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An efficient hybrid evolutionary algorithm based on PSO and HBMO algorithms


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Energy Conversion and Management xxx (2009) xxx–xxx

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Energy Conversion and Management
journal homepage: www.elsevier.com/locate/enconman

An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration
Taher Niknam
Electronic and Electrical Engineering Department, Shiraz University of Technology, Shiraz, Iran

a r t i c l e

i n f o

a b s t r a c t
This paper introduces a robust searching hybrid evolutionary algorithm to solve the multi-objective Distribution Feeder Recon?guration (DFR). The main objective of the DFR is to minimize the real power loss, deviation of the nodes’ voltage, the number of switching operations, and balance the loads on the feeders. Because of the fact that the objectives are different and no commensurable, it is dif?cult to solve the problem by conventional approaches that may optimize a single objective. This paper presents a new approach based on norm3 for the DFR problem. In the proposed method, the objective functions are considered as a vector and the aim is to maximize the distance (norm2) between the objective function vector and the worst objective function vector while the constraints are met. Since the proposed DFR is a multi objective and non-differentiable optimization problem, a new hybrid evolutionary algorithm (EA) based on the combination of the Honey Bee Mating Optimization (HBMO) and the Discrete Particle Swarm Optimization (DPSO), called DPSO–HBMO, is implied to solve it. The results of the proposed recon?guration method are compared with the solutions obtained by other approaches, the original DPSO and HBMO over different distribution test systems. ? 2009 Elsevier Ltd. All rights reserved.

Article history: Received 25 August 2008 Accepted 21 March 2009 Available online xxxx Keywords: Honey Bee Mating Optimization (HBMO) Distribution Feeder Recon?guration (DFR) Discrete Particle Swarm Optimization (DPSO)

1. Introduction Utilities have constantly been looking for new technologies that may enhance power delivery performance. One of several important issues is the control of power loss. The recon?guration of a distribution network is the process that alters feeder topological structure by changing the open/close status of sectionalizers and interrupters in a system. Under normal operational conditions, the objectives avoid excessive transformer load, conductor overheating, and minimize abnormal voltages and at the same time minimize the active power losses of the system. Because there are many candidate switching combinations in a distribution system, network recon?guration is a complicated combinatorial, non-differentiable constrained optimization problem [1–24]. In recent years, many researchers have investigated losses minimization in the area of network recon?guration of distribution systems. One of the ?rst papers on this topic was presented by Merlin and Back [2]. The discrete branch and bound optimization technique was taken from a meshed distribution network. However, its application to real systems is not very easy due to the required signi?cant computer effort. Since then, many methods have been proposed. For instance, Civanlar et al. conducted the early research on feeder recon?guration for loss reduction [3]. Baran and Wu modeled the problem of loss reduction and load balancing as

E-mail address: taher_nik@yahoo.com 0196-8904/$ - see front matter ? 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2009.03.029

an integer programming problem [4]. Nara et al. and Prasad and Ranjan used a genetic algorithm to look for the minimum loss con?guration [5,12]. Shirmohammadi and Hong presented the use of the power ?ow method based on a heuristic algorithm to determine the minimum loss con?guration for radial distribution networks [6,13]. Chiang and Rene proposed a solution procedure which used simulated annealing to search for an acceptable non inferior solution [7,8]. Miguel and Hernan proposed an economic operation model to solve distribution network con?guration [9]. Delbem et al. proposed a tree encoding and two genetic operators to improve the EA performance of network recon?guration problems [10]. Das proposed a fuzzy multi-objective approach to solve the network recon?guration problem [11]. Niknam et al. presented an ef?cient hybrid algorithm for multi-objective distribution feeder recon?guration based on Honey Bee Mating Optimization (HBMO) and fuzzy multi-objective approach [14]. Olamaie et al. proposed a cost based on compensation methodology for distribution feeder recon?guration considering distributed generators [15– 17]. Since the objective functions of the distribution feeder recon?guration are not the same and commensurable, it is dif?cult to solve the problem of this sort by conventional approaches that are utilized to optimize single objective problems. Therefore, in this paper a new formulation based on norm2 method is proposed for the multi-objective distribution feeder recon?guration. In the proposed approach, the objective functions are employed to minimize the real power loss, the number of switching operations, and

