re-defining the reactive power dispatch problem in the context of competitive electricity markets

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Published in IET Generation, Transmission & DistributionReceived on 26th February 2009Revised on 5th June 2009doi: 10.1049/iet-gtd.2009.0099

Special Issue – selected papers on Electricity Markets:Analysis & Operations

ISSN 1751-8687

Re-defining the reactive power dispatchproblem in the context of competitiveelectricity marketsC.A. Canizares1 K. Bhattacharya1 I. El-Samahy2

H. Haghighat1 J. Pan3 C. Tang2

1E&CE Department, University of Waterloo, Waterloo, ON, Canada2IESO, ON, Canada3ABB Inc. Raleigh, NC, USE-mail: [email protected]

Abstract: This study proposes a novel reactive power dispatch model that takes into account both the technicaland economical aspects associated with reactive power dispatch in the context of the new operating paradigms incompetitive electricity markets. The main objective of the proposed model is to minimise the total amount ofdollars paid by the system operator to the generators for providing the required reactive power support. Thereal power generation is decoupled and assumed fixed during the reactive power dispatch procedures;however, because of the effect of reactive power on real power, a re-schedule in the real power generation isallowed within given limits. The 32-bus CIGRE benchmark system is used to illustrate the proposed reactivepower dispatch technique. The developed model is generic in nature and designed to be adopted by systemoperators in any electricity market structure, as demonstrated by its application to Ontario’s grid consideringits market rules for reactive power payments.

NomenclaturePGR real power rating of a synchronous generator

QGR reactive power rating of a synchronous generator

QGmin minimum reactive power limit of a generator

QGblead base leading reactive power of a generator

QGblag base lagging reactive power of a generator

QGA maximum reactive power limit of a generatorwithout reduction in real power generation

QGB maximum allowable reactive power limit of agenerator with reduction in real power generation

r0 availability price for generator g in $

r1 price of losses in the under-excitation region forgenerator g in $/Mvar

r2 price of losses in the over-excitation region forgenerator g in $/Mvar

r3 loss of opportunity price for generator g in $/Mvar2

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QG1 under-excitation reactive power of a generator inp.u.

QG2 over-excitation reactive power of a generator inp.u.

QG3 reactive power of a generator operating in theopportunity region in p.u.

m1, m2 binary variables associated with Regions I and IIof reactive power operation, respectively

m3r, m3f binary variables associated with armature and fieldlimits on reactive power generation, respectively

X set of procured generators for reactive power

Y set of available generators from market clearing

j available set of generators for reactive powerdispatch

J total payment associated with the reactive powerdispatch in $/h

J1 reactive power payment component in $/h

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J2 balance services payment component in $/h

J3 loss payment/credit component in $/h

rB1 price of upward balance services PB in $/MW

rB2 price of downward balance services PB in $/MW

PB1i upward balance service at bus i in p.u.

PB2i downward balance service at bus i in p.u.

PL total system losses with proposed Q-dispatch, p.u.

PLo pre-determined total system losses after marketclearing, p.u.

DPGi reduction in real power at bus i due to increase inreactive power beyond heating limits, in p.u.

PGoi market clearing pre-determined real powerdispatch at bus i in p.u.

QGi reactive power generation at bus i in p.u.

PDi real power demand at bus i in p.u.

QDi reactive power demand at bus i in p.u.

Vi voltage magnitude in p.u. at bus i.

di voltage angle in radians at bus i

Pij power flow from bus i to bus j in p.u.

Yij element ij of admittance matrix in p.u.;Yij ¼ Gijþ jBij ¼ jYijj/uij

ci maximum allowed level of real power reduction atbus i

PGxg new real power dispatch for generator g

Z the set of generators in zone z

Kz the amount of reactive power reserves in zone z

1 IntroductionReactive power dispatch has been of great interest toresearchers as well as system operators, especially after therestructuring of the power industry. This interest is mainlybecause of the significant effect that reactive power has onsystem security given its close relationship with the busvoltages throughout the power network. Insufficientreactive power supply can result in voltage collapse, whichhas been one of the reasons for some recent majorblackouts; for example, the US–Canada Power SystemOutage Task Force states in its report that insufficientreactive power was an issue in the August 2003 blackout,and has recommended strengthening the reactive powerand voltage control practices in all North American ElectricReliability Council (NERC) regions [1].

Traditionally, reactive power dispatch has always beenviewed by researchers as a loss minimisation problem,subject to various system constraints such as nodal real andreactive power balance, bus voltage limits and powergeneration limits [2–5]. Another approach has been todispatch reactive power with the objective of maximisingthe system loadability in order to minimise the risk of

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voltage collapse [6, 7]. Multi-objective optimisation modelshave also been proposed for the reactive power dispatchproblem; in these models, reactive power is dispatched toachieve other objectives, in addition to the traditional lossminimisation, such as maximising voltage stability margin[8], or minimising voltage and transformer tap deviations [9].

In deregulated electricity markets, the Independent SystemOperator (ISO) is responsible for the provision of ancillaryservices that are necessary to support the transmission ofelectrical energy while maintaining secure and reliableoperation of the power system. According to the FederalEnergy Regulatory Commission’s (FERC) Order No. 888,reactive power supply and voltage control from generators isone of six ancillary services that transmission providers mustinclude in an open access transmission tariff [10]. FERCOrder 2003 further states that a reactive power providershould not be financially compensated when operating withina power factor range of 0.95 lagging and 0.95 leading, but anISO may change this range at its discretion [11].

Reactive power ancillary services in deregulated electricitymarkets can be provided based on a two-stage approach,namely, reactive power procurement and reactive powerdispatch, as proposed by the authors in [12]. Reactivepower procurement is essentially a long-term issue, wherethe ISO signs seasonal contracts with possible serviceproviders that would best suit its needs and constraints inthe given season [13, 14]. Reactive power dispatch, on theother hand, corresponds to the short-term allocation ofreactive power generation required from already contractedsuppliers based on ‘real-time’ operating conditions [15, 16].The issues associated with the first level of the proposedframework, that is, reactive power procurement, werediscussed by the authors in [17], where appropriatemechanisms were proposed for management and pricing ofreactive power in the long term. The current paper, on theother hand, concentrates on the issues associated with thesecond level of the proposed framework, that is, reactivepower dispatch in the short term.

The ‘traditional’ dispatch approaches do not consider thecost incurred by the system operator to provide reactivepower. One of the reasons for this is that, in a verticallyintegrated system, all generators were under the directownership and control of the central operator, and hencereactive power payments were bundled in the energy price.The problem of reactive power dispatch in the contextof competitive electricity markets, therefore, needs to bere-defined, since reactive power has been recognised as anancillary service to be purchased separately by the ISO[10, 11], and hence has an economic effect on the market,playing an important role in the way it is operated.

