Applied Energy 86 (2009) 1555–1564

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Applied Energy

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

Multiobjective clearing of reactive power market in deregulated power systems

A. Rabiee a,*, H. Shayanfar a, N. Amjady b

a Center of Excellence for Power System Automation and Operation, Electrical Engineering Department, Iran University of Science and Technology (IUST), Narmak, Tehran 16846, Iranb Department of Electrical Engineering, Semnan University, Semnan, Iran

a r t i c l e i n f o

Article history:Received 1 May 2008Received in revised form 28 October 2008Accepted 6 November 2008Available online 19 December 2008

Keywords:Reactive power marketTotal payment function (TPF)Multiobjective Mathematical ProgrammingVoltage security margin (VSM)

0306-2619/$ - see front matter � 2008 Elsevier Ltd. Adoi:10.1016/j.apenergy.2008.11.006

* Corresponding author. Tel.: +98 912 2586069; faxE-mail address: rabiee@iust.ac.ir (A. Rabiee).

a b s t r a c t

This paper presents a day-ahead reactive power market which is cleared in the form of multiobjectivecontext. Total payment function (TPF) of generators, representing the payment paid to the generatorsfor their reactive power compensation, is considered as the main objective function of reactive powermarket. Besides that, voltage security margin, overload index, and also voltage drop index are the otherobjective functions of the optimal power flow (OPF) problem to clear the reactive power market. AMultiobjective Mathematical Programming (MMP) formulation is implemented to solve the problem ofreactive power market clearing using a fuzzy approach to choose the best compromise solution accordingto the specific preference among various non-dominated (pareto optimal) solutions. The effectiveness ofthe proposed method is examined based on the IEEE 24-bus reliability test system (IEEE 24-bus RTS).

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction

Reactive power is tightly related to bus voltages throughout apower network, and hence it has a significant effect on systemsecurity. One of the main reasons for some of recently major black-outs in the power systems around the world such as those occurredin September 23, 2003 in Sweden and Denmark, September 28,2003 in Italy and also the United State and Canada blackout (Au-gust 2003) was reported as insufficient reactive power of systemresulting in the voltage collapse [1–3].

This paper proposes a new reactive power market framework.So, the literature review discusses about previous reactive powermarkets presented in the literature. Moreover, some referencesabout technical and economical aspects of reactive power in thederegulated systems are also discussed in this section.

In recent years, some papers are published in the area of opti-mal pricing of reactive power, using the well-known marginalprice theory to determine optimal prices for reactive power [4–9]. All of these papers assume that the consumer of reactive powershould pay for the reactive power service and the producers ofreactive power are remunerated. The differences among variousworks are mainly in the formulation of the optimization. Also,some of more recent research works on designing reactive powermarket consider technical issues of the power system in additionto the economical aspects [10–14].

Bhattacharya et al. have designed a competitive reactivepower market [15–17]. In order to compensate a generator

ll rights reserved.

: +98 021 77240490.

financially for its reactive power support, generator expectedpayment function (EPF) is defined and formulated so that inde-pendent system operator (ISO) can easily call for reactive bidsfrom all parties [15]. The ISO administers transmission tariffs,maintains the system security, coordinates maintenance schedul-ing, and has a role in coordinating long-term planning. The ISOshould function independent of any market participants, suchas transmission owners, generators, distribution companies, andend-users, and should provide nondiscriminatory open accessto all transmission system users. The ISO has the authority tocommit and dispatch some or all system resources and to curtailloads for maintaining the system security (i.e., remove transmis-sion violations, balance supply and demand, and maintain theacceptable system frequency). Also, the ISO ensures that propereconomic signals are sent to all market participants, which inturn, should encourage efficient use and motivate investmentin resources capable of alleviating constraints. More details aboutISO can be found in [18]. Consequently, according to the gener-ator EPF, a two-part reactive bid structure is suggested. In [16],a four-component bidding framework is proposed for synchro-nous generators. Mitigating market power, a localized reactivepower market is proposed in [17]. It is observed that the local-ized reactive power market restricted the market power of eachgenerator to its area and it no longer affects the reactive powerprices of the other zones.

In [19], a pricing mechanism has been proposed for the othercompensators of reactive power in the competitive market. In theproposed market, the owners of network devices (e.g. shunt capac-itors, SVCs), like generators, bid the price for supplying or absorb-ing reactive power. Nevertheless, this approach could be applicable

B

A

base

Min

B A Rated Max

Fig. 1. Synchronous generator capability curve.

1556 A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564

only if the ISO accepts these devices as the other resources of reac-tive power ancillary service and compensate them financially.

In the context of deregulated electricity markets, reactive powerdispatch corresponds to short-term allocation of reactive power re-quired from suppliers based on current operating conditions [20].The ISO is concerned with determining the optimal reactive powerschedule for all providers based on a given objective that dependson system operating criteria [21]. Different objective functions canbe used by the ISO, besides the traditional transmission losses min-imization, such as minimization of reactive power cost [22,23],minimization of deviation from contracted transactions [16], min-imization of the cost of adjusting reactive power control devices[24], or maximization of the system load ability to minimize therisk of voltage collapse [25–27].