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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voltage deviation of the buses and to balance the loads on the feeders. In this approach, the objective functions have been considered as a vector and the aim is to maximize the distance between the present objective function vector and the worst objective function vector. The former vector is calculated based on the results obtained before recon?guration. The control variables are the status of the tie and sectionalizing switches. Since the con?guration of the distribution systems should remain radial, while numerous switches are being used, control variables are de?ned so that when a tie switch is closed, one sectionalizing switch that forms a loop must be opened. As the proposed DFR is a nonlinear and non-differentiable optimization problem, to solve such a problem, classical methods, e.g. linear programming, mixed-integer programming, quadratic programming, etc., can be used. However, in some cases, the mentioned methods fail to provide the global minima and only reach local minima. Moreover, some classical methods cannot handle the integer problems [1,14] and [15]. The two aforementioned shortcomings can be overcome by utilizing an evolutionary method. The reason is its independence from the type of the objective function and constraints. In this article, a new hybrid evolutionary optimization algorithm based on combining of the HBMO and DPSO algorithms, called DPSO–HBMO, is employed to solve DFR problem. The main contributions of this paper are as follows: (i) present a new approach for multi-objective DFR, (ii) present a hybrid evolutionary optimization based on the combination of PSO and HBMO algorithms to solve the DFR problem. The paper has been organized as follows: In Section 2, the proposed DFR has been formulated. In Sections 3 and 4, the basic principles of the DPSO and HBMO algorithms are presented, respectively. In Section 5, the application of DPSO–HBMO algorithm in the DFR is shown. In Section 6, the feasibility of the DPSO–HBMO algorithm and the proposed DFR is demonstrated and compared with the results obtained by other works and other evolutionary methods such as HBMO and DPSO, over different distribution test systems. Finally, Section 7 includes summary and the conclusion.

1 = close). Swi is the sectionalizing switch number that forms a loop with Tiei. Ntie is the number of the tie switches. 2.1.2. Minimizing the deviation of the bus voltage Bus voltage is one the most signi?cant security and service quality indices, which can be described as follows:

f2 ?X? ? max jV i ? V rate j;
i

i ? 1; 2; 3; . . . ; N bus

?2?

where Nbus is total number of the buses. Vi and Vrate are the real and rated voltages on the ith bus, respectively. 2.1.3. Minimizing switching operation Minimization of the number of switching operations can be shown as the following model:

f3 ?X? ?

Ns X i?1

jSi ? So;i j

?3?

where Si and So,i are the new and original states of the switch i, respectively. Ns is the number of the switches. 2.1.4. Feeder load balancing Load balancing is one of the major objectives of the feeder recon?guration. An effective strategy to increase the loading margin of heavily loaded feeders is to transfer part of their loads to lightly loaded feeders. Feeder load balancing can be described as follows:

f4 ?X? ? ? min jIi;rate ? Ii j; i ? 1; 2; 3; . . . ; Nbr
i

?4?

where Ii and Ii,rate are the actual loading current and rated current of the ith branch, respectively. 2.2. Formulation of the distribution feeder recon?guration based on norm2 Formulation of the multi-objective distribution feeder recon?guration including the mentioned objective functions can be written as below:

2. Distribution feeder recon?guration problem The related formulation of the proposed distribution feeder recon?guration is developed as below. 2.1. Objective function With the proposed DFR, the objective function consists of four terms: (i) total active power losses; (ii) load balancing among the feeders; (iii) the number of switching operations; and (iv) deviation of the bus voltage. Objective functions can be described as below: 2.1.1. Minimization of the power losses Minimizing the real power loss of the feeders is selected as the ?rst objective function for the feeder recon?guration because reduction of the real power loss of the distribution feeders is of great importance in feeder recon?guration. The minimization of the total real power losses arising from feeders can be calculated as follows:

maxJ?X? ? kf ?X? ? f0 k2 q?????????????????????????????????????????????????????????????????????????????????????????????????????????????????????????? ? ?f1 ?X? ? fo1 ?2 ? ?f2 ?X? ? fo2 ?2 ? ?f3 ?X? ? fo3 ?2 ? ?f4 ?X? ? fo4 ?2 f01 f1 ?X? 6 f ?X? 7 6f 7 62 7 6 02 7 f ?X? ? 6 7; f ? 6 7 4 f3 ?X? 5 0 4 f03 5 f4 ?X? f04
where f01 and f02 are the real power loss and the maximum voltage deviation before recon?guration, f03 equals with the worst switching operations, which equals to 2?Ntie, f04 equals with the worst load balancing before recon?guration, and f(X) and f0 are the objective function vector and the worst objective function vector, respectively. 2.3. Constraints The constraints can be listed as follows:  Distribution line limits

2

3

2

3

?5?

f1 ?X? ?

