stage,the whole DFIG is considered to be able to inject reactive power,both from the electrical machine and from the grid side con-verter. The reactive power injection capability from the DFIG(electrical machine+GSC) is included in the optimization strat-egy. There are not related works following the sameformulation.

5.2. Simulations

Table 4 shows the optimum results obtained by the geneticalgorithm and the proposed RPP methodology. It should be re-marked that in stages 2, 3 and 4 the optimum location of SVC’sare always the same. Just in the first case (without wind farms)it will be necessary to change the location of one SVC (from bus#14 to bus #2) in order to improve the voltage profile and voltagestability of the whole network. However, in stage 1, the two SVC’sare the only responsible units for injecting reactive power as wellas the only ones in charge of improving the voltage stability ofthe whole network.

DFIG1 are optimally located in bus #13, and DFIG2 are located inbus #31. The optimum location points are not influenced by thereactive power regulation capability available. Fixed speed windfarm is optimally located at bus #11 in all cases.

Furthermore, it should be pointed out that as soon as the reac-tive power capability of the DFIG starts increasing (stages 2, 3 and4), the external SVC’s located at buses #14, #21 need to inject few-er reactive power. Consequently, the SVC’s sizes and cost arereduced.

As can be noted in stages 3-4, the optimization algorithm triesto maximize the reactive power injection available from the DFIG.For that reason, the reactive power injected by the wind farmsmeets the maximum available value in all cases.

5.3. Analysis of results

First of all it should be remarked that the obtained results inprevious stages (1–3) are in line with the conclusions publishedby authors [5,8,18,22]. The work done in these stages (hypothesis,models, constrains and optimization algorithm) allows us todeduce its validation.

Table 4Optimal bus location and reactive power injection.

Case Three wind farms Two SVC’s

DFIG1 Fixed speed wind farm DFIG2 SVC1 SVC2

# bus Q (Mvar) # bus Q (Mvar) # bus QSVC (Mvar) #bus Q (Mvar) # bus Q (Mvar)

1 – – – – – – 2 8 21 102 13 0 11 1 31 0 14 8 21 93 13 1.45 11 0.53 31 1.45 14 7 21 94 13 3.06 11 0.66 31 3.06 14 4 21 9

0 20 40 60 80 100 120 140

Stage 0Stage 1Stage 2Stage 3Stage 4

0.9

0.95

1

1.05

Volta

ge (p

.u.)

# Buses

Fig. 16. Optimum voltage profile.

Table 5Optimal and critical loadability parameter.

Stage klimit (p.u.) kcritical (p.u.)

Stage 1 0.04 1.2Stage 2 0.17 1.2Stage 3 0.18 1.4Stage 4 0.20 1.4

H. Amaris, M. Alonso / Energy Conversion and Management 52 (2011) 2575–2586 2585

Stage 4 focuses on the proposed novelty of the paper, i.e.: toinclude the DFIG reactive power capability of the wind turbine(electrical machine and grid side converter) in the optimizationprocess. This is the novel keystone of the paper and the main con-tribution to the scientific community and its justification could beeasily inferred from the fulfillment of the starting hypothesis.

Fig. 16 shows the voltage profile curve for the 140-buses in allthe analyzed stages. It should be noted that an increase of the reac-tive power capability from wind turbines (stages 3 and 4) allows toobtain a voltage profile closer to the voltage reference level (1[p.u]) with lower voltage dispersion in the whole network.

Table 5 shows the voltage stability results: klimit where the min-imum voltage allowable is reached Umin; and the kcritical where thepoint of voltage collapse is reached. Results have proved that an in-crease of the reactive power capability from DFIG wind turbines in-creases the loadalability parameter and the Voltage StabilityMargin too. As a consequence of this widespread increase thewhole network improves the voltage stability.

It can be seen that incorporation of reactive power capability ofthe DFIG (stage 3) improve Voltage Stability Margin (VSM) from1.2 p.u. to 1.4 p.u., so that critical loading correspond to the pointof voltage collapse is increased in a 116.7%. Furthermore, if thereactive power capability of grid side converter is tacking into ac-count in the optimization process (stage 4), maximum loading ofthe system, under planning operator voltage limits, is improvedin a 111.1%.

6. Conclusions

This paper presents a new approach for the coordinated reactivepower planning among wind farms and FACTS devices.The pro-posed optimization algorithm includes directly in its formulationthe DFIG physical limits to compute the optimum reactive powerthat should be injected by the variable speed wind turbines.

In contrast to DFIG models reported in the open literature, theproposed models are not constrained to the classic PQ or PV repre-sentation usually employed in power flow or reactive power anal-ysis. On the contrary, they consider the reactive power capabilityfrom the DFIG induction machine and from the grid side converterin the optimization process. This has the advantage of representingcurrent DFIG operations more realistically.

Furthermore this paper introduces the mathematical model ofthe DFIG and analyses the influence of each limiting DFIG parame-ter in the power delivery to the grid. Thus the methodology pro-posed could be extended to any other VAR sources and variablespeed wind turbines.

In order to validate the effectiveness of the methodology, it hasbeen applied in an existing power network with 140 buses and ithas been efficiently approved. The formulation could be very usefulto systems operators in the planning, operation and any other sit-uations where reactive power reserve would be needed.

Acknowledgement

This work has been partially supported by the Spanish Ministerof Science and Innovation under Contract ENE2009-13883-CO2-01.

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