an investigation of reactive power planning based on chance constrained programming

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Electrical Power and Energy Systems 29 (2007) 650–656

An investigation of reactive power planning basedon chance constrained programming

Ning Yang a,b, C.W. Yu a,*, Fushuan Wen b, C.Y. Chung a

a Department of Electrical Engineering, The Hong Kong Polytechnic University, Hong Kongb Department of Electrical Engineering, Zhejiang University, Hangzhou, China

Received 23 March 2005; received in revised form 4 August 2006; accepted 5 September 2006

Abstract

Deregulation in the electricity supply industry has brought many new challenges to the problem of reactive power planning. Althoughthe problem has been extensively studied, available standard optimization models and methods do not offer 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 first formulated under the presumption that the generator outputs and load demandscan be modeled as specified 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.

Keywords: Reactive power planning; Uncertainties; Chance constrained programming; Monte Carlo simulation; Genetic algorithm

1. Introduction

Reactive power planning is well recognized to be one ofthe most complex problems for power systems engineers, asit requires the simultaneous minimization of two objectivefunctions [1]. The first objective deals with the minimiza-tion of operating costs by reducing the loss of real powerand improving the voltage profile. 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 manydifferent ways such as through the use of linear [2], nonlin-ear [3], or mixed integer programming [4], evolutionaryprogramming [5], and methods of decomposition [6].

As the electricity supply industry all over the world ismoving towards general deregulation, the philosophy of

0142-0615/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.ijepes.2006.09.008

* Corresponding author. Tel.: +852 2766 6184; fax: +852 2330 1544.E-mail address: [email protected] (C.W. Yu).

reactive power management and power system operationis expected to change greatly, so that the significance ofcompetition, coordination between market participants,and security requirements can be identified. 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 affect 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].

In a deregulated electricity market, there is generally nocentrally 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

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N. Yang et al. / Electrical Power and Energy Systems 29 (2007) 650–656 651

management. These are all uncertainties in reactive powerplanning and are very difficult 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.

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 field 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 effectivestochastic 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.

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 classified into three timeperiods (peak, shoulder, and off-peak periods); and thegenerator outputs and the future load demands have spe-cific 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 powerflow patterns to some extent. This will pose higher require-ments for flexibility 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 flexibility and robustnessare required for reactive power planning and CCP couldwell meet such requirements by imposing appropriate con-fidence levels.

After building the CCP-based reactive power planningmodel, 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 specified 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.

2. Brief introduction of chance constrained programming

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 satisfied with acertain probability at the optimum solution point [8,9].

A typical chance constrained programming problem hasthe following form:

min �f

s:t:

P r ff ðx; nÞ 6 �f gP b

P r fgjðx; nÞ 6 0; j ¼ 1; 2; . . . ; kgP a

8>>><>>>:

ð1Þ

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{Æ} denotes the probability ofthe event {Æ}; and a and b are the specified confidence levelsof the constraint functions and the objective function,respectively.

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 confidence 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.

For any given decision vector x, the procedure of usingMonte Carlo simulation to check the constraint

P rfgjðx; nÞ 6 0; j ¼ 1; 2; . . . ; kgP a ð2Þ

is shown as follows.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

gjðx; niÞ 6 0 i ¼ 1; 2; . . . ;N ð3Þ

is satisfied, i.e., the number of random vectors satisfyingthe constraints. Mathematically, Pr{gj(x,n) 6 0} can be

652 N. Yang et al. / Electrical Power and Energy Systems 29 (2007) 650–656

estimated by N 0/N if N is large enough. This means that thechance constraint in Eq. (2) holds if and only if N 0/N P a.

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 rff ðx; nÞ 6 �f gP b ð4ÞFrom 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 definition of probability, the N 0thlargest element in the series {f1, f2, . . ., fN} can be taken tobe the estimation of �f .

3. Reactive power planning model

3.1. Modeling of uncertain factors

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.

Suppose that the operating time of a day is classifiedinto three time periods (peak, shoulder, and off-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 off-peak period isfrom 11 p.m. to 8 a.m.

Suppose that the generator outputs at a generation nodei is PGik (k = 1,2,3. 1, peak period; 2, shoulder period; 3,off-peak period). PGik is a random variable and is normallydistributed, i.e. P Gik � Nðlik; r

2ikÞ, where lik and rik are the

corresponding mean and standard deviation for the timeperiod k.