Nbr X i?1

Ri ? jIi j2

?1?

X ? ?Tie1 ; Tie2 ; . . . ; TieNtie ; Sw1 ; Sw2 ; . . . ; SwNtie ?
where Ri and Ii are resistance and actual current of the ith branch, respectively. Nbr is the number of the branches. X is the control variables vector. Tiei is the state of the ith tie switch (0 = open and

jPLine j < PLine ij ij;max
jP Line j and PLine ij ij;max

?6?

where are the absolute power ?owing over the distribution lines and the maximum power transmitted between the nodes i and j, respectively.

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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 Distribution power ?ow equations

Pi ? Qi ?

Nbus X i?1

V i V j Y ij cos?hij ? di ? dj ? ?7? V i V j Y ij sin?hij ? deltai ? dj ?

N bus X i?1

where Pi and Qi are the net injected active and reactive powers at the ith bus. Vi and di are the amplitude and angle of the voltage at the ith bus, respectively. And Yij and h ij are the amplitude and angle of the branch admittance between the ith and jth buses.  Objective function limit

fi ?X?

f0i

i ? 1; 2; 3; 4

?8?

with the results calculated for each agent of particles, and Gbest is calculated based on all previous iterations. Then, in the next iteration, two partial probability values (DVi) are added on or subtracted from the previous probability of each particle. Therefore, the values DV i;1 ? Pbesti ? X t and DV i;2 ? Gbest ? X t in Eq. (10) i i are small changes of probability in each iteration that can be ?1, 0, or 1. So sigmoid transformation only transforms V t from the i interval [?1, + 1] to [0, 1]. It is clear that by adding or subtracting the previous experiences to, the previous experiences and histories are tracked and ?nally based on the V t index, where X t becomes 0 i i or 1. To apply the DPSO algorithm in DFR, the following steps should be taken [15]: Step 1: The input data should be de?ned. Step 2: The constraint optimization problem should be transferred to an unconstraint one. Step 3: The initial population and initial velocity for each particle should be generated randomly. Step 4: The objective function should be evaluated for each individual. Step 5: The individual that has the maximum objective function should be selected as the global position (Gbest). Step 6: The ith individual is selected. Step 7: The best local position (Pbest) should be selected for the ith individual. Step 8: The modi?ed velocity for the ith individual should be calculated in accordance with the local and global positions and Eq. (10). Step 9: The modi?ed position for the ith individual should be calculated based on Eq. (11). Step 10: If all individuals are selected, go to the next step, otherwise i = i+1 and go to Step 6. Step 11: If the current iteration number reaches the predetermined maximum iteration number, the search procedure is stopped, otherwise go to Step 4. The last Gbest can be regarded as a solution for the problem. 4. Original HBMO algorithm The honey bee is a social insect that can only survive as a member of a community, or colony. The colony inhabits an enclosed cavity. The honey bee community structurally consists of three different forms: the queen (reproductive female), the drone (male), and the worker (non reproductive female). These castes are associated with different functions in the colony; each caste possesses its own special instincts geared to the needs of the colony. The behavior of honey-bees shows many features like cooperation and communication, so honey-bees have aroused great interests in modeling intelligent behavior these years. Marriage in Honey-Bees Optimization (MBO) is a kind of swarm-intelligence method. And such swarm intelligence has some successful applications. Ant colony is an example and the search algorithm is inspired by its behavior. Mating behavior of honey-bees is also considered as a typical swarm-based optimization approach. The behavior of honey-bees is related to the product of their genetic potentiality, ecological and physiological environments, the social conditions of the colony, and various prior and ongoing interactions among these three [14,32–35]. The HBMO algorithm combines a number of different procedures. Each of them corresponds to a different phase of the mating process of the honey bee presented in Fig. 1. A drone mates with a queen probabilistically using an annealing function as follows [14,32,34]:

