Optimal placement of capacitors in radial distribution system using a Fuzzy-GA method

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Electrical Power and Energy Systems 30 (2008) 361–367 www.elsevier.com/locate/ijepes

Optimal placement of capacitors in radial distribution system using a Fuzzy-GA method D. Das Electrical Engineering Department, Indian Institute of Technology, Kharagpur 721 302, India Received 31 October 2006; received in revised form 22 August 2007; accepted 27 August 2007

Abstract The paper presents a genetic algorithm (GA) based fuzzy multi-objective approach for determining the optimum values of fixed and switched shunt capacitors to improve the voltage profile and maximize the net savings in a radial distribution system. The two objectives, i.e. maximization of net savings and minimization of the nodes voltage deviation are first fuzzified and, then, dealt with by integrating them into a fuzzy satisfaction objective function through appropriate weighting factors. The optimization technique of the GA is then adopted to solve the fuzzy multi-objective problem for obtaining the optimum values of shunt capacitors. The effectiveness of the proposed technique is demonstrated through an example. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Capacitor placement; Radial feeders; Genetic algorithm; Fuzzy set theory

1. Introduction Capacitors are widely used in distribution systems for reactive power compensation, to achieve power and energy loss reduction, to improve service quality via voltage regulation and to achieve deferral of construction, if possible, via system capacity release. The problem is to choose the optimum capacitor allocation and the capacitor control in order to maximize the benefits against the cost of capacitors. In the past, considerable efforts were put into the capacitor placement problem. The early approaches to this problem include (i) are based on the dynamic programming technique to handle the discrete nature of capacitor size [1] and (ii) one that uses analytical methods in conjunction with heuristics [2–4]. Recently, the growing need for distribution automation and control has regenerated interest in the capacitor placement problem. Grainger and Lee pioneered in reformulating the problem as a non-linear pro-

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gramming problem and in developing computationally, efficient solution methods. In [5], they treated the capacitor sizes as continuous variables and a computationally simple, iterative solution scheme has been proposed. They later introduced switched type capacitors with simultaneous switching [6] and a voltage dependent model for loss reduction [7]. In [8], together with El-kib, they proposed a solution method to determine the optimal design and real time control scheme for switched capacitors with non-simultaneous switching under certain assumptions. Grainger et al. [9] considered the optimal design and control scheme for continuous capacitive compensation case. Civanlar and Grainger [10], combined capacitor placement and voltage regulator problems for a general distribution system and proposed a decoupled solution methodology. Ponnavaikko and Rao [11] considered load growth as well as system capacity release and voltage raise at light load condition and used a local optimization technique called the method of local variations. Baran and Wu [12,13] presented a problem formulation similar to that of Grainger et al., a non-linear optimization problem, but incorporated the following directly into the model: (i) the distribution power

