A new hybrid evolutionary algorithm based on new fuzzy adaptive PSO and NM algorithms for Distribution Feeder Reconfiguration

Energy Conversion and Management 54 (2012) 7–16

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

A new hybrid evolutionary algorithm based on new fuzzy adaptive PSO and NM algorithms for Distribution Feeder Reconfiguration Taher Niknam a, Ehsan Azadfarsani b, Masoud Jabbari c,⇑ a

Department of Electrical and Electronics Engineering, Shiraz University of Technology, Shiraz, Iran Young Researchers club, Science and Research Branch, Islamic Azad University, Tehran, Iran c Department of Electrical Engineering, Marvdasht Branch, Islamic Azad University, Marvdasht, Iran b

a r t i c l e

i n f o

Article history: Received 6 October 2009 Received in revised form 18 September 2011 Accepted 19 September 2011 Available online 4 November 2011 Keywords: New Fuzzy Adaptive Particle Swarm Optimization (NFAPSO) Fuzzy Adaptive Discrete Particle Swarm Optimization (FADPSO) Fuzzy Adaptive Binary Particle Swarm Optimization (FABPSO) Nelder–Mead (NM) Distribution Feeder Reconfiguration (DFR)

a b s t r a c t Network reconfiguration for loss reduction in distribution system is a very important way to save the electrical energy. This paper proposes a new hybrid evolutionary algorithm to solve the Distribution Feeder Reconfiguration problem (DFR). The algorithm is based on combination of a New Fuzzy Adaptive Particle Swarm Optimization (NFAPSO) and Nelder–Mead simplex search method (NM) called NFAPSO–NM. In the proposed algorithm, a new fuzzy adaptive particle swarm optimization includes two parts. The first part is Fuzzy Adaptive Binary Particle Swarm Optimization (FABPSO) that determines the status of tie switches (open or close) and second part is Fuzzy Adaptive Discrete Particle Swarm Optimization (FADPSO) that determines the sectionalizing switch number. In other side, due to the results of binary PSO(BPSO) and discrete PSO(DPSO) algorithms highly depends on the values of their parameters such as the inertia weight and learning factors, a fuzzy system is employed to adaptively adjust the parameters during the search process. Moreover, the Nelder–Mead simplex search method is combined with the NFAPSO algorithm to improve its performance. Finally, the proposed algorithm is tested on two distribution test feeders. The results of simulation show that the proposed method is very powerful and guarantees to obtain the global optimization. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction In the radially distribution system, the configuration may be varied to obtain a new network structure to reduce power loss, increase system security and enhance power quality. This change (reconfiguration) is performed by opening sectionalizing (normally close) and closing tie (normally open) switches of network so that the radiallity of network is maintained and all of the loads are energized. The discrete nature of the switch values and radiallity constraint prevent the use of classical optimization techniques to solve the DFR problem. Therefore, most of the algorithms in the literature are based on heuristic search techniques by using either analytical or knowledge-based engines. In recent years, considerable researches have been conducted for loss minimization in the DFR. Kim et al. [1] proposed a neural network-based method to identify network configurations corresponding to different load levels. Taylor and Lubkeman [2] presented an expert system using heuristic rules to shrink the search space. Hsiao and Chen [3] proposed the problem as a multiobjective programming that the objectives are considering power

⇑ Corresponding author. E-mail addresses: [email protected], [email protected] (M. Jabbari). 0196-8904/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.enconman.2011.09.014

loss, system security and power quality. These performances of the system were expressed in fuzzy sets to represent their inaccurate nature. An evolutionary programming was then introduced to determine the optimal solution. Kashem et al. [4] proposed ‘‘distance measurement technique algorithm’’ that found a loop first and then to improve load balancing a switching option was determined in that loop. Jeon incorporated the simulated annealing algorithm with Tabu search for loss reduction in [5]. The Tabu search attempted to determine a better solution in the manner of a greatest-descent algorithm but it could not give any guarantee for the convergence property. Lin et al. [6] presented a refined genetic algorithm (RGA) to reduce losses. Morton and Mareels presented a brute-force solution for determining a minimal-loss radial configuration [7]. The graph theory involving semi sparse transformations of a current sensitivity matrix was used, which guaranteed a globally optimal solution but needed an exhaustive search. Goswami and Basu proposed a power-flow-minimum heuristic algorithm for the DER problem [8]. Lopez et al. proposed a method for online reconfiguration [9]. Debaprya presented a fuzzy multi-objective approach to solve DFR [10]. Niknam presented two approaches based on norm2 for multi-objective Distribution Feeder Reconfiguration [11,12]. In [13], a new reconfiguration based on differential evolution algorithm was proposed for DSTATCOM allocation in distribution networks. Niknam et al. proposed a hy-