 Radial structure of the network

3. Original DPSO algorithm PSO is an evolutionary computational algorithm inspired by a natural system. On a given iteration, a set of solutions called ‘‘particles” move around the search space from one iteration to another according to rules that depend on three factors: inertia (the particles tend to move in the direction they have previously moved), memory (the particles tend to move in the direction of the best solution found so far in their trajectory), and cooperation (the particles tend to move in the direction of the global best solution) [26– 31]. The movement rule of each particle can be expressed by

V it?1 ? x ? V t ? c1 ? rand1 ??? ? ?Pbesti ? X t ? i i ? c2 ? rand2 ??? ? ?Gbest ? X t ? i X t?1 i ? Xt i ? V it?1 ?9?

x

?t?1?

? xmax ?

xmax ? xmin
t max

?t

In these equations, i = 1, 2, . . . , NSwarm is the index of each particle, t is iteration number, rand1(.) and rand2(.) are random numbers between 0 and 1. Pbesti is the best previous experience of the ith particle that is recorded. Gbest is the best particle among the entire population. NSwarm is number of the swarms. Constants c1 and c2 are the weighting factors of the stochastic acceleration terms, which pull each particle towards Pbesti and Gbest positions. tmax is the maximum number of iterations. xmax and xmin are the maximum and minimum of the inertia weights, respectively. In the discrete binary version, PSO is modi?ed as

V it?1 ? V t ? c1 ? rand?:? ? DV i;1 ? c2 ? rand?:? ? DV i;2 i

?10?

DV i;1 ? DV i;2 ?

Pbesti ? X t i Gbest ? X t i

While parameters Pbesti, Gbest and X t can take any real value in Eq. i (9), these parameters are integers in {0, 1} in Eq. (10). V t is limited to i the interval [0, 1]as it is a probability not velocity. A logical transformation S?V t ? can be used to accomplish this i last modi?cation. The resulting change in the position is then de?ned by the following rule:

( Xt ? i

1; 0;

rand??? 6 S?V t ? i otherwise

?11?

S?V t ? ? i

1 1 ? exp??V t ? i

First, each particle of a swarm is randomly initiated in state 0 or 1 and the objective function value is calculated according to this state arrangement. For each iteration, Pbesti is found in accordance

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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List of Drones Queen Select a Drone at Random Selected Drone

Mate

Brood Brood Apply Local Search

Replace the queen if the best brood is better than the queen

Selected Brood

Step 1: De?ne the input data. In this step, the input data including the network con?guration, line impedance and status of switches, the speed of queen at the start of a mating ?ight (Smax), the speed of queen at the end of a mating ?ight (Smin), the speed reduction schema (a), the number of iteration, the number of workers (NWorker), the number of drones (NDreone), the size of the queen’s spermatheca (NSperm) and the number of broods (NBrood) are de?ned. Step 2: Transfer the constraint optimization problem to an unconstraint one. Step 3: Generate the initial population. Step 4: Calculate the objective function value by using results of the distribution load ?ow. Step 5: Sort the initial population based on the objective function values. Step 6: Select the queen. The individual that has the maximum objective function should be considered as the queen. Step 7: Generate the queen speed. The queen speed is randomly generated as:

Selected Best Solution
Fig. 1. The HBMO algorithm.

Squeen ? rand??? ? ?Smax ? Smin ? ? Smin
where rand(?) is a random function generator.

?14?

Prob?D? ? exp??D?f ?=S?t??

?12?

where Prob(D) is the probability of adding the sperm of drone D to the spermatheca of the queen, D(f) is the absolute difference between the ?tness of D and the ?tness of the queen and S(t) is the speed of the queen at time t. The probability of mating is high when the queen is with the high speed level, or when the ?tness of the drone is as good as the queen’s. After each transition in space, the queen’s speed decreases according to the following equations:

Step 8: Select the population of the drones. The population of drones is selected from the sorted initial population as:

2

6 7 Drone Population ? 4 D2 5 DNDrone
where Di is the ith drone.

D1

3 ?15?

Di ? ?Tie1 ; Tie2 ; . . . ; TieNtie ; Sw1 ; Sw2 ; . . . ; SwNtie ?; i ? 1; 2; . . . ; N Drone

S?t ? 1? ? a ? S?t?

