A NEW HYBRID EVOLUTIONARY OPTIMIZATION ALGORITHM FOR DISTRIBUTION FEEDER RECONFIGURATION

Abstract

This article extends a hybrid evolutionary algorithm to cope with the feeder reconfiguration problem in distribution networks. The proposed method combines the Self-Adaptive Modified Particle Swarm Optimization (SAMPSO) with Modified Shuffled Frog Leaping Algorithm (MSFLA) to proceed toward the global solution. As with other population-based algorithms, PSO has parameters which should be tuned to have a suitable performance. Thus, a self-adaptive framework is proposed to adjust the parameters dynamically. In SAMPSO, the PSO learning factors are considered to be the new control variables and are changed in the evolutionary process. To enhance the quality of the solutions, the SAMPSO is combined with MSFLA and a new hybrid algorithm is proposed to minimize the electrical energy losses of the distribution system by feeder reconfiguration. The effectiveness of the proposed method is demonstrated through two test systems.

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INTRODUCTION

Usually, electric distribution networks are configured as radial networks to improve protection coordination. Generally, the feeder reconfiguration is implemented to supply all loads in distribution networks, decrease the electric energy losses, enhance the system security, and improve the power quality. In addition, after applying a suitable reconfiguration, the overloading is relieved from the system components. Reconfiguration is explained simply by opening sectionalizing (normally closed) and closing tie (normally open) switches of the network. These switching operations are performed in such a way that the radiality of the network is maintained and the loads are energized. Obviously, the larger the number of switches is, the greater the number of possibilities for reconfiguration and the better the effects are.

Recently, many researchers have investigated the distribution feeder reconfiguration (DFR) problem in distribution networks. For example: Kim, Ko, and Jung (1993) proposed a new method based on neural networks to solve the DFR problem, including time intervals. Taylor and Lubkeman (1990) suggested an expert system using heuristic rules to shrink the search space. Kashem, Ganapathy, and Jasmon (1999) proposed a new technique based on the distance measurement that first finds a loop and then improves the load balancing by a switching plan that was determined in that loop. Jeon and Kim (2000) integrated the simulated annealing algorithm with Tabu search by aiming at the loss reduction. The Tabu search attempted to determine a better solution in the manner of the greatest-descent algorithm. The big drawback of this algorithm is that the proposed algorithm may not be converged. Lin, Cheng, and Tsay (2000) proposed a refined genetic algorithm (RGA) to reduce the active power losses. Morton and Mareels (2000) presented a brute-force solution for determining a minimal-loss radial configuration. They used a method based on graph theory including semi-sparse transformations of a current sensitivity matrix, which guarantees the globally optimal solution through a complete search. Goswami and Basu (1992) proposed a power-flow-minimum heuristic algorithm for the DFR problem. Lopez and Opaso (2004) suggested a new method for solving the online DFR problem. Das (2006) proposed the DFR problem as a multiobjective problem, and a new method based on fuzzy mechanism was introduced to solve it. Niknam (2009) presented an approach based on the norm 2 to solve the multiobjective DFR problem, and also proposed a hybrid approach based on dissipative particle swarm optimization (DPSO), ant colony optimization, and fuzzy system for reconfiguration of distribution systems (Niknam 2010). Liu and Chen (2000) proposed a modified evolutionary algorithm, namely the fuzzy genetic algorithm, for the DFR problem. Bi, Liu, and Liu (2002) proposed another version of the genetic algorithm called the refined genetic algorithm for distribution network reconfiguration. Shirmohammadi and Hong (1998) proposed the reconfiguration in the electric distribution networks for resistive line loss reduction. Li et al. (2007) proposed another hybrid algorithm, namely hybrid particle swarm optimization algorithm, for the DFR problem. Yu et al. (2009) presented a modified genetic algorithm with infeasible solution disposing for the DFR problem. Bosi, Kaigui, and Jiaqi (2005) applied a dynamic programming method on the DFR problem. Shaoyun, Zifa, and Yixin (2004) presented a new version of Tabu search for the DFR problem. In their proposed method, the performance of the algorithm was modified by a new mechanism. Xiufan, Lizi, and Lingyu (2004) proposed a modified algorithm, namely a family eugenics-based evolution algorithm, for the DFR problem. Wenchuan and Jiaju (2006) proposed an artificial immune algorithm for distribution network reconfiguration. Chiou, Chang, and Su (2005) proposed a feeder reconfiguration scheme by aiming the power loss reduction and voltage profile enhancement of distribution systems, and a method based on variable scaling hybrid differential evolution (VSHDE) employed to do that. Su and Lee (2003) proposed an improved mixed-integer hybrid differential evolution (MIHDE) method for distribution system reconfiguration. They solved the problem by aiming the reducing power loss and enhancing the voltage profile. Cheng and Kou (1994) used simulated annealing for network reconfiguration in the distribution system. Raju and Bijwe (2008) proposed an algorithm based on sensitivity and heuristics for minimum-loss reconfiguration of the distribution system. Ahuja, Das, and Pahwa (2007) proposed an AIS-ACO hybrid approach for a multiobjective DFR problem.