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T. Niknam et al. / Energy Conversion and Management 54 (2012) 7–16

brid algorithm based on Honey Bee Mating Optimization (HBMO) and a fuzzy set for the multi-objective Distribution Feeder Reconfiguration [14]. Gomes proposed an algorithm based on simple heuristic rules for optimal network reconfiguration in distribution systems [15]. McDermott et al. presented a heuristic nonlinear constructive method for the DFR problem [16]. Shirmohammadi and Hong proposed a heuristic method for the reconfiguration of distribution networks in order to reduce their resistive line losses under normal operating conditions [17]. Su and Lee proposed a method to reduce power loss and enhance the voltage profile by the improved mixed-integer hybrid differential evolution (MIHDE) method for distribution systems [18]. Chiou et al. presented a method based on the variable scaling hybrid differential evolution (VSHDE) for solving the network reconfiguration for power loss reduction and voltage profit enhancement of distribution systems [19]. Viswanadha Raju and Bijwe presented an algorithm based on sensitivity and heuristics for minimum loss reconfiguration of distribution system [20]. In [21], a new reconfiguration scheme was considered for voltage stability enhancement of radial distribution systems. Ahuja et al. presented a hybrid artificial immune systems and ant colony optimization (AIS–ACO) approach for the multi-objective DFR problem [22]. Assadian et al. proposed a guaranteed convergence particle swarm optimization in cooperation with graph theory to distribution network reconfiguration for minimization of power losses [23]. Arun and Aravindhababu proposed a fuzzy genetic based approach for reconfiguration of radial distribution systems to maximize the voltage stability for a specific set of loads [24]. Cheng and Kou used simulated annealing to solve the DFR problem in distribution system [25]. As shown in mentioned references, saving electrical energy is one of the most important problems in electrical networks. Feeder reconfiguration is a very important and usable operation to reduce distribution feeder losses and improve system security. Therefore, this paper presents a new algorithm for the DFR problem, which its aim is to minimize the electrical power losses. In the distribution system, since there are many candidate switching combinations, the DFR problem is modeled as a mixed integer nonlinear optimization problem. Therefore, it is difficult to solve the problem by conventional approaches. Also most optimal algorithms cannot effectively solve this kind of problem and they usually achieve local optimal solutions rather than global optimal solutions. Most of the mentioned methods in above have low accuracy and slow convergence rate. In this paper, a new hybrid algorithm is presented which its accuracy and convergence rate is very high. The proposed algorithm is based on the combination of the New Fuzzy Adaptive Particle Swarm Optimization (NFAPSO), with Nelder–Mead (NM). In the proposed algorithm, we use of discrete Particle Swarm Optimization (DPSO) and Binary Particle Swarm Optimization (BPSO). The BPSO determines the status of tie switch (open or close) while DPSO determines the sectionalizing switch number. In other side, PSO algorithm is a powerful and effective optimization method [26]. Although PSO eventually determines the desired solution [28] but its convergence rate is slow. To solve this drawback of PSO, it should be noted that its parameters should be carefully selected for efficient performance. In order to find a good set of parameters, the algorithm has to be run several times with different parameter sets [29]. However, any set of static parameters seems to be inappropriate. The use of rigid parameters that do not change their values may not be optimal since different values of parameters may work better/ worse at different stages of the evolutionary process [30]. Some attempts have been made to define an adaptive PSO. Shi and Eberhart presented a promising technique. They used a fuzzy controller for adapting one of the parameters dynamically [26]. In this paper, a fuzzy system based on some heuristics is designed to adaptively adjust the parameters of DPSO and BPSO

during the optimization process to improve the overall performance. In other side, the Nelder–Mead is a simplex search method that has been widely used in unconstrained optimization problem [27]. The NM is not always available since it is very sensitive to the choice of initial points and not guaranteed to obtain the global optimization solution. The proposed algorithm (NFAPSO–NM) guarantees that the final solution converges to the global solution. Two distribution test feeders are used to demonstrate the accuracy of the algorithm. Main contributions of the proposed algorithm are as follows: (i) Discrete nature of the switch values prevents the use of classical optimization techniques to solve the reconfiguration problem, therefore in this paper we use Discrete Particle Swarm Optimization (DPSO) and Binary Particle Swarm Optimization (BPSO) to determine the status of the tie switches (open or closed) and determine the sectionalizing switch number, respectively. (ii) In the original PSO algorithm, there are two tuning parameters (C1, C2) and an inertia weight (W) that greatly influence the algorithm performance. Proper selection of these parameters increases the performance efficiency of algorithm. However, as in other evolutionary algorithms, appropriate adjustment of PSO’s parameters is cumbersome and usually requires a lot of time. Thus, in this paper, a fuzzy-adaptive framework is proposed for adjusting the PSO algorithm’s parameters. (iii) The PSO algorithm is a relatively weak local search procedure and its convergence rate is slow. In the proposed algorithm, in order to increase the ability of PSO in local search and its convergence rate, we combine PSO with NM. The paper is organized as follows: In Section 2, the proposed DFR is formulated. In Sections 3 and 4, the basic principles of the NM and NFAPSO algorithms are introduced, respectively. In Section 5, the application of the NFAPSO–NM to solve the proposed DFR is shown. In Section 6, the feasibility of the NFAPSO–NM method and the proposed DFR is demonstrated and compared with the solution results by other works and other evolutionary methods such as the DPSO algorithm and NM over different distribution test systems. Finally, a summary and the conclusion is presented in Section 7. 2. Distribution Feeder Reconfiguration problem The DFR problem is a mixed integer nonlinear optimization problem and is a multi-objective problem. In the multi-objective DFR, there are many different objectives including loss minimization, balancing load on transformers, balancing load on feeders, maximum load on feeders, and deviation of voltages from nominal. In this paper, loss minimization is considered as the objective while the other objectives are considered as constraints. The DFR problem is described as below: 2.1. Objective function In this paper the objective function for the DFR problem is to minimize the power losses, which can be calculated as follows:

f ðXÞ ¼

N br X

Ri  jIi j2

ð1Þ

i¼1

X ¼ ½Tie1 ; 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

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1 = close). Swi is the sectionalizing switch number that forms a loop with Tiei. Ntie is the number of the tie switches.