?13?

where a is a factor 2 (0, 1) and is the amount of speed and energy reduction after each transition and each step. Initially, the speed of the queen is generated at random. A number of mating ?ights are realized. At the start of a mating ?ight drones are generated randomly and the queen selects a drone using the probabilistic rule in Eq. (12). If the mating is successful (i.e. the drone passes the probabilistic decision rule), the drone’s sperm is stored in the queen’s spermatheca. By using the crossover of the drone’s and the queen’s genotypes, a new brood (trial solution) is generated, which can be improved later by employing workers to conduct local search. One of the major differences of the HBMO algorithm from the classic evolutionary algorithms is that since the queen stores a number of different drone’s sperm in her spermatheca, she can use parts of the genotype of different drones to create a new solution which gives the possibility to have ?ttest broods more. In real life, the role of the workers is restricted to brood care and for this reason the workers are not separate members of the population and they are used as local search procedures in order to improve the broods produced by the mating ?ight of the queen. Each of the workers has different capabilities and the choice of two different workers may produce different solutions. This is realized with the use of a number of single local search heuristics (Nworker1) and combinations of them (Nworker2). Thus, the sum of these two numbers (NWorker = Nworker1 + Nworker2) gives the number of workers. Each of the broods is randomly chosen to be feed by worker, which is also randomly selected. If the new brood is better than the current queen, it takes the place of the queen. If the brood fails to replace the queen, then in the next mating ?ight of the queen this brood will be one of the drones. To apply the HBMO algorithm in the DFR, the following steps have to be taken [14,34]:

Step 9: Generate the queen’s spermatheca matrix (Mating ?ight). At the start of the mating ?ight, the queen ?ies with her maximum speed. A drone is randomly selected from the population of drones. The mating probability is calculated based on the objective function values of the queen and the selected drone. A number between 0 and 1 is randomly generated and compared with the calculated probability. If it is less than the calculated probability, the drone’s sperm is sorted in the queen’s spermatheca and the queen speed is decreased. Otherwise, the queen speed is decreased and another drone from the population of drones is selected until the speed of the queen reaches to her minimum speed or the queen’s spermatheca is full.

2

7 6 6 Sp2 7 7 6 Spermactheca matrix ? 6 7 7 6 5 4 SpNSperm Spi ? ?sj ?1?n ? ?Tie1 ; Tie2 ; :::; TieNtie ; Sw1 ; Sw2 ; . . . ; SwNtie ? i ? 1; 2; 3; . . . ; NSperm

Sp1

3 ?16?

where Spi is the ith individual in the queen’s spermatheca. Step 10: Breeding process. In this step, a population of broods is generated based on mating between the queen and the drones stored in the queen’s spermatheca. The ith individual is generated as:

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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X best ? x1 x2 ? ? ? xn ? 1 best2 best n ? best Spi ? si si ? ? ? si

?

?

?17? j ? 1; 2; 3; . . . ; NBrood

Broodj ? round?X best ? b ? ?X best ? Spi ??;

Step 13: Check the termination criteria. If the termination criteria satis?ed ?nish the algorithm, else discard all previous trial solutions and go to step 3 until convergence criteria met.

where b is a random number between 0 and 1. Broodj is the jth brood. Step 11: Feeding selected broods and queen with the royal jelly by workers. Improve the newly generated set of solutions employing different heuristic functions and mutation operators according to their ?tness values. Step 12: Calculate the objective function value for the new generated solutions.

5. Application of DPSO–HBMO to distribution feeder recon?guration Although PSO has the advantages of good convergent property and is effective on solving optimization, after some generations the population diversity would be greatly reduced and the PSO algorithm might lead to a premature convergence to a local optimum. The combination of HBMO and PSO enables the algorithm

Define the DPSO & HBMO parameters, network configuration, line impedance and state of switches Generate N individuals randomly Transfer the constraint objective function to unconstraint one Calculate the objective function for each individual by using the results of distribution load flow and sort the initial population according to their fitness and select the best one as the Queen and Gbest

B
Select the N best Particles Sort N+K particles according to their fitness Consider K broods generated by HBMO and add them to the N new particles generated by DPSO No The best value among Queen and Gbest is considered as Queen in HBMO and Gbest in DPSO

A
Calculate the objective function for new population

Generate N new solutions randomly Is convergence condition satisfied? Yes No

B
Calculate the objective function for each individual Select the ith individual

Stop and print the results

A

Generate the speed Queen and select the population of drones

Generate the queen's spermatheca matrix (Mating flight) by using simulated annealing (eq(16)) Select the local position for the i individual Use crossover operator to generate new set of solutions (Breeding process) Feeding selected broods and queen with the royal jelly by workers Calculate the next position for each individual based on eq. (11) Calculate the objective function value for the new generated solutions
th

Calculate the modified velocity for each individual based on eq. (10)

Check the new position with its limits i=i+1 Are all individuals Yes No Yes Substitute the best solution
Is the new best solution better than the previous one?

