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CHARGING REACTIVE POWER CONSIDERING SYSTEM SECURITY ASPECTS
R.C. Leme, A.C.Z. de Souza, J.C.S. de Souza, and K.L. Lo
References
[1] M. Shahidehpour, H. Yamin, & Z. Li, Market operationsin electric power systems: Forecasting, scheduling, and riskmanagement, (Wiley-IEEE Press, New York, USA, 2002).
[2] F.L. Alvarado, J. Meng, C.L. DeMarco, & W.S. Mota, Stabilityanalysis of interconnected power systems coupled with marketdynamics, IEEE Transactions on Power Systems, 16(4), 2001,695–701.
[4] proposes a cost function for each reactivepower supplier. The problem of bringing reactive marketpower along with system security is explicitly proposed in
[5], which works on the procurement market model thatconsiders a seasonal time horizon. The idea of the availabletransfer capability is discussed with the help of the contin-uation method
[6]. Besides considering the continuationmethod, Ref. [5] proposes an optimal power ﬂow (OPF) fordetermining the available transfer capability, having theload margin as the objective function. The reactive powerpricing is discussed in details, including an optimizationmodel to deal with the whole formulation. Because of thenature of reactive power, zonal pricing is used. BesidesOPF, an optimal solution may also be found with the helpof expert systems. In this sense, particle swarm optimiza-tion may provide good results, as described in
[7], wherethe total system reactive power reserve is maximized, while
[8], executes the reactive power control by means of aparticle swarm optimization. The control actions deter-mined are considered and a PV curve is drawn, so the newload margin may be calculated and compared with the oneassociated with the previous operating point.The ideas above, however, have been under discussionby the academic and industry community. In this sense, theidea of a reactive market power as proposed for active powerseems to be inappropriate. For example, Ref.
[9] presentssome diﬃculties to meet the standards for reactive powersupport. If a reactive power model is not a consensus,paying the reactive power suppliers is acknowledged as afair treatment, because the whole system is beneﬁted byits voltage control and security enhancement. This paper,thus, deals with the problem of pricing reactive powersupply. As a common sense by System Operators aroundthe world, the amount of reactive power to be supplied byeach generator is previously determined. This scenario stillpermits the Operator to assign a reactive power redispatch,55if necessary. The payment in this stage may be determinedaccording to the beneﬁts to the system security and couldalso consider the possibility of loss revenue, in case agenerator is assigned to reduce its active power to satisfythe reactive power control requirements. Such a focusis not analyzed in this paper. Rather than that, thispaper concentrates on the operating scenario following acontingency, so some measures need to take place. For thissake, the reactive power responsibility must be addressed.A good attempt toward this issue is found in [10, 11],where the amount of reactive power support following atransaction is studied. According to those references, agenerator may deserve a credit or to be associated witha debit, depending on its generation. The reactive powerpricing, however, is not a concern in those references.Pricing reactive power could be primarily addressedas usually carried out in active power market. However,because of the local nature of reactive power, nodal pricewould not give adequate signals for the agents. Because ofthat, a zonal methodology is used in this paper. The localnature of reactive power imposes another barrier, becausereactive power cannot travel far. To identify the generatorsmost likely to play reactive power market, a sensitivitytechnique is used. The reactive power market, however, isnot carried out in the strict manner, because the amount ofreactive power is also determined by the methodology. Asa consequence, the paper focuses on pricing reactive powerafter a redispatch takes place. The discussion is carriedout with the help of a sample real system with 39 buses,where the reactive power limits are considered.The main idea of the paper, therefore, consists inplaying reactive power redispatch following a contingency.For this sake, the most critical contingencies must beevaluated and the load margin assessed for each case.Because of this, continuation method is brieﬂy explained,because this is the tool used for load margin calculation.QV curve is also described as the tool for contingencyscreening. The interior point algorithm is also presented,because this is used for reactive power pricing.2. Techniques Used to Assess the System SecuritySystem security is analyzed in this paper by means ofload margin calculation through the continuation method[6] and the determination of the reactive reserve at thegenerators by QV curve [12]. These techniques are brieﬂydescribed next.2.1 The Continuation MethodContinuation methods may be used to trace the path ofa power system from a stable equilibrium point up to abifurcation point. These methods operate on the followingsystem model:f(x, λ) = 0 (1)where x represents the state variables and λ is a systemparameter, used to drive a dynamic system from oneequilibrium point to another. This type of model hasbeen used in numerous voltage security studies, with