Because of several candidate-switching combinations in distribution networks, the DFR problem is modeled as a complicated combinatorial, nondifferentiable, constrained optimization problem. So, conventional approaches are unable to solve this sort of problem, and most of the optimization algorithms reach to local minima dealing with these problems. Therefore, this article presents a new hybrid algorithm to find the optimal operating condition of the distribution networks. The algorithm is based on the combination of Self-Adaptive Modified Particle Swarm Optimization (SAMPSO) and Modified Shuffled Frog Leaping Algorithm (MSFLA).

PSO is one of the most recent population-based algorithms inspired from natural occurrence (Niknam 2010). Similar to other traditional evolutionary algorithms, PSO has some parameters. Learning factors are two parameters of the PSO that control the exploration and exploitation of this algorithm. Self-adaptive parameter control is a way of adjusting the parameter that considers the state of the algorithm to control the parameter dynamically. In this control method, parameters are used as variables and are added to each particle.

Shuffled Frog Leaping Algorithm (SFLA) is a population-based algorithm, which is inspired from the behaviors of frogs located in swamps, searching around stones to find more food (Elbeltagi, Hegazy, and Grierson 2005). However, some limitations with regard to the frog's jump in local exploration may slow down the convergence speed or even cause premature convergence. To improve the quality of the population in each evolutionary process, a new mutation is proposed and used in each memplex of the frog population, such that Modified SFLA (MSFLA) is proposed.

Taking advantage of the compensatory properties of SAMPSO and MSFLA, we propose a new algorithm that combines the evolutionary natures of both algorithms (denoted as SAMPSO-MSFLA). In order to test and ensure the performance of the proposed method, two test distribution feeders are used and the results are compared with corresponding values obtained by other methods.

PROBLEM FORMULATION

As a result of the nonlinear equation for the electrical energy losses, presence of elements with nonlinear behavior—such as transformers—and presence of control variables, which determine the status of the switches, the DFR problem is modeled as a mixed-integer nonlinear and multiobjective optimization problem. In the multiobjective DFR, there are many different objectives, such as loss minimization, balancing load on transformers, balancing load on feeders, maximum load on feeders, and deviation of voltages from their nominal values. In this article, the loss minimization is considered to be the main objective whereas the others are formulated in the constraints. To define the DFR problem, the objective functions and constraints are explained as follows:

Objective Function

Total electrical losses in distribution networks:

where R _i and I _i are resistance and actual current, respectively, of the ith branch. N _br is the number of the branches. X is the vector of control variables. Tie _i is the state of the i th tie switch with 0 and 1 corresponding to open and closed states, respectively. Sw _i is the sectionalizing switch number that forms a loop with Tie _i . N _tie is the number of the tie switches.