In the context of competitive electricity markets, reactivepower dispatch essentially refers to short-term allocation ofreactive power required from suppliers (e.g. generators),based on current system operating conditions. The ISO’s

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problem is to determine the optimal reactive power schedulefor all providers based on a given objective that depends onsystem operating criteria. Different objective functions canbe used by the ISO, besides the traditional transmissionloss minimisation, such as minimisation of reactive powercost [15, 16, 18] or minimisation of deviations fromcontracted transactions [19]. Any of the aforementionedobjectives can be adopted, but since some of them are of aconflicting nature, the ISO needs to choose a criterion thatbest suits the market structure.

In this paper, a novel framework that re-defines thereactive power dispatch problem to suit the ISOrequirements in the context of competitive electricitymarkets is proposed, based on the preliminary reactivepower dispatch model proposed and discussed in [20]. Themodel seeks to minimise the ISO’s total payments thatinclude payments for reactive power dispatched fromservice providers, payments for balance services needed tocompensate for the deficit in real power supply because ofpossible changes in generation dispatch levels andpayments/credits associated with the increase/decrease intotal system losses. The effect of scheduling reactive poweron real power is also considered, by allowing limited re-scheduling of real power, as well as accounting for theavailability of balance services (referred to operating reservesin some markets such as Ontario), with the objective ofminimising the overall payments for the ISO.

It is important to highlight the fact that, in this paper, andadhering to existing FERC regulations, only reactive powersupport from generators is considered, as one of the sixancillary services eligible for financial compensation [10].However, the proposed dispatch scheme is fairly generic andhence could be in principle extended to include other reactivepower resources in addition to generators, such as capacitorbanks and FACTS controllers, as recommended in [21]. Thiswill be the focus of future research, considering that thesereactive power sources are essentially different fromgenerators. It is also relevant to mention that it is assumedhere, as in previous Q-dispatch papers (e.g. [2–5]), that thegenerators have the means to maintain reactive power at therequired dispatch levels, and that the system conditions donot change significantly between dispatch intervals. This iscertainly not necessarily the case in practice, as ‘special’generator Q-controls would be required to accomplish this.Thus, the current practice in most ISOs, that typically do nothave generator Q-control capabilities, is to use the results ofthe Q-dispatch processes, which in most cases are simplybased on a power flow run for the desired P-dispatch levels asdiscussed in more detail below, to actually define the setpoints of the voltage regulators that resulted from the Q-dispatch procedure. These voltage levels are then maintained,with the generator Q-output changing according to thesystem conditions, until new Q-dispatch levels, that is, newvoltage regulator set points, are provided by the ISO. Theactual generator Q-output is then paid ex-post according tothe contractual agreement, which in this paper corresponds to


the contracted prices resulting from the procurement process[12, 17].

The rest of paper is organised as follows: Section 3provides a background review for reactive power productionand associated costs from a synchronous generator. Theproposed reactive power dispatch model is presented inSection 4, including detailed discussions on how thedispatch problem is re-defined in the context ofcompetitive electricity markets. In Section 5, the proposedmodel is tested on the CIGRE 32-bus system, and severalcase studies are presented and discussed. Section 6demonstrates the application of the proposed dispatchmethodology to the Ontario grid, in view of the Ontario’smarket rules for reactive power payments. Finally, inSection 7, the main conclusions and contributions of thepresented work are summarised and highlighted.

2 Reactive power fromsynchronous generatorThe reactive power capacity of a synchronous generator isdetermined from its capability curve, which defines therelationship between real and reactive power generation fromthis generator. When real power and terminal voltage arefixed, the armature and field winding heating limits determinethe reactive power capability of the generator, as shown inFig. 1. The generator’s MVA rating is the point of intersectionof the two curves, and therefore its MW rating is given byPGR. At an operating point A, with real power output PGA

such that PGA , PGR, the limit on QG is imposed by thegenerator’s field heating limit; whereas, when PGA . PGR, thelimit on QG is imposed by the generator’s armature heating limit.

There is a mandatory amount of reactive power that eachgenerator has to provide, which is shown by the shaded

Figure 1 Synchronous generator capability curve [12, 13]


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area in Fig. 1. If the generator is called upon by the ISO foradditional reactive power provision beyond this area, it is theneligible for payment to compensate for the increased costsassociated with losses in the windings. Such mandatory andancillary classification of reactive power capability is in linewith what most system operators have in place for reactivepower management nowadays.

According to the capability curves in Fig. 1, the generatorcan provide reactive power until it reaches its heating limits(point A in Fig. 1); any further increase in reactive powerprovision from the generator will be at the expense of areduction in its real power generation. Hence, the generatoris expected to receive an opportunity cost payment forproviding reactive power beyond QGA in Fig. 1, whichaccounts for the lost opportunity to sell its real power inthe energy market and the associated revenue loss. Thus,the following three regions for reactive power generationcan be identified in Fig. 1 [12, 13]:

† Region I (QminG �QG ¼ QG1 � 0) refers to the under-

excitation region, in which the generator is required toabsorb reactive power.

† Region II (0 � QG ¼ QG2 � QGA) refers to the over-excitation region, in which the generator is required to supplyreactive power within its reactive power capability limits.

† Region III (QGA � QG ¼ QG3 � QGB) refers to the lossof opportunity region, in which the generator is asked toreduce its real power production in order to meet thesystem reactive power requirements. It is assumed here thatPGB would be the minimum amount of real power that thegenerator is able/willing to produce.

Based on the above three regions of reactive power operationand the associated costs, a reactive power payment function(QPF) is formulated, as shown in Fig. 2, including the

Figure 2 Reactive power payment function [19]

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following payment components: an availability paymentcomponent (with a price r0), which is a fixed component toaccount for that portion of a supplier’s capital cost that canbe attributed to reactive power production; two lossespayment components (with prices r1 and r2), which areassumed as linearly varying components to account for theincreased winding losses as reactive power output increases,in the under- and over-excitation regions, respectively; andan opportunity payment component (with a price r3) toaccount for the lost opportunity cost associated with theoperation in Region III. This opportunity component isassumed to be quadratic based on the assumption that realpower costs are parabolic functions of output power, whichmay be considered to approximately change linearly withreactive power in Region III (from point A to B in Fig. 1).Accordingly, QPF for each generator g in the system can bemathematically represented by the following equation [19]

QPFg ¼ r0g þm2gr2g(QG2g �QlagGbg)

�m1gr1g(QG1g �QleadGbg)

þ r2g(m3f g þm3rg)(QG3g �QlagGbg)

þ 0:5r3g(m3f g þm3rg)(QG3g �QGA)2

m1g ¼1 if Qmin

Gg �QGg ¼QG1g � 0

0 otherwise

(

m2g ¼1 if 0�QGg ¼QG2g �QGAg

0 otherwise

�

m3f g ¼1 if QGAg �QGg ¼QG3g �QGBg &PGgo , PGRg

0 otherwise

�

m3rg ¼1 if QGAg �QGg ¼QG3g �QGBg &PGgo . PGRg

0 otherwise

�(1)

In this equation, the binary variables are needed to reflect thefact that the generator operates in only one of the three regionsdefined in Fig. 1.