Δλbeing considered as the system loading/generation level,tap changes, power transfer level or other parameter ofinterest. Two steps move the system along the bifurcationpath:(a) Predictor step, which deﬁnes a direction for load andgeneration increase. Tangent vector may be used forthis purpose, and is given by:TV =⎡⎢⎢⎣ΔθΔλΔVΔλ⎤⎥⎥⎦ = J−1⎡⎣P0Q0⎤⎦ (2)where J denotes the power ﬂow Jacobian. TV is theacronym for tangent vector. The predictor step is given by:Δλ =1TVwhere · stands for the Euclidean norm.The method takes larger steps when the system isfar away from the bifurcation point and smaller stepsas the bifurcation is approached. The actual solution isobtained at each loading level with the help of the correctoralgorithm.Tangent vector converges to the zero-right eigenvector(the eigenvector associated with the zero eigenvalue at thebifurcation point), as reported in [13]. Because of this, sucha vector may anticipate the critical buses identiﬁcation,enabling the operator to take some preventive measures.Because this vector is easily calculated, Ref. [14] proposesthis vector as a tool for local compensation with themeaning of loss reduction. These approaches are also usedhere.(b) Corrector step, obtained by the inclusion of an extraequation that imposes an orthogonality condition be-tween the predictor and corrector vectors. An alterna-tive to this approach is to consider the predictor stepresults as the initial guess in a conventional power ﬂow.In general, the corrector algorithm converges rapidlyto the desired operating point. When a continuationmethod is implemented with the help of the power ﬂowequations, such a code is named continuation powerﬂow.2.2 QV CurveThe QV curve method has been used as a planning toolby many utilities, a practise that should often be comple-mented by dynamic studies. The QV curve analysis shouldbe performed in conjunction with PV curves. Using theQV curve may help engineers to identify critical buses inthe system as well as the reactive power injections neededat those buses to ensure voltage security. The possibilityof reducing the computational cost associated with the cal-culation of QV curves is analyzed here. As stressed for thePV curve, the whole curve is not the focus. Rather thanthat, the point of minimum is meant, yielding the reactivepower margin for the bus analyzed.56Indeed, in this paper, the approach proposed in [12]is used, so the QV curve is obtained with the help ofa continuation power ﬂow, yielding the QV continuationmethod described next:QV Continuation MethodThe idea is to trace the QV curve with the help of controlledsteps until reaching the minimum. The step size is given by:Δλ =kTVQ(3)where TVQ is calculated as:TVQ = J−1Q1 (4)In (4), J is the power ﬂow Jacobian. Q1 is a vectorwith all zeros except for the reactive power associated withthe bus under study. It is important to mention that thereactive power limits for all other PV buses are taken intoaccount.Computing (3) provides the step length. The voltagelevel at the PQ buses is updated as:ΔV = kTVQTVQ(5)During the power ﬂow convergence process, the busunder study is considered as a PV bus. However, thestep length calculation and the voltage level correction areexecuted considering this bus as a PQ bus. This happensregardless the original type of the bus under study. Thecomputational cost for obtaining the step length and thecorrection term is, however, very low because it is onlyexecuted after convergence of the Newton method. At thisstage, the Jacobian matrix is known, and has already beenfactorized.3. Reactive Power Redispatch and PricingThe technical aspects previously presented are meant toassess the system operating conditions with respect to itsload margin and reactive power margin. These aspectsplay a key role in this paper, because they are consideredas the ﬂag to take some measures. Such measures consistof reactive power redispatch and its pricing, as brieﬂydescribed next.3.1 Reactive Power RedispatchThe approach proposed here seeks the reactive power re-dispatch value by means of loss reduction in the criticalarea. This area is deﬁned considering the critical bus, iden-tiﬁed by tangent vector in (2), as its seed. Buses directlylinked to this bus form the level-1 neighbourhood. Busesconnected to buses belonging to level-1 form the level-2,and so on. In this paper, the level of neighbourhood isdeﬁned as a function of the tangent vector entries. In thissense, for each level, a voltage sensitivity rate is computedby comparing the voltage sensitivity entry of each neigh-bour bus with the entry related to the critical one. Thecritical area includes all buses presenting sensitivity ratesgreater than a speciﬁed value. Once this area is deﬁned, asub-optimal solution is pursued according to:⎡⎢⎢⎢⎣ΔPΔQΔloss⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣H NgenM LPD 0⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎣ΔθΔVΔR⎤⎥⎥⎥⎦(6)The row vector PD is formed by the partial derivativesof the active power loss equations (regarding the criticalarea) with respect to the system state variables. Thegoal of (6) is to bring the local loss to the pre-transactionobserved value, so that Δloss is the diﬀerence between thepre-transaction and post-transaction losses. As for columngen, it contains the generators to play redispatch. Thevalues of the elements associated with these generatorsare set to their normalized sensitivity