Considering voltage security in the reactive power pricing, in[28] a cost-based reactive power pricing is proposed, which inte-grates the production cost of reactive power and voltage stabilitymargin requirement of pre- and post-contingencies into the OPFproblem. In [29], a two-level framework is proposed for theoperation of a competitive reactive power market taking into ac-count system security aspects. The first level, i.e., procurement,is on a seasonal basis while the second level, i.e., dispatch, isclose to real-time operation. In that work, reactive power pro-curement is considered as an essentially long-term issue, i.e., aproblem in which the independent system operator or ISO seeksoptimal reactive power ‘‘allocation” from possible suppliers thatwould be best suited to its needs and constraints in a given sea-son [29]. This optimal set should ideally be determined based ondemand forecast and system conditions expected over theseason [29].

However, seasonal market for reactive power encounters prob-lems. First, the reactive power consumption of system is volatilethat its forecasting over a season becomes very hard. Second, inspite of active power, the reactive power requirement of systemstrongly depends on the loading condition of network. In the heavyload conditions of system, some of transmission lines are loadedmore than their surge impedance loading (SIL) and become as sinksof reactive power in the over SIL loading conditions. On the otherhand, in the light loading conditions, the transmission lines areusually loaded in the under SIL conditions, and become as sourcesof reactive power. This further complicates the prediction of reac-tive power requirement of the power system over a long horizon.Third, the occurrence of different planned/unplanned outagesand effects of maintenance scheduling (such as generators andtransmission lines entering to circuit after their maintenance per-iod) in a season can change the configuration of the power system,leading to more complexity of designing a seasonal reactive powermarket. Fourth, over the long time of a season, the ISO can handlethe reactive power requirements of the system only with the se-lected generators of the network that have contract with them tobecome available for reactive power compensation and theremaining generators that are not selected at the beginning ofthe season are no longer participated in the reactive power com-pensation. In other words, the available sources of reactive poweris limited to the selected generators over a long time which is tosome extent in contradiction with the local nature of reactivepower. Considering the above mentioned problems of the seasonalprocurement model for reactive power market, this paper presentsa day-ahead reactive power market model, including both econom-ical issues and security aspects.

It is mentioned that, in this paper, adhering to existing FERC(Federal Energy Regulatory Commission) regulation, only reactivepower support from generators is considered as one of six ancillaryservices eligible for financial compensation. However, the pro-posed market is generic enough to be readily extended to includeother reactive power devices such as capacitor banks, reactors,

and FACTS devise. Consequently, these devices, like generators,can be considered as independent VAR sources which lead toimproving the level of competition and also limiting the exerciseof market power from privilege located generators [30]. In [31] adiscussion about considering LTC among the reactive power ancil-lary services has been presented and it has been concluded thatLTC transformers should be separated from SVCs, synchronouscondensers, and shunt compensators. However, to the best of ourknowledge, no research work in the area proposes bidding struc-ture for LTC transformers in order to participate in the reactivepower market. Due to the reasons described above, in this paperwe only consider generators participating in the reactive powermarket. However, participation of the other reactive power marketcompensators will be considered in the future works. Contributionof this work can be summarized as follows:

(a) A day-ahead reactive power market is proposed, whichincorporates system security aspects in the clearing of reac-tive power market. The clearing process of the proposedreactive power market is formulated in the form of a multi-objective optimization problem.

(b) The multiobjective optimization problem of the reactivepower market clearing is solved by the e-constraint method.Besides, a fuzzy decision making technique is proposed toefficiently select the best compromise solution among thepareto optimal solutions based on the ISO preference.

2. Reactive power market

For the clearing of the reactive power market, in the proposedmethod the following four objective functions are used which willbe described in detail in the next subsections:

Multiobjective functions :

MinðF1ÞMaxðF2ÞMinðF3ÞMinðF4Þ

8>>><>>>:

ð1Þ

where F1: minimization of the TPF, F2: maximization of voltagesecurity margin, F3: minimization of overload index, and F4: mini-mization of voltage deviation index.

2.1. Minimization of the TPF (F1)

The reactive power capability curve of a generator is shown inFig. 1 [16]. Qbase is the reactive power required by the generatorfor its auxiliary equipment. If the operating point lies inside the

A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564 1557

limiting curve, e.g. (PA, Qbase), then the unit can increase its reactivegeneration from Qbase to QA without requiring the adjustment ofPA. However, this results in increased loss of winding and, hence,increases the cost of loss. If the generator operates on the limitingcurve, any increase in Q will require a decrease in PA to adhere tothe winding heating limit. Consider the operating point ‘‘A” onthe curve defined by (PA, QA). If more reactive power is requiredfrom the unit, for example QB, the operating point requires shiftingback along the curve to point (PB, QB), where PB < PA. This indicatesthat the unit has to reduce its active power output to adhere to thefield heating limits when higher reactive power is demanded. Thelost in the revenue of generator due to the reduced production ofactive power is termed lost opportunity cost (LOC).