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flow equations and (ii) the constraints of node voltage magnitudes at different load levels. They developed a solution algorithm by using decomposition techniques and the phase-I and phase-II feasible direction method. Salama et al. [14] presented a heuristics strategy to reduce system losses by identifying sensitive nodes at which capacitors should be placed. This method does not guarantee a minimization in the cost function or maximizing in the net saving function. Chis et al. [15] modifies the method of [14] to overcome this disadvantage, so their technique obtained good results but does not compromise between maximum net saving and voltage constraints. Salama and Chikhani [16] proposed an analytical method based on differentiating a well-defined net saving function of power and energy losses with respect to capacitor size, thus obtaining the capacitor size. Haque [17] proposes a method of minimizing the loss associated with the reactive component of branch currents by placing optimal capacitors at proper locations. The disadvantage of this method is that it neglects the cost-benefit analysis which in turn, depends on the cost of capacitors and energy saving. Safigianni and Salis [18] proposed a complex method for the optimum VAr control of radial primary power distribution networks. Chiang et al. [19] formulated capacitor placement problem as a combinatorial optimization problem and used simulated annealing to search the global optimal solution. Huang et al. [20] proposed a tabu search based solution algorithm and used sensitivity analysis method to select the candidate installation locations of capacitors to reduce the search space. Huang [21] applied immune algorithm to capacitor placement problem by representing objective function and constraints as antigens. Also many other researchers proposed genetic algorithm (GA) application to search global optimal solution of capacitor placement problem [22–27]. Boone and Chiang [22] proposed fixed capacitor placement problem using GA. Sundhararajan and Pahwa [23] used GA to determine the size of the capacitors, which somewhat depends on experiences in the selection of the probability parameters. Miu et al. [24] suggested the two-stage algorithms that combine the good qualities of GA and a fast sensitivity based heuristic. Levitin et al. [25], Kim et al. [26], Milosevic and Begovic [27] and Das [28] also proposed shunt capacitor placement problem using GA. This study presents a fuzzy multi-objective problem formulation to satisfy realistic objectives. The fuzzy set theory provides an excellent framework for integrating the mathematical and heuristic approach into a more realistic formulation of capacitor placement problem. Owing to the non-commensurable characteristic of the objectives, a traditional approach that optimizes a single objective function is inappropriate for this problem. The fuzzy approach is therefore adopted to simultaneously consider the multiple objectives and to obtain a fuzzy satisfaction maximizing decision. GA is considered to be a method for solving the large-scale combinatorial optimization problem due to the ability to search global or near global

optimal solutions and its appropriateness for parallel computing. In this work, a combination of Fuzzy-GA method is implemented to determine the optimal sizes of fixed and switched capacitors. The objectives are to maximize the net savings and minimize the nodes voltage deviation in the distribution network by taking the cost of capacitors into account. 2. Traditional objective function With the delivery of customer load on the distribution system, there are demand (kW) losses which over a period of time will reflect as energy losses. Hence the objective function for optimization can be stated mathematically as to [12–16,19,27,28] Minimize

S ¼ Ke

L X

T jP j þ

ncap X

K c  Qci

ð1Þ

i¼1

j¼1

where Pj is the power loss at any load level j; Tj is the time duration of jth load level; Qci is the size of the capacitor at node i; ncap is the number of candidate locations for capacitor placement; L is the number of load level; Ke is the energy cost of losses; and Kc is the cost of capacitor per kVAr. Eq. (1) does not consider the voltage constraint. However, the voltage constraint can be formulated as a penalty function to the basic objective function of Eq. (1). That is, constraint objective function could be represented as [23,25,27,28] Minimize

S ¼ Ke

L X

T jP j þ

j¼1

ncap X

 2 K c  Qci þ k V lmin  Vsmin

i¼1

ð2Þ where V lmin is the minimum voltage limit; V smin is the minimum system voltage; and k is the constant multiplier. The value of k plays an important role, while handling with constrained optimization. A higher value of k, the system may evolve individuals that violates the constraint but are rated better than those that do not. Hence, suitable value of k, satisfying the constraint limit and also giving best possible result could only be found out by trial and error, which is extremely difficult task. 3. Problem formulation In fuzzy domain, each objective is associated with a membership function. The membership function indicates the degree of satisfaction of the objective. In the crisp domain, either the objective is satisfied or it is violated, implying membership values of unity and zero, respectively. On the contrary, fuzzy sets entertain varying degrees of membership function values from zero to unity. Thus fuzzy set theory is an extension of standard set theory [29]. The present work considers the following objectives for the capacitor placement problem.

D. Das / Electrical Power and Energy Systems 30 (2008) 361–367

(1) Maximizing the saving by minimizing the energy loss due to the placement of shunt capacitors. (2) Minimization of the deviation of nodes voltage. The membership function consists of a lower and upper bound value together with a strictly monotonically decreasing and continuous function for above mentioned objectives are described below. 3.1. Membership function for the net saving The net saving at jth load level due to capacitor placement in a distribution system is given as N S ¼ K e T j P j  K e T j P cj 

ncap X

K c  Qci;j

ð3Þ

363

1.0 μVj

0

ymin

ymax yj

Fig. 2. Membership function for maximum node voltage deviation.