In this step, N + 1 vertex points are randomly generated in their respective search range or space. The fitness of the objective function is evaluated. The generated population is sorted based on the objective function values as follows:

2.2. Constraints The constraints can be listed as follows:

2

 Distribution line limits: Line jPLine ij j < P ij;max

ð2Þ

Line jP Line ij j and P ij;max are the absolute power flowing over the distribution lines and the maximum power transmitted between the nodes i and j, respectively.  Distribution power flow equations:

Pi ¼

N bus X

V i V j Y ij cosðhij  di þ dj Þ

i¼1

Qi ¼

Nbus X

V i V j Y ij sinðhij  di þ dj Þ

Step 1: Initialization

ð3Þ

Xl 6  6 NM Population ¼ 6 4 Xs Xh

Fl

3

 7 7 7 Fs 5 Fh

ð9Þ ðNþ1Þð2Ntie þ1Þ

X i ¼ ½Tie1 ; Tie2 ; . . . ; TieNtie ; Sw1 ; Sw2 ; . . . ; SwNtie  i ¼ 1; 2; . . . ; N þ 1 where Xi is the ith vertex points. Xs is the vertex with the second highest objective function value, and fs represent the corresponding observed objective function. Xh and Xl, vertices are the vertices with the highest and the lowest, respectively while fh and fl represent the corresponding observed function values.

i¼1

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. Yij and hij are the amplitude and angle of the branch admittance between the ith and jth buses.  Maximum number of switching operations: Minimizing number of switching operations can be modeled as follows: Ns X

Step 2: Reflection Find Xc (Eq. (10)), the center of the simplex without Xh in the minimization case. Generate a new vertex Xo, (Fig. 1) by reflecting the worst point according to Eq. (11).

Xc ¼

Nþ1 1 X Xj N j¼1

ð10Þ

j–h

jSi  Soi j 6 N switch

ð4Þ

i¼1

where Si and So,i are the new and original states of the switch i, respectively. Ns is the number of the switches. Nswitch is the maximum number of switching for switches.  Bus voltage limit: Bus voltage can be described as follows:

V min 6 V i 6 V max

ð5Þ

where Vmin and Vmax are the minimum and maximum values of bus voltages, respectively.  Radial structure of the network:

M ¼ Nbus  N f

ð6Þ

where M is the number of branches, Nbus is the number of nodes and Nf is the number of sources.  Transformers limits:

jIt;i j 6 Imax t;i

i ¼ 1; 2; . . . ; Nt

ð7Þ

Imax t;i

where |It,i| and are the current amplitude and maximum current of the ith transformer, respectively. Nt is the number of transformers.  Feeders limits:

jIf ;i j 6 Imax f ;i

i ¼ 1; 2; . . . ; Nfeeder

X o ¼ ð1 þ aÞX c  aX h

ða > 0Þ

ð11Þ

If fl 6 fo 6 fs , replace Xh by Xo and return to step 2. Step 3: Expansion If fo 6 fl , expand the simplex (Fig. 2) using an expansion factor

c > 1 and find Xoo such that: X oo ¼ ð1  cÞX c þ cX o

ðc > 1Þ

ð12Þ

(a) If foo < fl replace Xh by Xoo and return to step 2. (b) If foo > fl replace Xh by Xo and return to step 2. Step 4: Contraction If fo P fs contract the simplex by using a contraction factor bð0 < b < 1Þ. There are two cases to consider: (a) If fo < fh (Fig. 3), find Xoo such that,

X oo ¼ ð1  bÞX c þ bX o

ð0 < b < 1Þ

ð13Þ

(b) If fo > fh (Fig. 4), find Xoo such that,

X oo ¼ ð1  bÞX c þ bX h

ð0 < b < 1Þ

ð14Þ

ð8Þ

where |If,i| and Imax are the current amplitude and maximum f ;i current of the ith feeder, respectively. Nfeeder is the number of feeders.

Whether (step 4a) or (step 4b) is used, there are again two cases to consider: (c) If foo < fh and foo < fo , replace Xh by Xoo and return to step 2.

3. The Nelder–Mead simplex search method (NM) The Nelder–Mead simplex search method is proposed by Nelder and Mead in 1965 designed for unconstrained optimization without using gradient information [27]. The operation of this method rescales the simplex based on the local behavior of the function. To apply the NM algorithm to solve the DFR problem, the following steps should be repeated:

Fig. 1. Reflection of vertex Xh in non-regular simplex.

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T. Niknam et al. / Energy Conversion and Management 54 (2012) 7–16

(pbesti1, . . . , pbestid, . . . , pbestin). The best global position of the swarm found so far is denoted by Gbest = (gbest1, . . . , gbestd, . . . , gbestn). The modified velocity of each particle can be first calculated regarding the personal initial velocity, the distance from personal (local) best position and the distance from global best position as expressed by Eq. (15). ðtþ1Þ

Fig. 2. Expansion of non-regular simplex.