Accordingly, three operating regions for a generator on thereactive power coordinate are defined. In region-I (0 to Qbase),the provided reactive power of generator is necessary for thegenerator own requirements to maintain its auxiliary equipment.Therefore, the generated reactive power in this region is not con-sidered as an ancillary service to be remunerated nor the gener-ator entitled to payments. In region-II (Qbase to QA), and (0 toQmin), because of generating or absorbing reactive power, lossesof generator increase and therefore it can expect to be paid forits service. Thus, the EPF contains the cost of loss componentin addition to availability component. Finally in region-III (QA

to QB), the generator is managed to reduce its active power togenerate the required reactive power. Thus, the generator incursloss of revenue cost and consequently, the EPF will contain allcomponents of cost (availability cost, cost of loss and opportu-nity cost). Accordingly, the EPF can be determined in any operat-ing condition of synchronous generator. The EPF of a generatoras a function of the amount of generator reactive power produc-tion is illustrated in Fig. 2 [16]. According to the classification ofreactive power production cost, an offer structure is formulated[16]

EPFi;uG ¼ ai;u

0 þZ 0

Qi;umin

mi;u1 dQ i;u

G þZ Qi;u

A

Qi;ubase

mi;u2 dQi;u

G þZ Qi;u

B

Qi;uA

mi;u3 Q i;u

G dQi;uG

ð2Þ

The coefficients in (2) represent the various components ofreactive power cost incurred by the uth provider connected tothe ith bus needed to be offered in the reactive power marketwhere ai;u

0 is availability price offer in dollars, mi;u1 is cost of loss

price offer for operating in under excited mode (Qmin < Q 6 0) in$/MVAr-h, mi;u

2 is cost of loss price offer for operating in region(Qbase to QA) in $/MVAr-h, and mi;u

3 is opportunity price offer foroperating in region (QA to QB) in $/MVAr-h/MVAr-h (Fig. 2). Inthe proposed multiobjective clearing of reactive power market,the minimization of total payment to the participants of reactivepower market is considered as the main objective function (F1).The total payment function (TPF) is formulated as follows [16,17]:

Fig. 2. Reactive power offer structure of provider.

TPF ¼XNB

i¼1

XNUi

u¼1q0Wi;u

0 � q1Wi;u1 Q i;u

1G þ q2Wi;u2 ðQ

i;u2G � Qiu

baseGÞh

þq2Wi;u3 ðQ

i;u3G � Q i;u

baseGÞ þ12q3Wi;u

3 ðQ i;u3GÞ

2 � ðQ i;uAGÞ

2� ��

ð3Þ

where Qi;u1G, Qi;u

2G, and Qi;u3G represent the regions (Qmin < Q 6 0), (Qbase

to QA), and (QA to QB), respectively, for the uth provider connected tothe ith bus; Wi;u

1 , Wi;u2 , and Wi;u

3 are binary variables, showing thecompensation region of the unit. Also q0 is the uniform auctionavailability price, q1 and q2 are the uniform auction loss price forabsorption and production of reactive power, respectively, and q3

is the uniform auction opportunity price. If the uth unit connectedto ith bus is selected by the ISO, Wi;u

0 will be equal to one and it willreceive the availability price, regardless of its reactive power out-put. However, only one of the binary variables Wi;u

1 , Wi;u2 , and Wi;u

3

can be selected. In other words, Wi;u1 þWi;u

2 þWi;u3 ¼ 1.

As discussed in [14], a generator may produce reactive power,but cannot support the system or even the generator could needthe reactive power support for transmission of its own activepower. For a generator, the minimum amount of reactive power re-quired to transfer its own active power is named as Qneed. The valueof Qneed depends strongly on the loading condition of the system sothat in the heavy load condition this value may become even morethan the maximum reactive power output of the generator. In or-der to consider Qneed of generators in the reactive power market,the boundaries of the integral terms in the EPF in Eq. (2) of the pa-per should change from Qi;u

base, and Q i;uA to Qi;u

base1 and Qi;uA1, respec-

tively, where

Qi;ubase1 ¼ maxðQ i;u

base;Qi;uneedÞ ð4Þ

Qi;uA1 ¼ maxðQ i;u

A ;Qi;uneedÞ ð5Þ

Accordingly the Eq. (2) of paper will be changed as follows:

EPFi;uG ¼ ai;u

01 þZ 0

Qi;umin

mi;u1 dQi;u

G þZ Qi;u

A

Qi;ubase1

mi;u2 dQ i;u

G þZ Qi;u

B

Qi;uA1

mi;u3 Q i;u

G dQi;uG

ð6Þ

where ai;u01 is

ai;u01 ¼ ai;u

0 � HðQi;uB � Qi;u

needÞ ð7Þ

where H(.) is the heavy side (unit step) function. It is equal to 1 if itsargument is positive; otherwise it is equal to zero. Eq. (7) indicatesthat if a generator’s Qneed becomes more than its QB it should not bepaid for availability, as it only produces reactive power to transferits own active power. In this case, the generator cannot be consid-ered a reactive power compensator for the system.

In [14], it is mentioned that Qneed is determined only for heavyload conditions. In the light load conditions however the concernfor the system security may be different and a generator mayrather absorb reactive power. But, Qneed is not determined in [14]for the light load condition and therefore the Qmin in Eq. (2) isnot changed accordingly. However, modification of Qneed for lightload conditions (like the modification of Qbase and QA for heavy loadconditions) will be considered in our future work.

The proposed reactive power market is cleared sequential to theday-ahead active power market like the reactive power marketsproposed in [16,17], [19,21,29]. In other words, the active poweroutput of generating units is considered as the input of reactivepower market and assumed fixed during the reactive power dis-patch procedure. Active and reactive powers are however coupledwith together through the AC power flow equations and branchloading limits as well as the synchronous generators capabilitycurve. So, the solution obtained from a coupled OPF model simul-taneously dispatching active and reactive powers may be theoret-

P

Vmax

Vmin

Pn Pcr

VCurrent OperatingPoint

Security Loding Point

Voltage Collapse PointVoltage Security

Margin

VSM

Fig. 4. Representation of current and security loading points.