values, obtainedfrom the tangent vector (generators assigned to increasegeneration), while the rest of vector gen entries equals zero.Setting column gen entries to the normalized sensitivityvalues ensures that the generators closest to the criticalarea will contribute with a larger share of reactive powerredispatch. As a consequence, the unique value of ΔRmultiplied by the sensitivity of each generator in the gencolumn provides the redispatch of each machine. However,the reactive power limits are continuously monitored, sothat when a generator reaches its limit, it is not consideredfor redispatch any further.Note that (6) is dependant on the choice of the gen-erators to play redispatch. Such generators are chosenaccording to the proposal below:Tangent vector calculated in (2) is used here as a toolto identify the generators likely to act in redispatch. Thisapproach is based on the way the state variables vary as afunction of the system parameter. In this case, the systemparameter is the variation of the reactive power generationat each generator, one by one. The total system activepower loss is given by:PsystemLoss =nlk=1{VikVjk[Gk(cos θ(ij)k + cos θ(ji)k)]− Gk(V 2ik + V 2jk)} (7)where nl is the number of transmission lines, Vik and Vjkare the voltage level at ends (i) and (j) of transmissionline k, Gk is the transmission line k conductance, θ(ij)kand θ(ji)k represent the diﬀerences between voltage phaseangles at terminal buses i and j.If (7) is derived in relation to system parameter λ, oneobtains:dPsystemLossdλ=nlk=1GkdVikdλVjk +dVjkdλVik A + VikVjk×dAdλ− 2GkdVikdλVik +dVjkdλVjk (8)57where A = Gk(cos(θ(ij)k) + cos(θ(ji)k) and dA/dλ = Gk(2 sin(θ(ji)k) (dθ(i)k/dλ − dθ(j)k/dλ))Equation (8) shows how the active power loss variesas a function of the system parameter Δλ. All its partialderivatives consist of tangent vector components knownfrom (2). Therefore, such a computation is not timeconsuming.Assuming that the active power loss is computed onlyfor the system critical area and that the right-hand sideof (2) is slightly perturbed by a generation increase at agiven bus “g”, the new tangent vector may be obtainedwithout the need to calculate the new operating point. If(8) is calculated, the active power loss variation in thearea of interest, as a function of parameter λ (reactivepower generation increase at bus “g”), is known. Takingall system generators, one by one, computing (2) and (8)indicates the generators whose redispatch reduce at mostthe system power loss.The points described so far are related to the problemof reactive power redispatch. The next step consists ofpricing the reactive power redispatch calculated from (6).3.2 Charging Reactive PowerThe scenario considered in this paper assumes bilateraltransactions. In this sense, the active power generationprices have already been established. As a consequence,for each operating point, it is assumed that the SystemOperator has already taken all the measures to supplythe demand in a stable and safe manner. Therefore, thispaper does not address the important fundamentals ofenergy pricing and tariﬀs, where discussing zonal and nodalpricings becomes an issue. Rather than that, because ofthe bilateral contracts considered, the active power priceconsidered is the spot price of the generator scheduled toplay redispatch.4. MethodologyThe focus of the paper is to enhance the system securityby means of reactive power redispatch. Such a measureis considered following a system contingency. At thispoint, it is important to emphasize an important aspectregarding the active and reactive power pricing. References[10, 11] propose a way of identifying the reactive powerresponsibility of each agent associated with a transaction.Such a feature allows one to allocate costs, not encouragingan agent of playing market, and consequently, reducingthe ﬁnal price for the costumer. In the context analyzedhere, however, the price tends to increase, because thesystem security is the focus. In this sense, the proposedmethodology seeks the minimum price elevation, becausethe most eﬀective generators are set for redispatch. Themethodology may be summarized as follows:1. The most critical contingencies are identiﬁed using QVcurve and conﬁrmed by the load margin assessed bythe continuation method (Section 2).2. If the load margin associated with the most criticalcontingencies is larger than 10%, no action is required.Otherwise:3. Identify the critical area and the generators to playredispatch (Section 3).4. Charges as a direct function of the generators identiﬁedin (2), i.e., the larger the contribution, the bigger thecharge. The value of each MVAr is given by:a. For a generator assigned to increase its reactivepower generation, it should be paid the very sameamount as collected for each MW generated.b. If the reactive power reaches its upper limit, theactive power generated should be reduced. Inthis case, the generator should be paid for thereactive power generated as well as the activepower curtailed.Note that a post-contingency load margin threshold of10%, deﬁned in step 2, has been arbitrarily chosen in thiswork. However, diﬀerent threshold values may be used.The next section describes how the ideas proposed abovemay work in an operational scenario with the help of asample real system.5. Test ResultsThe 39 bus New England system, whose data are availablein [15], is used here to investigate the approaches proposedin