membership value is assigned and xmax = 1.0 means if the savings is zero percent or has a negative value, zero membership value is assigned. 3.2. Membership function for the maximum node voltage deviation

i¼1

where Ke is the energy loss cost; Tj is the duration of load at jth load level; Pj is the power loss at jth load level before compensation; P cj is the power loss at jth load level after compensation; Qci,j is the value of capacitor at ith node for jth load level; and ncap is the number of candidate locations for capacitor placement. Now for net saving, in Eq. (3), Ns > 0, i.e. K e T j P j  K e T j P cj 

ncap X

K c  Qci;j > 0

i¼1

K e T j P cj þ

ncap P

ð4Þ

K c  Qci;j

i¼1

<1

K eT jP j

Basic purpose of this membership function is that the deviation of nodes voltage should be less. At jth load level, let us define y j ¼ max jV s  V i;j j for i ¼ 2; 3; . . . ; NB

where Vi,j is the voltage magnitude of node i at jth load level in per unit and Vs is the voltage magnitude of substation in per unit. If maximum value of nodes voltage deviation is less, then higher membership value is assigned and if deviation is more, then a lower membership value is assigned. Fig. 2 shows the membership function for maximum node voltage deviation. From Fig. 2, we can write

Let us define K e T j P cj þ

ncap P

lV j ¼

K c  Qci;j

i¼1

xj ¼

ðy max  y j Þ ðy max  y min Þ

for y min < y j < y max

ð5Þ

lV j ¼ 1

for y j 6 y min

Eq. (5) indicates that if xj is high saving is low and if xj is low, saving is high. Membership function for net saving is given in Fig. 1. From Fig. 1, lsj can be written as

lV j ¼ 0

for y j P y max

lsj ¼

K eT jP j

ðxmax  xj Þ ðxmax  xmin Þ

lsj ¼ 1 lsj ¼ 0

for xmin < xj < xmax ð6Þ

for xj 6 xmin for xj P xmax

In the present work, it has been assumed that xmin = 0.50 and xmax = 1.0. xmin = 0.50 means if the savings is 50% or more of the energy loss cost before compensation, unity

ð8Þ

In the present work, ymin = 0.05 and ymax = 0.10 have been considered. ymin = 0.05 means if the substation voltage is 1.0 pu, then minimum system voltage will be 0.95 pu and if the minimum system voltage greater than or equal to 0.95 pu, unity membership value is assigned. Similarly, if ymax = 0.10, minimum system voltage will be 0.90 pu and if the minimum system voltage is less than or equal to 0.90 pu, zero membership value is assigned. 4. Fuzzy multi-objective formulation The two objectives described in the previous section are first fuzzified and, then, dealt with by integrating them into a fuzzy satisfaction objective function Fj through appropriate weighting factors as given below:

1.0 μsj

0

ð7Þ

xmin

xmax xj

Fig. 1. Membership function for maximizing saving.

Max

F j ¼ W 1 :lsj þ W 2  lV j

ð9Þ

The proper weighting factors used are W1 = W2 = 0.50 in which these two objectives are assumed to be equally