1558 A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564

ically closer to the optimal. However, as discussed in [29], in addi-tion to the market power and price volatility problems associatedwith handling active and reactive power markets simultaneously,computational burden becomes an issue for practical sized powersystems, since it would require solving a rather complex andlarge-scale non-linear programming (NLP) model. Decoupling theOPF problem provides the required flexibility for spot marketapplications, and avoids having to deal with the coupled modelcomplexity, while retaining an acceptable level of accuracy [29].

2.2. Loading margin maximization (F2)

For secure operation of the power system, it is required to main-tain adequate voltage stability margin not only under normal con-ditions but also under contingency cases. Voltage stability marginin the load domain, which measures the distance from the currentoperating point to the voltage collapse point in terms of load incre-ment, is considered as the voltage stability index [32]. In steadystate voltage stability studies, the P–V curve, as shown in Fig. 3,has been usually used and its nose point considered as the systemvoltage collapse point. However, in literature it has been shownthat in the systems with inconstant-power loads, the real voltagecollapse point is the SNB of the bifurcation curve (or point B00 onP–V curve in Fig. 3) instead of the nose point (NP) of P–V curve(point B0) [33]. Nevertheless, when all the loads are constant-power type, nose point just coincides with the saddle point node[34]. It should be mentioned that in this work the loads are as-sumed to be constant-power type.

Referring to Fig. 3 as a typical P–V curve of a power system, thevoltage stability margin (VSM) is the horizontal distance betweenthe current operating point (B) and the voltage collapse point(B0). The VSM in the load domain indicates the power system max-imum load-ability in terms of voltage stability [35], which is de-fined as follows:

VSM ¼MVAL �MVAN

MVAN ¼MVAðB0 Þ �MVAðBÞ

MVAðBÞð8Þ

In Fig. 4 both voltage security margin and VSM are shown. Thesecurity loading point refers to the maximum allowable load incre-ment at which overload and voltage drop constraints (which willbe mathematically described in Section 2.4) are satisfied. The volt-age security margin is the distance between the current operatingpoint and security loading point (Fig. 4). The conventional voltagestability indices, like (8), indicate the stability border at which thepower system has stable solution without considering the qualityof the operating point. Besides, the operator usually determines aproper voltage range, as shown in Fig. 4, in order to keep high qual-

P ,

NosePoint

SNB.

P 2

.Bifurcation curve

P-V curve

P 1

B.B’.B’’.

Fig. 3. Bifurcation and P–V curves of an inconstant-power load.

ity voltages and to prevent the electric power devices from dam-ages in addition to active power losses reduction [36]. In otherwords, the security constraint should be satisfied not only at thecurrent operating point but also at the system security loadingpoint (Fig. 4).

Therefore, to maintain both adequate VSM and high qualityvoltages, the maximization of voltage security margin is consid-ered as the second objective function of the proposed market clear-ing process

F2 ¼ k ð9Þ

2.3. Overload and voltage deviation minimization

In order to maintain power system security, it is further re-quired that all bus voltages be in the allowable ranges and alsoany of the system lines should not be overloaded. So, to considerthese concerns in the market clearing procedure, one alternativeis including voltage drop and overload of lines as the extra objec-tive functions in the multiobjective optimization problem asfollows:

F3 ¼XNB

i¼1

jVi � Vref jVref

ð10Þ

F4 ¼XNB

i¼1

XNB

j¼1j–i

Sij

�Sij

!2

ð11Þ

Lower values of F3 and F4 result in less voltage deviations andbranch overloads, respectively, which in turn lead to a more securepower system. Therefore, these indices must be minimized in theoptimization problem.

2.4. Constraints of multiobjective reactive power market clearing

(1) Load flow constrains:

XNUi

u¼1

Pi;uG � PDi ¼

XNB

j¼1

Vi � Vj � Yij � cosðdi � dj � hijÞ ð12Þ

XNUi

u¼1

Q i;uG � Q Di ¼

XNB

j¼1

Vi � Vj � Yij � sinðdi � dj � hijÞ ð13Þ

XNUi

u¼1

Pi;uG � PDi ¼

XNB

j¼1

V i � V j � Yij � cosðdi � dj � hijÞ ð14Þ

A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564 1559

XNUi

u¼1

Q_

i;uG � Q Di ¼

XNB

j¼1

V i:V j:Yij: sinðdi � dj � hijÞ ð15Þ

i, j is the buses indices; Pi;uG is active power generation of the uth unit

of the ith bus at current operating point; PDi is active power demandat bus i at current operating point; Qi;u

G is reactive power generationof the uth unit of the ith bus at current operating point; QDi is reac-tive power demand at bus i at current operating point; V is the mag-nitude of voltage at current operating point; d is the angle of voltageat current operating point; ‘^’ is a symbol indicating security load-ing point; Sb,i is apparent power of branch i at current operatingpoint; Yij is magnitude of element i and j of admittance matrix; hij

is angle of element i and j of network admittance matrix.Eqs. (12) and (13) are power flow constraints at the current

operating point and (14) and (15) are power flow constraints atthe security loading point.