the previous sections. The diagram of this system isdepicted in Fig. 1, so the reader may visualize the resultsobtained. The default system topology (47 transmissionlines operating) in a bulk operating condition is used asthe base case, obtaining the benchmark load margin withthe limits considered. The continuation method yields aload margin about 45.7%. The critical area identiﬁed fromthe base case is associated with Buses 4, 5, 6, 7, 8, 9 and39, where Bus 8 is the most critical. Finally, the criticalarea loss is 13.55 MW.For the base case, the most critical contingencies aredetected using the QV curve, and the load margin varia-tions are obtained. The three most critical contingenciesare shown in Table 1.According to the methodology proposed in Section 4,reactive power redispatch should take place for systemsecurity with respect to Contingency 1, because the systemload margin is less than 10%.It is important to state that when a contingency takesplace, automatic voltage regulators (AVRs) act in the gen-erators ﬁelds to increase the reactive power supply, avoid-ing the system voltage collapse. This automatic action isnot charged in this paper. Rather than that, the post-contingency case is considered as the new base case fromwhich sensitivity analysis is carried out. This, explicitly,divides the problem into two: ﬁrst, the automatic systemresponse provided by the AVRs. Charging this action bythe methodology proposed here is straightforward, but willnot be addressed. Secondly, the reactive power redispatchto reduce the local loss, consequence of further adjustmentsdemanded by the proposed methodology.Contingency 1 is then analyzed, because of the low loadmargin it provides. Table 2 presents the results for thiscase. In the second column of Table 2 the reactive powergenerated at the base case is presented. The third columnof that table presents the reactive power generated after58Figure 1. New England system with 39 buses.Table 1Some of Critical ContingenciesContingency Bus From Bus To Load Margin (%)Number1 5 6 9.42 16 17 10.93 17 27 44.9the contingency is considered. The automatic responseprovided by the AVRs result in an output deviation withrespect to the base case for most of the generators. Asstressed before, however, pricing this variation is not thefocus of this paper.To reduce the local loss, the technique described inSection 3.1 is used. For this sake, the sensitivity factorsare obtained according to the methodology presented inSection 3.2. The number of generators designated to re-dispatch reactive power is based on the sensitivity factorsobtained by (8). Thus, the sensitivity technique assignsthree generators/condensers to play reactive power redis-patch. These results are summarized in the fourth columnof Table 2.Note, from Table 2 that Generators 33, 34 and 37should be paid for their reactive power redispatch. Asproposed in the foregoing sections, the amount to be paidfor each generator should be equal to their active powerTable 2Reactive Power Generations (MVAr)Generator Base Case Contingency 1 Redispatch30 380.0 380.0 –31 464.0 590.2 –32 462.7 500.0 –33 229.0 309.2 80.234 286.4 320.5 34.235 562.2 600.0 –36 500.0 500.0 –37 197.3 276.0 78.738 391.9 391.9 –39 669.7 669.7 –marginal price. The last column of Table 2 indicates atotal reactive power variation about 193.1 MVAr. This isenough to take the local loss as close as possible to thepre-contingency value, while increasing the system loadmargin from 9.4% to 27.8%.Charging reactive power when the system load marginis lower than a benchmark value stimulates market players59to consider stability indexes during a transaction (bilateralor pool), connecting the technical and economical opera-tion of power system. The charges may be assigned togenerators and loads according to their responsibility inthe system power ﬂows and loadability. Such responsibilitycan be obtained, for example, using current adjustmentfactor (CAF) proposed in
[10]. It is important to statethat ﬁnancial implication is not the focus of this paper.Hence, the marginal costs themselves are not taken into ac-count. The reactive power is only charged if a contingencydeteriorates the system stability margin.6. ConclusionsThis paper presented a methodology to charge reactivepower having in mind system security. The results ob-tained show that the consideration of some system secu-rity aspects make the market problem analysis more in-teresting, because some issues usually not addressed maynow be focused. In this paper, reactive market analysisis performed when a contingency takes place. However,the methodology proposed here may be applied after theoccurrence of any disturbance that drives the system to adeteriorated security margin condition.Redispatch is executed to maintain the system loadmargin at least in 10%, so a further load variation maytake place. Some options for charging reactive powerare proposed. It is acknowledged that the energy pricetends to increase under the circumstances analyzed, so theproposed methodology is aimed to minimize the generationactive power cost increase whereas enhancing the systemoperating point.Because the methodology is based on an augmentedpower ﬂow Jacobian, a sub-optimal solution is obtained.On the other hand, as the solution may be obtained bymeans of the Newton’s method, a short computational loadis necessary. The tests were executed using a well-knownsample test system, so the results may be reproduced.7. AcknowledgementsThe authors thank CNPq, CAPES (project 023/05),FAPEMIG and FAPERJ.References[1] M. Shahidehpour, H. Yamin, & Z. Li, Market operationsin electric power systems: Forecasting, scheduling, and riskmanagement, (Wiley-IEEE Press, New York, USA, 2002).