No Keep on the previous best solution

B

A

Fig. 2. Flowchart of proposed hybrid evolutionary DPSO–HBMO algorithm.

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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incorporating constraints:

penalty factors for

any

value

violating the

F?X? ? J?X? ? k1

! ! Neq Nueq X X 2 2 ?hj ?X? ? ? k2 ?Max?0; ?g j ?X???
j?1 j?1

?18?

J(X) is the objective function values of the DFR problem. Neq and Nueq are the number of equality and inequality constraints of the DFR problem, respectively. hi(X) and gi(X) are the equal and unequal constraints, respectively. k1 and k2 are penalty factors. X is the control variable vector including status of tie and sectionalizing switches. As can be seen, HBMO and DPSO both work with the same initial population. The hybrid approach takes N individuals that are randomly generated, which can mathematically be modeled as follows:
Fig. 3. A single line diagram of Baran and Wu distribution test system.

to maintain the population diversity and prevent leading to misleading local optima. This section discusses the infrastructure and rationale of the hybrid algorithm and also presents its application in the DFR problem. Fig. 2 depicts the schematic representation of the proposed hybrid DPSO–HBMO to solve the DFR problem. At ?st to apply the hybrid evolutionary algorithm in the DFR problem, it is required to be transformed into an unconstrained one by constructing an augmented objective function

3 X1 6X 7 6 27 Population ? 6 7 4...5 XN X i ? ?xi ?1?n ? ?Tie1 ; Tie2 ; . . . ; TieNtie ; Sw1 ; Sw2 ; . . . ; SwNtie ?; i ? 1; 2; 3; . . . ; N n ? 2 ? Ntie

2

?19?

where xj is the jth control variable. Xi is position the ith individual. n is the number of control variables. In the above equations, Tiei is

Table 1 Results for different methods. Method Optimum [14] The proposed algorithm Goswami [25] MeDemott [24] Shirmohammadi [6] Vanderson Gomes [23] Niknam [14] Power losses (kW) 139.53 139.53 139.53 139.53 140.26 139.53 139.53 Minimum voltage (pu) 0.938 0.938 0.938 0.938 0.9378 0.938 0.938 Saving (%) 31.14 31.14 31.14 31.14 30.78 31.14 31.14 CPU time (s) 647.03 6 0.87 1.99 0.14 1.66 8 Open switches s7, s7, s7, s7, s7, s7, s7, s9, s14, s32, and s37 s9, s14, s32, and s37 s9, s14, s32, and s37 s9, s14, s32, and s37 s10, s14, s32, and s37 s9, s14, s32, and s37 s9, s14, s32, and s37

Table 2 Comparison of average and standard deviation for 100 trials. Method DPSO–HBMO HBMO DPSO Average of objective function value 99648.0810 98469.26 98006.81 Standard deviation 0 1194.888 1717.162 Worst solution 99648.0810 96443.6910 95694.146 Best solution 99648.0810 99648.0810 99648.0810 CPU time (s) $8 $13 $10 No. of global solution 100 62 50

Table 3 Results for different methods. Method Optimum [14] The proposed algorithm Goswami [25] MeDemott [24] Shirmohammadi [6] Vanderson Gomes [23] Niknam [14] Power losses (kW) 139.53 139.53 143.69 139.53 140.26 139.53 139.53 Minimum voltage (pu) 0.938 0.938 0.9397 0.938 0.9378 0.938 0.938 Saving (%) 31.14 31.14 29.08 31.14 30.78 31.14 31.14 CPU time (s) 647.03 8 0.65 1.99 0.14 1.66 9 Open switches s7, s7, s7, s7, s7, s7, s7, s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s10, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37