(2) The operation constraints of generators for reactive powercompensation:

Wi;u0 ;W

i;u1 ;W

i;u2 ;W

i;u3 2 f0;1g ð16Þ

Q i;uG ¼ Q i;u

1G þ Qi;u2G þ Q i;u

3G ð17Þ

Wi;u1 � Q

Gi;umin 6 Q i;u

1G 6 0 ð18Þ

Wi;u2 � Q

i;ubaseG 6 Q i;u

2G 6Wi;u2 � Q

i;uAG ð19Þ

Wi;u3 � Q

i;uAG 6 Q i;u

3G 6Wi;u3 � Q

i;uBG ð20Þ

Wi;u1 þWi;u

2 þWi;u3 6Wi;u

0 ð21Þ

Q i;uG 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðVi;u

t � Ii;ua Þ

2 � ðPi;uG Þ

2q

capability curve limit

ðarmature current limitÞ ð22Þ

Q i;uG 6

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVi;u

t � Ei;uaf

Xi;us

!2

� ðPi;uG Þ

2

vuut

� ðVi;ut Þ

2

Xi;us

capability curve limit ðfield current limitÞ ð23Þ

When the uth unit of ith bus is not selected or is selected and oper-ated in the absorption region [QminG, 0] or production region [Qbase,QB] then the constraint (21) is satisfied in the equality form (0 = 0and 1 = 1, respectively). However, when the unit is selected for reac-tive reserve then this constraint will be satisfied in the inequalityform (0 < 1).

(3) Constraints related to determination of MCPs in reactivepower market

Wi;u0 � a

i;u0 6 q0 ð24Þ

Wi;u1 �m

i;u1 6 q1 ð25Þ

ðWi;u2 þWi;u

3 Þ �mi;u2 6 q2 ð26Þ

Wi;u3 �m

i;u3 6 q3 ð27Þ

(4) Security constrains:

Sb;l 6 Smaxb;l ð28Þ

Vmini 6 Vi 6 Vmax

i ð29Þ

Sb;l 6 Smaxb;l ð30Þ

Vmini 6 V i 6 Vmax

i ð31Þ

VSM P VSMspec ðvoltage stability margin constraintÞ ð32Þ

k P kmin ðvoltage security margin constraintÞ ð33Þ

Pi;uG ¼ ð1þ kþ kGÞ � Pi;u

G ð34Þ

PDi ¼ ð1þ kÞ � PDi ð35Þ

QDi ¼ ð1þ kÞ � Q Di ð36Þ

Eqs. (28)–(33) include security constraints at the current operatingpoint, i.e., (28) and (29), security constraints at security loadingpoint, i.e., (30) and (31), Voltage Stability Margin (VSM) constraint,i.e., (32), and voltage security margin constraint, i.e., (33)–(36). Inorder to consider Qneed of generators in the reactive power marketin the above equations, ai;u

0 Qi;ubase and Qi;u

A are substituted byai;u

01;Qi;ubase1, and Qi;u

A1, respectively which are defined according toEqs. (4), (5), and (7).

2.5. Multiobjective Mathematical Programming

In Multiobjective Mathematical Programming (MMP) there ismore than one objective function and there is no single optimalsolution that simultaneously optimizes all the objective functions.In these cases, the decision makers are looking for the ‘‘most pre-ferred” solution. In MMP the concept of optimality is replaced withthat of efficiency or Pareto optimality. The efficient (or Pareto opti-mal, nondominated, non-inferior) solution is the solution that can-not be improved in one objective function without deteriorating itsperformance in at least one of the rest. In order to deal with theMMP problem of reactive power market clearing, four objectivefunctions F1, F2, F3 and F4 are taken into account, described in (3),(9), (10) and (11), respectively. A well-organized technique to solveMMP problems owning one main objective function among allobjective functions is the e-constraint method. This method is se-lected to solve the MMP problem of reactive power market clearingconsidering F1 as the main objective function of the problem.

In general, the e-constraint technique optimizes the main objec-tive function F1 considering the other objective functions as con-straints [37]:

min F1ðxÞsubject to F2ðxÞ 6 e2 F3ðxÞ 6 e3 � � � FpðxÞ 6 ep

ð37Þ

where subscript p indicates the number of competing objectivesfunctions of MMP problem. In order to properly apply the e-con-straint method, the range of at least p�1 objective functions areneeded that will be used as the constraints. The most common ap-proach is to calculate these ranges from the payoff table [37] as de-scribed below.

In general, to calculate the payoff table for a MMP problem withp competing objective functions, at first, the individual optima ofthe objective functions is calculated. Then with the solution thatoptimizes the objective function Fi, i = 1, . . . ,p, (F�i indicates theoptimum value of Fi) the value of the other objective functionsF1, . . . ,Fi�1,Fi+1, . . . ,Fp is calculated, which are represented byFi

1; . . . ; Fii�1; F

iiþ1; . . . ; Fi

p. The ith row of the payoff table includesFi

1; . . . ; Fii�1; F

�i ; F

iiþ1; . . . ; Fi

p. In this way all rows of the payoff ta-ble are calculated. The jth column of the payoff table includes theobtained values for the objective function Fj among which the min-imum and maximum values indicate the range of the objectivefunction Fj for the e-constraint method.

It is noted that in the MMP problem of reactive power marketclearing, only the range of the objective functions F2, F3, and F4

1 2 7

8

6

13

23

201916

17

1821 22

39 10

54

2411 12

1514

Fig. 5. IEEE 24-bus RTS.

