# An investigation of reactive power planning based on chance constrained programming

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<ul><li><p>ivin</p><p>Fu</p><p>Hon</p><p>g, Z</p><p>form</p><p>Deregulation in the electricity supply industry has brought many new challenges to the problem of reactive power planning. Although</p><p>Reactive power planning is well recognized to be one of</p><p>programming [5], and methods of decomposition [6].As the electricity supply industry all over the world is</p><p>moving towards general deregulation, the philosophy of</p><p>competition, coordination between market participants,</p><p>centrally coordinated planning of new generation or of theclosure of existing plants. The locations, capacities, andcommissioning time of new power plants, as well as the clo-sure of old generators or power plants are basically at thediscretion of the power generation companies, who usuallyonly need to give short notice to the market operation</p><p>* Corresponding author. Tel.: +852 2766 6184; fax: +852 2330 1544.E-mail address: eecwyu@polyu.edu.hk (C.W. Yu).</p><p>Electrical Power and Energy Systethe most complex problems for power systems engineers, asit requires the simultaneous minimization of two objectivefunctions [1]. The rst objective deals with the minimiza-tion of operating costs by reducing the loss of real powerand improving the voltage prole. The second objectiveminimizes the cost of investing in additional sources ofreactive power. Reactive power planning is a nonlinearoptimization problem for large-scale systems with manyuncertainties. The problem has been approached in manydierent ways such as through the use of linear [2], nonlin-ear [3], or mixed integer programming [4], evolutionary</p><p>and security requirements can be identied. In a verticallyintegrated utility, reactive power facilities are owned andoperated by a single utility. The costs and contribution ofa reactive power supply are not precisely evaluated. Undera deregulated environment, the obligations and rights ofthe owners of reactive power facilities become essentialissues that aect not only the investment returns of thepower industry but also the security of the power system.The situation is even more complicated when several self-supported systems are interconnected [7].</p><p>In a deregulated electricity market, there is generally nothe problem has been extensively studied, available standard optimization models and methods do not oer good solutions to this prob-lem, especially in a competitive electricity market environment where many factors are uncertain. Given this background, a novel methodfor reactive power planning based on chance constrained programming is presented in this paper, with uncertain factors taken intoaccount. A stochastic optimization model is rst formulated under the presumption that the generator outputs and load demandscan be modeled as specied probability distributions. A method is then presented for solving the optimization problem using the MonteCarlo simulation method and genetic algorithm. Finally, a case study is used to illustrate the validity and essential features of the pro-posed model and methodology. 2007 Elsevier Ltd. All rights reserved.</p><p>Keywords: Reactive power planning; Uncertainties; Chance constrained programming; Monte Carlo simulation; Genetic algorithm</p><p>1. Introduction reactive power management and power system operationis expected to change greatly, so that the signicance ofAn investigation of reacton chance constra</p><p>Ning Yang a,b, C.W. Yu a,*,a Department of Electrical Engineering, The</p><p>b Department of Electrical Engineerin</p><p>Received 23 March 2005; received in revised</p><p>Abstract0142-0615/$ - see front matter 2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.ijepes.2006.09.008e power planning baseded programming</p><p>shuan Wen b, C.Y. Chung a</p><p>g Kong Polytechnic University, Hong Kong</p><p>hejiang University, Hangzhou, China</p><p>4 August 2006; accepted 5 September 2006</p><p>www.elsevier.com/locate/ijepes</p><p>ms 29 (2007) 650656</p></li><li><p>d Emanagement. These are all uncertainties in reactive powerplanning and are very dicult to manage. Moreover, thereare many other uncertainties such as those of future loadchanging, of cooperation and competition in the electricitymarket in the future, and uncertainties resulting from eco-nomic development, the environment, or policy regulation.All of these can bring new challenges to reactive powerplanners.</p><p>Given this background, a novel method for optimalreactive power planning based on chance constrained pro-gramming is presented in this paper. Chance constrainedprogramming (CCP) [8,9], which has gained increasinginterest in various eld as well as in power systems, is akind of stochastic programming and is especially suitablefor solving optimal problems with uncertain factors. Xiaoet al. [10], recognizing the uncertainties of power systemsand taking the advantage of CCP, formulated an eectivestochastic model to evaluate available transfer capabilityof prescribed interfaces in interconnected power networks,in which the availability of generators and circuits as wellas load forecast errors are considered as random variables.In Ref. [11], a chance constrained programming formula-tion was developed for scheduling units of a power gener-ating system by taking into consideration thestochasticity of the hourly load and its correlation struc-ture. The deterministic form of the stochastic constraintwas used in solving the unit commitment problemiteratively.