The above four reactive power price components areusually determined from a procurement stage, where theISO signs long-term contracts with reactive power serviceproviders, in which both parties agree on the prices and thepayment mechanism. For example, in Ontario, theIndependent Electricity System Operator (IESO) signs 36-month contracts with generators that are willing to providereactive power services. Prices are based on costs ofproviding reactive power, which include additional costsfrom energy losses incurred by operating at non-unitypower factor, and cost of running the generating units assynchronous condensers if requested by the IESO [22].Generators that are asked to reduce their real power outputin order to meet the reactive power requirements arefurther paid a lost opportunity component at the marketclearing price. The ISO-New England, on the other hand,

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pays a capacity component for qualified generator reactiveresources for the capability to provide reactive powerservices, in addition to the energy and lost opportunitycomponents [23]; a ‘base VAR rate’ of 2.32 $/kvar yr hasbeen newly incorporated for qualified generators availablefor reactive power provision below 0.95 leading or 0.95lagging power factors.

In [12], the authors propose a two-settlement reactivepower market structure, which segregates reactive powermanagement in two distinct time frames: a reactive powerprocurement stage carried out on a seasonal basis, and areactive power dispatch stage that determines the reactivepower levels in ‘real-time’. The authors propose an optimalprocurement model in [17] that considers both thetechnical and economical issues associated with reactivepower provision in the long term. The solution of thisOPF-based model yields a set of contracted generators andthe four reactive power price components, namelyavailability, under- and over-excitation and opportunitycomponents, which reflect current reactive power pricingpractices. Hence, regardless of the procurement procedure,it is assumed here that these four price components areknown to the ISO during the dispatch process.

3 Proposed reactive powerdispatch schemeIn practice, real power and reactive power have generally beenhandled separately by most power system operators.Typically, real power dispatch is carried out using a linearprogramming model associated with an economic loaddispatch (ELD) calculation that maximises social welfare,while guaranteeing that system security constraints are met[24]. Reactive power, on the other hand, is dispatchedbased on power flow studies and operational experience.However, there are several complex issues involved inreactive power service provision in deregulated electricitymarkets that call for further systematic procedures to arriveat better solutions [12]. Ideally, reactive power should bedispatched from generators in an economical manner thatminimises the ISO’s payment burden, while consideringsystem security constraints.

Fig. 3 illustrates the proposed schematic procedure forshort-term dispatch of reactive power. The scheme is basedon the assumption that a pre-determined set of contracted(or procured) generators with corresponding reactive powerprice components, which should be obtained within a long-term framework to avoid the adverse impacts of energyprice volatility on these prices [12, 17], is available aspreviously discussed. In line with current reactive powerdispatch approaches, the ISO would carry out the dispatchprocedure several minutes (e.g. one hour to half-hour)ahead of real time. Based on the set of procured/contractedgenerators and the generating units available from short-term energy market clearing, the ISO would determine the


available units for reactive power dispatch, and thenproperly dispatch the units using an OPF-based model thatminimises total payments associated with reactive powerdispatch, subject to appropriate system security constraints.Finally, the payments would be calculated post-real-timeoperation, based on the actual usage and dispatch requestedby the ISO. The following sub-sections provide detaileddiscussion on each of the blocks in Fig. 3.

3.1 Reactive power procurement

As explained earlier in Section 3, an ISO typically makeslong-term plans for reactive power procurement, in whichit determines the expected reactive power capacity requiredto ensure a secure and reliable operation of the powersystem. Currently, most system operators depend on powerflow studies and operational experience in determining a setof contracted generators for reactive power serviceprovision. The ISO then signs seasonal contracts withthese generators, in which both parties agree on reactivepower prices as well as the payment mechanism. It isassumed here that the procurement process determines theprices needed in the proposed real-time dispatchprocedures. This is discussed in more detail for theapplications presented in Sections 5 and 6.

It is important to highlight the main differences betweenthe long-term Q-procurement methodology proposed in[17], and the short-term Q-dispatch procedure proposedherein. Thus:

† The time horizons for the two problems are different, with[17] discussing the first level of Q-management (months inadvance), whereas the current paper discusses the secondlevel (real-time), as defined in [12].

† The mathematical optimisation models are significantlydifferent, with the model in [17] concentrating on themaximisation of a Societal Advantage Function that

Figure 3 Flowchart for the proposed reactive powerdispatch scheme


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comprises the marginal benefits of reactive power fromgenerators with respect to system security, and Q-offersfrom generators, subject to appropriate security constraints.The model here, on the other hand, is based on aminimisation of the ISO’s total payments associated withreal-time reactive power dispatch, consisting of reactivepower payments to generators, balance service paymentsand credit/payments for system losses, and subject toseveral additional constraints not considered in [17], asdiscussed in detail below.

† In [17], the reactive power price components are outcomesof the optimisation process, whereas these prices areconsidered as inputs to the Q-dispatch model discussedherein.

† Zonal reserves and the effect of reactive power on realpower and system losses are considered here in the dispatchmodel, whereas these issues are not considered at all in [17].

3.2 Real power market clearing

Most ISOs use dc-OPF models for real power marketclearing and dispatch, with iterative mechanisms toguarantee system security [24]. In this paper, because theemphasis is on reactive power dispatch, it can safely beassumed that the energy market clearing (and hence themarket price), and the resulting P-dispatch are available tothe ISO.

It should be noted that an ac-OPF can also be used in lieuof the dc-OPF to arrive at the real power market clearing anddispatch. However, the computational burden of such OPF islarge for practical-sized power systems, since it requiressolving a rather complex and large-scale non-linearprogramming (NLP) model every few minutes (e.g. every5 min in Ontario). Thus, decoupling the OPF problem andhandling reactive power dispatch separately [15, 25, 26],helps alleviate market power and price volatility whichmay come into play if reactive power dispatch is handledin the same time-frame as that of real power marketclearing [12].

