J. Cent. South Univ. (2018) 25: 600-615

DOI: https://doi.org/10.1007/s11771-018-3764-1

A compound objective reconfiguration of distribution networks using hierarchical encoded particle swarm optimization

WEN Juan(文娟)^{1, 2}, TAN Yang-hong(谭阳红)¹, JIANG Lin(蒋林)², XU Zu-hua(徐祖华)²

1. College of Electrical and Information Engineering, Hunan University, Changsha 410082, China;

2. College of Electrical Engineering, University of South China, Hengyang 421000, China

Central South University Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Abstract: With the development of automation in smart grids, network reconfiguration is becoming a feasible approach for improving the operation of distribution systems. A novel reconfiguration strategy was presented to get the optimal configuration of improving economy of the system, and then identifying the important nodes. In this strategy, the objectives increase the node importance degree and decrease the active power loss subjected to operational constraints. A compound objective function with weight coefficients is formulated to balance the conflict of the objectives. Then a novel quantum particle swarm optimization based on loop switches hierarchical encoded was employed to address the compound objective reconfiguration problem. Its main contribution is the presentation of the hierarchical encoded scheme which is used to generate the population swarm particles of representing only radial connected solutions. Because the candidate solutions are feasible, the search efficiency would improve dramatically during the optimization process without tedious topology verification. To validate the proposed strategy, simulations are carried out on the test systems. The results are compared with other techniques in order to evaluate the performance of the proposed method.

Key words: distribution network reconfiguration; node importance degree; compound objective function; hierarchical encoded

Cite this article as: WEN Juan, TAN Yang-hong, JIANG Lin, XU Zu-hua. A compound objective reconfiguration of distribution networks using hierarchical encoded particle swarm optimization [J]. Journal of Central South University, 2018, 25(3): 600–615. DOI: https://doi.org/10.1007/s11771-018-3764-1.

1 Introduction

Distribution network is an important part of the power system infrastructure that acts as a link between the high-voltage transmission system and the end-users of electric energy. A distribution network is composed of many inter-connected feeders with tie switches and sectionalized switches. Distribution network reconfiguration (DNR) is the process of adjusting the structure by altering the status of some tie and sectionalized switches. The aim of DNR is obtaining a radial topology that satisfies some objectives of optimally utilizing the power system and improving the reliability and efficiency [1–5].

In DNR, traditional goal of the minimum system loss has gained much attention as it reflects the cost of electrical energy [6]. Other objectives have focused on the reliability and usage life of equipment [4, 7–11]. With the rapid development of power industry, the structures of distribution systems become more and more complex. The network survivability, as one of the vital factors affecting the analysis and design of topology structure, the enhancement of the network invulnerability stands for improving the ability to maintain power supply or restoring its performance in the situation of power interruptions [12].

According to the complex network theory, the node importance degree (NID) is considered as an effective evaluation index of reconfiguration as it can help us to find the topology structure of enhancing the invulnerability and robustness of a distribution network [13, 14]. DENG et al [15] and LI et al [16] give an evaluation approach of node importance degree from the perspective of system science. And then it is used to analyze the relationship between nodes and cascade failures in Ref. [17].

In Refs. [18, 19], the NID is regarded as an index of transmission line network reconfiguration. ZHANG et al [20] employ maximization restoration paths as the reconfiguration objective of finding the optimal restoration path through analyzing the node importance of each node.

However, these reviews have focused on the contributions related on the skeleton-network reconfiguration of transmission network [17–20]. The index of the NID is disregard during distribution network reconfiguration.Furthermore, no reference exists to indicate the relation of node importance degree and system power loss. In this paper, the DNR problem is finding the radial configurations that optimize the objectives of the node importance degree and power loss as satisfying the optimum operating constraints. This problem is a complex combinatorial and non-linear optimization problem that makes it impossible to use traditional mathematical optimization method to address such a problem.

Recently, intelligent search algorithms are used to find an optimal topology [21–27]. For example, simulated annealing [21], evolutionary programming [22], genetic algorithm [23, 24], particle swarm optimization [25, 26], ant optimization [27], and so on. In applying the algorithms, the feasible candidate solutions obtained is the key to deal with the DNR.