[2] F.L. Alvarado, J. Meng, C.L. DeMarco, & W.S. Mota, Stabilityanalysis of interconnected power systems coupled with marketdynamics, IEEE Transactions on Power Systems, 16(4), 2001,695–701.[3] R. Fetea & A. Petroianu, Reactive power: A strange concept?,Second European Conf. in Engineering Education, Budapest,Hungary, 2000.[4] R. Hirvonen, R. Beune, L. Mogridge, R. Martinez, K. Roud´en,& O. Vatshelle, Is there market for reactive power services –Possibilities and problems, Session 2000, Cigr´e, 9–213.[5] I. El-Samahy, K. Bhattacharya, C. Ca˜nizares, M.F. Anjos,& J. Pan, A procurement market model for reactive powerservices considering system security, IEEE Transactions onPower Systems, 23(1), 2008, 137–149.[6] H.D. Chiang, A. Fluak, K.S. Shah, & N. Balu, A practicaltool for tracing power system steady-state stationary behaviordue to load and generation variations, IEEE Transactions onPower Systems, 10(2), 1995, 623–634.[7] L.S. Titare & L.D. Arya, A particle swarm optimization forimprovement of voltage stability by reactive power reservemanagement, Journal – Institution of Engineers India Part ElElectrical Engineering Division, 87, 2006, 3–7.[8] H. Yoshida, K. Kawata, Y. Fukuyama, & Y. Nakanishi, Aparticle swarm optimization for reactive power and voltagecontrol considering voltage stability, IEEE-ISAP’99, Rio deJaneiro, Brazil, 1999.[9] M. Ilic & C.-N. Yu, A possible framework for voltage/reactivepower markets, Proc. of the IEEE Power Engineering. SocietyWinter Meeting, New York, NY, 1999.[10] K.L. Lo & Y.A. Alturki, Towards reactive power markets.Part1: Reactive power allocation, IET Proceedings–GenerationTransmission and Distribution, 153(1), 2006, 59–70.
[12]. These techniques are brieﬂydescribed next.2.1 The Continuation MethodContinuation methods may be used to trace the path ofa power system from a stable equilibrium point up to abifurcation point. These methods operate on the followingsystem model:f(x, λ) = 0 (1)where x represents the state variables and λ is a systemparameter, used to drive a dynamic system from oneequilibrium point to another. This type of model hasbeen used in numerous voltage security studies, with Δλbeing considered as the system loading/generation level,tap changes, power transfer level or other parameter ofinterest. Two steps move the system along the bifurcationpath:(a) Predictor step, which deﬁnes a direction for load andgeneration increase. Tangent vector may be used forthis purpose, and is given by:TV =⎡⎢⎢⎣ΔθΔλΔVΔλ⎤⎥⎥⎦ = J−1⎡⎣P0Q0⎤⎦ (2)where J denotes the power ﬂow Jacobian. TV is theacronym for tangent vector. The predictor step is given by:Δλ =1TVwhere · stands for the Euclidean norm.The method takes larger steps when the system isfar away from the bifurcation point and smaller stepsas the bifurcation is approached. The actual solution isobtained at each loading level with the help of the correctoralgorithm.Tangent vector converges to the zero-right eigenvector(the eigenvector associated with the zero eigenvalue at thebifurcation point), as reported in
[13]. Because of this, sucha vector may anticipate the critical buses identiﬁcation,enabling the operator to take some preventive measures.Because this vector is easily calculated, Ref.
[14] proposesthis vector as a tool for local compensation with themeaning of loss reduction. These approaches are also usedhere.(b) Corrector step, obtained by the inclusion of an extraequation that imposes an orthogonality condition be-tween the predictor and corrector vectors. An alterna-tive to this approach is to consider the predictor stepresults as the initial guess in a conventional power ﬂow.In general, the corrector algorithm converges rapidlyto the desired operating point. When a continuationmethod is implemented with the help of the power ﬂowequations, such a code is named continuation powerﬂow.2.2 QV CurveThe QV curve method has been used as a planning toolby many utilities, a practise that should often be comple-mented by dynamic studies. The QV curve analysis shouldbe performed in conjunction with PV curves. Using theQV curve may help engineers to identify critical buses inthe system as well as the reactive power injections neededat those buses to ensure voltage security. The possibilityof reducing the computational cost associated with the cal-culation of QV curves is analyzed here. As stressed for thePV curve, the whole curve is not the focus. Rather thanthat, the point of minimum is meant, yielding the reactivepower margin for the bus analyzed.56Indeed, in this paper, the approach proposed in [12]is used, so the QV curve is obtained with the help ofa continuation power ﬂow, yielding the QV continuationmethod described next:QV Continuation MethodThe idea is to trace the QV curve with the help of controlledsteps until reaching the minimum. The step size is given by:Δλ =kTVQ(3)where TVQ is calculated as:TVQ = J−1Q1 (4)In (4), J is the power ﬂow Jacobian. Q1 is a vectorwith all zeros except for the reactive power associated withthe bus under study. It is important to mention that thereactive power limits for all other PV buses are taken intoaccount.Computing (3) provides the step length. The voltagelevel at the PQ buses is updated as:ΔV = kTVQTVQ(5)During the power ﬂow convergence process, the busunder study is considered as a PV bus. However, thestep length calculation and the voltage level correction areexecuted considering this bus as a PQ bus. This happensregardless the original type of the bus under study. Thecomputational cost for obtaining the step length and thecorrection term is, however, very low because it is onlyexecuted after convergence of the Newton method. At thisstage, the Jacobian matrix is known, and has already beenfactorized.3. Reactive Power Redispatch and PricingThe technical aspects previously presented are meant toassess the system operating conditions with respect to itsload margin and reactive power margin. These aspectsplay a key role in this paper, because they are consideredas the ﬂag to take some measures. Such measures consistof reactive power redispatch and its pricing, as brieﬂydescribed next.3.1 Reactive Power RedispatchThe approach proposed here seeks the reactive power re-dispatch value by means of loss reduction in the criticalarea. This area is deﬁned considering the critical bus, iden-tiﬁed by tangent vector in (2), as its seed. Buses directlylinked to this bus form the level-1 neighbourhood. Busesconnected to buses belonging to level-1 form the level-2,and so on. In this paper, the level of neighbourhood isdeﬁned as a function of the tangent vector entries. In thissense, for each level, a voltage sensitivity rate is computedby comparing the voltage sensitivity entry of each neigh-bour bus with the entry related to the critical one. Thecritical area includes all buses presenting sensitivity ratesgreater than a speciﬁed value. Once this area is deﬁned, asub-optimal solution is pursued according to:⎡⎢⎢⎢⎣ΔPΔQΔloss⎤⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎣H NgenM LPD 0⎤⎥⎥⎥⎥⎦⎡⎢⎢⎢⎣ΔθΔVΔR⎤⎥⎥⎥⎦(6)The row vector PD is formed by the partial derivativesof the active power loss equations (regarding the criticalarea) with respect to the system state variables. Thegoal of (6) is to bring the local loss to the pre-transactionobserved value, so that Δloss is the diﬀerence between thepre-transaction and post-transaction losses. As for columngen, it contains the generators to play redispatch. Thevalues of the elements associated with these generatorsare set to their normalized sensitivity values, obtainedfrom the tangent vector (generators assigned to increasegeneration), while the rest of vector gen entries equals zero.Setting column gen entries to the normalized sensitivityvalues ensures that the generators closest to the criticalarea will contribute with a larger share of reactive powerredispatch. As a consequence, the unique value of ΔRmultiplied by the sensitivity of each generator in the gencolumn provides the redispatch of each machine. However,the reactive power limits are continuously monitored, sothat when a generator reaches its limit, it is not consideredfor redispatch any further.Note that (6) is dependant on the choice of the gen-erators to play redispatch. Such generators are chosenaccording to the proposal below:Tangent vector calculated in (2) is used here as a toolto identify the generators likely to act in redispatch. Thisapproach is based on the way the state variables vary as afunction of the system parameter. In this case, the systemparameter is the variation of the reactive power generationat each generator, one by one. The total system activepower loss is given by:PsystemLoss =nlk=1{VikVjk[Gk(cos θ(ij)k + cos θ(ji)k)]− Gk(V 2ik + V 2jk)} (7)where nl is the number of transmission lines, Vik and Vjkare the voltage level at ends (i) and (j) of transmissionline k, Gk is the transmission line k conductance, θ(ij)kand θ(ji)k represent the diﬀerences between voltage phaseangles at terminal buses i and j.If (7) is derived in relation to system parameter λ, oneobtains:dPsystemLossdλ=nlk=1GkdVikdλVjk +dVjkdλVik A + VikVjk×dAdλ− 2GkdVikdλVik +dVjkdλVjk (8)57where A = Gk(cos(θ(ij)k) + cos(θ(ji)k) and dA/dλ = Gk(2 sin(θ(ji)k) (dθ(i)k/dλ − dθ(j)k/dλ))Equation (8) shows how the active power loss variesas a function of the system parameter Δλ. All its partialderivatives consist of tangent vector components knownfrom (2). Therefore, such a computation is not timeconsuming.Assuming that the active power loss is computed onlyfor the system critical area and that the right-hand sideof (2) is slightly perturbed by a generation increase at agiven bus “g”, the new tangent vector may be obtainedwithout the need to calculate the new operating point. If(8) is calculated, the active power loss variation in thearea of interest, as a function of parameter λ (reactivepower generation increase at bus “g”), is known. Takingall system generators, one by one, computing (2) and (8)indicates the generators whose redispatch reduce at mostthe system power loss.The points described so far are related to the problemof reactive power redispatch. The next step consists ofpricing the reactive power redispatch calculated from (6).3.2 Charging Reactive PowerThe scenario considered in this paper assumes bilateraltransactions. In this sense, the active power generationprices have already been established. As a consequence,for each operating point, it is assumed that the SystemOperator has already taken all the measures to supplythe demand in a stable and safe manner. Therefore, thispaper