1560 A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564

are calculated, since F1 is the main objective function. Then, we di-vide the range of the objective functions F2, F3, and F4 to q2, q3, andq4 equal intervals using (q2 � 1), (q3 � 1), and (q4 � 1) intermediateequidistant grid points, respectively. Considering the minimumand maximum values of the range, we have in total (q2 + 1),(q3 + 1), and (q4 + 1) grid points for F2, F3, and F4, respectively. So,we should solve (q2 + 1) � (q3 + 1) � (q4 + 1) optimization subprob-lems where the subproblem(i,j,l) has the following form:

min F1ðxÞsubject to F2ðxÞP e2i F3ðxÞ 6 e3j F4ðxÞ 6 e4j

ð38Þ

e2i ¼ MinðF2Þ þMaxðF2Þ �MinðF2Þ

q2

� �� i i ¼ 0;1; . . . ; q2 ð39Þ

e3j ¼ MaxðF3Þ �MaxðF3Þ �MinðF3Þ

q3

� �� j j ¼ 0;1; . . . ; q3 ð40Þ

e4i ¼ MaxðF4Þ �MaxðF4Þ �MinðF4Þ

q4

� �� l l ¼ 0;1; . . . ; q4 ð41Þ

where Max(.) and Min(.) represent maximum and minimum valuesof the individual objective function based on the payoff table,respectively. It is noted that the constraints of the MMP problemof reactive power market clearing, described in Section 2.4, shouldbe also considered in each of the optimization subproblems. Thus,each optimization subproblem is in the form of Mixed IntegerNon-Linear Programming (MINLP) optimization problem. By solv-ing each optimization subproblem, one Pareto optimal solution isobtained. Some of these (q2 + 1) � (q3 + 1) � (q4 + 1) optimizationsubproblems may have infeasible solution space, which will bediscarded.

A desirable characteristic of the e-constraint method is that wecan control the density of the efficient set representation by prop-erly assigning the values to the q2, q3, and q4 [38]. The higher thenumber of grid points the more dense is the representation ofthe efficient set but with the cost of higher computation times. Atrade off between the density of the efficient set and the computa-tion time is always advisable. In this paper, the number of intervalsfor the objective functions F2, F3, and F4 is considered to be equal to4, i.e., q2 = q3 = q4 = 4.

After obtaining the Pareto optimal solutions by solving the opti-mization subproblems, the decision-maker needs to choose onebest compromise solution according to the specific preference fordifferent applications. In this paper a fuzzy approach is proposedfor the decision making process wherein a linear membershipfunction (li) is defined for each of the objective functions, i.e., F1,F2, F3, and F4, according to (42) and (43)

lki

i¼1;3;4¼

1 Fki 6 MinðFiÞ

MaxðFiÞ�Fki

MaxðFiÞ�MinðFiÞMinðFiÞ 6 Fk

i 6 MaxðFiÞ

0 Fki P MaxðFiÞ

8>><>>:

ðfor minimization objective functionsÞ ð42Þ

lki

i¼2¼

0 Fki 6 MinðFiÞ

Fki �MinðFiÞ

MaxðFiÞ�MinðFiÞMinðFiÞ 6 Fk

i 6 MaxðFiÞ

1 Fki P MaxðFiÞ

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

ðfor maximization objective functionÞ ð43Þ

where Fki and lk

i represent value of the ith objective function in thekth Pareto optimal solution and its membership function, respec-tively. The fuzzification process described in (42) and (43) are usedfor the objective functions that should be minimized or maximized,

respectively. The defined membership function lki indicates the de-

gree of optimality for the ith objective function in the kth Paretooptimal solution. The whole membership function of the kth Paretooptimal solution (lk) is calculated based on its individual member-ship functions lk

i as follows:

lk ¼Pp

i¼1wi � lkiPM

k¼1

Ppi¼1wi � lk

i

ð44Þ

where wi is the weight value of the ith objective function in theMMP problem and M is the number of Pareto optimal solutions.In our MMP problem p = 4. The weight values wi can be selectedby the ISO based on the importance of economical issue and differ-ent security aspects. We selected w1 = 0.5 and w2 = w3 = w4 = 0.5/3for the case study (IEEE RTS 24-bus). In other words, the sameimportance is considered for the economic and security aspects inthe market clearing process. Besides, the same weight is assignedto different security indicators F2, F3, and F4. The solution with themaximum membership function lk is the most preferred compro-mise solution based on the adopted weight factors and so is selectedas the best Pareto optimal solution or the final solution of the MMPproblem.

3. Numerical results

The IEEE reliability test system (RTS 24-bus) [39] is used toexamine the proposed multiobjective reactive power market clear-ing method. The test system is illustrated in Fig. 5, which consistsof 32 synchronous generators, 1 synchronous condenser (located atbus 14), and 17 constant-power type loads. The system total activeand reactive loads are 2850 MW and 580 MVAr, respectively. Thesystem data can be found in [39]. The participants of reactivepower market are supposed to submit their four components of of-fer prices (a0, m1, m2, m3). In this examination, a uniform random

A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564 1561

number generator is used to simulate the offer prices of generatorsshown in Table 1 [16]. The participants are also required to sendtheir Qbase, QA, and QB (Fig. 1). In this study, like [16], it is assumedthat QB = 1.5 � QA, and Qbase = 0.10 � Qmax, and QB = Qmax. As men-tioned before, QA is restricted either by the field or the armatureheating limit, with respect to operating condition. From Table 1,it can be seen that the synchronous condenser, is participated inthe reactive power market with its opportunity cost (m3) equalto zero. In [40], a minimum 10% value has been proposed for thevoltage security margin in the IEEE RTS, which is also consideredin this paper. The lower and upper bound of voltage are taken0.95 pu and 1.05 pu, respectively. The flows of lines are limitedto their continuous MVA rating. Each optimization subproblem ofthe MMP problem of reactive power market clearing (in the formof MINLP) is modeled in GAMS software using DICOPT solver[41]. The results of single objective and multiobjective reactivepower market are compared in two cases.