SIVANAGARAJU et al [28] attempts to choose all the switches as a set of variables expression which is used to represent the topologies of the network [28]. With such encoding scheme, each element of a solution vector represents one switch. The binary value 0 or 1 indicates that the status of corresponding switch is open or closed. Since there are many switches in a distribution network, the length of a code string is very long. The scheme has proved to be very time consuming because of the extremely larger number of candidate non-radial configurations appearing.

Based on fundamental loop and loop cut set, integer encoding schemes are suggested to determine the possible solutions of the optimization algorithms [25, 26, 29–32]. A code string in these schemes represents the identity of a set of open switches. In a complex distribution network with containing cross loops, non-radial configurations would still appear because there are public switches between the loops. Thus, the analysis of such systems would result in an excessive computational burden because the tedious mesh verification for all candidate solutions could not avoid. A large number of infeasible solutions would disturb the optimization process as one solution is identified from numeric possible solutions.

To avoid tedious mesh check of candidate solutions, an encoding scheme based on matroid theory is presented in Ref. [33]. BRAZ et al [23] proposed a sequential encoding which uses the fundamental loop matrix and the XOR operator to express a switch opened if, and only if, and this switch belongs to a current loop. In both, the codes are capable of representing only radial configurations. Due to there are many public switches between the loops, a configuration may correspond to different codes. The redundant codes would reduce the evolutionary efficiency of optimization algorithms.

This work first presents a new reconfiguration mathematical model considering the network invulnerability of node importance degree and economy of power loss. We establish a compound objective function which is used to solve the conflict of the objectives by weight coefficient method. According to the loop switches hierarchical encoded scheme, a novel quantum particle swarm optimization is applied to solve the DNR problem of compound objective function. The tedious mesh checks for the topology constraint validation is no longer required during the optimization because the candidate solutions generated are fixed in the feasible solutions. Moreover, a code string represents uniquely a feasible solution, thus the codes do not appear duplication. Instead of providing an optimal topology network in detail, the reconfiguration strategy intends to obtain several reconfiguration schemes with better performance as the guidance of dispatching operation. And the relationship among the indices of evaluating the network performance is also reflected. The application examples are employed to demonstrate the essential features of the evaluation model. The obtained reconfiguration results are more in accord with the system practice comparing with other reconfiguration schemes.

2 Evaluation model of node importance degree for distribution networks

The structure model of a real distribution network can be represented by an abstract scale-free topology graph, whose nodes and branches are corresponding to bus bars and electric elements, respectively. Assume a distribution system with n nodes and m branches, that is, G=(V, E) where V={v₁, v₂, …, v_n} is a set of labeled nodes and E={e₁, e₂, …, e_m} is a set of branches. The core elements of the topology are nodes which directly impact the reliability and invulnerability of a network. An undirected branch exists between two nodes if there is a direct link between the associated nodes. An adjacency matrix A of the graph G is defined as:

               (1)

where a_ij is a binary variable, if there is a branch connecting to node i and node j, then a_ij=1; otherwise, a_ij=0.

The degree of node i is defined as the maximum number of branches emanating from a node. It is denoted by Eq. (2).

                              (2)

where D_i represents the degree value of node i.

In studying conventional approaches of the complex network topological structure, the node degree is employed to measure the importance of nodes. The node with more branches connected corresponds to the node with more important [15].

In fact, some critical nodes are not with large node degrees. To address the inconsistency, this paper introduces the network node cohesion to explain the concept of node importance degree.

The node cohesion reflects the location of the node in whole network [16]. Let G=(V,E) be an original distribution network, where V is the set of nodes and E is the set of branches. Let G′=(V′, E′) represent the topological structure obtained after contracting node i, namely node i and nodes connected with the node are replaced by new node i′. The node i importance degree (NID) is formulated as:

                            (3)

where δ_i represents the importance degree of node i, n′ is the number of nodes in new network G′, l_ave is the average distance. It is described as:

               (4)

where d_min represents the minimum distance between node j and node k; V′ represents the set of nodes in the graph G′.

Obviously, the high NID depends on the node degree D_i and the number of connecting nodes. An initial distribution network is shown in Figure 1, there are 10 nodes and 9 branches. For obtaining the NID of node 6, the nodes directly jointed node 6 will merger into one new node. Thus, nodes 1, 2, 3, 8 and 6 are replaced by node 6′. Figure 2 shows the network after node contraction. Following the Table 1 gives the calculation results of the NID.

