Network reconfiguration is an alternative to reduce power losses and optimize the operation of power distribution systems. In this paper, an encoding scheme for evolutionary algorithms is proposed in order to search efficiently for the Paretooptimal solutions during the reconfiguration of power distribution systems considering multiobjective optimization. The encoding scheme is based on the edge window decoder (EWD) technique, which was embedded in the Strength Pareto Evolutionary Algorithm 2 (SPEA2) and the Nondominated Sorting Genetic Algorithm II (NSGAII). The effectiveness of the encoding scheme was proved by solving a test problem for which the true Paretooptimal solutions are known in advance. In order to prove the practicability of the encoding scheme, a real distribution system was used to find the near Paretooptimal solutions for different objective functions to optimize.
Modern societies require a complex system of generating plants, interconnected transmission lines, and distribution systems. The overall power losses in the generation, transmission, and distribution of electrical energy are estimated in 8–15% [
An alternative to reduce power losses in distribution systems is network reconfiguration [
Considering that modern distribution system may have thousands of possible topologies and the nonlinear nature of power losses, the distribution system reconfiguration (DSR) problem can be defined as a highly complex, combinatorial, and nondifferentiable optimization problem. Furthermore, the radiality constraint introduces additional complexity to the problem, especially in large size distribution networks. Because of this, new algorithms are emerging continuously to deal with the complexity of optimizing radial power distribution system operation.
Metaheuristic algorithms using a multiobjective approach for solving the DSR problem have been very popular in the last decade [
One of the main challenges in EA design for loss reduction in DSR problems is how to encode the possible solutions or system topologies in order to make the search efficient and effective. A good encoding strategy should be capable of representing all possible solutions, must facilitate that genetic operators are being implemented in an efficient way, and also should be inexpensive in evaluating the fitness function and constraints while moving easily between the encoded solution and its representation. In addition, the encoding strategy must generate only solutions with radial topologies; otherwise an excessive number of unfeasible solutions may be generated, reducing the efficiency and effectiveness of the search process. According to research results presented in [
The problem of finding the Paretooptimal reconfigurations in distribution networks considering several objective functions is quite similar to the multiobjective minimum spanning tree (MOMST) problem. This analogy is relevant because, in [
In this paper, an encoding scheme for representing the system configuration during DSR problems is presented. The proposed encoding scheme was embedded in the NSGAII [
The main contributions of the paper are as follows: (1) the encoding technique proposed, combined with specialized genetic operators, can explore the search space efficiently, finding the true Paretooptimal solutions for MOMST problems and near Paretooptimal solutions for DSR problems; (2) the proposed encoding techniques and genetic operators are capable of dealing with the radiality constraint in a multiobjective search space; (3) the encoding technique enables the search process to find welldispersed near Paretooptimal solutions in largescale power distributions systems.
The DSR problem basically consists in determining a new topology that minimizes different objective functions. Since the main concern during normal distribution system operation is efficiency and power quality, in this paper three objectives are considered: minimizing power losses, minimizing bus voltage deviations, and minimizing the number of operated switches during the DSR process. The objective function for loss reduction due to Joule effect in the different line sections is
The objective function to optimize voltage deviations at every node in the distribution system is
Finally, it is also desirable that during the reconfiguration process the number of operated switches is as small as possible in order to reduce the reconfiguration time, the probability of human error, and operation costs. The objective function used to minimize the number of operated switches in the reconfiguration process is
All new topologies generated during the DSR process must also comply with the following system constraints: (a) a radial network structure must be obtained after network reconfiguration in which all loads are energized, (b) the apparent power on each line section must be smaller than the maximum apparent power allowed, (c) the voltage level in a given node must be within allowed limits, and (d) the apparent power at the substations transformers must be within allowed limits. These constraints can be formally expressed as follows:
In addition, the multiobjective optimization process for solving the DSR problem has two goals that must be fulfilled [
The proposed encoding scheme based on the EWD technique, but modified and adapted to DSR problems, was implemented in two multiobjective EAs: the Strength Pareto Evolutionary Algorithm 2 (SPEA2) and Nondominated Sorting Genetic Algorithm II (NSGAII). Also, specialized recombination and mutation operators were developed in order to guarantee a radial configuration in all the new possible solutions generated along the evolutionary process. The salient steps of the developed encoding scheme and the evolutionary operators for the DSR problem are described in the next sections.