Suppose that at a load node j, the active power is PDjk

and the reactive power is QDjk. They are random variablesand are normally distributed, i.e. P Djk � NðlPjk; r

2PjkÞ;

QDjk � NðlQjk; r2QjkÞ ðk ¼ 1; 2; 3Þ.

3.2. Chance constrained programming model for reactive

power planning

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 thevoltage constraints are considered and represented as softconstraints, i.e., the constraints can be violated in some

exceptional cases, but the probability of such a violationmust be less than a specified level. However, the proposedframework can be easily extended to include otherconstraints.

The reactive power planning based on chance con-strained programming can be formulated as follows:

min �f ðSÞs:t: P rff ðSÞ6 �f ðSÞgP b ð5Þ

P rfV mini 6 V i6 V max

i gP a i2NB

Qminci 6Qci6Qmax

ci i2Nc

Qmingi 6Qgi6Qmax

gi i2N g

0¼Qi�V i

Xj2Ni

V jðGij sinhij�Bij coshijÞ i2NPQ

where

f ðSÞ ¼ hX3

k¼1

tkP loss;k þXi2Nc

ðei þ CciQciÞ ð6Þ

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; tk

is the duration of time period k (peak, shoulder, and off-peak periods); Ploss,k is the network real power loss duringtime period k; ei is the fixed 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 difference 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 specified confidence level forthe constraint function; b is the specified confidence levelfor the objective function.

In Eq. (6), the first term represents the total cost ofenergy loss while the second term represents the cost ofreactive power source installation which has two compo-nents, a fixed installation cost and a purchase cost.

4. Solution method

4.1. Checking constraints

It is very difficult to convert the stochastic constraint inEq. (5) to its deterministic equivalents according to the pre-determined confidence level. Monte Carlo simulation pro-vides a general approach for solving this problem.

Consider the following chance constraint:

P rfV mini 6 V i 6 V max

i gP a: ð7ÞVi is a function of PG and PD, where PG is the vector of thepower outputs of generators and PD is the vector of nodal


load 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:

a. Specify the number of Monte Carlo simulationsallowed, N.

b. Set counter t = 0.c. Set N 0 = 0.d. Sample the generator output randomly. A random

number PGik is generated with P Gik � Nðlik; r2ikÞ.

e. Sample the load demands randomly. P Djk � NðlPjk; r

2PjkÞ, QDjk � NðlQjk; r

2QjkÞ.

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 step d.

g. Calculate Vi.h. If V min

i 6 V i 6 V maxi , N 0 = N 0 + 1.

i. Set t = t + 1.j. If t < N, return to step d, otherwise go to step k.

k. If N 0/N P a, the chance constraintP rfV min

i 6 V i 6 V maxi gP a is satisfied.

4.2. Computing the objective value

Similarly, for a given vector S, we can use the MonteCarlo technique to compute the maximum value �f satisfy-ing the objective function in Eq. (5), i.e.,

P rff ðSÞ 6 �f ðSÞgP b: ð8ÞThe procedure of using Monte Carlo simulation to com-

pute the objective value �f is as follows:

a. Calculate f from the generated vectorsP ð1ÞGik; P

ð2ÞGik; . . . ; P ðNÞGik and P ð1ÞDi ; P

ð2ÞDi ; . . . ; P ðNÞDi , and get

the series {f1, f2, . . ., fN}.b. Let N 0 be the integer part of (1 � b)N.c. The N 0th largest element in the series {f1, f2, . . ., fN} is

the objective value.

4.3. Monte Carlo simulation based genetic algorithm for

solving the chance constrained programming model of

reactive power planning

A genetic algorithm based on Monte Carlo simulation isused to solve the chance constrained programmingdescribed in Eq. (5). Since genetic algorithms have beenintroduced well in numerous studies [14–17], they are notdiscussed in detail here. We will only discuss specific issuessuch as the representation of structure, the handling of con-straints, and evaluation of fitness.

A two-part vector X = (X(1), X(2)) is used as a chromo-some to represent a solution to the optimization problem,

where X(1) is an NPV dimension vector representing Vi

(i 2 NPV, NPV is the set of PV-bus numbers), and X(2) isan Nc dimension vector representing Qci (i 2 Nc).