Although node 8 has a lower degree than nodes 6 and 7, it is observed that this node is the most critical node from Figure 1. According to the node cohesion method, the NID values of nodes 6, 7 and 8 are 0.0781, 0.0652 and 0.0833, respectively. It is concluded that node 8 is more important than nodes 7 and 6. These results illustrate that some key nodes may not have large node degree. Moreover, nodes 7 and 8 have the same node degree D_i=3, but node 8 is much more important in comparison to node 7 through analyzing the index δ_i. Thus, the index of the NID is more distinct to distinguish than node degree.

Figure 1 Initial distribution network

Figure 2 Network after node contraction

Table 1 Comparison of NID and node degree

3 DNR problem formulation

The DNR problem is a mixed nonlinear compound objectives programming problem. The mathematic function model of the compound objective consists of two sub-objectives. The first one is node importance degree which is related with the structure invulnerability of the distribution network. By protecting the nodes with high importance degree, the system’s invulnerability will be improved. The second one represents the total power loss. The sub-objectives that are optimized for the study are as follows.

3.1 Node importance degree

As mentioned in Section 2, the NID reflects the invulnerability of the network itself. It could provide a robust assessment index for whole network structure. If the importance degree of a node after the contraction is high, the node is important. The maximum node importance degree is 1 and the range is 0<δ_i≤1. According to the relation of δ_i and maximum value can identify the key nodes of the system, the corresponding objective function is described as:

                (5)

where n is the number of nodes in the distribution network; δ_i represents the node i importance degree.

3.2 Power loss

Apart from the important node assessment index defined in the previous section, the active power loss can be considered as another objective because excessive power loss increases overheating of the electric components and additional costs. Therefore, the total active power loss in a network is expressed as:

            (6)

where m is the number of branches; f_ploss, p_lossb represent the total and b-branch active power loss, respectively; P_b, Q_b are respectively active and reactive power flows through the b-branch. The impedance of b-branch is r_b. k_b is the status of b-branch, k_b=1 if b-branch is closed, and k_b =0 if b-branch is open. U_b is the voltage amplitude of b-branch terminal.

3.3 Compound objective function

The initial configuration formulation is assumed to a multi-objective, the fitness assignment is performed by a sum of objectives [19]. However, the optimized objectives may be in conflict during the process of performing DNR. To coordinate the conflict, a compound objective function is formulated as a weighted sum where each term represents one of the sub-objectives as Eq. (7). Due to the NID and power loss have different dimensions, they should be normalized.

              (7)

where w₁ and w₂ represent the weight coefficients, w₁+w₂=1; x₁ and x₂ are the penalty factors which are used to improve the searching speed; f ′_ploss, f ′_δ represent the normalized power loss and normalized NID, respectively.

Generally, the value of the penalty factor lies on the problem itself. The normalized active power loss f ′_ploss, NID f ′_δ, penalty factors x₁ and x₂ can be calculated by using Eqs. (8) and (9).

                           (8)

                               (9)

where f_plossin and f_δin are the power loss and node importance degree in the initial network. If f ′_ploss≥1, ≥1 represents x₁=x₂=N; otherwise, x₁=x₂=1. N is the decimal integer and N=10 here.

In the case of the final solution, the optimal intermediate distribution system should satisfy the system operation requirements. All the constraints are given by the set of Eqs. (10)–(13).

                             (10)

                             (11)

                        (12)

                               (13)

where I_ij and S_ij are the current magnitude and the power flow of branch ij, respectively; V_i is the voltage magnitude at node i; Subscripts of max and min are upper and lower limits; φ(k) deals with the radial topology constraint. Equation (10) represents the current thermal constraint. Equation (11) refers to the feeder capacity limits for each branch. Equation (12) considers the node voltage limits for different buses in the network. Equation (13) keeps the system radiality.

4 Proposed reconfiguration strategy of avoiding infeasible solutions

4.1 Loop switches hierarchical encoded scheme

Consider a distribution system with a set of n nodes and m branches, the topology graph is described as G=(V, E). A diagrammatical representation of a distribution system is illustrated, using a 33-bus system as shown in Figure 3.

