1. Introduction
With the increasing demand for electricity, the size and complexity of modern distribution networks have increased significantly [
1]. In practice, most power outages are due to damage caused by distribution network failures, which can have serious consequences for both utilities and customers [
2]. At the same time, medium-voltage direct current (MVDC) distribution networks are becoming increasingly important in view of the large-scale application of DC technology [
3,
4,
5]. Compared with the traditional AC distribution network, the flexible DC distribution network based on voltage source converter (VSC) technology has the advantages of easy access to distributed energy and multiple loads, and flexible control [
6]. Therefore, to improve power supply reliability and customer satisfaction, it is necessary to study the fault recovery reconfiguration of the DC distribution network.
Reconfiguration of the distribution network aims to find the optimal combination of all switches in the distribution, and achieve objects as follows: (1) maximizing demand, (2) minimizing the number of switching operations, (3) giving priority to automatic switching operations over manual operations, and (4) giving priority to special loads [
7]. Distribution network reconfiguration (DNR) is essentially a multi-objective nonlinear hybrid optimization problem, which is mathematically an NP-hard problem [
8].
There are several common reconfiguration methods, including heuristic algorithms, artificial intelligence (AI), and mathematical modeling. Heuristic algorithm is an intuitive analysis algorithm. It iterates step-by-step, according to certain principles, to obtain satisfactory results, and mainly includes branch-and-swap and optimal flow methods. In [
9], the branch switching pair that caused the most network loss reduction is selected for branch switching in a simple way. In [
10], all switches are closed, and the switches with the least optimal flow are subsequently opened sequentially until the reconfiguration solution is obtained. In addition, inspired by the fractal theory for solving optimization problems, implementation of a stochastic fractal search (SFS) algorithm is proposed to solve the distribution network reconstruction problem [
11]. Although the method of [
11] is an improvement on [
9,
10], it still cannot guarantee that the solution is the global optimal solution. In addition, ref [
12] applies a heuristic algorithm based on Harris Hawks optimization to DNR, which can ensure global convergence performance. Ref [
13] uses the discrete network reconfiguration of the data set method, which can significantly improve the effectiveness of the distribution network. This type of method has good convergence performance, and can obtain a global optimal solution for single-objective optimization problems.
AI algorithms for solving reconstruction problems are mainly based on swarm optimization methods. In [
14], a genetic algorithm (GA) based on a standard benchmark problem is used to solve single-objective optimization problems. PSO has the advantages of simple principle, robustness, and easy implementation [
15]. A binary particle swarm algorithm (BPSO) is proposed in [
16] to solve the DNR problem. A new optimization algorithm called the Salp swarm algorithm can effectively solve the optimization problem. The beetle antennae search (BAS) algorithm is an intelligent optimization algorithm, which has the advantages of a simple principle, fewer parameters, and less calculation required [
17]. The cuckoo algorithm (CA) can effectively solve the optimization problem of reconstruction by simulating the parasitic brooding of some species of cuckoo, and using the related Lévy flight search mechanism [
18,
19]. The above methods have been successfully applied to the field of DNR with better results than heuristic algorithms. However, a common problem is that they tend to fall into premature maturity and still cannot guarantee global optimality.
In addition, the reconfiguration problem can also be solved using mathematical modeling. In [
20], a mixed-integer linear programming (MILP) model was developed to solve the distribution network fault recovery–reconstruction problem. Based on MILP, a mixed-integer second-order cone programming formulation for service restoration in distributed generation distribution networks was proposed, which relaxes the original non-convex tidal equation to quadratic form [
21]. Meanwhile, the problem of uncertainty for renewable energy or load can be solved in the mathematical model by stochastic optimization [
22] or robust optimization [
23] to obtain the theoretically optimal solution. Mathematical modeling can be proved by formulas, but solving non-convex problems takes too long to solve. Therefore, linearized mathematical models and convex transformation of the original non-convex problems are further difficulties to be overcome for this kind of problem.