3.3 Available generators and Q-capacity

As mentioned earlier, an important input to the dispatchmodel is the set of available generators for real-timereactive power dispatch. Letting X be the set of contracted/procured generators for reactive power obtained at theprocurement stage, and Y be the set of available generatorsfrom market clearing, then the available set of generatorsfor reactive power dispatch will be given by j ¼ X > Y.

The reactive power capacity of the available generatingunits is predetermined based on the capability curves ofthese generators (Fig. 1). Accordingly, for each generator,the upper and lower limits of the three reactive poweroperating regions are assumed to be known.


3.4 Reactive power dispatch model

Equipped with the information regarding the set of availablegenerators, reactive power limits of the three operatingregions, and the four reactive power price components, theISO should now be able to execute an optimal dispatchprogram to arrive at the required amount of reactive powerin real-time operation stage. A reactive power dispatchmodel is proposed as follows, taking into account both theeconomic and technical issues associated with serviceprovisions in a competitive electricity market:

3.4.1 Objective functions: The proposed model seeksto minimise the following objective function J, whichrepresents the total payments associated with reactive powerdispatch

J ¼ J1 þ J2 þ J3

J1 ¼X

g

r0g þ m2gr2g(QG2g � QlagGbg)

�m1gr1g(QG1g � QleadGbg)

þr2g(m3f g þ m3rg)(QG3g � QlagGbg)

þ0:5r3g(m3f g þ m3rg)(QG3g � QGA)2

0BBBBB@

1CCCCCA

J2 ¼X

i

(rB1 PB1i þ rB2 PB2i)

J3 ¼ rMC (PL � PLo)

(2)

where all components are defined in Section 1. Thesepayments can be divided into the following three maincategories:

† Payment associated with reactive power provided fromgenerators ( J1), which is a function of the predeterminedprice components associated with each region of operation(r0, r1, r2 and r3), as explained in Section 3. Accordingly,payment components are determined for QG1, QG2 andQG3 corresponding to operation in Regions I, II and III,respectively, plus an availability payment.

† Payment associated with energy balance services(operating reserves) that are required to compensate forre-scheduling of real power ( J2), that is, the effect ofQ-dispatch on P-dispatch. This will apply only underconditions when some generators are required to supplyreactive power in Region III, where they need to reducetheir real power generation in order to meet the systemreactive power requirement. Consequently, there is a needfor rescheduling of their real power (DPG), and a balanceservice (PB) is required at certain buses to compensate forreal power deviations from already dispatched values (PGo).The energy balance services from available providers mightbe an upward or downward service, that is, PB1 and PB2,respectively; the corresponding prices rB1 and rB2 areassumed to be predetermined from an energy balancemarket [27].

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† Payment/credit associated with the change in the totalsystem losses due to reactive power dispatch and the re-scheduling of real power generation ( J3). This componentcan be positive (payment) or negative (credit) depending onthe difference between the value of the total lossescalculated within the Q-dispatch model (PL), and the valueassociated with the real power market clearing dispatch(PLo). The payment/credit can be calculated by multiplyingthis difference by the market clearing price (rMC).

It is important to highlight the fact that the proposedpayment function J given in (2) is of a generic nature, andcan be modified to fit other payment schemes adopted bysystem operators. For example, it was mentioned earlierthat both the IESO in Ontario and the ISO-New Englandcompensate the generators operating in the opportunityregion by directly paying them a lost opportunitycomponent at the market clearing price; hence, in this case,the objective function can be readily modified as follows torepresent such a payment mechanism

J � ¼X

g

r0 g þ m2 gr2 g(QG2 g � QlagGbg)

�m1 gr1 g(QG1 g � QleadGbg)

þr2 g(m3f g þ m3rg)(QG3 g � QlagGbg)

0BB@

1CCA

þX

g

rMC DPGg þX

i

(rB1 PB1i þ rB2 PB2i)

(3)

Observe here that the quadratic opportunity cost term in (2)has been replaced by a direct payment for re-scheduled powerrMCDPG. Furthermore, loss payments/credits are notincluded, since this is currently not a common practice in‘standard’ reactive power dispatch approaches.

3.4.2 Power balance equations: The following nodalreal power balance equation is modified to include DPG, PB1

and PB2

PGoi � DPGi þ PB1i � PB2i � PDi

¼X

j

ViVjYij cos(uij þ dj � di) 8i (4)

QGi � QDi ¼ �X

j

ViVjYij sin(uij þ dj � di) 8i (5)

3.4.3 System security limits: These limits include busvoltage limits and line power transfer capacity constraints

V mini � Vi � V max

i 8i (6)

jPij(V , d)j � Pmaxij 8i, j (7)

Note that the line limits are being represented by real powerlimits, instead of apparent power or current limits, which is inline with current practices by most system operators, whotypically represent system security limits by defining powertransfer limits determined as the minimum of thermallimits and voltage and angle stability limits for the ‘worst’


contingencies (N 2 1 contingency criterion) [28]. However,the apparent power or current limits could also be readilyconsidered in the present model.

3.4.4 Reactive power limits: The three regions ofreactive power production identified from the generator’scapability curves (Fig. 1) are introduced as follows

m1 gQminGg � QG1 g � 0

0 � QG2 g � m2 gQGAg 8g (8)

(m3f g þ m3rg)QGAg � QG3 g � (m3f g þ m3rg)QGBg

m1 g þ m2 g þ m3f g þ m3rg � 1 8g (9)

These constraints guarantee that only one of the threeregions, out of QG1, QG2 and QG3, is selected at a time foreach generator.

3.4.5 Effect of Q-dispatch on P-dispatch: Therequired reduction in real power dispatch (DPG) isdetermined as follows

PGxg ¼ m3f g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVtEaf

Xs

� �2

� QG þV 2

t

Xs

� �2s

þ m3rg

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi(VtIa)2

� Q2Gg

qþ (m1 g þ m2 g)PGog (10)

PGog � PGxg ¼ DPGg (11)

DPGi � ciPGoi (12)

Observe that DPG will have a non-zero value only if thegenerator is operating in Region III, that is, if thegenerator hits its field limit (PGA , PGR) or armature limit(PGA . PGR); otherwise, PGx in (9) will be equal to PGo

and hence, according to (10), DPG will be zero. In order tominimise the effect on real power dispatch, a ‘cap’ onreduction in real power is imposed (e.g. between 5 and15%) as per constraint (12).