Constraints

The constraints are listed as follows:

	Limits of distribution lines
	where and are the active power flow of line from bus i to bus j and its corresponding upper limits, respectively.
	Distribution power flow equations
	where P i and Q i are the injected active and reactive power components at the i th bus, V i and δi are the amplitude and angle of the voltage at the i th bus, respectively, and Y ij and θ ij are the amplitude and angle of the branch admittance between the i th and j th buses.
	Limits on the number of switching operations
	where S i and S o,i are the new and original states of the switch i, respectively, N s is the number of the switches, and N switch is the maximum allowable number of switching operations.
	Magnitude of bus voltages
	where V minand V maxare the minimum and maximum values, respectively, of bus voltage amplitudes.
	Radial structure of the network
	where M is the number of branches, N bus is the number of nodes, and N f is the number of sources.
	Transformers’ limits
	where |I t, i | and are the current amplitude and its maximum allowable value, respectively, of the i th transformer, and N t is the number of transformers.
	Feeders limits
	where |I f, i | and are the current amplitude and its maximum allowable value, respectively, of the i th feeder, and N f is the number of feeders.

PARTICLE SWARM OPTIMIZATION ALGORITHM

In this section, PSO, MPSO, and DPSO are briefly explained as follows:

Original PSO

Particle swarm optimization, proposed by Eberhart and Kennedy (1995), is an evolutionary algorithm that is closely related to trends that imitate natural evolutions. PSO draws inspiration from collective behavior and emergent intelligence that resulted from socially organized populations (Carlisle and Dozier 2001; Sakthivel, Bhuvaneswari, and Subramanian 2010; Das and Dulger 2009). In this article, the vector of control variables is considered as follows:

This vector can be divided into the two below vectors:

1.
2.

The status of the control variables is defined as follows:

1.	Status of the tie switch is open: in this case sectionalizing switches are close
2.	Tie switch is close: in this case, the sectionalizing switch that forms a loop with the tie switch is open.

For example, if two sections of vector X are as follows,

it means that the tie switches of the 1st, 2nd, and 4th loops must remain unchanged and the tie switches of the 3rd and 5th loops must be replaced by the 2nd and 7th sectionalizing switches of the loops, respectively. In this article, the vector 1 from the control variables is obtained by MPSO and the vector 2 of the control variables is obtained by self-adaptive PSO (SADPSO). Now, suppose that the search space has n-dimension. Therefore, the i th particle in the swarm can be represented as X _i  = (x _i1 , …,x _id , …,x _in ), that in this article and the velocity vector is V _i  = (v _i1 , … v _id , … v _in ). The best encountered position by the ith particle is Pbest _i  = (pbest _i1 , … pbest _id , … pbest _in ). The best global position of the swarm found so far is denoted by Gbest = (gbest₁, …,gbest _d , … gbest _n ). The best encountered position and the best position encountered by any particle is used to update the velocity of each particle as follows:

After updating the velocity vector, the position vector of each particle is updated as follows:

In the above equations, i = 1,2, …,N _swarm, is the index of each particle, N _swarm is number of the swarms, t is the iteration number, rand ₁(0) and rand ₂(0) are random numbers between 0 and 1. The inertia parameter (W) in the PSO algorithm determines the impact of the velocity history on the new position. This parameter is used to adjust the global and the local search abilities. In other words, a large inertia weight tends the particles to the global optimal while a small inertia factor simplifies a local search. The acceleration parameters including C ₁ and C ₂ are called cognitive and social parameters, respectively. Cognitive parameter controls the effect of memory of each particle on the new velocity while social parameter controls the effect of Gbest on the new velocity.

The Modified PSO (MPSO)

In the population-based meta-heuristic optimization method, premature convergence may occur for various reasons such as: the population is converged to local optimum of the objective functions, or the diversity of the population is lost, or the search algorithm proceeds slowly or has not proceeded at all. There are several methods that can mitigate this deficiency. Mutation is a powerful strategy among them, which can keep the diversity of the population and extend the domain of the exploration. So, in this article, a new mutation method is suggested that is described as follows:

where is the j th mutated vector in the t th iteration, is the population from the swarm in the t th iteration, z ₁, z ₂, z ₃ are three integer numbers in the range [1, N _swarm] that are generated randomly, and z ₁ ≠ z ₂ ≠ z ₃ ≠ j. In the next step, should be mixed with (from the swarm) as follows:

In Equation (12), rand is a random number between 0 and 1, andcross is a constant value between 0.1 and 0.9. After generation of , the fitness function of this particle is compared with the ; each of them define a better solution, which is selected and stored as j th particle in the new population. The above process is repeated for all particles in the swarm and the new population is generated with the same size as the initial one. Here, the position vector of the initial population is updated based on Equations(9)–(10) and the N _swarm best particles between the updated population and new population is selected as the population for the next iteration. In this article, the value of C ₁ and C ₂ in MPSO algorithm is 1.49445 (Chiou, Chang, and Su 2005).