AI is a stochastic class of algorithms, and this class of methods has been successfully applied to solve DNR. Since traversing the feasible solution space takes a lot of computational time, the efficiency of the algorithm becomes crucial. The key to improving the efficiency of the algorithm lies in the encoding method, and the corresponding evolutionary approach. In traditional distribution networks, most distribution systems are designed as weak meshes, but usually operate in a radial topology to efficiently coordinate their protection systems [
24,
25]. In [
26], a loop-based coding method was proposed to disconnect a branch in an independent loop to satisfy radial constraints, and increase the proportion of feasible solutions. The resulting solutions are all feasible, but the repetition rate is too high, reducing evolutionary efficiency. In [
27], a new graph theory-based method is proposed for restoring the distribution system after multiple simultaneous faults due to extreme weather conditions. All of the above different encoding methods and evolutionary strategies improve the encoding efficiency to a certain extent, but still need to carry out the feasibility testing process of the solution, and the computation time is still long.
In summary, the reconstruction method based on the stochastic algorithm can solve the optimal solution more efficiently when solving large-scale distribution network problems. However, the stochastic algorithm generates many invalid solutions, and needs to judge whether the generated solutions are valid, which reduces the evolutionary efficiency. Meanwhile, the stochastic algorithm falls into local optimum easily. Therefore, a reconfiguration strategy based on HPSO is proposed. The main contributions of this paper are summarized as follows:
- (1)
Considering the topological characteristics of DC distribution, the network structure is equivalently simplified to improve the search efficiency in the DNR process;
- (2)
An adaptive coding strategy is designed to make the generated solutions satisfy the topological constraints. This strategy can improve the evolution efficiency of particles in the particle search process without judging infeasible solutions;
- (3)
To avoid the algorithm from falling into premature, the idea of Lévy flight (LF) is introduced to improve the global search ability and convergence speed of the particles.
The paper is organized as follows. The mathematical modeling of DNR is in
Section 2. In
Section 3, the simplification and coding method of the distribution network is proposed. In
Section 4, an HPSO combining adaptive coding strategy and improved discrete particle swarm optimization (DPSO) is proposed. In
Section 5, the proposed algorithm is compared with other algorithms such as GA, PSO, and CA to verify the efficiency and accuracy of HPSO.
Section 6 summarizes the paper.
5. Case Study
The structure of the original IEEE 33-node arithmetic example is shown in the original diagram in
Figure 2. Since the original example is an AC system, a VSC converter is added between node 1 and node 2, and the reactive load of the node load and the reactance of the branch circuits are ignored to make it a DC distribution network. To prove the effectiveness of the algorithm, we take the same approach for the IEEE 69-node improvement as a test case. In the modified IEEE 33-node, branch 7 is set as a faulty branch; in the modified IEEE 69-node example, branch 14 is set as a faulty branch. Since the current study is a static reconstruction problem, the test is focused on a single time section.
Table 1 shows the parameters of different algorithms. This paper mainly considers the proposed HPSO in comparison with PSO, CA, GA, and BAS. The population size and maximum number of iterations are kept consistent for all algorithms. In addition to the variables that have already appeared, it is necessary to account for the added parameters. In GA,
pc, pm,
pw, and
pn denote the crossover probability, the exchange variation probability, the inverse variation probability, and the addition probability, respectively. The
η in BAS denotes the coefficient of variable step size.
5.1. Comparison of Differen Metaheuristic t Algorithms
The metaheuristic algorithm seeks the optimal solution mainly through iterations of pseudo-probabilities, leading to differences in the results of each run. Therefore, each algorithm is set to run 50× to compare the mean value of the optimal solution and running time. In addition, the number of successful convergences to the global optimal solution is used as a comparison metric.
Table 2 shows the comparison results of different algorithms. Among all algorithms, HPSO has the highest number of successful solutions, which results in the smallest mean value of all solutions. The convergence performance of HPSO is faster than other algorithms, because the adaptive solution of HPSO reduces the judgment process of feasible solutions, which further improves the operation efficiency. PSO and BAS have simpler algorithm principles and faster iterations, thus the running time is smaller than the other two algorithms. In addition, the network architecture of 69 nodes is more complex than that of 33 nodes, thus the algorithms take longer to solve 69 nodes and have fewer successful solutions.