In the genetic algorithm, violations of constraints arehandled using a penalty function approach. Penalty termsare incorporated into the fitness function. The fitness func-tion is taken as

F fitness ¼ �f ðSÞ þX

i2NPQ

kViDV 2i þ

Xi2NPV

kQgiDQ2gi; ð9Þ

where kVi and kQgi are penalty factors, and kVi is defined asfollows:

kVi ¼0 V min

i 6 V i 6 V maxi

W 1 ð1� 5%ÞV mini 6 V i < V min

i or V maxi < V i 6 ð1þ 5%ÞV max

i

W 2 V i < ð1� 5%ÞV mini orV i > ð1þ 5%ÞV max

i

8><>:

ð10Þ

where W1�W2.DVi and DQgi are defined by the following equations:

DV i ¼V i � V max

i ðV i > V maxi Þ

0 ðV mini 6 V i 6 V max

i ÞV min

i � V i ðV i < V mini Þ

8><>:

DQgi ¼Qgi � Qmax

gi ðQgi > Qmaxgi Þ

0 ðQmingi 6 Qgi 6 Qmax

gi ÞQmin

gi � Qgi ðQgi < Qmingi Þ

8>><>>:

ð11Þ

The procedure for solving the model is as follows:

(a) Input the original data.(b) Input the parameters associated with the genetic

algorithm.(c) Randomly generate a group of initial planning

schemes and take them as the initial population.(d) For each given chromosome, check the constraints

in Eq. (7) using the method described in Section4.1.

(e) Compute �f ðSÞ using the method described in Section4.2.

(f) Evaluate the chromosome fitness using Eq. (9). If theconstraints are violated, calculate the fitness functionusing the penalty function approach based on theobjective function obtained in step d. Otherwise, theobjective function obtained in step d is directly takenas the fitness function.

(g) Use the roulette wheel selection method to the cur-rent population and apply crossover and mutationoperators to produce new offspring.

(h) Repeat steps d–g for L times, where L is the maxi-mum number of iteration.

(i) Take the best chromosome found in the solving pro-cess as the final planning scheme.

The flowchart depicted in Fig. 1 summarizes the aboveprocedure.

Fig. 1. Flowchart of the algorithm.

G

G

G

G

G

1

6

7

11 10 9

8

5

32

13

12

14

4

Fig. 2. IEEE 14-bus system.

Table 1Base power and parameters of cost

SB (MVA) h ($/MWh) ei ($) Cci ($/MVAr) t1 (h) t2 (h) t3 (h)

100 60 1000 6000 3650 1825 3285

Table 2Variable limits (p.u.)

Qg1 Qg2 Qg3 Qg6 Qg8 Vg Vload Qc

Max 0.8 0.6 0.6 0.5 0.5 1.1 1.05 0.2Min �0.2 �0.15 �0.15 �0.1 �0.1 0.95 0.95 0

Table 3Probability distributions of generations

Nodes 8 a.m. to 6 p.m. 6 p.m. to 11 p.m. 11 p.m. to 8 a.m.

li1

(MW)ri1

(MW)li2

(MW)ri2

(MW)li3

(MW)ri3

(MW)

2 70 7 60 6 50 53 70 7 50 5 30 36 50 5 30 3 10 18 50 5 30 3 10 1


5. Case study

The IEEE 14-bus system is used to show the effective-ness of the proposed algorithm. The system consists of aslack bus (node 1), four PV buses (nodes 2, 3, 6, and 8),nine PQ buses, and 20 branches. The system is shown inFig. 2. The branches data and the initial bus data can beobtained in [18]. The base power and costs parametersare given in Table 1 and the limits of the variables are givenin Table 2. Table 3 gives the generation data. The data ofall load nodes are listed in Table 4.

Five buses, namely buses 4, 10, 11, 13, and 14, areselected as the possible VAR source installation buses. Sup-pose that 10 capacitor banks each rated at 2 MVAr may beinstalled at each of these nodes. The per-unit energy cost is$60/MWh and the per-unit cost is $6000/MVAr. The per-iod of operation is 1 year.

Four detailed planning schemes are listed in Table 5 fordifferent confidence levels a = 0.7, 0.8, 0.9, and 1.00,respectively when b = 0.8.

Four other detailed planning schemes are listed in Table6 for different confidence levels b = 0.7, 0.8, 0.9, and 1.00,respectively when a = 0.8.

The following can be observed from the comparisons.

(1) When b = 0.8, as the confidence level a is set higher,the total investment and operation cost will increase.When a = 1.0, this means that all of the voltage limitswill not be violated. The total cost is $2,194,150.When a = 0.7, the total cost will be reduced to$2,009,313.