does not address the important fundamentals ofenergy pricing and tariﬀs, where discussing zonal and nodalpricings becomes an issue. Rather than that, because ofthe bilateral contracts considered, the active power priceconsidered is the spot price of the generator scheduled toplay redispatch.4. MethodologyThe focus of the paper is to enhance the system securityby means of reactive power redispatch. Such a measureis considered following a system contingency. At thispoint, it is important to emphasize an important aspectregarding the active and reactive power pricing. References[10, 11] propose a way of identifying the reactive powerresponsibility of each agent associated with a transaction.Such a feature allows one to allocate costs, not encouragingan agent of playing market, and consequently, reducingthe ﬁnal price for the costumer. In the context analyzedhere, however, the price tends to increase, because thesystem security is the focus. In this sense, the proposedmethodology seeks the minimum price elevation, becausethe most eﬀective generators are set for redispatch. Themethodology may be summarized as follows:1. The most critical contingencies are identiﬁed using QVcurve and conﬁrmed by the load margin assessed bythe continuation method (Section 2).2. If the load margin associated with the most criticalcontingencies is larger than 10%, no action is required.Otherwise:3. Identify the critical area and the generators to playredispatch (Section 3).4. Charges as a direct function of the generators identiﬁedin (2), i.e., the larger the contribution, the bigger thecharge. The value of each MVAr is given by:a. For a generator assigned to increase its reactivepower generation, it should be paid the very sameamount as collected for each MW generated.b. If the reactive power reaches its upper limit, theactive power generated should be reduced. Inthis case, the generator should be paid for thereactive power generated as well as the activepower curtailed.Note that a post-contingency load margin threshold of10%, deﬁned in step 2, has been arbitrarily chosen in thiswork. However, diﬀerent threshold values may be used.The next section describes how the ideas proposed abovemay work in an operational scenario with the help of asample real system.5. Test ResultsThe 39 bus New England system, whose data are availablein
[15], is used here to investigate the approaches proposedin the previous sections. The diagram of this system isdepicted in Fig. 1, so the reader may visualize the resultsobtained. The default system topology (47 transmissionlines operating) in a bulk operating condition is used asthe base case, obtaining the benchmark load margin withthe limits considered. The continuation method yields aload margin about 45.7%. The critical area identiﬁed fromthe base case is associated with Buses 4, 5, 6, 7, 8, 9 and39, where Bus 8 is the most critical. Finally, the criticalarea loss is 13.55 MW.For the base case, the most critical contingencies aredetected using the QV curve, and the load margin varia-tions are obtained. The three most critical contingenciesare shown in Table 1.According to the methodology proposed in Section 4,reactive power redispatch should take place for systemsecurity with respect to Contingency 1, because the systemload margin is less than 10%.It is important to state that when a contingency takesplace, automatic voltage regulators (AVRs) act in the gen-erators ﬁelds to increase the reactive power supply, avoid-ing the system voltage collapse. This automatic action isnot charged in this paper. Rather than that, the post-contingency case is considered as the new base case fromwhich sensitivity analysis is carried out. This, explicitly,divides the problem into two: ﬁrst, the automatic systemresponse provided by the AVRs. Charging this action bythe methodology proposed here is straightforward, but willnot be addressed. Secondly, the reactive power redispatchto reduce the local loss, consequence of further adjustmentsdemanded by the proposed methodology.Contingency 1 is then analyzed, because of the low loadmargin it provides. Table 2 presents the results for thiscase. In the second column of Table 2 the reactive powergenerated at the base case is presented. The third columnof that table presents the reactive power generated after58Figure 1. New England system with 39 buses.Table 1Some of Critical ContingenciesContingency Bus From Bus To Load Margin (%)Number1 5 6 9.42 16 17 10.93 17 27 44.9the contingency is considered. The automatic responseprovided by the AVRs result in an output deviation withrespect to the base case for most of the generators. Asstressed before, however, pricing this variation is not thefocus of this paper.To reduce the local loss, the technique described inSection 3.1 is used. For this sake, the sensitivity factorsare obtained according to the methodology presented inSection 3.2. The number of generators designated to re-dispatch reactive power is based on the sensitivity factorsobtained by (8). Thus, the sensitivity technique assignsthree generators/condensers to play reactive power redis-patch. These results are summarized in the fourth columnof Table 2.Note, from Table 2 that Generators 33, 34 and 37should be paid for their reactive power redispatch. Asproposed in the foregoing sections, the amount to be paidfor each generator should be equal to their active powerTable 2Reactive Power