</p><p>CCP can be used in reactive power planning under acompetitive electricity market environment. Consideringthe randomness of generator outputs and load changing,without loss of generality, the following presumptions aremade: the daily operating time is classied into three timeperiods (peak, shoulder, and o-peak periods); and thegenerator outputs and the future load demands have spe-cic probability distributions such as normal probabilitydistributions. It should be noted that the method proposedbelow is also good for other probability distributions. Inthis paper the reactive power planning time is one yearand the optimal objective is to minimize the operationand investment cost of the candidate compensationdevices. With advancements in technology and improve-ments in management as well as the deepening of therestructuring of the electricity industry, engineers havethe capability and need to intervene and regulate powerow patterns to some extent. This will pose higher require-ments for exibility in reactive power planning. Therefore,the optimal result will be on the conservative side if theconstraints are handled by deterministic methods. CCP isa good means of solving this kind of problem. In the elec-tricity market environment more exibility and robustnessare required for reactive power planning and CCP couldwell meet such requirements by imposing appropriate con-dence levels.</p><p>After building the CCP-based reactive power planning</p><p>N. Yang et al. / Electrical Power anmodel, an approach based on Monte Carlo simulationand a genetic algorithm is developed to solve the problem.The paper is organized as follows: A brief introduction tochance constrained programming and stochastic simula-tion is given in Section 2. A chance constrained program-ming model is formulated in Section 3 under thepresumption that the generator outputs and the future loaddemand can be modeled as specied probability distribu-tions. The formulation of an approach based on MonteCarlo simulation and a genetic algorithm to solve the prob-lem of optimization is outlined in Section 4. In Section 5, acase study is used to illustrate the validity and essential fea-tures of the proposed method. Finally, conclusions aregiven in Section 6.</p><p>2. Brief introduction of chance constrained programming</p><p>Chance constrained programming is a kind of stochasticprogramming in which the constraints or objective functionof an optimization problem contain stochastic parameters.The programming attempts to reconcile optimization overuncertain constraints. The constraints, which contain sto-chastic parameters, are guaranteed to be satised with acertain probability at the optimum solution point [8,9].</p><p>A typical chance constrained programming problem hasthe following form:</p><p>min f</p><p>s:t:</p><p>P r ff x; n 6 f gP bP r fgjx; n 6 0; j 1; 2; . . . ; kgP a</p><p>8>>>>>:</p><p>1</p><p>where x 2 Rn is a decision variable vector; n is a stochasticvector with a given joint probability density function U(n);f(x,n) is the objective function; gj(x,n) (j = 1,2, . . .,k) arethe constraint functions; Pr{} denotes the probability ofthe event {}; and a and b are the specied condence levelsof the constraint functions and the objective function,respectively.</p><p>The traditional technique to solve the chance con-strained programming is to convert the stochastic con-straints to their respective deterministic equivalentsaccording to the predetermined condence level. Unfortu-nately, this approach is not successful for some compli-cated cases. In this paper, Monte Carlo simulation [12,13]provides a general approach to solving this problem.</p><p>For any given decision vector x, the procedure of usingMonte Carlo simulation to check the constraint</p><p>P rfgjx; n 6 0; j 1; 2; . . . ; kgP a 2is shown as follows.</p><p>First, we generate N independent random vectors n1,n2, . . . ,nN from a given joint probability distributionU(n). Let N 0 be the number of occasions in which theinequality</p><p>gjx; ni 6 0 i 1; 2; . . . ;N 3</p><p>nergy Systems 29 (2007) 650656 651is satised, i.e., the number of random vectors satisfyingthe constraints. Mathematically, Pr{gj(x,n) 6 0} can be</p></li><li><p>d Eestimated by N 0/N if N is large enough. This means that thechance constraint in Eq. (2) holds if and only if N 0/NP a.</p><p>Similarly, for a given vector x, we can use the MonteCarlo technique to compute the maximum value f , satisfy-ing the objective function in Eq. (1).</p><p>P rff x; n 6 f gP b 4From the generated N independent random vectorsn1,n2, . . .,nN, we can obtain a series {f1, f2, . . ., fN}, wherefi = f(x,ni), i = 1,2, . . .,N. Let N 0 be the integer part of(1 b)N. By the basic denition of probability, the N 0thlargest element in the series {f1, f2, . . ., fN} can be taken tobe the estimation of f .