Fig. 3. Contraction of non-regular simplex when fs < fo 6 fh .

Fig. 4. Contraction of non-regular simplex fo P fh .

(d) If foo P fh or foo > fo , replace the size of the simplex by halving the distances from Xl and return to step 2. Step 5: Exit the algorithm if the stopping criteria are satisfied; otherwise go to step 2.

Vi

ðtÞ

ðtÞ

¼ W  V i þ C 1  rand1 ðÞ  ðPbesti  X i Þ   ðtÞ þ C 2  rand2 ðÞ  Gbest  X i

ð15Þ

Eq. (15) determines the direction in which the ith particle should be taken along. In this equation, i = 1, 2, . . . , Nswarm is the index of each particle, Nswarm is number of the swarms, t is iteration number, rand1() and rand2() are random numbers between 0 and 1. Constants C1 and C2 are learning factors of the stochastic acceleration terms, which determine the influence of personal best Pbesti and global best Gbest, respectively. Authors have introduced the parameter W into the BPSO’s equation to control the impact of the previous history of velocities on the current velocity to improve its performance. The appropriate selection of inertia weight W in (15) provides a balance between global and local explorations, requiring less iteration on average to find a sufficiently optimal solution. Therefore, the new position of that particle can be determined by applying Eq. (16). ðtþ1Þ

Xi

ðtÞ

ðtþ1Þ

¼ roundðX i þ V i

Þ

ð16Þ

In this equation, i = 1, 2, . . . , Nswarm is the index of each particle, Nswarm is number of the swarms, t is iteration number, round(X) rounds the elements of X to the nearest integers.

4.2. The original BPSO

4. The original BPSO, DPSO, FABPSO and FADPSO PSO in the binary version is modified as The Particle Swarm Optimization algorithm (PSO) was firstly proposed by Eberhart and Kennedy [26] and has been deserved some attention during the last years in the global optimization field [28]. PSO is based on the population of agents or particles and tries to simulate its social behavior in optimal exploration of problem space. In this paper, the control variables are composed of two parts: 1. Tie1 ; Tie2 ; . . . ; TieNtie 2. Sw1 ; Sw2 ; . . . ; SwNtie Tiei is the state of the ith tie switch (0 = open and 1 = close). Swi is the sectionalizing switch number that forms a loop with Tiei. Ntie is the number of the tie switches. If the status of the tie switch is open, all sectionalizing switches are close and if the tie switch is close, a sectionalizing switch is open that it forms a loop with the tie switch. For example, assume that X is [1 0 1 0 1 4 0 3 0 9] where Tie1 = 1, Sw1 = 4 and Tie2 = 0, Sw2 = 0. It means that switch number 4 must be open (Tie1 = 1) and Sw2 = 0 means that the switch must remain close (Tie2 = 0). In this paper Tiei is obtained by BPSO and Swi is obtained by DPSO. In following we briefly describe BPSO and DPSO algorithms. 4.1. The original DPSO In an n-dimensional search space, the position and velocity of the ith individual are represented as vectors Xi = (Xi1, . . . , Xid, . . . , Xin) and Vi = (vi1, . . . , vid, . . . , vin), respectively. The best previous experience of the ith particle is recorded and represented by Pbesti =

V tþ1 ¼ V ti þ C 1  randðÞ  DV i;1 þ C 2  randðÞ  DV i;2 i DV i;1 ¼ Pbesti  X ti

ð17Þ

DV i;2 ¼ Gbest  X ti while parameters Pbesti, Gbest and X ti can take any real value in (15), these parameters are integers from {0, 1} in (17). V ti is limited to the interval [0, 1]as it is a probability not velocity. A logical transformation SðV ti Þ can be used to accomplish this last modification. The resulting change in the position is then defined by the following rule:

( X ti

¼

1;

randðÞ 6 SðV ti Þ

0;

otherwise

SðV ti Þ ¼

1 1 þ expðV ti Þ

:

ð18Þ

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 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 probability of each particle. Therefore, the values DV i;1 ¼ Pbesti  X ti and DV i;2 ¼ Gbest  X ti in (17) are small changes of probability in each iteration that can be 1, 0, or 1. Therefore, the sigmoid transformation only transforms V ti from the interval [1, 1] to [0, 1].

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T. Niknam et al. / Energy Conversion and Management 54 (2012) 7–16