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important. The weighting factors can be varied according to the preferences of the operators. 5. Genetic algorithm To solve the optimization problem formulated in Eq. (9), genetic algorithm (GA) is used. Over the last few years, it is becoming popular to solve a wide range of search, optimization and machine learning problems. A GA (multi path search scheme) is an iterative procedure which maintains a constant size population p(t) of candidate solutions. The initial population p(0) can be chosen heuristically or at random [30,31]. The structure of the population p(t + 1) (i.e. for next iteration called generation) are chosen from p(t) by randomized selection procedure that ensures that the expected number of times a structure is chosen is approximately proportional to that structures performance relative to the rest of the population. In order to search other points in a search space, some variation is introduced into the new population by means of genetic operators (crossover and mutation). A GA is an iterative procedure which maintains a constant size population of candidate solutions. The algorithm begins with a randomly selected population of function inputs represented by strings of bits. During each iteration step, called a generation, the structure in the current population is evaluated and on the basis of those evaluations, a new population of candidate solution is formed. That is GA uses the current population of string to create a new population such that the string in the new population are, on average ‘‘better” than those in the current population. Three processes selection, crossover and mutation are used to make the transition from one population generation to the next. Those three steps are repeated to create each new generation and it continues in this fashion until some stopping condition is reached (such as maximum number of generations). The performance of each structure of string is evaluated according to its fitness, which is defined as non-negative figure of merit to be maximized. It is directly associated with the objective function value [30,31] in the optimization. GA proceeds only according to the fitness and not to other information, the properties of the fitness will influence the GAs performance. In the present work, experiments were carried out for different combination of four crossover probabilities (0.6, 0.8, 0.9, and 1.0) and four mutation probabilities (0.0001, 0.001, 0.006 and 0.01). It has been found that, crossover probability = 1.0 and mutation probability 0.006 gives the better performance. 6. Implementation of GA for capacitor optimization problem Here, the purpose of the GA is to determine the capacitor sizes at the candidate locations for each load level. As mentioned in Section 5, in GA, the ‘‘fitness” is defined as non-negative figure of merit to be maximized which is directly associated with the objective function. In the pres-

ent work, fitness function is Fj as defined in Eq. (9). Since GA is multipath search technique, the number of functions evaluation are greater. One of the major tasks to be carried out for each objective function calculation is ‘‘load flow”. In the present work, a load flow algorithm for solving radial distribution networks developed in [32] is used. Complete evaluation procedure is given below: Step 1: Set load level j = 1. Step 2: Form initial binary strings (chromosomes) equal to population size as ½capj;1 ; capj;2 ; . . . ; capj;ncap 

Step 3: (i)

(ii) (iii) (iv)

where capj,i, i = 1, 2 , . . ., ncap, represents the binary substrings (0 and 1) representing the value of shunt capacitors banks at ith candidate location for jth load level. Evaluation of the fitness for each string is as follows: Convert each binary substring (genes) in a string into decimal number. Multiply this decimal number with bank size, this gives the sizes of each capacitor at particular location. Carry out load flow to find out power loss, energy loss and voltage magnitudes at each node. Compute lsj and lVj using Eqs. (6) and (8), respectively, and then compute Fj using Eq. (9). Fitness = Fj.

Step 4: (i) Find out cumulative fitness of each string by adding its fitness to the fitness of the preceding population members. (ii) Sum the fitness of all the population members. (iii) Find the best fitted string and send it to the solution vector. Step 5: Set generation number = 1. Step 6: For i = 1 to i = ‘‘population size  crossover rate”, Do, (i) Select two parents from population (based on roulette wheeling method). (ii) Perform crossover and hence generate two offspring. (iv) Mutate these two offspring based on mutation probability. Step 7: Calculate fitness of each offspring (as in Step 3 and Step 4). Step 8: Combine the old population and new population to create a single population containing best of both. Step 9: Repeat Step 6–Step 8, till the maximum number of generation is reached. Step 10: Increment load level, i.e. j = j + 1;If (j 6 L), go to Step 2. Step 11: Stop.

D. Das / Electrical Power and Energy Systems 30 (2008) 361–367

Note that constraints are imposed on selecting fixed and switched type of capacitors sizes and it is given as Qci;1 ¼ Qci;2 . . . ¼ Q