Generally, distribution system incorporates a few kinds of nodes, i.e., power supply source nodes, terminal nodes, electric T-nodes and junction nodes. So the different set of degree values D_i=1, D_i=2 and D_i≥3 are corresponding to the terminal nodes, junction nodes, and T-nodes. Power source nodes, terminal nodes, and T-nodes are called special nodes. The complex distribution network is simplified as a topology which incorporates special nodes and branch chains. Moreover, distribution feeders connect in a network structure by sectionalized switches and tie switches. If all the switches are closed, the feeders would be formed the loops. The encoding strategy considers the loop switches to represent the radial topologies of the distribution systems. Figure 4 shows the simplified generation of from Figure 3. Note that the number of nodes reduces from 33 to 8. The new switch chains are expressed as L1–L12.

According to the survey of the encoding scheme, traditional intelligent reconfiguration algorithms are commonly based on fundamental loops vectors and loop cut set [25–28, 30, 39–40]. However, numerical non-radial solutions would generate because cross loops exist in the distribution network. To reduce the number of public switches, we propose the rules to determine the loops. It is required that there are a minimum number of switches in each loop. For convenience, a serial number is assigned to each loop. The determination of the loops and corresponding switch chains or switches combination are illustrated in Table 2.

Figure 3 Initial network topology of 33-bus system

Figure 4 Simplified graph of 33-bus system

Table 2 Determination of loops for 33-bus system

Since there are a large number of switches in the distribution network, the code string would be very long. Each number of the encoding scheme represents the identifier of an open switch to reduce the length of the coding string. And only the statuses of switches that belong to loops are regarded as the optimization variables. Each loop can be considered as a control variable. All the switches in a loop assign an integer as a label. Integers of the beginning and the end are to be as the lower and upper limits of a control variable. However, if a complex distribution system contains multiple cross loops, unfeasible non-radial solutions would appear. To avoid these solutions, the loop switches hierarchical encoded scheme is proposed and the flowchart is expressed as Figure 5.

The initial topology network is simplified as a graph that reserves special nodes and branch chains. According to the table of switches combination of the loops, the loop matrix Lre is created. The rows of Lre matrix represent the number of loop layer and the columns represent the number of switches in a loop. If the loop is assigned to a label of layers, all the switches in this loop belong to this layer.

Assuming that Lre matrix has k rows indicates that the number of the layers is equal to k. Let n=1 layer, any switch in the first layer loop is disconnected. In n+1 layer, the set of public branches Gs between n layer and n–1 layer is found. If there are switches in the set of Gs and the open switch exist in the set, the elements in n row of Lre matrix should be updated. Otherwise, the elements of the Lre remain unchanged. Then any s-switch is chosen to disconnect in n line of Lre matrix. Finally,the open switches combinations are determined. The advantage of this scheme is that generated switches combinations are corresponding radial connected configurations in a distribution network with lots of nested loops. Therefore, the solution space size is reduced and the performance of the scheme is improved.

Figure 5 Flowchart of loop switches hierarchical encoded scheme

4.2 Hierarchical encoded particle swarm optimization

According to the mathematical objective function and constraints, the DNR problem is described as a mixed integer and nonlinear combinatorial optimization problem. In order to find the optimal solutions, many intelligent search algorithms are frequently used to solve the optimization problem because of their main advantages.

One of these algorithms, the quantum particle swarm optimization (QPSO), is highlighted in this paper. Because QPSO adapts strategy of biological groups sharing information, this optimization method converges fast, especially in the early evolution and need less adjustable parameters [34–36]. Then the QPSO is used to handle a wide class of complex optimization problems involved in engineering and science [28, 37, 38]. Furthermore, the QPSO can hold the best solution of each particle along with the global best solution of the whole population. It is less sensitive to the nature of the objective function and local minima problem. In the reconfiguration strategies of QPSO, a population of particles is randomly generated and each particle corresponds to a possible configuration for the distribution network topology. The large number of candidate solutions would lead to a long computing time before reaching an optimal solution. And a number of unfeasible solutions would decrease the efficiency of the search process as it needs tedious mesh checks for the topology constraint validation. Based on the QPSO, we propose a hierarchical encoded particle optimization method (HEPSO) which uses the loop switches hierarchical encoded scheme to obtain the candidate solutions.