4.3. Parameters of BPSO and DPSO There are three tuning parameters W, C1, C2 as shown in (15), (17) that greatly influence the algorithm performance. The inertia weight W is employed to improve performance of PSO and DPSO. A larger inertia weight W facilitates global exploration while a smaller inertia weight tends to facilitate local exploration to finetune the current search area. A suitable selection of the inertia weight W can provide a balance between global and local exploration abilities, thus require less iterations on average to find the optimum [30]. The learning factors [12] C1 and C2 pull each particle towards Pbesti and Gbest positions as shown in (15). Since C1 expresses how much the particle trusts its own past experience, it is called cognitive parameter. While C2 expresses how much it trusts the swarm, it is called social parameter. If C1 P C2 the particle will be much more attracted to the best position found by itself Pbesti, rather than the best position found by the population (or the neighborhood) Gbest, and vice versa. If C1 = C2 = 2, each particle will be attracted to the average of Pbesti and Gbest. Recent work reports that it might be even better to choose a larger cognitive parameter C1 than a social parameter C2, but with C1 + C2 6 4 [26]. 4.3.1. Fuzzy adaptive BPSO and DPSO The parameters W, C1, C2 are often held constant [28] or linearly changed for the entire run of a PSO and DPSO, but this approach will not produce optimal results in many cases. From experience, it is known that (i) when the best fitness is low at the end of the run, e.g., in the optimization of a minimum function, low inertia weight and high learning factors are often preferred; (ii) when the best fitness is in local, number of generations for unchanged best fitness is large. The system is often stuck at a local minimum so the system should probably concentrate on exploiting rather than exploring. Therefore, the inertia weight should be increased and learning factors should be decreased. Based on this kind of knowledge, a fuzzy system is developed to adjust the inertia weight and learning factors with best fitness (BF) and number of generations for unchanged best fitness (NU) as the input variables, and the inertia weight (W) and learning factors (C1 and C2) as output variables. The BF measures the performance of the best candidate solution found so far. Different optimization problems have different ranges of BF value. To design a FAPSO and FADPSO applicable to a wide range of problems, the ranges of BF and NU are normalized into [0, 1.0]. One example of converting BF to be a normalized BF format NBF is shown in (19).

NBF ¼

BF  BF min BF max  BF min

ð19Þ

where BFmin and BFmax are the estimated or real minimum fitness value and the fitness value greater or equal to maximum fitness value, respectively. NU may be converted into [0, 1.0] in similar way. Other converting methods are possible, of course. The values for W, C1 and C2 are bounded in .2 6 w 6 1.2 and 1 6 C1 6 2, 1 6 C2 6 2. The fuzzy system consists of four principal components: fuzzification, fuzzy rules, fuzzy reasoning and defuzzification, which are described as following. 4.3.2. Fuzzification Among a set of membership functions, left-triangle, triangle and right-triangle membership functions are used for every input and output as shown in Fig. 5. PS (positive small), PM (positive medium), PB (positive big) and PR (positive bigger) are the linguist variables for the inputs and outputs. 4.3.3. Fuzzy rules The Mamdani-type fuzzy rule is used to formulate the conditional statements that comprise fuzzy logic. For example:

Fig. 5. Membership functions of inputs and outputs: (a) NBF or NU, (b) W, and (c) C1 or C2.

Ri : IF ðNBF is PRÞ and ðNU is PBÞ; THEN ðC 1 is PSÞ; ðC 2 is PSÞ; ðW is PRÞ: The fuzzy rules in Tables 1–3 are used to adjust the inertia weight (W), learning factors (C1 andC2), respectively. Each rule represents a mapping from the input space to the output space. 4.3.4. Fuzzy reasoning The fuzzy control strategy is used to map from the given inputs to the outputs. Mamdani’s fuzzy inference method is used in this paper. The AND operator is typically used to combine the membership values for each fired rule to generate the membership values for the fuzzy sets of output variables in the consequent part of the rule. Since there may be several rules fired in the rule sets, for some fuzzy sets of the output variables there may be different membership values obtained from different fired rules. These output fuzzy sets are then aggregated into a single output fuzzy set by OR operator. That is to take the maximum value as the membership value of that fuzzy set. 4.3.5. Defuzzification In this paper for defuzzification, the method of centroid (centerof-sums) is used. To apply the NFAPSO algorithm to solve the DFR problem, the following steps should be taken: Step 1: The initial population and initial velocity for each particle are generated randomly.

Table 1 Fuzzy rules for inertia weight W. W

NU

NBF PS PM PB PR

PS PS PM PB PB

PM PM PM PB PB

PB PB PB PB PR

PR PB PR PR PR

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Table 2 Fuzzy rules for factor C1. C1