Table 4 Comparison of average and standard deviation for 100 trials. Method DPSO–HBMO HBMO DPSO Average of objective function value 99648.0810 98902.3115 98284.6551 Standard deviation 0 1175.271964 1912.61832 Worst solution 99648.0810 96443.6910 95694.1460 Best solution 99648.0810 99648.0810 99648.0810 CPU time (s) $8 $13 $10 No. of global solution 100 67 49

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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T. Niknam / Energy Conversion and Management xxx (2009) xxx–xxx Table 6 Feeder current before and after recon?guration. Method IF1 IF2 IF3 IF4 Before recon?guration (A) 99.9 109.01 162.3 148.86 After recon?guration (A) 117.01 131.57 133.93 135.13 Capacity of feeder (A) 270 270 270 270 7

for implementation of the DPSO–HBMO algorithm are Smax, Smin, a, NWorker, NDreone, NSperm, NBrood, c1, c2, xmin and xmax. In this paper, the best values for the aforementioned parameters are Smax = 1, Smin = 02, a = 0.93, NWorker = 10, NDreone = 16, NSperm = 15, NBrood = 15, c1 = c2 = 2, xmin = 0.4 and xmax = 0.9 determined by 100 runs of the algorithm [14,15]. Number of the initial population is 16. 6.1. Case 1 – Baran and Wu test system [4] The Baran and Wu distribution test system is a hypothetical 12.66 kV system with a two-feeder substation, 32 buses, and 5 looping branches. The number of ties and sectionalizing switches are 5 and 32, respectively. The system data is given in [4] and the single line diagram of this system is shown in Fig. 3. The total load conditions are 5058.25 kW and 2547.32 kVar. The normally open switches, s33, s34, s35, s36 and s37, are illustrated by doted lines. The normally closed switches, s1 to s32, are represented by solid lines. Before recon?guration, the initial losses and minimum per unit voltage are 202.67 kW and 0.913, respectively. Table 1 shows a comparison between the proposed algorithm and some other algorithms. It is seen that the proposed algorithm leads to the global optimum con?guration. Table 2 presents a comparison among the proposed DPSO– HBMO, the results of HBMO [14,34] and the original DPSO [15] for 100 random trials. To demonstrate that the proposed algorithm does not depend on the initial switching con?guration, the initial con?guration has been changed, closing the normally open switches s33 and s37 and opening the normally closed s3 and s6. The initial losses and minimum voltage in per unit are 208.15 kW and 0.9212, respectively. The simulation results are shown in Table 3, where it can be seen that the proposed algorithm gives the global optimum con?guration, which means that the proposed approach does not depend on the initial switching con?guration. A comparison between the DPSO–HBMO and the original DPSO [15], and HBMO [14,34] for 100 random trials is shown in Table 4. The simulation results in Tables 2 and 4 show that the execution time of the proposed method is signi?cantly short in respect with DPSO and HBMO and provides a general idea that the method can be implemented without any restriction in practical networks, while it is of a good precision. In other words, this method not only reaches a better optimal solution compared with

Fig. 4. A single line diagram of 11 kV distribution test system.

randomly generated and is 0 or 1. It should be noted that the value of Swi is randomly generated so that the distribution system remains radial. On the other hand, when the value of Tiei is equal to zero, the value of Swi is equal to zero, too. And, when the value of Tiei is equal to 1, that of the sectionalizing switch, which forms a loop with Tiei is randomly selected. These individuals are considered as drones in the case of HBMO and as particles in the case of DPSO. The augmented objective function (Eq. (18)) is evaluated for each individual by using the result of the distribution load ?ow. The N individuals are sorted by ?tness, and the best individual is selected as the queen (in the HBMO algorithm) and Gbest (in the DPSO algorithm). By using the DPSO and HBMO algorithms as shown in Fig. 2 at each iteration, the new particles, broods and queen are generated. The best solution is considered as the queen and Gbest for the next iteration. If the convergence condition is not satis?ed, for HBMO algorithm generate the new drones are randomly generated and for the DPSO algorithm the broods and new particles are sorted by ?tness and the number of N, which are the best, are selected as the new particles. Otherwise the process is stopped and printed the best solution. 6. Simulation and results In this section, the DPSO–HBMO algorithm is employed to solve the DFR for two distribution test feeders. The parameters required