The EWD encoding scheme was used to codify each possible solution in the DSR problem. In order to describe the encoding scheme, let us consider the distribution network shown in Figure
Power distribution system for optimal reconfiguration.
The encoding process consists basically of the following two steps. First, an initial string is built up by visiting every node and branch in the network without leaving the paper. For example, for the network in Figure
This process is continued through the end of the string. Observe that edge sets
New configuration for the distribution system.
The edge set
The crossover operator used in this study is presented in [
Offspring generated during crossover.
This crossover operator generates only legal offspring solutions (radial configuration), avoiding problems of low heritability and topological unfeasibility. Figure
Distribution network after crossover.
Like the crossover operator, the mutation operator implemented in this research was presented in [
Distribution network after mutation.
The proposed encoding technique was embedded in two of the most representative stateoftheart multiobjective evolutionary algorithms: the Strength Pareto Evolutionary Algorithm 2 (SPEA2) and the Nondominated Sorting Genetic Algorithm
Evolutionary algorithms for solving single or multiobjective optimization problems are often criticized for their lack of theoretical foundation. In the case of multiobjective optimization problems a question frequently arises: How close are the obtained solutions to the Paretooptimal front? In the EA literature such questions are often addressed by first solving a set of test problems for which the optimal solutions are known in advance. Such exercises provide confidence about the efficacy of the proposed procedure before being applied to a real problem where the optimal solutions are not known. Therefore, testing the proposed encoding strategy in solving a given MOMST problem is appropriate in order to assess how effective the encoding strategy can be for solving multiobjective DSR problems.
The MOMST problem used to assess the proposed encoded strategy is described as follows [
Figure
Paretooptimal front for the multiobjective minimum spanning tree problem.
A real distribution system with 84 nodes and 531.99 kW of power losses in its initial configuration is used for practical validation purposes. The system has been used by many researchers for comparative purposes and the one line diagram and system data can be found in [
In this case, the problem consists in simultaneously minimizing power losses,
From Figure
Table
Characteristics of the approximate Paretooptimal solutions in Figure
Solution number  Line sections open  Power losses (MW)  Switching operations 

1  84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96  0.5320  0 
2  34, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96  0.5096  1 
3  7, 34, 84, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96  0.4900  2 
4  7, 34, 63, 84, 86, 87, 88, 89, 90, 91, 92, 93, 95  0.4771  3 
5  7, 34, 63, 72, 84, 86, 88, 89, 90, 91, 92, 93, 95  0.4731  4 
6  7, 34, 63, 72, 83, 84, 86, 88, 89, 90, 92, 93, 95  0.4707  5 
7  7, 13, 34, 63, 72, 83, 84, 86, 89, 90, 92, 93, 95  0.4704  6 
8  7, 13, 34, 55, 62, 72, 83, 86, 89, 90, 92, 93, 95  0.4701  7 
9  7, 13, 34, 39, 55, 62, 72, 83, 86, 89, 90, 92, 95  0.4700  8 
10  7, 13, 34, 39, 42, 55, 62, 72, 83, 86, 89, 90, 92  0.4699  9 
Approximate Paretooptimal solutions to the problem of optimizing power losses
A second study in the same distribution network was carried out. It is now required to optimize using a multiobjective approach the following objective functions: power losses,
This behavior is explained by the fact that, for this particular study, during the optimization process improvements in
In this particular case there exist only two configurations for which both objective functions are competing between them and both configurations are nondominated solutions. Figure
Table
Characteristics of the approximate Paretooptimal solutions in Figure
Solution number  Line sections open  Power losses (Mw)  Voltage deviation 

1  7, 34, 39, 42, 55, 62, 72, 83, 86, 88, 89, 90, 92  0.4702  0.002931 
2  7, 13, 34, 39, 42, 55, 62, 72, 83, 86, 89, 90, 92  0.4699  0.002905 
Approximate Paretooptimal solutions to the problem of optimizing power losses
Solutions in the search space for optimizing power losses and voltage deviations.
Now, the DSR problem is solved optimizing voltage deviations in the different system nodes,
In general, the form of the approximate Paretooptimal front obtained with NSGAII and SPEA2 algorithms using the encoding strategy is basically the same except one set of solutions on which both algorithms achieve different values. However, it should be remembered that these are metaheuristics methods, not analytical ones, and there exists the probability of some small difference in the results in each simulation run.