3.4.6 Energy balance limits: The ISO will requiresome balancing mechanism to compensate for thereschedule in real power and changes in system losses as aresult of the reactive power dispatch. In this work, it isassumed that such a market mechanism, that is, upwardand downward balance services (operating reserves), isalready in place as an ancillary service, which is typically thecase in most markets, and thus can be used by the ISOwithin the proposed reactive power dispatch framework. Alimit on the maximum upward and downward balanceservice available at each system bus is assumed as follows

PB1,2i � PmaxB1,2i (13)


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3.4.7 Zonal reactive power constraints: Based ontypical voltage control approaches, the following constraintsare included to ensure sufficient reactive power reserveswithin a voltage control zone

Xg

QGg � Kz

Xg

QGAg 8g [ Z, 8z (14)

where Z denotes the set of generators in zone z. The amountof reactive power reserves in zone z is denoted by Kz; forexample, Kz ¼ 0.9 implies a 10% reactive power reserve inthe zone. The sum of QGAg in any zone defines themaximum total amount of reactive power available in thiszone. Observe that the value of QGA instead of QGB wasused to define the zonal reactive power reserve to be moreconservative, that is, extra reactive power coming from theoperation in the opportunity region is not considered in thezonal reserve constraints, as this is unknown at the start ofthe dispatch process.

3.4.8 Losses: The total system losses PL in (2) arecalculated as follows

PL ¼ 0:5X

i, j

(Gi, j(V2

i þ V 2j � 2ViVj cos(dj � di))) (15)

3.5 Computational issues

The proposed dispatch model (2)–(15) captures both thetechnical and economical aspects of reactive powerdispatch. However, from the optimisation point of view,this model represents a difficult optimisation problem,since it is essentially a mixed-integer NLP (MINLP)problem, because of the presence of binary variablesrequired to properly select only one out of the three regionsof reactive power operation in the model.

One approach to solve the proposed model (2)–(15) is tosolve the problem as an explicit non-convex MINLPproblem; however, the solution of these types of problems isvery challenging because of the presence of both the integervariables and the non-convexities of the model itself[29–31]. The capability of available solvers for MINLPproblems is still rather limited and most of them require asubstantial amount of computational time even for small casestudies, and might not yield an optimal solution withinmany CPU hours for a large case study [30]. Moreover, mostof the available MINLP solvers (e.g. DICOPT) are not ableto handle line flow limits, which are of great importance torepresent system security, in various test cases tried by theauthors. Hence, solving (2)–(15) using non-convex MINLPtechniques is not the most appropriate choice, especiallywhen realistic-sized power systems are considered.

To address these issues, a generator reactive powerclassification (GRPC) algorithm proposed by the authors in[17] was used here. This GRPC algorithm solves theproposed optimisation problem by re-formulating the


MINLP problem (2)–(14) into a series of NLP sub-problems, and hence alleviating the need for binaryvariables. The algorithm starts with reactive power valuesobtained from an initial feasible power flow solution, toidentify an initial region of reactive power operation foreach generator. The solution process then goes into a seriesof sequential updates until no further improvement in theobjective function can be achieved. The main steps of theGRPC algorithm are given in Appendix for the readers’convenience; more details can be found in [17].

The solution of the proposed dispatch model yields therequired reactive power support from each generator; theamount of real power to be rescheduled in order to meetthe system reactive power requirements; the amount ofenergy balance services needed to compensate for thechange in real power resulting from the reactive powerdispatch; and the total payment of the ISO to the serviceproviders.

4 Implementation and test resultsThe proposed reactive power dispatch model (2)–(15) istested on the CIGRE 32-bus system (Fig. 4), and theassociated results are presented and discussed in thissection. The optimisation models, which are transformedinto NLP problems as previously discussed, are modelled inGAMS and solved using the MINOS solver [32]. Withoutany loss of generality, the power flow limits are consideredto be dependent on the transmission voltage levels; thus,limits of 2000 MW for the 400 kV lines, 350 MW for220 kV lines, and 250 MW for 130 kV lines are assumed.The test system has 20 generators and a total demand of10 940 MW. The system is split into three zones or voltagecontrol areas as reported in [33]. To simplify the analysis

Figure 4 CIGRE 32-bus system

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and without loss of generality, all generators are assumed tobe eligible for payments in all the three regions ofoperation, which implies that Qlead

Gb and QlagGb in Fig. 1 are

equal to zero for all generators. Finally, a 30% reactivepower reserve is guaranteed for each zone, that is, Kz ¼ 0.7for all three zones.

For brevity of presentation, only two different operatingscenarios are presented and discussed here, illustrating theperformance, validity and robustness of the proposedreactive power dispatch model. The first scenario is a baseloading condition, whereas the second is a ‘stressed’operating condition in which the system loading level isincreased by 10% with respect to base load, with onegenerating unit out of service. For each of these scenarios,the following test cases are studied:

† Case I (‘standard’ reactive power dispatch): In this case,common ISO-practices are applied, where the real powerdispatch from energy market clearing is used to solve anac-power flow to determine the required reactive powerdispatch. In most cases, operators use their own experienceto ‘tune’ the ac-power flow until a feasible and securesolution that does not violate voltage and line flow limits isachieved. Hence, this approach is simulated here to obtaina ‘standard’ reactive power dispatch, and the associatedpayments are then calculated based on the proposedpayment function J1 in (2).

† Case II (proposed reactive power dispatch): In this case, thereal power dispatch from energy market clearing, togetherwith the set of contracted generators and the four reactivepower price components from a procurement stage, areused to solve the proposed reactive power dispatch model.The solution of this model simultaneously yields therequired reactive power dispatch and the associatedpayment components. In order to demonstrate thegenerality of the proposed dispatch model, the results arecompared for two objectives, one with the function J givenin (2), denoted here by Case II.a, and the second with themore ‘realistic’ objective J� given in (3), denoted here byCase II.b.

† Case III (‘ideal’ reactive power dispatch): This case simulatesan ideal scenario in which a ‘typical’ security-constrained ac-OPF, minimising the total real power cost, is used tosimultaneously dispatch P and Q [34]. This would be anideal solution because it achieves the least-cost solution;however, such an approach is not used by ISOs in practice,because of the complexity associated with solving acoupled, large-scale, non-convex NLP model every fewminutes. Furthermore, possible adverse effects on marketprices associated with simultaneous dispatch of real andreactive power within a competitive market environmentcould be a significant problem [14]. Once the reactivepower dispatch is obtained for this ideal scenario, theassociated reactive power payment is calculated based onthe proposed J1 in (2).