Table 5 Results obtained by different methods. Method The proposed algorithm Das [11] Niknam [14] Power losses (kW) 205.32 205.32 205.32 Minimum Voltage (pu) 0.9268 0.9268 0.9268 Saving (%) 9.76 9.76 9.76 CPU time (s) 8 3 10 Open sectionalizing switches s26–27, s14–15, s37–38, s49–50, s44–45, and s65–66 s26–27, s14–15, s37–38, s49–50, s44–45, and s65–66 s26–27, s14–15, s37–38, s49–50, s44–45, and s65–66 Closed tie switches tie 1, tie 3, tie 4, tie 7, tie 8, and tie 9 tie 1, tie 3, tie 4, tie 7, tie 8, and tie 9 tie 1, tie 3, tie 4, tie 7, tie 8, and tie 9

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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8 T. Niknam / Energy Conversion and Management xxx (2009) xxx–xxx

Table 7 Comparison of average and standard deviation for 100 trials. Method DPSO–HBMO HBMO DPSO Average of objective function value 149472.1 147773.3 145676.3 Standard deviation 0 2178.864 2861.175 Worst solution 149472.1 144630.5 143520.9 Best solution 149472.1 149472.1 149472.1 CPU time (s) $12 $18 $15 No. of global solution 100 68 52

other methods but also has zero standard deviation for different trials. 6.2. Case 2 – A 70-Bus 11 kV radial distribution system [11] A single line diagram of the 11 kV radial distribution system having two substations, four feeders, 70 nodes, and 78 branches (including tie branches) is shown in Fig. 4. The system data is given in [11]. Before recon?guration, the initial losses and the minimum voltage in per unit are 227.53 kW and 0.9052 kW, respectively. The simulation results are presented in Table 5. As shown in the Table, only 6 out of 11 tie switches has changed. Also, after recon?guration the currents of the feeders are more balanced than before recon?guration. Table 6 shows the feeders’ currents before and after recon?guration. It is seen that the current of each feeder is more balanced after recon?guration. A comparison between the DPSO–HBMO, the original DPSO methods [15] and HBMO [14,34] for 100 random trials is shown in Table 7. The simulation results show that the DPSO–HBMO method has a short run time and zero standard deviation for different trials. On the other hand, the best, worst and average values of objective function are the same for 100 trials. 7. Conclusions This paper proposes a hybrid evolutionary algorithm based on the combination of DPSO and HBMO, called DPSO–HBMO, to optimally recon?gure radial distribution system. This evolutionary algorithm is very suitable for solving large-scale integer optimization problems such as the optimal feeder recon?guration problem; also it does not need complex mathematical programming. It has been shown that this combination leads to better results in contrast with the original ones. In addition to what mentioned before, DPSO–HBMO reaches a much better optimal solution in comparison with the others and has zero standard deviation for different trails. Also, the paper presents a new formulation based on norm2 for multi-objective DFR. The simulation results for several tests have shown that global or close to global optimum solutions for the system losses, the voltage deviation and load balancing were attained. Also, the proposed method has minimized the number of switching operations. The introduced method takes the advantage of being independent on the initial status of network switches. References
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Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029

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T. Niknam / Energy Conversion and Management xxx (2009) xxx–xxx [31] Niknam T, Amiri B, Olamaie J, Are? A. An ef?cient hybrid evolutionary optimization algorithm based on PSO and SA for clustering. J Zhejiang Univ Sci A 2009;10(4):512–9. [32] Fathian M, Amiri B, Maroosi A. Application of honey-bee mating optimization algorithm on clustering. Appl Math Comput 2007;190(2):1502–13. [33] Fathian M, Amiri B. A honey-bee mating approach on clustering. Int J Adv Manufact Technol 2007. doi:10.1007/s00170-007-1132-7. 9

[34] Niknam T. Application of honey bee mating optimization on distribution state estimation including distributed generators. J Zhejiang Univ Sci A 2008;9(12):1753–64. [35] Afshar A, Bozorg Haddad O, Mari?o MA, Adams BJ. Honey-bee mating optimization (HBMO) algorithm for optimal reservoir operation. J Franklin Inst 2007;344(5):452–62.

Please cite this article in press as: Niknam T. An ef?cient hybrid evolutionary algorithm based on PSO and HBMO algorithms for multi-objective Distribution Feeder Recon?guration. Energy Convers Manage (2009), doi:10.1016/j.enconman.2009.03.029


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