Self-Adaptive DPSO (SADPSO)

In the DPSO algorithm, the velocity of each particle is updated based on Equation (9), but the situation of particles are updated as follows:

In the self-adaptive framework the value of the parameters are considered the new control variables. So, they can change in the evolutionary process. This subject caused the value of the parameters to be adjusted based on the state of the algorithm. In SADPSO, the new control variables are as follows:

So, the improvement of each particle is affected not only by their cognition of individual thinking and social cooperation, but also by improving the way that they do it by accommodating themselves to the best known conditions: namely, their conditions when getting the best so far position and the leader's conditions.

The inertia weight, (W), plays a major role for the PSO algorithm's convergence behavior. The constant value for W has no suitable performance. Decreasing W in a linear trend for the entire run of PSO will not produce an optimal result in many cases. This control method has a linear transition of searching ability from global to local search. So, for better performance of the PSO search process, the inertia weight should be nonlinearly, dynamically adapted to obtain better dynamics of balance between global and local search abilities. A significant improvement in the performance of SADPSO, with decreasing inertia weight over the generations, is achieved using an equation proposed in He and Wang (2007).

This parameter is adaptively controlled by using Equation (14) in the proposed algorithm.

SHUFFLED FROG-LEAPING ALGORITHM (SFLA)

The Shuffled Frog Leaping Algorithm (SFLA) proposed by Eusuff and Lansey (2003) is a memetic meta-heuristic method that is inspired from the behavior of frogs seeking for food laid on separated stones that are randomly located in a swamp (Zhang et al. 2008). SFLA has been designed as a meta-heuristic method to perform an informed heuristic search using a heuristic function (any mathematical function) to seek a solution of a combinatorial optimization problem. The SFLA progresses in two steps: (1) memetic evolution by infection of ideas among themselves, (2) a shuffling technique that can cause information to be exchanged among local searches to improve toward global optimum. According to this abstract, the SFLA draws the concept of PSO as a local search tool and the idea of competitiveness and mixing information from parallel local searches to move toward global optimum from the Shuffled Complex Evolution (SCE) algorithm (Duan, Sorooshian, and Gupta 1992). In SFLA, each possible solution Xi = (x _i1 , …,x _id , …,x _in ), that in this article , where N _tie is the number of the tie variables (switches), is considered to be a frog. The steps of the algorithm are as follows:

Step 1: Create an initial population of k frogs generated randomly. The frogs are then sorted in descending order according to their fitness.

Step 2: Divide the frogs into p memplexes, each holding q frogs, such that k = pq. The method of this division is described as follows: the first frog with the best fitness value goes to the first memplex; the second memplex holds the second one. The p th frog goes to the p th memplex and (p + 1)th frog returns to the first memplex.

Step 3: Update the position of the worst frog in each memplex.

For each memplex, the frogs with the best fitness and worst fitness are identified as X _w and X _b , respectively. Also the frog with the global best fitness as Xg is identified. Then the position of the worst frog X _w in each memplex is adjusted as follows:

where, rand(.) is a random number between 1 and 0, and B _max is the maximum allowable change in the frog's position. If the evolutions produce a better frog (solution), it replaces the last frog. Otherwise, X _b is replaced by X _g in (15) and the process is repeated. If no improvement becomes possible, in this case a random frog is generated that replaces the old frog.

Step 4: Continue the calculation of Step 3 for a specific number of iterations.

Step 5: Reshuffle the frogs and sort them again.

Step 6: Return back to Step 2, if the termination criterion is not met, else stop.