Table
Characteristics of the approximate Paretooptimal solutions in Figure
Solution number  Line sections open  Voltage deviation  Operated switches 

1  84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96  0.0033805  0 
2  34, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96  0.0032245  1 
3  34, 72, 84, 85, 86, 88, 89, 90, 91, 92, 93, 95, 96  0.0031690  2 
4  7, 34, 62, 84, 86, 87, 88, 89, 90, 91, 92, 93, 95  0.0030271  3 
5  7, 34, 62, 72, 84, 86, 88, 89, 90, 91, 92, 93, 95  0.0029678  4 
6  7, 34, 62, 72, 83, 84, 86, 88, 89, 90, 92, 93, 95  0.0029392  5 
7  7, 34, 55, 62, 72, 83, 86, 88, 89, 90, 92, 93, 95  0.0029166  6 
8  7, 34, 39, 62, 72, 83, 84, 86, 88, 89, 90, 92, 95  0.0029314  6 
9  7, 34, 39, 55, 62, 72, 83, 86, 88, 89, 90, 92, 95  0.0029087  7 
10  7, 34, 39, 42, 55, 62, 72, 83, 86, 88, 89, 90, 92  0.0029046  8 
Approximate Paretooptimal solutions to the problem of optimizing voltage deviations
In this case it is desired to optimize simultaneously the three objective functions,
A close comparison with previous results shows that this set of solutions contains the best solutions found in Figures
Table
Characteristics of the Paretooptimal solutions in Figure
Solution number  Line sections open  Power losses (Mw)  Voltage deviation  Operated switches 

1  84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96  0.5320  0.0033805  0 
2  34, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96  0.50957  0.0032245  1 
3  7, 63, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95  0.49949  0.0032055  2 
4  7, 62, 84, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95  0.49992  0.0031927  2 
5  7, 34, 84, 86, 87, 88, 89, 90, 91, 92, 93, 95, 96  0.48997  0.003276  2 
6  34, 72, 84, 85, 86, 88, 89, 90, 91, 92, 93, 95, 96  0.50562  0.003169  2 
7  7, 34, 62, 84, 86, 87, 88, 89, 90, 91, 92, 93, 95  0.4775  0.0030271  3 
8  7, 34, 63, 84, 86, 87, 88, 89, 90, 91, 92, 93, 95  0.47707  0.0030406  3 
9  7, 34, 62, 72, 84, 86, 88, 89, 90, 91, 92, 93, 95  0.47354  0.0029678  4 
10  7, 34, 63, 72, 84, 86, 88, 89, 90, 91, 92, 93, 95  0.47311  0.0029816  4 
11  7, 34, 63, 72, 83, 84, 86, 88, 89, 90, 92, 93, 95  0.47066  0.0029532  5 
12  7, 34, 62, 72, 83, 84, 86, 88, 89, 90, 92, 93, 95  0.47109  0.0029392  5 
13  7, 13, 34, 63, 72, 83, 84, 86, 89, 90, 92, 93, 95  0.47035  0.0029792  6 
14  7, 34, 55, 62, 72, 83, 86, 88, 89, 90, 92, 93, 95  0.47046  0.0029166  6 
15  7, 13, 34, 55, 62, 72, 83, 86, 89, 90, 92, 93, 95  0.47015  0.002943  7 
16  7, 34, 39, 55, 62, 72, 83, 86, 88, 89, 90, 92, 95  0.47032  0.0029087  7 
17  7, 34, 39, 42, 55, 62, 72, 83, 86, 88, 89, 90, 92  0.47019  0.0029046  8 
18  7, 13, 34, 39, 55, 62, 72, 83, 86, 89, 90, 92, 95  0.47001  0.0029351  8 
19  7, 13, 34, 39, 42, 55, 62, 72, 83, 86, 89, 90, 92  0.46988  0.0029311  9 
Approximate Paretooptimal solutions to the problem of optimizing
One of the main challenges in designing an EA to solve DSR problems is how to encode the possible solutions in order to make the EA process efficient and effective. The demand for solutions with radial topologies makes developing a good encoding strategy for solving the DSR problem more difficult, since this constraint can make encoding methods and their genetic operators generate an excessive number of unfeasible solutions, reducing the efficiency and effectiveness of the search process. Additionally, in a multiobjective optimization context, the effectiveness of an encoding scheme affects the quality of the Paretooptimal solutions obtained, expressed in terms of a set of solutions as diverse and close as possible to the true Paretooptimal solutions.