3.1. Case I: single objective model

In this case only total payment function (TPF), i.e., F1, is consid-ered as the single objective function for clearing of reactive powermarket. The optimization problem takes 1.038 s of CPU time and11.203 s of model solution time (indicated by the GAMS softwarepackage) on a Pentium IV, 512 MB RAM computer. The results ofthe single objective market clearing are presented in Tables 2and 3. The value of the F1 (TPF) in addition to the values of securityindices (overloading index, voltage drop index, voltage security

Table 1Units’ offer prices in reactive power market.

Bus no. Unit no. ai;u0 ($) mi;u

1 ($

1 1 0.96 0.862 0.94 0.823 0.85 0.794 0.83 0.82

2 1 0.50 0.542 0.42 0.423 0.69 0.684 0.65 0.62

7 1 0.75 0.612 0.80 0.753 0.70 0.65

13 1 0.68 0.502 0.70 0.543 0.75 0.60

14a 1 0.94 0.81

15 1 0.65 0.602 0.50 0.583 0.60 0.734 0.55 0.615 0.52 0.506 0.51 0.51

16 1 0.50 0.5018 1 0.90 0.8521 1 0.80 0.75

22 1 0.42 0.422 0.50 0.483 0.45 0.424 0.48 0.445 0.49 0.456 0.55 0.46

23 1 0.90 0.852 0.95 0.893 0.86 0.80

a Synchronous condenser (SC).

margin or k and voltage stability margin or VSM) are shown in Ta-ble 2. The reactive power outputs of units are represented in theTable 3.

3.2. Case II: multiobjective model

This case, studies the multiobjective optimization for clearing ofreactive power market. In addition to the TPF, i.e., F1, three securityrelated indices, viz. voltage security margin F2, voltage deviationindex F3 and overloading index F4, are considered as the additionalobjective functions. The whole optimization problem takes 15.45 sof CPU time and 19 min and 55.26 s of model solution time to gen-erate and solve (q2 + 1) � (q3 + 1) � (q4 + 1) = (5 � 5 � 5) = 125optimization subproblems. Out of these 125 optimization subprob-lems, six subproblems have infeasible solution space and so arediscarded. The remaining subproblems produce 125 � 6 = 119 Par-eto optimal solutions. Using the fuzzy decision making process, themost preferred compromise solution is selected among these Par-eto optimal solutions according to the adopted weight factors(w1 = 0.5 and w2 = w3 = w4 = 0.5/3). The results of the multiobjec-tive optimization are shown in Tables 2 and 3. Some interestingobservations can be seen from these tables as follows:

– Both voltage deviation index F3 and overloading index F4 in thiscase are lower than case I, while voltage security margin F2 (andconsequently voltage stability margin or VSM) in this case aregreater than case I, indicating enhancement of system securitylevel considering the security indices as extra objective functions

/MVAr-h) mi;u2 ($/MVAr-h) mi;u

3 ($/(MVAr-h)2)

0.86 0.460.82 0.450.79 0.390.82 0.40

0.54 0.280.42 0.350.68 0.390.62 0.37

0.61 0.430.75 0.360.65 0.32

0.50 0.310.54 0.390.60 0.50

0.81 0.00

0.60 0.300.58 0.250.73 0.380.61 0.270.50 0.260.51 0.27

0.50 0.300.85 0.480.75 0.41

0.42 0.170.48 0.200.42 0.380.44 0.350.45 0.330.46 0.32

0.85 0.480.89 0.550.80 0.45

Table 2Objective function values and MCPs of price components in cases I and II.

Objectives Case I single objective Case II multiobjective

TPF (F1) 436.12$ 496.55$Overloading index (F3) 8.665 8.49Voltage drop index (F4) 0.610 0.393Voltage security margin (k) (F2) 0.145 0.16Voltage stability margin (VSM) 0.161 0.18q0 0.96 0.96q1 0.00a 0.00q2 0.86 0.86q3 0.00 0.46

a Zero indicates that there is no need for that component in the power system and so no unit is selected in the corresponding region.

1562 A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564

in the multiobjective optimization problem. It is noted that themarket clearing in the form of multiobjective optimization frame-work changes the output of generating units as indicated in Table3.

– The TPF (F1) in this case is more than that of case I, reflectingthe additional cost of considering security indices as the extraobjective functions. In other words, the additional cost is relatedto enhancing the security level and loading capability of the powersystem in this case.

To better demonstrate the effect of weighting factors on themultiobjective framework, three cases with different weighting

Table 3Reactive power output and payment of units in cases I and II.