</p><p>3. Reactive power planning model</p><p>3.1. Modeling of uncertain factors</p><p>Probabilistic measures such as probability density func-tions can be used to model many kinds of uncertainties.The parameters associated with probability density func-tions can be obtained based on historical data and predic-tions of future development. Among the uncertaintiesencountered in reactive power planning, the generator out-puts and load demands are the most important factors andare therefore included in this paper. For simplicity, otheruncertain factors are not modeled in this paper. However,it must be emphasized that the framework described in thispaper does not preclude the inclusion of other uncertainties.</p><p>Suppose that the operating time of a day is classiedinto three time periods (peak, shoulder, and o-peak peri-ods). The peak period is from 8 a.m. to 6 p.m., the shoulderperiod is from 6 p.m. to 11 p.m., and the o-peak period isfrom 11 p.m. to 8 a.m.</p><p>Suppose that the generator outputs at a generation nodei is PGik (k = 1,2,3. 1, peak period; 2, shoulder period; 3,o-peak period). PGik is a random variable and is normallydistributed, i.e. PGik Nlik; r2ik, where lik and rik are thecorresponding mean and standard deviation for the timeperiod k.</p><p>Suppose that at a load node j, the active power is PDjkand the reactive power is QDjk. They are random variablesand are normally distributed, i.e. PDjk NlPjk; r2Pjk;QDjk NlQjk; r2Qjk k 1; 2; 3.</p><p>3.2. Chance constrained programming model for reactive</p><p>power planning</p><p>Because of the uncertainties in generator outputs as wellas load demands for the planning period, the reactivepower planning problem can be formulated as a stochasticoptimization problem. CCP provides a good means forsolving this problem. The objective is to minimize the oper-ation cost and investment cost of candidate reactive com-pensation devices. For illustrative purposes, only the</p><p>652 N. Yang et al. / Electrical Power anvoltage constraints are considered and represented as softconstraints, i.e., the constraints can be violated in someexceptional cases, but the probability of such a violationmust be less than a specied level. However, the proposedframework can be easily extended to include otherconstraints.</p><p>The reactive power planning based on chance con-strained programming can be formulated as follows:</p><p>min f Ss:t: P rff S6 f SgP b 5</p><p>P rfV mini 6 V i6 V maxi gP a i2NBQminci 6Qci6Qmaxci i2NcQmingi 6Qgi6Qmaxgi i2Ng0QiV i</p><p>Xj2Ni</p><p>V jGij sinhijBij coshij i2NPQ</p><p>where</p><p>f S hX3k1</p><p>tkP loss;k Xi2Nc</p><p>ei CciQci 6</p><p>S an n-dimension solution vector to the optimal planningproblem; n is the number of candidate reactive compensa-tion devices; Nc is the set of numbers of possible VARsource installation buses; NB is the set of numbers of totalbuses; Ng is the set of generator bus numbers; NPQ is the setof PQ-bus numbers; Ni is the set of numbers of buses adja-cent to bus i, including bus i; h is the per-unit energy cost; tkis the duration of time period k (peak, shoulder, and o-peak periods); Ploss,k is the network real power loss duringtime period k; ei is the xed VAR source installation cost atbus i; Cci is the per-unit VAR source purchase cost at bus i;Qci is the VAR source installed at bus i; Vi is the voltagemagnitude at bus i; hij is the voltage angle dierence be-tween bus i and bus j; Qgi is the reactive power generationat bus i; Gij,Bij is the mutual conductance and susceptancebetween bus i and bus j; Gii,Bii are the self-conductance andsusceptance of bus i; a is the specied condence level forthe constraint function; b is the specied condence levelfor the objective function.</p><p>In Eq. (6), the rst term represents the total cost ofenergy loss while the second term represents the cost ofreactive power source installation which has two compo-nents, a xed installation cost and a purchase cost.</p><p>4. Solution method</p><p>4.1. Checking constraints</p><p>It is very dicult to convert the stochastic constraint inEq. (5) to its deterministic equivalents according to the pre-determined condence level. Monte Carlo simulation pro-vides a general approach for solving this problem.</p><p>Consider the following chance constraint:</p><p>P rfV mini 6 V i 6 V maxi gP a: 7</p><p>nergy Systems 29 (2007) 650656Vi is a function of PG and PD, where PG is the vector of thepower outputs of generators and PD is the vector of nodal</p></li><li><p>d Eload demands. As mentioned in the last section, there arestochastic parameters in PG and PD; hence, Vi is a stochas-tic parameter. For any given decision vector S, the proce-dure of using Monte Carlo simulation to check theconstraints is as follows:</p><p>a. Specify the number of Monte Carlo simulationsallowed, N.</p><p>b. Set counter t = 0.c. Set N 0 = 0.d. Sample the generator output randomly. A random</p><p>number PGik is generated with PGik Nlik; r2ik.e. Sample the load demands randomly. PDjk NlPjk; r2Pjk, QDjk NlQjk; r2Qjk.</p><p>f. Check whether the total power generated is largerthan the total load. If yes this means that the sup-ply and demand are feasible and then go to step g,otherwise go to ste...</p></li></ul>