Mutated Population in Each Memplex

In this section, computation steps of generating mutated population in each memplex are described. This new population tries to move the mean of the solution toward the best solution in each memplex. The mutated population is generated as follows:

In the above equations, M is the vector which holds the mean of each control variable for all frogs in each memplex, T _F is an integer number that equals 1 or 2, and rand is a vector of random numbers between 0 and 1.

After implementing Equations (17) and (18) for all frogs in each memplex, a new set of population is generated. Here, q best frog between new mutated population and updated population achieved from the Equations (15) and (16) are selected as the new population.

HYBRID SELF-ADAPTIVE MODIFIED PSO AND MODIFIED SHUFFLED FROG-LEAPING ALGORITHM (SAMPSO-MSFLA)

The goal of integrating Self-Adaptive Modified Particle Swarm Optimization (SAMPSO) and Modified Shuffled Frog-Leaping Algorithm (MSFLA) is to combine their advantages and avoid the disadvantages. For example, SFLA is a very efficient procedure but there exist some limitations that not only slow down the convergence speed, but also cause premature convergence. Furthermore, MPSO and SADPSO algorithms belong to the class of global search procedures, but require much computational effort; also, accuracy of MPSO and SADPSO isn't very high, but with the combination of these algorithms (MSFLA and SAMPSO), we obtain a new algorithm (SAMPSO-MSFLA), that not only isn't sensitive to the choice of initial points, but it has a faster rate and more accurate convergence than SAMPSO and MSFLA and other algorithms. This section begins with recollecting the procedures of SAMPSO and MSFLA that will be used for the DFR problem. The origins and literatures of these algorithms can be found in previous sections.

Figure 1 depicts the schematic representation of the proposed hybrid SAMPSO-MSFLA. The population size of this hybrid SAMPSO-MSFLA approach is set at 3N when solving an N-dimensional problem. The initial 3N particles are randomly generated and sorted by fitness, and the top N particles are then fed into the MSFLA method to improve the (N) particles. The other 2N particles are adjusted by the SAMPSO method. The procedure of adjusting the 2N particles in the SAMPSO method involves selection of the global best particle, selection of the neighborhood best particles, and, finally, update of the velocity. 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. By Equations (9)–(12), velocity and position are updated for each of the 2N particles and are then carried out. The global best particle of the population is determined according to the sorted fitness values. The 3N particles are sorted again in preparation for repeating the entire run.

FIGURE 1 Schematic representation of the SAMPSO-MSFLA. (Figure is provided in color online.)

To apply the SAMPSO-MSFLA algorithm in the DFR, the following steps have to be taken:

Step 1: import the basic data

In this step, the input data—including the network configuration, line impedance and status of switches, number of memplexes (p), number of frogs in each memplex (q), number of iterations, and the number of populations—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, which this operation performed by Equation (19):

f(X) is the objective function values of the DFR problem. N _eq and N _ueq are the number of equality and inequality constraints, respectively, of the DFR problem. The equality and inequality constraints, respectively, are h _i (x) and g _i (x), and k ₁ and k ₂ are penalty factors. Because the constraints should be met, the value of the k ₁ and k ₂ parameters should be high. In the article, these values are 10^8.

Step 3: Generate the initial population.

The population is in the following form:

where Tie _i is the status of the i th tie switch, which is zero or one and Sw _i is the switch number of the i th sectionalizing switch, and Ci ₁, Ci ₂ are tuning parameters in the PSO algorithm.

Step 4: Calculate the augmented objective function value for each individual by using the results of the distribution load flow. 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 based on the results of the distribution load flow. Then, the augment objective function is calculated by using the values of objective function, constraints, and penalty factors.

Step 5: Sort the initial population based on the augmented objective function values.

Step 6: Select the N generated population that has the best objective function value for the MSFLA, and select the 2N remaining generated population for the SAMPSO.

Step 7: Go to SFLA algorithm.

Step 8: Go to SAPSO algorithm and MPSO

Step 9: In this step, results of SAMPSO and MSFLA are combined.

Step 10: Check the termination criteria

If the termination criteria satisfied, finish the algorithm, else discard all previous trial solutions and go to S 6 until convergence criteria are met.

Simulation and Results

In this section, the SAMPSO-MSFLA algorithm is employed to solve the DFR problem. Two networks from different distribution systems were used to evaluate the approach proposed.