Generations (MVAr)Generator Base Case Contingency 1 Redispatch30 380.0 380.0 –31 464.0 590.2 –32 462.7 500.0 –33 229.0 309.2 80.234 286.4 320.5 34.235 562.2 600.0 –36 500.0 500.0 –37 197.3 276.0 78.738 391.9 391.9 –39 669.7 669.7 –marginal price. The last column of Table 2 indicates atotal reactive power variation about 193.1 MVAr. This isenough to take the local loss as close as possible to thepre-contingency value, while increasing the system loadmargin from 9.4% to 27.8%.Charging reactive power when the system load marginis lower than a benchmark value stimulates market players59to consider stability indexes during a transaction (bilateralor pool), connecting the technical and economical opera-tion of power system. The charges may be assigned togenerators and loads according to their responsibility inthe system power ﬂows and loadability. Such responsibilitycan be obtained, for example, using current adjustmentfactor (CAF) proposed in [10]. It is important to statethat ﬁnancial implication is not the focus of this paper.Hence, the marginal costs themselves are not taken into ac-count. The reactive power is only charged if a contingencydeteriorates the system stability margin.6. ConclusionsThis paper presented a methodology to charge reactivepower having in mind system security. The results ob-tained show that the consideration of some system secu-rity aspects make the market problem analysis more in-teresting, because some issues usually not addressed maynow be focused. In this paper, reactive market analysisis performed when a contingency takes place. However,the methodology proposed here may be applied after theoccurrence of any disturbance that drives the system to adeteriorated security margin condition.Redispatch is executed to maintain the system loadmargin at least in 10%, so a further load variation maytake place. Some options for charging reactive powerare proposed. It is acknowledged that the energy pricetends to increase under the circumstances analyzed, so theproposed methodology is aimed to minimize the generationactive power cost increase whereas enhancing the systemoperating point.Because the methodology is based on an augmentedpower ﬂow Jacobian, a sub-optimal solution is obtained.On the other hand, as the solution may be obtained bymeans of the Newton’s method, a short computational loadis necessary. The tests were executed using a well-knownsample test system, so the results may be reproduced.7. AcknowledgementsThe authors thank CNPq, CAPES (project 023/05),FAPEMIG and FAPERJ.References[1] M. Shahidehpour, H. Yamin, & Z. Li, Market operationsin electric power systems: Forecasting, scheduling, and riskmanagement, (Wiley-IEEE Press, New York, USA, 2002).[2] F.L. Alvarado, J. Meng, C.L. DeMarco, & W.S. Mota, Stabilityanalysis of interconnected power systems coupled with marketdynamics, IEEE Transactions on Power Systems, 16(4), 2001,695–701.[3] R. Fetea & A. Petroianu, Reactive power: A strange concept?,Second European Conf. in Engineering Education, Budapest,Hungary, 2000.[4] R. Hirvonen, R. Beune, L. Mogridge, R. Martinez, K. Roud´en,& O. Vatshelle, Is there market for reactive power services –Possibilities and problems, Session 2000, Cigr´e, 9–213.[5] I. El-Samahy, K. Bhattacharya, C. Ca˜nizares, M.F. Anjos,& J. Pan, A procurement market model for reactive powerservices considering system security, IEEE Transactions onPower Systems, 23(1), 2008, 137–149.[6] H.D. Chiang, A. Fluak, K.S. Shah, & N. Balu, A practicaltool for tracing power system steady-state stationary behaviordue to load and generation variations, IEEE Transactions onPower Systems, 10(2), 1995, 623–634.[7] L.S. Titare & L.D. Arya, A particle swarm optimization forimprovement of voltage stability by reactive power reservemanagement, Journal – Institution of Engineers India Part ElElectrical Engineering Division, 87, 2006, 3–7.[8] H. Yoshida, K. Kawata, Y. Fukuyama, & Y. Nakanishi, Aparticle swarm optimization for reactive power and voltagecontrol considering voltage stability, IEEE-ISAP’99, Rio deJaneiro, Brazil, 1999.[9] M. Ilic & C.-N. Yu, A possible framework for voltage/reactivepower markets, Proc. of the IEEE Power Engineering. SocietyWinter Meeting, New York, NY, 1999.[10] K.L. Lo & Y.A. Alturki, Towards reactive power markets.Part1: Reactive power allocation, IET Proceedings–GenerationTransmission and Distribution, 153(1), 2006, 59–70.[11] K.L. Lo & Y.A. Alturki, Towards reactive power markets. Part2: Diﬀerentiated market reactive power requirements, IETProceedings-Generation, Transmission and Distribution, 2(4),2008, 516–529.[12] F.W. Mohn & A.C.Z. de Souza, Tracing PV and QV curves withthe help of a CRIC continuation method, IEEE Transactionson Power Systems, 21, 2006, 1104–1114.[13] A.C.Z. de Souza, Discussion on some voltage collapse indices,Electric Power Systems Research, 53, 2000, 53–58.[14] A.C.Z. de Souza, Tangent vector applied to voltage collapseand losses sensitivity studies, Electric Power Systems Research,47(1), 1998, 65–70.[15] V. Ajjarapu, Computational techniques for voltage stabilityassessment and control power electronics and power systemsseries (New York, USA: Springer 2006).
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10.2316/Journal.203.2010.1.203-4604
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(203) International Journal of Power and Energy Systems - 2010
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