NU

NBF PS PM PB PR

PS PR PB PB PM

PM PB PM PM PM

PB PB PM PS PS

PR PB PR PR PR

PS PR PB PM PM

PM PB PM PM PS

PB PM PS PS PS

PR PM PS PS PS

Table 3 Fuzzy rules for factor C2. C2

NU

NBF PS PM PB PR

Step 2: The objective function is to be evaluated for each individual. Step 3: The individual that has the minimum objective function should be selected as the global position. Step 4: The ith individual is selected. Step 5: The best local position (Pbesti) is selected for the ith individual. Step 6: Update the NFAPSO parameters as shown in the previous subsections. Step 7: Calculate the next position for each individual based on the NFAPSO parameters and Eqs. (15)–(18) and then check with its limit. Step 8: If all individuals are selected, go to the next step, otherwise i = i + 1 and go to step 4. Step 9: If the current iteration number reaches the predetermined maximum iteration number, the search procedure is stopped, otherwise return to step 2. The last Gbest is the solution of the problem. 5. Hybrid New Fuzzy Adaptive Particle Swarm Optimization and Nelder–Mead (NFAPSO–NM) The goal of integrating New Fuzzy Adaptive Particle Swarm Optimization (NFAPSO) and Nelder–Mead (NM) simplex search method is to combine their advantages and avoid disadvantages, For example, NM simplex method is a very efficient local search procedure but the NM is not always available since it is very sensitive to the choice of initial points and not guarantees to obtain the global optimization. Also, FABPSO, FADPSO algorithms belong to the class of global search procedures but require much computational effort also accuracy of FABPSO and FADPSO is not very high. We obtain a new algorithm (NFAPSO–NM) that not only is not sensitive to the choice of initial points bus also has faster rate and more accurate convergence rather than NM and NFAPSO and other algorithms. This section starts from recollecting the procedures of NFAPSO and NM that will be used for the DFR problem. The originals and literatures of these algorithms can be found in Sections 3 and 4. Fig. 6 depicts the schematic representation of the proposed hybrid NFAPSO–NM. The population size of this hybrid NFAPSO–NM approach is set at 3N + 1 when solving an N-dimensional problem. The initial 3N + 1 particles are randomly generated and sorted by fitness, and the top N + 1 particles are then fed into the Nelder– Mead method to improve their position. The other 2N particles are adjusted by the NFAPSO method. The procedure of adjusting the 2N particles in the NFAPSO method involves selection of the global best particle, selection of the neighborhood best particles,

and finally velocity updates. The global best particle of the population is determined according to the sorted fitness values. The neighborhood best particles are selected by first evenly dividing the 2N particles into N neighborhoods and designating the particle with the better fitness value in each neighborhood as the neighborhood best particle. Moreover, by Eqs. (15)–(18) velocity and position are updated for each of the 2N particles. In this step (velocity updates) the inertia weight and learning factors are adjusted by the fuzzy system. Finally, the global best particle of updated population is determined according to the sorted fitness values and the 3N + 1 particles are sorted again in preparation for repeating the entire run. To apply the NFAPSO–NM algorithm in the DFR, the following steps have to be taken: Step 1: In this step, the input data including the network configuration, line impedance and status of switches, the reflection factor (a), the expansion factor (c), the contraction factor (b), the primer values of inertia weight and learning factors, the number of iteration and the number of population are defined. Step 2: Transfer the constraint optimization problem to an unconstraint one. In this step, a constraint optimization problem must be transferred to an unconstraint optimization problem, that this operation performed by (20):

FðXÞ ¼ f ðXÞ  k1

! ! N eq Nueq X X ðhj ðXÞÞ2  k2 ðMax½0;g j ðXÞÞ2 j¼1

j¼1

ð20Þ f(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 ðXÞ and g i ðXÞ are the equality and inequality constraints, respectively. k1 and k2 are penalty factors. Due to the constraints should be met; the value of the k1 and k2 parameters should be high. In the paper, these values are 1000,000. The augment objective function value is calculated as follows. Distribution load flow is run for the control variables vector (status of the tie and sectionalizing switches). The objective function value (f(X)), equality and inequality constraints are calculated and based on the results of distribution load flow. Then the augment objective function is calculated by using the values of objective function, constraints and penalty factors. Step 3: Generate the initial population. The population is at following form:

2

X1 6 X 6 2 Population ¼ 6 4

3 7 7 7 5

X 3Nþ1 X i ¼ ½Tie1 ; Tie2 ; . . . ; TieNtie ; Sw1 ; Sw2 ; . . . ; SwNtie ; ¼ 1; 2; . . . ; 3N þ 1 where Tiei is the statues of the ith tie switch which is zero or one and Swi is the switch number of the ith sectionalizing switch. Step 4: Calculate the augmented objective function value for each individual by using results of the distribution load flow. Step 5: Sort the initial population based on the augmented objective function values. Step 6: Select the N + 1 generated population that have best objective function value for the NM and select the 2N remained generated population for the FABPSO and FADPSO.

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T. Niknam et al. / Energy Conversion and Management 54 (2012) 7–16

3N+1 from random generation

Best

Worst

Sorted Population

2N

N

FABPSO and FADPSO Initial population of BPSO and DPSO

Fitness function evaluation

(NM) Modified Simplex Fuzzy system of BPSO and DPSO parameters

Updating particle’s velocity and position by (15) and (17)

Sort the new population

New Population

2N

N

Fig. 6. Schematic representation of proposed NFAPSO–NM algorithm.

Step 7: Go to NM algorithm. Step 8: Go to NFAPSO algorithm. Step 9: In this step, results of NFAPSO and NM are combined. Step 10: Check the termination criteria, (in the proposed algorithm, number of iteration is proposed for termination criteria). If the termination criteria satisfied finish the algorithm, else go to step 6 until convergence criteria met. 6. Simulation and results In this section, the NFAPSO–NM algorithm is employed to solve the DFR problem for two distribution test feeders. The parameters required for implementation of the NFAPSO–NM algorithm are a, c, b, N, W, C1, C2. In this paper, the best values for the aforementioned parameters are a = 0.5, c = 1.2, b = 1, N = 50 and the parameters W, C1, C2 are determined by the fuzzy system. 6.1. Case study 1 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 [31] and the single line diagram of this system is shown in Fig. 7. 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 reconfiguration, the initial loss is 202.67 kW. Table 4 illustrates a comparison of the proposed algorithm and some other algorithms in terms of computational efficiency and performance. We observe that there are incongruities between the results reported by several authors with regard to total active

Fig. 7. A single line diagram of distribution system for case study 1.