In this paper, an encoding technique for representing the distribution system topology during DSR problems has been presented. The technique is based on the EWD technique but modified and adapted for solving DSR problems. The encoding scheme was embedded in the NSGAII and SPEA2 algorithms and applied to obtain the true Paretooptimal solutions in a MOMST problem. The encoding scheme was also applied to obtain the approximate Paretooptimal solutions in a real distribution system during system reconfiguration considering a multiobjective approach.
The obtained results show that the proposed encoding technique enables the NSGAII and SPEA2 algorithms to find the true Paretooptimal solutions, which means that the proposed encoding technique and their genetic operators are suitable to be used in multiobjective evolutionary algorithms to solve MOMST based problems. Also, to the best knowledge of the authors, this is the first time that an encoding technique proposed for DSR problems is tested in a MOMST problem.
In order to prove the practicability of the encoding scheme, a real distribution system was used to find the near Paretooptimal solutions for different scenarios. The objective functions to optimize used in this analysis are aimed at minimizing power losses, voltage deviation on system nodes, and the number of operated switches during the reconfiguration process. However, any other objective function can be incorporated into the analysis.
According to the results of this analysis, the encoding technique is suitable to find the near Paretooptimal solutions in different scenarios of DSR. Also, the obtained Paretooptimal solutions showed a welldispersed characteristic, and some of these solutions correspond to the optimal solutions presented in the literature for monoobjective formulations used for optimizing a single variable in DSR problems.
From all the obtained results, it can be concluded that the encoding strategy is valid and can be successfully used for solving singleobjective optimization problems as well as multiobjective optimization problems. This is relevant because it has been proved that the good performance of some encoding techniques for solving singleobjective optimization problems is not necessarily the same, or approximately the same, when they are applied for solving multiobjective optimization problems [
In addition, the versatility of the encoding strategy allows for the use of efficient operators for crossover and mutation which guarantees an excellent global and local search, generating only legal solutions. These genetic operators can also be adapted easily to the problem to be solved.
Since the performance of proposed encoding strategy was analyzed in two multiobjective evolutionary algorithms, research should be done on other encoding techniques to be able to draw more general conclusions related to the encoding strategy proposed in this paper. Possible interesting directions would be comparing different encoding techniques tested on distributions networks with different characteristics such as level of power demand, total length of the feeders, and average conductor size of the feeders.
Table
Cost associated with each line.
Nr 


Nr 



0  0.0000  0.0000  23  17.3314  34.806 
1  31.6776  32.9922  24  72.6631  46.7469 
2  65.3549  35.8537  25  68.0291  44.9918 
3  44.9595  23.4323  26  51.6128  22.2112 
4  16.1865  23.6162  27  16.5226  34.7425 
5  21.971  38.0799  28  96.3964  17.0508 
6  60.7701  42.5447  29  25.0725  16.6977 
7  20.1846  39.823  30  37.0809  11.0776 
8  15.6921  31.8862  31  74.4496  28.6943 
9  12.3896  43.8891  32  66.3899  35.9456 
10  38.2169  42.847  33  22.3423  31.9243 
11  87.851  18.5533  34  85.3415  26.3923 
12  27.4456  44.4433  35  75.702  30.5113 
13  52.9695  34.3833  36  52.4566  15.5317 
14  24.7319  23.5492  37  18.4093  32.6586 
15  18.9167  28.3563  38  39.4639  39.5681 
16  38.8376  36.9862  39  46.0833  28.0846 
17  69.9022  36.481  40  54.0409  37.3085 
18  56.6392  27.931  41  39.8044  20.8134 
19  81.37  39.3791  42  69.8945  46.8211 
20  53.6563  34.8405  43  41.2821  11.2217 
21  47.2281  15.6442  44  39.8463  12.9746 
22  99.8769  30.3881  45  90.4494  21.9358 
The authors declare no conflict of interests.
The authors acknowledge the support provided by CONACYT and DGEST of México.