Case Study 1

The case study 1 is a 69-bus test system consisting of one source transformer, 68 load points and 73 branches. The five initially open switches are S11-66, S13-20, S15-69, S27-54 and S39-48. The total system load is 3802 kW, while the initial system power loss is 225.0 kW. The system base is Vb = 12.66 kV and Sb = 10 MVA. The system data is given in Su and Lee (2003) and the single line diagram of the system is shown in Figure 2.

FIGURE 2 A single line diagram of distribution system for case study 1.

Table 1 illustrates a comparison of the proposed algorithm and other methods in terms of computational efficiency and performance. It is observed that there are incongruities between the results reported by several authors with regard to the total active power losses; however, with respect to the open tie line, we can ensure that our results are equal to the results obtained by Li et al. (2007) – Yu et al. (2009).

TABLE 1 Results for Different Methods

	Initial network	Liu (2000)	Bi (2002)	Shirmohammadi (1998)	Li (2007)	Yu (2009)	Proposed method
Power loss (kW)	225.0	102.6	102.1	106.632	99.62	99.62	99.62
Saving %	–	54.513	54.588	52.487	55.656	55.656	55.656
Open switches	S11-66	S11-66	S11-66	S11-66	S11-66	S11-66	S11-66
	S13-20	S13-20	S13-20	S17-18	S13-20	S13-20	S13-20
	S15-69	S15-69	S15-69	S67-68	S14-15	S14-15	S14-15
	S27-54	S50-51	S50-51	S21-22	S50-51	S50-51	S50-51
	S39-48	S46-47	S48-49	S47-48	S47-48	S47-48	S47-48

Table 2 compares the proposed SAMPSO-MSFLA with the results of immune algorithm (Wenchuan and Jiaju 2006), family eugenics-based evolution algorithms (Xiufan, Lizi, and Lingyu 2004), genetic algorithm (Liu and Chen 2000), Tabu search (Shaoyun, Zifa, and Yixin 2004), Hybrid PSO (Li et al. 2007), and dynamic programming (Bosi, Kaigui, and Jiaqi 2005). According to Table 2, we have found that the power loss is almost the same as dynamic programming and Hybrid PSO, and is less than the other methods. The result validates the effectiveness of the proposed algorithm.

TABLE 2 Results for Different Algorithm

	Wenchuan (2006)	Xiufan (2004)	Liu (2000)	Shaoyun (2004)	Li (2007)	Bosi (2005)	Proposed method
Power loss (KW)	101.01	101.01	100.81	103.86	99.62	99.62	99.62

Table 3 illustrates the results of the proposed SAMPSO-MSFLA, the original SAPSO, and original SFLA for 50 random trials. According to Table 3, the result of SAPSO and original SFLA methods are rather weak, whereas the proposed method 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 as short as possible among the different methods.

TABLE 3 Comparison of Average and Standard Deviation for 50 Trials

			Worst solution		Best solution
Method	Average of objective function value	Standard deviation	Power losses[Kw]	Open Switches	Power losses[Kw]	Open Switches	CPU time [Sec] For best solution	No. of global solution
SAMPSO-MSFLA	99.62	0	99.62	S11-66, S13-20S14-15, S50-51S47-48	99.62	S11-66, S13-20S14-15, S50-51S47-48	∼4	50
SAPSO	102.3891	1.99732	115.90	S11-66, S17-18S67-68, S21-22S39-48	99.62	S11-66, S13-20S14-15, S50-51S47-48	∼10	34
SFLA	101.9489	1.192661	106.632	S11-66, S17-18S67-68, S21-22S47-48	99.62	S11-66, S13-20S14-15, S50-51S47-48	∼4	37

Figure 3 shows the comparisons between the proposed algorithm (SAMPSO-MSFLA) with SAPSO and SFLA methods in the convergence process for the case study 1, which illuminates the relationship between the iteration times and the best value of the objective function. It is obvious the value of the objective function decreases faster than the others when SAMPSO-MSFLA is applied.