The proposed method uses the loop switches hierarchical encoded scheme to update the population particles during each generation. The flowchart is described as Figure 6. The computational procedure for reconfiguration is summarized as follows:

Figure 6 Flowchart of proposed method

1) Input original distribution network data and parameters of the algorithm. The forward-backward sweep method is used to calculate the initial power loss f_plossin.

2) Simplify the original topology network. By analyzing the topological structure, we obtain the loop matrix Lre which the rows represent the number of loop layer.

3) Generate random population particles. Through observing the Lre, the dimension of control variables is equal to the rows of the matrix. The sequence of section switches in each loop is recorded as an integer code. Then the sequence numbers at both ends of each loop are represented as the upper and lower range of each dimension particle. According to the flowchart of the hierarchical encoded scheme, a feasible solution vector is described as x=(x₁, x₂, …, x_i).

4) Calculate and evaluate the fitness of each particle by applying the results of the compound objective function to Eq. (7).

5) Update the population particles. A solution vector x_id=(x_i1, x_i2, …, x_iD) represents the position of a particle in D-dimensional search space. The current best position of particle i and the globally best position are represented as the vector p_id=(p_i1, p_i2, …, p_iD) and p_gd=(p_g1, p_g2, …, p_gD), respectively. The updated position of particle i is modified under the following equation that calculates t+1 iteration position x_id(t+1) based on its previous position x_id(t).

      (14)

                    (15)

              (16)

where x_id(t) and x_id(t+1) are the original position and new position of particle respectively; p_id(t) and p_gd(t) represent the current best position and globally best position of particle i, respectively; mbest(t) represents the average best position of all particles; u and φ are the uniformly distributed random number in [0,1] respectively. The integral function uses round(). βis the factor of controlling the convergence speed of the algorithm.

6) Apply the hierarchical encoded scheme to repair the population particles that ensure updated population particles are satisfied to the network topology constraints.

7) Increase the iteration counter by 1.

8) Analyze the algorithm stopping criteria. If the current iteration t reaches to the maximum iterations t_max then output reconfiguration results. Otherwise, return to step 4).

5 Case studies

In order to evaluate the performance of the proposed method, IEEE 33-bus and PG&E 69-bus distribution systems are simulated with MATLAB.

The first test case is a 33-bus, 12.66 kV, radial distribution system [26] as shown in Figure 1. It consists of 33 nodes, 32 sectionalizing switches, and 5 tie-switches. The normally closed switches are 1 to 32, and normally open switches are 33 to 37. The active and reactive power loads of the system are 3715 kW and 2300 kVar, respectively. Section 2 gives the simplified topology and the branches look-up table of each loop.

The PG&E 69-bus test system is a hypothetical 12.66 kV radial distribution network with five loops. GUAN et al [31] gave the line and load data of initial configuration. The active and reactive initial system loads are 3802.19 kW and 2694.60 kVar, respectively. The normally open switches are 69, 70, 71, 72, and 73. Figure 7 represents the abstract graph of the initial original network. Figure 8 gives the simplified graph of this system by above the section. Table 3 shows the determined loops and switches look-up table of each loop.

In these cases, the standard substation voltage is considered as 1.0 pu, and the switches in the loops are candidate switches for network reconfiguration. A computer program is developed to implement the loop layered coding scheme and the proposed algorithm using MATLAB 2015. The population size is 30 and the particles’ dimension is equal to the number of loops. The maximum and minimum of weighting values are 0.9 and 0.4. The maximum number of iterations is 200, and the learning factor, c₁ and c₂, are both equal to 2.0. In order to evaluate the performance fairly, 100 runs for the algorithm are performed. The obtained test results and discussions are described as the following section.

6 Test results and discussions

In this section, the ability of the loop switches hierarchical encoded scheme to generate the feasible solutions is verified. The performance of the proposed reconfiguration method of avoiding infeasible solutions is tested on two radial distribution systems. Our method is compared with other optimization algorithms to demonstrate the superiority of the proposed work.

Figure 7 Initial network of 69-bus system

Figure 8 Simplified graph of 69-bus system

Table 3 Determination of loops for 69-bus system

6.1 Encoding scheme performance tests

1) Encoding space

As the survey of the existing network codifications, the ability of encoding scheme has an impact on the efficiency of the particle swarm optimization. Several coding schemes are applied to generating the trial solutions. To evaluate the performance of the encoding scheme, one of the parameters to compare code strategies is the size of solution space. Usually, if all possible configurations are represented all feasible solutions of a system, then the smaller the search space, the better the technique. Table 4 shows a summary of encoding spaces using binary coding [18], improved binary encoding [32], decimal encoding [39], and the proposed encoding.