Figure 3 shows the comparison of evolution curves of different algorithms. Among the results of the runs for the improved IEEE 69-node, the one with the fastest convergence is selected for comparison. In
Figure 3, the maximum number of iterations for each algorithm is 50. In the enlarged local plot, HPSO converges to the global optimal solution most quickly after 9 iterations. Although the running time of GA is longer, the optimal solution can be obtained in fewer iterations than other methods, and it only took 12 iterations to converge successfully. Therefore, the fastest number of iterations and running time are a pair of contradictory metrics in the DNR, and HPSO is highly compatible with them.
5.2. Comparison of Different Algorithms
HPSO is divided into two main parts: branched group selection optimization based on adaptive solving, and the internal optimization of branched groups based on LF. In order to reflect the advantages of HPSO, the following three algorithms are set up separately for comparison:
Improved binary particle swarm optimization (IBPSO):The branch group selection optimization uses the improved BPSO algorithm based on adaptive coding strategy proposed in this paper, and the conventional DPSO is used for the internal optimization of the branch group.
Improved discrete particle swarm optimization (IDPSO): The internal optimization of the branch group adopts the improved DPSO based on LF proposed in this paper, and the optimization of branch group selection uses the conventional BPSO.
HPSO: The branch group selection optimization adopts the improved BPSO based on adaptive solution proposed in this paper, and the internal optimization of the branch group adopts the improved DPSO based on LF.
Table 3 shows the results of different algorithms. On the one hand, IBPSO has the fastest running speed, since LF is not included in the intra-branch group search, which saves computation time. On the other hand, the lack of LF also reduces the diversity of particles, which makes IBPSO the weakest in global search. IDPSO has the same global search ability asHPSO, which is due to the inclusion of the LF strategy in the intra-branch group search. The particle search space becomes wider, which improves the population’s merit-seeking ability. However, the average running time is longer than that ofHPSO. The lack of adaptive solution strategy causes IDPSO to require the inclusion of infeasible solution judgment, which increases the total running time of the intelligent algorithm. HPSO combines the advantages of both. It can not only ensure the global merit-seeking ability of the population by improving DPSO, but also improve the algorithm’s operation efficiency by adaptive solution method. In summary, HPSOhas a better global search capability than Algorithm 1, and also can converge faster than IDPSO. The test verifies the effectiveness of the proposed algorithm HPSO.
Figure 4 shows the comparison of evolution curves of different algorithms. Three algorithms obtained the global optimal solution. HPSO achieves convergence the fastest and obtains the global optimal solution at the ninth iteration. The convergence performance of IDPSO is basically the same as that of HPSO, but the convergence speed of IBPSO is much smaller than those of IDPSO andHPSO. Therefore, the convergence performance of Algorithm 1 is weaker than those of the other algorithms, which proves the necessity of the intra-branch group optimization search strategy.
5.3. Validity Analysis of Conclusions
Table 4 compares the network losses and minimum voltage of the distribution network at different periods.
Figure 5 shows the voltage comparison of each node before and after DNR. To verify the effectiveness of the reconfiguration scheme, a comparative analysis is performed for the distribution network before and after reconfiguration.
Both the network loss and voltage excursion of the reconfigured system were significantly reduced. In the 33-node case, the network loss value is reduced by 34.4%, and the minimum voltage of the distribution network is increased by 3.1%, which improves the economic performance of the system. In the 66-node case, the network loss value is reduced by 55.8%, the system loss is reduced, and the minimum voltage of the distribution network is increased by 4.0%. In
Figure 5, the voltage magnitudes after reconfiguration are all improved, which improves the reliability of the system. The above diagram proves the effectiveness of the HPSO, which can be successfully applied to practical engineering.