power loss; however, with respect to open tie line, we can ensure that our results are equal to the results obtained by others. Table 5 illustrates the comparison between the proposed NFAPSO–NM with the results of NM and the original DPSO and PSO– NM for 20 random trials. According to Table 5, the results of DPSO and NM methods are rather weak in comparison with the proposed hybrid NFAPSO–NM algorithm. Also, the NFAPSO–NM algorithm is very powerful and in all trials the best solution is obtained (standard deviation for different trials is zero) and execution time of proposed method is short. The reason the proposed algorithm is faster than the others is that the number of iteration in the proposed algorithm is less than the others. In Table 6 an error analysis has been done and the simulation results of the student T-test between the results of the proposed algorithm and the others are shown. The purpose of T-test is to test

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Table 4 Results for different methods. Method

Power losses (kW)

Saving (%)

Open switches

Goswami and Basu [8] McDermott et al. [16] Shirmohammadi and Hong [17] Gomes et al. [15] DPSO–HBMO [11] DPSO [13] PSO–ACO [11] HBMO [14] PSO–HBMO [12] The proposed algorithm

139.53 139.53 140.26 139.53 139.53 139.53 139.53 139.53 139.53 139.53

31.14 31.14 30.78 31.14 31.14 31.14 31.14 31.14 31.14 31.14

s7, s7, s7, s7, s7, s7, s7, s7, s7, s7,

s9, s14, s32, s37 s9, s14, s32, s37 s10, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37

Table 5 Comparison of average and standard deviation for 20 trials. Method

NFAPSO–NM PSO–NM DPSO NM

Average of objective function value

Standard deviation

Worst solution Power losses (kW)

Open switches

Power losses (kW)

Best solution Open switches

139.53 139.8404 140.4097 142.5762

0 0.094710 0.989149 1.592661

139.53 140.26 142.90 143.4966

s7, s7, s6, s7,

139.53 139.53 139.53 139.53

s7, s7, s7, s7,

s9, s14, s32, s37 s10, s14, s32, s37 s9, s14, s32, s37 s11, s14, s36, s37

the null hypothesis whether there are no differences between the results of two different statistical experiments. In the table, the value of H indicates that the means are equal or not equal. The H = 1 shows a rejection of the null hypothesis at 5% significance level. The H = 0 indicates a failure to reject the null hypothesis at 5% significance level. ci is a 95% confidence interval for the true difference in the means. As shown in all data set, the value of H is equal 1. It means that we can reject the null hypothesis. To demonstrate that the proposed algorithm does not depend on the initial switching configuration, the initial configuration has been changed by closing the normally open switches s35 and s37 and opening the normally closed s3 and s6. The initial loss is 208.15 kW. The results in Table 7 illustrate the comparison between the proposed NFAPSO–NM with the other methods when the initial configuration has been changed. Simulation results show

s9, s9, s9, s9,

s14, s14, s14, s14,

s32, s32, s32, s32,

s37 s37 s37 s37

CPU time (s)

No of global solution

6 6.5 9 9

20 18 10 7

that the proposed approach does not depend on the initial switching configuration. 6.2. Case study 2 The case study 2 is a practical distribution network [18]. It is a three-phase, 11.4 kV system. The test system consists of 11 feeders, 83 normally closed switches, and 13 normally open switches. Three-phase balance and the constant load are assumed and the single line diagram of this system is shown in Fig. 8. The normally open switches, s84 to s96, are illustrated by doted lines. The normally closed switches, s1 to s83, are represented by solid lines. Before reconfiguration, the initial loss is 531.99 kW. Table 8 illustrates a comparison of the proposed algorithm and some other algorithms in terms of computational efficiency and

Table 6 Results of T-test for case study 1. Method

H

Significance

NFAPSO–NM PSO–NM DPSO NM

1 1 1 1

0 0 0 0

95% Confidence interval of the difference (ci) Lower

Upper

0.3291 1.0748 3.3603 11.0616

0.2917 0.6846 2.7321 9.8784

Table 7 Results of proposed method with other initial configuration. Method

Power losses (kW)

Saving (%)

Open switches

Goswami and Basu [8] McDermott et al. [16] Shirmohammadi and Hong [17] Gomes et al. [15] Niknam et al. [14] The proposed algorithm

143.69 139.53 140.26 139.53 139.53 139.53

31.14 29.08 30.78 31.14 31.14 31.14

s6, s7, s7, s7, s7, s7,

s10, s14, s32, s37 s9, s14, s32, s37 s10, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37 s9, s14, s32, s37

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T. Niknam et al. / Energy Conversion and Management 54 (2012) 7–16

Fig. 8. A single line diagram of distribution system for case study 2.