Bus no. Unit no. Case I single objective

Qg (MVAr)

1 1 3.22 6.73 20.04 20.0

2 1 0.02 0.03 20.04 13.4

7 1 40.02 34.93 0.0

13 1 53.32 53.33 0.0

14a 1 133.3

15 1 0.02 0.03 0.04 0.05 0.06 12.7

16 1 53.318 1 0.021 1 0.0

22 1 0.02 10.73 10.74 0.05 0.06 0.0

23 1 0.02 0.03 100.0

Total 585.5MVAr

a Synchronous condenser (SC).

factors are considered and the obtained results are shown in Table4. Emphasizing on the system security, the ISO considers weightingfactor 0.4 (W1 = 0.4) for system TPF and 0.6 for the system securityindices (W2 = W2 = W3 = 0.6/3 = 0.2) and the results are shown inthe first row of Table 4. It can be seen that the system TPF is morethan the case with W1 = 0.5 and W2 = W2 = W3 = 0.5/3 (the secondrow of Table 4), while the system security indices becomes betterthan that case. Conversely, in the third case, W1 = 0.6 andW2 = W2 = W3 = 0.4/3 and the results are reported in the thirdrow of Table 4. It is observed that the system TPF is decreasedand becomes lower than the case W1 = 0.5 andW2 = W2 = W3 = 0.5/3, while the system security indices are worsethan that case. So the proposed reactive power market gives the

Case II multiobjective

Payment ($) Qg (MVAr) Payment ($)

2.85 10.0 21.475.86 10.0 21.4715.58 19.8 15.4115.58 19.8 15.41

0.0 9.9 20.930.0 9.9 20.9315.58 19.8 15.419.90 0.0 0.0

30.2 39.8 30.0325.81 39.8 30.030.0 0.0 0.0

39.92 28.4 18.5039.92 0.0 0.00.0 0.0 0.0

98.39 133.2 98.31

0.0 0.0 0.00.0 3.7 3.620.0 0.0 0.00.0 0.0 0.00.0 0.0 0.05.00 53.2 39.83

39.91 53.2 39.830.0 0.0 0.00.0 55.5 31.49

0.0 0.0 0.08.78 0.0 0.08.78 0.0 0.00.0 0.0 0.00.0 0.0 0.00.0 0.0 0.0

0.0 0.0 0.00.0 0.0 0.074.06 99.8 73.88

436.12$ 605.8 MVAr 496.55$

Table 4Results of reactive power market with different weight factors.

W1 W2 W3 W4 TPF (F1) Overload index (F3) Voltage deviation (F4) Voltage security (k) (F2)

Case 1 0.4 0.2 0.2 0.2 544 8.43 0.36 0.18Case 2 0.5 0.5/3 0.5/3 0.5/3 496.55 8.49 0.393 0.16Case 3 0.6 0.4/3 0.4/3 0.4/3 444 8.58 0.44 0.14

Table 5Results of reactive power market with and without considering generators’ Qneed

W1 W2 W3 W4 TPF (F1) Overload index (F3) Voltage deviation (F4) Voltage security (k) (F2)

Without Qneed 0.5 0.5/3 0.5/3 0.5/3 496.55 8.49 0.393 0.16With Qneed 0.5 0.5/3 0.5/3 0.5/3 243 8.51 0.41 0.16

A. Rabiee et al. / Applied Energy 86 (2009) 1555–1564 1563

ISO the chance to select different weighting factors for the conflict-ing objective functions based on the system conditions.

Finally, in order to study the effect of Qneed, the reactive powermarket is also cleared considering Qneed of the generators. In thesystem annual peak load, Qneed of the system generators areapproximately about their maximum reactive power capacity(QB) and the system TPF becomes very low. So, to better demon-strate the effect of Qneed and also to be closer to the reality, the sys-tem load is considered as 61.40% (annual load factor) of the systemannual peak load. The results are shown in the last row of Table 5.From this table it can be seen that, considering the generators’ Qneed

leads to remarkable decrease in the total payment of reactivepower market. In this case the generators are in fact paid for theamount of reactive power that is more than their Qneed (amountof their reactive power that is really provided for the system sup-port) and for this reason the TPF of reactive power market is de-creased. The security indices (objective functions F2, F3, and F4)does not change or change slightly by considering Qneed.

4. Conclusions

In this paper, security aspects of power system as one of mainresponsibilities of ISOs are incorporated in the clearing of reactivepower market based on the multiobjective optimization modelusing MINLP formulation. The proposed scheme permits the sys-tem operators in the day-ahead reactive power market clearingto consider security issues as extra objective functions in additionto the economical objective function of the reactive power market.In the suggested multiobjective approach, ISOs will be able to di-rectly manage the desired level of system security by controllingthe different objective functions, which is not possible in typicalsecurity constrained market clearing. The proposed method cancompromise the conflicting objectives of market clearing proce-dure in such a way that the ISO’s concerns about system securityare relieved with tolerable and reasonable total reactive powercompensation cost.The results show that the proposed reactivepower market is effective and the ISO is able to have priority onthe objectives, depending on the market conditions. If the systemsecurity is of a great importance for the ISO, it can easily choosehigh weighting factor for the security indices in the market model.Conversely, if the payment is the main concern of the ISO, it canchoose a high weighting factor for TPF in the market model. Alsothe proposed day-ahead reactive power market instead of long-term reactive power market gives more chance to reactive powerproviders to participate in the reactive power market. In day-aheadreactive power market, if a reactive power compensator is not se-lected in a day, it can modify its bid and participates in the next daymarket, while in the long term reactive power markets (such asseasonal and annual markets) this is not possible. In other words,a more competitive and efficient reactive power market can be

established within the day-ahead framework as the market partic-ipants are increased. Besides, inclusion of the other reactive powerproviders (such as FACTS devices) in the reactive power marketand modification of TPF for light load conditions (inclusion of Qneed)are the subject matters of the proposed reactive power marketstructure that demands further research.

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