FIGURE 3 The convergence process of the proposed algorithm (SAMPSO-MSFLA) in comparison with SAPSO and SFLA methods for case study 1. (Figure is provided in color online.)

Case Study 2

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 Figure 4. The system data is given in Das (2006). Before reconfiguration, the initial loss is 227.53 kW.

FIGURE 4 A single-line diagram of distribution system for case study 2.

Table 4 illustrates a comparison of the proposed algorithm and other methods in terms of computational efficiency and performance. It is observed that the obtained results by the proposed method are better than other methods.

TABLE 4 Comparison of the Proposed Algorithm with Other Methods

Method	Power losses [kW]	Saving in power [%]	Open switches
Initial network	227.53	–	s22-67, s15-67, s21-27, s9-50, s29-64, s9-38, s45-60, s38-43, s9-15, s39-59, s15-46
Das (2006)	205.32	9.76	s65-66, s15-67, s26-27, s49-50, s29-64, s9-38, s44-45, s37-38, s14-15, s39-59, s15-46
Niknam (2009)	205.32	9.76	s65-66, s15-67, s26-27, s49-50, s29-64, s9-38, s44-45, s37-38, s14-15, s39-59, s15-46
Proposed method	202.18	11.14	s62-65, s15-67, s21-27, s49-50, s28-29, s9-38, s44-45, s38-43, s9-15, s39-59, s15-46

Table 5 illustrates the results of the proposed SAMPSO-MSFLA, the original SAPSO, and the original SFLA for 50 random trials. According to Table 5, the results of SAPSO and SFLA methods are rather weak, whereas the proposed method is very powerful, and in 49 trials the best solution is obtained and execution time of proposed method is as short as possible among the different methods.

TABLE 5 Comparison of Average and Standard Deviation for 50 Trials

		Worst solution		Best solution
Method	Average of objective function value	Power losses [Kw]	Open Switches	Power losses [Kw]	Open Switches	No. of global solution
SAMPSO-MSFLA	202.267	205.05	s65-66, s15-67, s26-27, s49-50, s29-64, s37-38, s44-45, s36-38, s9-15, s39-59, s15-46	202.18	s62-65, s15-67, s21-27, s49-50, s28-29, s9-38, s44-45, s38-43, s9-15, s39-59, s15-46	49
SAPSO	204.471	209.04	s65-66, s15-67, s21-27, s9-50, s29-64, s37-38, s44-45, s36-38, s9-15, s39-59, s15-46	202.18	s62-65, s15-67, s21-27, s49-50, s28-29, s9-38, s44-45, s38-43, s9-15, s39-59, s15-46	35
SFLA	203.839	207.07	s65-66, s15-67, s21-27, s9-50, s29-64, s37-38, s44-45, s9-38, s9-15, s39-59, s15-46	202.18	s62-65, s15-67, s21-27, s49-50, s28-29, s9-38, s44-45, s38-43, s9-15, s39-59, s15-46	38

Figure 5 compares the proposed algorithm (SAMPSO-MSFLA) with SAPSO and SFLA methods in the convergence process for case study 2. From Figure 4, it is obvious that the proposed algorithm converges to the global solution faster than the others.

FIGURE 5 Convergence processes of the proposed algorithm (SAMPSO-MSFLA) in comparison with MSFLA and SAMPSO methods for case study 2. (Figure is provided in color online.)

CONCLUSION

This article proposes a hybrid evolutionary algorithm based on combining SAMPSO and MSFLA, called SAMPSO-MSFLA, to cope with the DFR problem. In the proposed algorithm, the PSO and SFLA are modified to extend the domain of the local searches and restrain the premature convergence. In the SAMPSO, learning factors of DPSO are coevolved with the particles during optimization process. In addition, for improving the performance of SAMPSO, we combined an MSFLA with SAMPSO. In this article the DFR problem is solved by aiming the minimization of the power losses in the distribution network while the load balancing on the transformers and feeders, maximum load on feeders, and the voltage deviations from the nominal values are considered as the constraints. Simulation results on two test feeders show that the proposed method is more effective and robust when compared with other methods. Since the proposed method reaches the optimal solution rapidly, it can be a good choice for solving practical problems.