(2) When a = 0.8, as the confidence level b is set higher,the total investment and operation cost will increase.When b = 1.0, the total cost is $2,170,125. Whenb = 0.7, the total cost will be reduced to $2,021,750.

Table 4Probability distribution of nodal loads

Nodes Loadclass

8 a.m. to 6 p.m. 6 p.m. to 11 p.m. 11 p.m. to 8 a.m.

lPj1

(MW)rPj1

(MW)lQj1

(MVAr)rQj1

(MVAr)lPj2

(MW)rPj2

(MW)lQj2

(MVAr)rQj2

(MVAr)lPj3

(MW)rPj3

(MW)lQj3

(MVAr)rQj3

(MVAr)

2 1 34 3.4 25 2.5 8 0.8 6 0.6 8 0.8 6 0.63 1 95 9.5 71 7.1 95 9.5 71 7.1 95 9.5 71 7.14 3 78 7.8 58 5.8 40 4.0 30 3.0 20 2.0 15 1.55 2 10 1.0 7 0.7 10 1.0 7 0.7 5 0.5 3 0.36 1 25 2.5 18 1.8 10 1.0 7 0.7 10 1.0 7 0.79 2 36 3.6 27 2.7 36 3.6 27 2.7 20 2.0 15 1.5

10 3 17 1.7 12 1.2 8 0.8 6 0.6 6 0.6 4 0.411 3 13 1.3 9 0.9 6 0.6 4 0.4 4 0.4 3 0.312 4 6 0.6 4 0.4 12 1.2 9 0.9 6 0.6 4 0.413 4 9 0.9 6 0.6 18 1.8 13 1.3 9 0.9 6 0.614 4 10 1.0 7 0.7 20 2.0 15 1.5 10 1.0 7 0.7

Load class: 1, industrial; 2, commercial; 3, enterprise; 4, residential.

Table 5Results of reactive power planning (b = 0.8)

�f ðSÞ ($) Vg2 Vg3 Vg6 Vg8 Qc4 (MVAr) Qc10 (MVAr) Qc11 (MVAr) Qc13 (MVAr) Qc14 (MVAr)

a = 0.7 2,009,313 1.05 1.03 1.02 1.08 0 0 10 2 6a = 0.8 2,075,875 1.05 1.03 1.02 1.09 12 2 8 0 10a = 0.9 2,081,500 1.04 1.02 1.01 1.10 4 0 0 0 12a = 1.0 2,194,150 1.05 1.02 1.03 1.07 2 20 0 0 10

Table 6Results of reactive power planning (a = 0.8)

�f ðSÞ ($) Vg2 Vg3 Vg6 Vg8 Qc4 (MVAr) Qc10 (MVAr) Qc11 (MVAr) Qc13 (MVAr) Qc14 (MVAr)

b = 0.7 2,021,750 1.04 1.02 1.02 1.10 0 0 2 0 8b = 0.8 2,075,875 1.05 1.03 1.02 1.09 12 2 8 0 10b = 0.9 2,092,625 1.05 1.02 1.01 1.10 4 2 2 0 8b = 1.0 2,170,125 1.05 1.02 1.03 1.09 2 4 2 8 10


(3) As the confidence levels a and b are set higher, thesecurity risk of the planning scheme is lower butthe cost is higher. As a and b are reduced, the secu-rity risk created by uncertainties will be higher whilethe total investment and operation cost willdecrease.

(4) Given the confidence levels a and b in advance, theproposed method could quantify the security risk cre-ated by uncertainties.

(5) Different planning schemes could be provided for dif-ferent choices of confidence levels for the objectiveand constraints. The planner could manage the riskby specifying the confidence levels in advance.

6. Conclusion

In this paper a chance constrained programming modelfor reactive power planning under a competitive electricitymarket environment was proposed. An approach based onMonte Carlo simulation and a genetic algorithm was devel-oped to solve the model. This method can accommodate

uncertain factors well and handle constraints flexibly.Deregulation introduces more uncertain factors for reac-tive power planning. Hence, a security risk will be inevita-bly incurred in the planning scheme produced under suchan uncertain environment. Given the confidence levels inadvance, the proposed framework could quantify the secu-rity risk created by uncertainties while minimizing the oper-ation cost and investment cost. A case study was used toillustrate the validity and essential features of the proposedmodel and methodology.

Acknowledgements

The authors are grateful to the financial support pro-vided by the Research Grants Council of Hong Kong(Grant number PolyU/5215/03E).

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an investigation of reactive power planning based on chance constrained programming

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