In the above cases, the four reactive power price componentsare assumed to be available from a given reactive powerprocurement stage (e.g. [17]); these are given in Table 1,and are assumed to remain unchanged for all test cases. Amarket clearing price rMC ¼ 100 $/MWh is assumed,which is a typical ‘high’ price figure in the Ontario electricitymarket. The prices of energy balance services rB1 ¼ 90 $/MWh and rB2 ¼ 110 $/MWh are assumed to be pre-determined from a given energy balance auction, which canbe assumed to be typical values, since these are usuallyaround the value of rMC.

It should be noted that reactive power limits for the‘standard’ reactive power dispatch Case I, and the ‘ideal’reactive power dispatch Case III do not include theopportunity region of operation. This is due to the fact thatin the ‘traditional’ reactive power dispatch approaches,which both of these cases represent, generators are typicallymodelled using fixed reactive power limits (QGR in Fig. 1).The re-defined reactive power dispatch proposed herein(Case II), on the other hand, is based on the concept ofoperating a generator in the opportunity region in returnfor adequate financial compensation. This allows forextended reactive power support from generators, which isimportant for power systems today, since grids areoperating in more stressed conditions and hence closer totheir limits.

4.1 Base load condition

The solution of the three case studies under base loadingconditions is given in Table 2. A set of 13 generators out

Table 1 Contracted generators and the associated reactivepower prices

Zone Contractedgenerators

Zonal reactive power prices

ro r1 r2 r3

a 4072 0.78 0.74 0.57 0.35

4011

1013

1012

b 4021 0.92 0.91 0.90 0.36

4031

2032

1022

1021

c 4063 0.85 0.53 0.81 0.26

4051

1043

1042


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Table 2 Solution for the three cases under base loading condition

BUS Case I Case II Case III

ac-PF Proposed dispatch model ac-OPF

PG (MW) QG (Mvar) a b PG (MW) QG (Mvar)

PG (MW) QG (Mvar) PG (MW) QG (Mvar)

4072 1380.6 484 1380.6 179.3 1380.6 179.6 1590.6 394.5

4011 900 2 5.7 2100 900 289.2 900 277.5 539.8 2100

4021 270 22.4 270 230 270 230 270 230

4031 315 223.1 315 240 315 240 315 240

4063 1035.4 103.4 1035.4 92.4 1035.4 97.1 1080 106.2

4051 630 97.4 630 83.6 630 90.7 630 92.9

2032 760 168.9 760 10.9 760 15 765 23.3

1013 275.5 250 275.5 0 275.5 250 383.9 250

1012 720 57.4 720 0 720 275.8 720 70

1022 225 225 225 0 225 0 225 225

1021 350.1 51.7 350.1 2 7.1 0 350.1 0 430.2 84.4

1043 180 69.2 180 55 180 63.9 180 65.1

1042 360 237.4 360 240 360 238.9 360 238.8

of 20 are assumed to be contracted for reactive powerprovision (Table 1). Generators with negative QG values areoperating in the under-excitation region (Region I), whilethose in bold are operating in the opportunity region(Region III), which there are none in this case. For thepurpose of the simulations presented here, the real powermarket clearing and dispatch, which is required to initiatethe proposed reactive power dispatch procedure, is obtainedusing a ‘standard’ dc-OPF model, which minimises thecost of energy production, subject to system securityconstraints. Transmission losses are modelled as a functionof generator shift factors and real power injections [35].

It can be seen from the results in Table 2 that in Case I,when reactive power is dispatched using an ac-power flow(ac-PF), a 5.7 MW reduction in the real power of the slackbus (4011) takes place to adjust for the lower losses. Notealso that the same PG dispatch (obtained from dc-OPF)applies to both Cases II.a and II.b. None of the generatorsare operating in Region III for these sub-cases, and henceDPG is zero for all the generators; however, because of thefact that real power generation levels are kept constant andthere is no slack bus in Case II, a downward balanceservice of 7.1 MW is required at generator Bus 1021 forCase II.a to account for the reduction in the total systemlosses. The difference in QG between these two sub-cases isdue to the difference in payment structure of the twoobjective functions. Finally, observe the significantlydifferent PG and QG values obtained from the security


constrained ac-OPF approach in Case III, which minimisesthe total real power generation costs.

Observe in Table 2 that there are significant differences inreactive power dispatch using the proposed approach (CaseII) with respect to more ‘traditional’ techniques (Cases Iand III). This is a notable change arising from the proposedphilosophy of reactive power dispatch, which is a basicparadigm shift.

The reactive power payment, the balance payment and thetotal system losses for all cases are given in Table 3. Observethat the reactive power payment resulting from the proposeddispatch model (Case II) is the lowest, since the objectivefunction is to minimise reactive power payments. It is to benoted that this base-load scenario does not induce anyreactive power dispatch of generators in the opportunityregion, and therefore the difference in opportunity paymentcomponents between Cases II.a and II.b is not clearlybrought out. A stressed operating condition is considerednext to clearly bring out this issue.

The balance payment associated with Case II.a is due tothe downward balance service required at generator Bus1021 illustrated in Table 2, which arises from the need toaccount for the change in system losses associated with thecorresponding reactive power dispatch. This balancepayment is lower in Case I, where a 5.7 MW reduction inthe losses is accounted for by Generator 4011 (slack bus).

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Note that Case III yields the lowest value of the systemlosses, since both PG and QG are simultaneously dispatchedusing an ac-OPF. On the other hand, the system losses inCase II.a are lower than that in Case II.b, since for theformer the objective function J explicitly includes a loss-payment component.

4.2 Stressed operating condition

Table 4 depicts the dispatch results obtained for the stressedsystem conditions. Observe in this case that the reactivepower requirements have significantly increased for all thestudied cases. In Case I, a re-scheduling in real powergeneration of three generators is needed to achieve afeasible power flow solution; in this case, the 9 MW(1.25%) reduction in PG from Generator 1022 is picked up

by Generators 4072, and an additional 9.7 MW is suppliedby the slack bus 4011 to account for the increase in systemlosses associated with the reactive power dispatch.

In Cases II.a and II.b, the generator at Bus 1012 isdispatched in the opportunity region (shown in bold), andconsequently a 11.7 MW (1.63%) reduction in its realpower output is necessary to maintain reactive powergeneration within its field limits. This re-scheduling in realpower is compensated by an upward balance service of11.4 MW in Case II.a and 12.1 MW in Case II.b, both atload Bus 1041 (in italics in Table 4). The differencebetween the MW reduction and the balance service isaccounted for by the change in the total system losses. InCase III, there is no re-scheduling of real powergeneration, since PG and QG are simultaneously dispatched.