Table 8 Comparison of the proposed algorithm with previous methods. Method

Power losses (kW)

Minimum voltage (p.u)

Saving (%)

Open switches

Chiou et al. [19] Su et al. [18] Ahuja et al. [22] Viswanadha Raju and Bijwe [20] SA [25] The proposed algorithm

469.88 469.88 463.2896 469.88 469.88 463.2896

0.9285 0.9285 0.9532 0.9285 0.9285 0.9532

11.68 11.68 12.92 11.68 11.68 12.92

s55, s55, s55, s55, s55, s55,

s7, s7, s7, s7, s7, s7,

s86, s86, s86, s86, s86, s86,

s72, s72, s72, s72, s72, s72,

s88, s88, s88, s88, s88, s88,

s89, s89, s14, s89, s89, s14,

s90, s90, s90, s90, s90, s90,

s83, s83, s83, s83, s83, s83,

s92, s92, s92, s92, s92, s92,

s39, s39, s39, s39, s39, s39,

s34, s34, s34, s34, s34, s34,

s41, s41, s42, s41, s41, s42,

s62 s62 s62 s62 s62 s62

s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88, s88,

s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14, s14,

s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90, s90,

s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83, s83,

s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92, s92,

s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39, s39,

s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34, s34,

s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42, s42,

s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62 s62

Table 9 Result of proposed algorithm in 20 trials. Number of trial

Power losses (kW)

Minimum voltage (p.u)

Open switches

1 2 4 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896 463.2896

0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532 0.9532

s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55, s55,

performance. It is seen that the proposed algorithm leads to the global optimum configuration. Table 9 illustrates results of proposed algorithm in 20 trials. According to Table 9 in the all trials global solution achieved.

s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7, s7,

s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86, s86,

s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72, s72,

Table 10 illustrates a comparison among the proposed NFAPSO– NM with the results of NM and the original DPSO for 20 random trials. Despite other methods the proposed algorithm (NFAPSO– NM) in all trials achieves to the best solution.

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Table 10 Comparison of average and standard deviation for 20 trials. Average of objective function value

Standard deviation

Worst solution

Best solution

Power losses (kW)

Open switches

Power losses (kW)

Open switches

NFAPSO– NM PSO–NM

463.2896

0.000

463.2896

463.2896

463.8320

0.401855

464.9552

DPSO

464.1634

1.026257

466.2741

NM

465.5611

1.951404

469.0376

s55, s83, s55, s91, s55, s83, s55, s83,

s55, s83, s55, s83, s55, s83, s55, s83,

Method

s7, s86, s72, s88, s14, s90, s92, s39, s34, s42, s62 s7, s86, s72, s88, s14, s90, s92, s39, s34, s40, s61 s7, s86, s72, s88, s14, s90, s92, s38, s33, s95, s61 s7, s86, s72, s88, s14, s90, s92, s35, s34, s95, s62

Based on obtained results, all reported methods are giving promising results but when the system becomes larger, finding the solution becomes more difficult or the running time will be high. However, not only running time of the proposed hybrid algorithm is low but also it converges to global solutions when the system is large. 7. Conclusion This paper proposes a hybrid evolutionary algorithm based on the combination of NFAPSO with NM, called NFAPSO–NM, to optimally reconfigure radial distribution system. In the NFAPSO all three parameters of BPSO and DPSO, the inertia weight and the two learning factors, are adapted by a fuzzy system based on the fitness values of particles during optimization process. In the proposed algorithm, minimizing power losses is considered as the objective function and balancing load on transformers, balancing load on feeders, maximum load on feeders, and deviation of voltages from nominal are considered as the constraints. In addition to what mentioned before, NFAPSO–NM reaches a much better optimal solution in comparison with the others and has zero standard deviation for different trails. The simulation results on a small-size and large-size distribution networks have proved the feasibility of the proposed method and the obtained results are quite good. Also the NFAPSO–NM takes the advantage of being independent on the initial status of network switches. The NFAPSO–NM is an effective technique for solving reconfiguration problem. Furthermore, the proposed method can also be used to solve other complex problems. The major disadvantage of the proposed algorithm is that its run time increases when the number of control variables increase. References [1] Kim H, Ko Y, Jung KH. Artificial neural-network based feeder reconfiguration for loss reduction in distribution systems. IEEE Trans Power Deliver 1993;8(3):1356–66. [2] Taylor T, Lubkeman D. Implementation of heuristic search strategies for distribution feeder reconfiguration. IEEE Trans Power Deliver 1990;5(3):239–45. [3] Hsiao YT, Chen CY. Multiobjective optimal feeder reconfiguration. Proc Inst Elect Eng – Gener Transm Distrib 2001;148:333–6. [4] Kashem MA, Ganapathy V, Jasmon GB. Network reconfiguration for load balancing in distribution networks. Proc Inst Elect Eng – Gener Transm Distrib 1999;146:563–7. [5] Jeon YJ, Kim JC. Network reconfiguration in radial distribution system using simulated annealing and tabu search. Proc IEEE Power Eng Soc Winter Meet 2000:2329–33. [6] Lin WM, Cheng FS, Tsay MT. Distribution feeder reconfiguration with refined genetic algorithm. Proc Inst Elect Eng – Gener Transm 2000;147:349–54. [7] Morton AB, Mareels IMY. An efficient brute-force solution to the network reconfiguration problem. IEEE Trans Power Deliver 2000;15(3):996–1000.

463.2896 463.2896 463.2896

s7, s86, s72, s88, s14, s90, s92, s39, s34, s42, s62 s7, s86, s72, s88, s14, s90, s92, s39, s34, s42, s62 s7, s86, s72, s88, s14, s90, s92, s39, s34, s42, s62 s7, s86, s72, s88, s14, s90, s92, s39, s34, s42, s62

CPU time (s)

No of global solution

14

20

17

14

21

9

33

6

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