ci;L

6 Qci;max

61 64 59

7. Numerical results In order to test the proposed method, a 69 node radial distribution system has been taken as an example network [12]. Line data and nominal load data for this system are given in Appendix. The cost constants and conditions are as follows [12]: Energy cost Ke = US $0.06/kWhr. Purchase cost of capacitor = Kc = US $3.0/kVAr. Three load levels and load duration time data for the system is given in Table 1. The power losses, minimum system voltage and total cost of energy loss for bare system (without shunt capacitors) at different load levels are given in Table 2. One bank is 100 kVAr/bank and maximum banks are 20. Substation voltage is 1.0 pu. Any voltage regulators are not included. Sensitivity analysis method [23,27] is used in this study to select the candidate installation locations of the capacitors to reduce the search space. The nodes are ordered according to their sensitivity value oPloss/oQ (i.e. nodes 61, 64, 59, 65, . . .). Top three nodes are selected as candidate locations (i.e. nodes 61, 64 and 59) to reduce the search space and then the amount to be injected in the selected nodes is optimized by GA. Table 3 shows the capacitor placement locations and sizes. Switched capacitors need to be installed at node 61. Thus 8 banks (800 kVAr, switched type) can be installed at node 61. At node 64, 3 banks (300 kVAr, fixed type) and 9 banks (900 kVAr switched type) can be installed. At node 59, 11 banks (1100 kVAr, switched type) can be installed. Table 4 shows the comparison of the results with and without considering capacitor placements. The total cost for the system without any compensation is found to be US $135,905. After compensation, the total cost is US Table 1 Load level and load duration time Load level

0.50 (light)

1.0 (nominal)

1.60 (peak)

Time duration (h)

2000

5260

1500

Table 2 Cost of energy loss and minimum system voltage for bare system Load level

0.50 (light)

1.0 (nominal)

1.60 (peak)

Energy loss cost Minimum system voltage

$6192 0.95668

$70,997 0.90919

$58,716 0.84449

Total energy loss cost = $135,905. Maximum voltage = V1 = Vsubstation = 1.0 pu.

Control setting (kVAr)

ð10Þ

ðswitched typeÞ

0 6 Qci;j 6 Qci;j;max

Table 3 Optimal capacitor placement location and size Optimal location

ðfixed typeÞ

365

1.60

1.0

0.50

800 1200 1100

700 800 100

0.0 300 0.0

Optimal size (kVAr)

800 1200 1100

Table 4 Comparison of the results without and with capacitor placement

Total cost Total losses cost Total capacitor cost

Without capacitors

With capacitors

$135,905 $135,905 0.0

$105,027 $95,727 $9300

Total annual savings = $30,878 Load level 1.6

V 65 min Losses (kW)

0.84449 652.40

0.90014 460.45

Load level 1.0

V 65 min Losses (kW)

0.90919 224.96

0.93693 156.62

Load level 0.50

V 65 min Losses (kW)

0.95668 51.60

0.96220 40.48

Maximum voltage = V1 = Vsubstation = 1.0 pu.

$105,027. From the results, it is seen that the annual saving is US $30,878. After capacitor placement, minimum voltage level of the system has improved and the system losses are reduced in each load level. It is worth mentioning here that total number of generation was set to 100 in each load level and population size was 60. However, at light load level (load level = 0.50), optimal solution was obtained in six generations. At nominal load level (load level = 1.0) optimal solution was obtained in 19 generations and at peak load level (load level = 1.80), optimal solution was obtained in 25 generations. The proposed method has been implemented on Pentium IV computer and CPU time was about 112 s. Attempt was also made to obtain the optimal solution using the objective function given by Eq. (2). Voltage constraint ðV lmin ¼ 0:90Þ was violated at peak load level and value for k at peak load level was selected by trial and error and it was found that a value of k = 9.31  105 works very well and it gives the same results with nearly same CPU time as mentioned in Tables 3 and 4. However, selection of the value of k is very difficult task. In the proposed method, the voltage constraint is translated into fuzzy membership function as given by Eq. (8) which makes the objective function formulation much simpler as given by Eq. (9) and proposed algorithm satisfy the voltage constraint. 8. Conclusion The study has presented a GA based fuzzy multi-objective approach for optimal capacitor placement while improving voltage profile and maximizing net savings in

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D. Das / Electrical Power and Energy Systems 30 (2008) 361–367

a radial distribution system. The objectives considered attempt to maximize the fuzzy satisfaction of maximization of net savings and minimization of nodes voltage deviations. The simulation on a medium size distribution network has proved the feasibility of the proposed approach and the obtained results are quite good and they encourage the implementation of the proposed strategy on a large size distribution network. However, future work needs to be carried out with regard to the following points: (1) installation and maintenance costs should be included for more realistic picture of the total savings for the capacitor placement and (2) the fixed and switched type capacitors should be reassessed following network reconfiguration to improve voltage profile and system losses. Appendix See Table A1.