Table 4 Solution for the three cases under stressed condition

BUS Case I Case II Case III

ac-PF Proposed dispatch model ac-OPF

PG (MW) QG (Mvar) a b a b PG (MW) QG (Mvar)

PG (MW) QG (Mvar) PG (MW) QG (Mvar)

4072 2310.3þ 9 303.5 2310.3 319.3 2310.3 297.4 2279.3 422.3

4011 900þ 9.7 2100 900 2100 900 232.5 804.1 2100

4021 270 230 270 230 270 230 270 230

4031 315 240 315 0 315 0 315 128.8

4063 1080 152.7 1080 156.8 1080 159.1 1080 163.6

4051 630 247.9 630 98.5 630 95.6 630 122.8

2032 765 175.6 765 201.1 765 179.9 765 212.5

1013 350.1 236.7 350.1 76.9 350.1 45.7 481.9 250

1012 720 2 9 349 720 2 11.7 367 720 2 11.7 367 720 348.7

1022 0 0 0 0 0 0 0 0

1021 391.4 88 391.4 156.5 391.4 157.8 540 213.6

1043 180 87.2 180 87.2 180 87.2 180 87.2

1042 360 16.3 360 0 360 0 360 27.8

1041 0 0 11.4 0 12.1 0 0 0

Table 3 Losses and payment components for the three cases under base loading condition

Case I

ac-PF

Case II: proposed dispatch model Case III

ac-OPFa b

Losses (MW) 433.5 432.2 439.2 424.4

Q-payment ($/h) 1900 1570 1780 1920

Balance payment ($/h) 515 640 0 0

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Table 5 Losses and payment components for the three cases under stressed condition

Case I

ac-PF

Case II: Proposed dispatch model Case III

ac-PFa b

Losses (MW) 567.5 557.4 558.2 575

Q-payment ($/h) 2420 2190 3340 2510

Balance payment ($/h) 2060 1250 1330 0

The reactive power and balance payments, as well as thetotal system losses for all cases are given in Table 5.Observe that the lowest reactive power payment is achievedin Case II.a, with the payment being significantly differentfrom that in Case II.b because of the difference in theopportunity payment component in the respective objectivefunctions. Note that the balance payment component ispresent in Cases I and II because of the required re-scheduling in real power generation. The difference in thebalance payment for Cases II.a and II.b can be attributedto the slight difference in the total system losses associatedwith the reactive power dispatch.

It is interesting to note that for the stressed systemconditions, the proposed dispatch of Case II.a reducessystem losses with respect to the ac-OPF dispatch of CaseIII. This could be attributed to the fact that the proposeddispatch model has the possibility to extend the reactivepower limits to a generator’s opportunity region, while anac-OPF-based reactive power dispatch constraints these tofixed rated limits.

Comparing the reactive power dispatch results of Case II.afor the two system conditions, it is observed that for base load(Table 2), four generators are operating in the under-excitation region, and four are not dispatched for reactivepower support, while none are required to operate inRegion III. On the other hand, from Table 4, a significantincrease is observed in reactive power output fromgenerators under stressed operating condition, as expected,since only two generators are operating in the under-excitation region, while the generator at Bus 1012 isoperating in Region III. The total amount of injectedreactive power into the system has increased from220 Mvar at base loading condition to 1333 Mvar understressed operating condition.

5 Application to OntarioThe main objective of this section is to describeimplementation and test the proposed competitiveframework for reactive power dispatch in the Ontarioelectricity market, considering the existent pricingregulations for reactive power services in the province.Thus, reactive power supply zones and associated priceswere first determined using the procurementmethodology proposed in [17]. These prices were then

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used to dispatch the IESO’s generators at minimumcosts using the approach described herein. The systemstudied is the dispatch model of the IESO-controlledgird, which comprises numerous generators located indifferent transmission zones in Ontario. It is shown herethat the proposed structure and models should enablethe IESO to properly determine zonal reactive powerprices ahead of real time, allowing it to develop aminimum-cost reactive power dispatch procedure in realtime that also guarantees system security.

The IESO-controlled grid includes all transmission linesat voltage levels equal to or greater than 50 kV locatedwithin the Ontario control area, including all distributionlines and loads in Ontario. The total length of transmissionlines is nearly 31 000 km and the installed generationcapacity present within the control area amounts to31 000 MW, with a peak system demand of approximately27 000 MW [36]. The Ontario system interconnects withMichigan, Minnesota, New York, Manitoba, and Quebec.A reduced version of the IESO-controlled grid utilised fordispatch purposes by the IESO was used for the studiespresented and discussed here. It consists of 2833 buses and4205 branches, including transmission lines and two- andthree-winding transformers.

The system peak load of 27 000 MW was used for thepresented studies. Normal system conditions and severalmain critical contingencies were analysed. The list of themost critical contingencies for the loading conditionsstudied here was provided by the IESO, comprising 13N 2 1 and N 2 2 cases; these include single and doublebus outages as well as transmission line and transformeroutages and special protection schemes.

The procurement procedure described in [17] was used toidentify the reactive power supply zones, independently fromthe IESO predefined zones, as well as the associated zonalprices and ‘contracted’ generators. From the 13 studiedcontingencies, the critical (worst) contingencycorresponding to the minimum value of the maximumloadability of the system was indentified to be a double-busoutage of buses located near the Bruce power station inWestern Ontario. The generators were then ranked withrespect to their effect on the system maximum loadabilityand the reactive power output levels, yielding the fiveV-zones for the IESO-controlled grid depicted in Fig. 5,

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where a few relevant generators are highlighted.Observe that the most important generators arelocated in Western Ontario, Southwestern Ontario and

the Greater Toronto Area (GTA), whereas the leastimportant are located in Northeastern and NorthwesternOntario.

Figure 5 V-zones of the IESO-controlled grid

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Two price components for reactive power services, that is,for under-excitation operation and for over-excitation, wereconsidered here, taking into consideration the IESO’sreactive power payment policies as per (3) with r0 ¼ 0 (noavailability price is recognised in Ontario). The price offersused for each generator were obtained randomly from auniformly distributed function in the range of 3-13 $/Mvar. Generators operating in the opportunity region wereassumed to also receive a payment of rMC ¼ $120/MWh,which is a typical Hourly Ontario Energy Price or HOEPfor peak-load periods. Table 6 shows the zonal reactivepower prices obtained for the under-excitation and over-excitation regions. It is interesting to note that there are 47generating units within Ontario that are forced to operatein the opportunity region in this case.

Based on the V-zones and the corresponding reactivepower prices obtained from the procurement studies, thedispatch optimisation problem was then solved. Since noslack buses are considered in this model, balancing servicesaccount for the need of any real power reschedulingassociated with system losses changes and generatorsoperating in the opportunity region (Region III). Theupward and downward balancing services were assumed tobe priced at rB1 ¼ $90/MWh, and rB2 ¼ $110/MWh,respectively, which are typical operating reserves costs inOntario (these costs are the result of the operating reserveprices plus the HOEP). A zonal reactive reserve limit of70% was assumed to ensure sufficient var reserves for eachV-zone, and a maximum 15% MW deviation of eachgenerator from the dispatched value was assumed.

Table 7 presents a summary of the Q-dispatch solutionobtained. In this case, 29 generators operate in Region III,and there is a total reduction of 143 MW in active power

Table 6 Zonal reactive power prices

V-zone Under-excitation pricesr1 ($/Mvar)

Over-excitation pricesr2 ($/Mvar)

1 10.21 10.32

2 10.67 10.24

3 9.71 10.52

4 11.07 9.98

5 10.98 11.24


with respect to the original system dispatch; the upwardbalancing services provide 88 MW and the system lossesare reduced by 55 MW.

The procurement and dispatch models were implementedin AMPL and solved using the KNITRO and IPOPTsolvers. The iterative solution process described inAppendix was applied to the MINLP procurement anddispatch models. This requires a reasonable initial guess;thus, an ac power flow solution was used to obtain theinitial reactive power outputs of all generators. The solutionof the MINLP dispatch model required more iterations perNLP problem and significantly more epochs, where anepoch refers to the stage where the algorithm checks if thesolution improves when the reactive power operating regionof a given generator changes; this is to be expected, sinceactive power needs to be re-dispatched in this case. Thefirst epoch required significant effort to solve, since manygenerators (up to 100) reached the reactive power limits ofthe initial reactive power operating region. The followingepochs ran faster since fewer generators (as low as 25)reached their limits. The average total CPU time needed toobtain a solution of the dispatch model was approximately60 min. Table 8 summarises the computational burden forboth models. Observe that this CPU times were obtainedusing ‘off-the-shelf ’ programs and solvers, which arecertainly not designed and optimised to solve the particularproblems at hand; furthermore, the studies were performedfor the worst system loading conditions. Hence, it isreasonable to expect that the CPU times could besignificantly reduced in an actual implementation of theproposed methodologies, so that their application should befeasible in practice.

Table 7 Dispatch problem main results

Variable Value

Active power reduction (MW) 143

Upward balance services (MW) 88

Downward balance services (MW) 0

Losses reduction (MW) 55

Reactive power payment ($) 18 681

Opportunity costs ($) 17 160

Balance service payment ($) 9685

Table 8 Typical computational burden for the solution of the MNLP optimisation models using AMPL

Model Range of number of iterationsper epoch

Range of CPU times perepoch (s)

Number ofepochs

Total average CPU time(min)

Procurement 74–94 13–16 1 0.25

Dispatch 91–205 660–1100 3–5 60

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6 ConclusionsThis paper has re-defined the reactive power dispatch problemfrom the perspective of an ISO operating in competitiveelectricity markets. A novel reactive power dispatch modelthat incorporates the ISO’s composite payment burdenassociated with the provision of reactive power support,while considering all operating aspects pertinent to acompetitive environment, was proposed and studied.

The proposed technique seeks to minimise the totalpayment by the ISO to reactive power providers, whileensuring a secure and reliable operation of the powersystem, and presents a reasonable compromise betweencurrent and ideal reactive power dispatch practices. Oneimportant contribution of this model is that it considers theeffect of reactive power on real power by internalising thecalculation of the reduction in real power output of agenerator due to an increase in its reactive power supply.Furthermore, in order to ensure that re-scheduling of realpower because of reactive power supply requirements fromgenerators is kept at a minimum, a payment component forbalance services is included in the objective function.

From the analysis of results obtained from various casestudies, it was shown that the proposed reactive powerdispatch model yields the lowest reactive power paymentsamong all cases considered. It was also observed that theproposed reactive power dispatch approach yields betteroverall results than current dispatch practices. However, anintegrated ac-OPF-based real and reactive power dispatchapproach was better, as expected, even though it might notbe practical at the moment. Finally, the application of theproposed procedure to the Ontario system and marketshowed its feasibility for practical applications.

7 References

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[2] EL-KADY M.A., BELL B.D., CARVALHO V.F., BURCHETT R.C., HAPP H.H.,VIERATH D.R.: ‘Assessment of real-time optimal voltagecontrol’, IEEE Trans. Power Syst., 1986, 1, pp. 98–107

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[16] LAMONT J.W., FU J.: ‘Cost analysis of reactive power support’,IEEE Trans. Power Syst., 1999, 14, pp. 890–898

[17] EL-SAMAHY I., BHATTACHARYA K., CANIZARES C.A., ANJOS M., PAN J.:‘A procurement market model for reactive power servicesconsidering system security’, IEEE Trans. Power Syst.,2008, 23, pp. 137–149

[18] HAO S.: ‘A reactive power management proposal fortransmission operators’, IEEE Trans. Power Syst., 2003, 18,pp. 1374–1381


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[21] FERC Staff Report: ‘Principles for efficient and reliablereactive power supply and consumption’, February 2005

[22] IESO: ‘Market rules for the Ontario electricity market’,March 2007

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8 AppendixThe steps of the proposed GRPC algorithm are as follows[17]:

1. Start with a feasible initial reactive power dispatch, whichcan be obtained from a power flow solution associated withthe real power dispatch obtained from an energy marketclearing process.

2. Use this initial reactive power dispatch to obtain an initialclassification for the regions of reactive power operation foreach generator. For example, if the initial QGg is less thanzero, the generator is operating in Region I, and hence,QG2 and QG3 for this generator will be zero, and QG1 willbe a variable within its lower and upper limits; similarly forRegions II and III.

3. Solve the NLP formulation of the proposed dispatchmodel (2)–(15) using this initial classification, where thebinary variables are no longer needed, since for eachgenerator only one region of reactive power operation isidentified.

4. Check if QG for any generator g has reached its limits forits region of operation. For example, if QG of a generator g inRegion I is at its upper limit, the problem is re-solved (Step3) with this generator operating in Region II, that is, QG1 andQG3 for this generator are zero, and QG2 is now a variablewithin its lower and upper limits. A similar updateprocedure is used for generators in Regions II and III,which are operating at their limits.

5. A new optimal value of the objective function (2) or (3) iscalculated. If this new value is higher than the previous one,this generator is now considered to be in Region II;otherwise, the generator is kept in Region I. The solutionis only updated if an improved value of the objectivefunction is achieved.

6. The algorithm terminates after a certain number ofiterations or if no further improvements in the objectivefunction are achieved.

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re-defining the reactive power dispatch problem in the context of competitive electricity markets

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