Table A1 Line and nominal load data Br. no. Send. end Recv. end R (X) (i) IS (i) IR (i)

X (X)

PL (IR) (kW)

QL (IR) (kVAr)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39

0.0012 0.0012 0.0036 0.0294 0.1864 0.1941 0.0470 0.0251 0.2707 0.0619 0.2351 0.3400 0.3400 0.3496 0.0650 0.1238 0.0016 0.1083 0.0690 0.1129 0.0046 0.0526 0.1145 0.2475 0.1021 0.0572 0.0108 0.1565 0.1315 0.0232 0.1160 0.2816 0.5646 0.4873 0.0108 0.1565 0.1230 0.0355 0.0021

0.0 0.0 0.0 0.0 2.6 40.40 75.0 30.0 28.0 145.0 145.0 8.0 8.0 0.0 45.0 60.0 60.0 0.0 1.0 114.0 5.0 0.0 28.0 0.0 14.0 14.0 26.0 26.0 0.0 0.0 0.0 14.0 19.50 6.0 26.0 26.0 0.0 24.0 24.0

0.0 0.0 0.0 0.0 2.2 30.0 54.0 22.0 19.0 104.0 104.0 5.0 5.0 0.0 30.0 35.0 35.0 0.0 0.60 81.0 3.50 0.0 20.0 0.0 10.0 10.0 18.6 18.6 0.0 0.0 0.0 10.0 14.0 4.0 18.55 18.55 0.0 17.0 17.0

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 3 28 29 30 31 32 33 34 3 36 37 38 39

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

0.0005 0.0005 0.0015 0.0251 0.3660 0.3810 0.0922 0.0493 0.8190 0.1872 0.7114 1.0300 1.0440 1.0580 0.1966 0.3744 0.0047 0.3276 0.2106 0.3416 0.0140 0.1591 0.3463 0.7488 0.3089 0.1732 0.0044 0.0640 0.3978 0.0702 0.3510 0.8390 1.7080 1.4740 0.0044 0.0640 0.1053 0.0304 0.0018

Table A1 (continued) Br. no. Send. end Recv. end R (X) (i) IS (i) IR (i)

X (X)

PL (IR) (kW)

QL (IR) (kVAr)

40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68

0.8509 0.3623 0.0478 0.0116 0.1373 0.0012 0.0084 0.2083 0.7091 0.2011 0.0473 0.114 0.0886 0.1034 0.1447 0.1433 0.5337 0.2630 0.1006 0.1172 0.2585 0.0496 0.0738 0.3619 0.5302 0.0611 0.0014 0.2444 0.0016

1.20 0.0 6.0 0.0 39.22 39.22 0.0 79.0 384.70 384.70 40.50 3.60 4.35 26.40 24.0 0.0 0.0 0.0 100.0 0.0 1244.0 32.0 0.0 227.0 59.0 18.0 18.0 28.0 28.0

1.0 0.0 4.30 0.0 26.30 26.30 0.0 56.40 274.50 274.50 28.30 2.70 3.50 19.0 17.20 0.0 0.0 0.0 72.0 0.0 888.0 23.0 0.0 162.0 42.0 13.0 13.0 20.0 20.0

40 41 42 43 44 45 4 47 48 49 8 51 9 53 54 55 56 57 58 59 60 61 62 63 64 11 66 12 68

41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69

0.7283 0.3100 0.0410 0.0092 0.1089 0.0009 0.0034 0.0851 0.2898 0.0822 0.0928 0.3319 0.1740 0.2030 0.2842 0.2813 1.5900 0.7837 0.3042 0.3861 0.5075 0.0974 0.1450 0.7105 1.0410 0.2012 0.0047 0.7394 0.0047

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