Table 4 Search spaces of different encodings

Based on the binary encoding scheme, any possible status of all switches can be described as a binary number. Due to extremely large number of possible solutions generated, the size of search space is very large. And the length of the code string is equal to the number of the switches. The improved binary coding scheme considers that the control variables are only the open switches. It can be observed that the length of codes would reduce. From Table 4, the search spaces for the 33-bus and 69-bus systems are 376992 and 4187106. The decimal coding scheme is to record the sequence number of each open switch in loops with one decimal number. Among these schemes of the references, the decimal coding scheme has minimized encoding space, i.e., 242550 in IEEE 33-bus system and 1775616 in PG&E 69-bus system. If using our scheme, the encoding spaces for two systems are 50751 and 327868, which are less than 242550 and 1775616, respectively. Compared with other coding schemes, the proposed scheme can significantly reduce the size of encoding space.

2）Feasible solutions(FS)

A possible solution has a corresponding code in the encoding space. In Table 4, the binary coding could generate extremely large number of possible solutions, i.e., 1.3744×10¹¹ in IEEE 33-bus system and 5.9030×10²⁰ in PG&E 69-bus system. However, it is found that the radio of efficient solutions is very lower as most solutions violate the topological restrictions. The maximum proportion of feasible solutions for 33-bus and 69-bus system obtained by the schemes of Ref. [18, 32, 39] are only 20.92% and 18.47%. Using the loop switches hierarchical encoded scheme, the number of candidate configurations is reduced drastically. For example, in the 69-bus system, the radio of feasible solutions is 100% which is significantly higher than 18.47%. Based on the results, the loop switches hierarchical encoded scheme improves the efficiency of optimization because of reducing the solution space and avoiding the verification of infeasible solutions.

6.2 Reconfiguration algorithm performance tests

The HEPSO algorithm is applied to solving the compound objective reconfiguration of distribution networks with considering node importance degree and power loss. To evaluate the performance of the algorithm, it is tested on 33-bus and 69-bus test distribution systems. In all cases, the reconfiguration results can be compared with some references in Refs. [23, 26, 31].

Following the steps of Section 4, we will obtain a set of solutions according to adjust values of weight coefficients. The open switches with the values of fitness for the two systems are shown in Tables 5 and 6.

Table 5 Reconfiguration results for 33-bus system

Table 6 Reconfiguration results for 69-bus system

From Tables 5 and 6, a set of solutions are obtained by adjusting the weight coefficients. The situation of w₁=1 and w₂=0 represents that only the minimization of power loss is considered as the reconfiguration objective. Under the circumstance of w₁=0 and w₂=1, the reconfiguration objective is only the node importance degree.

To verify the results presented in Tables 5 and 6, the extreme points the obtained optimal configurations are compared with results of other methods by optimizing the power loss that is shown in Table 7.

Table 7 shows a comparison of different approaches while the minimum power loss function of the 33-bus system is optimized. The found best open switch combination of this system is [7 14 9 32 37] by all the methods. The results represent that the proposed method is able to achieve the optimal configuration which is similar to other methods. Due to the method of calculating the power flow is different, the power loss of final configuration obtained is 139.4410 kW which is less than other approaches in Refs. [4, 25, 41]. In 69-bus system, the global solution is also obtained while the objective function of minimum power loss has been optimized. Having found the optimal solution, the system power loss is 99.6205 kW which is comparable to Refs. [42] and [43]. Since there are no similar references becoming for comparison, our method has not been compared with other algorithms in the view of the NID index.

Table 7 Obtained results by optimizing power loss

In order to reflect the relative importance of each node in the network, the NID values should be normalized further in terms of the ideal maximum value of nodes. If the NID index is considered as the only objective (w₁=0, w₂=1) in the evaluation model, the ideal maximum values of the node importance degree for 33-bus and 69-bus system are 0.0061 and 0.0016. The relative NIDs after normalizing are used to discuss the importance of each node.

To evaluate the system invulnerability and economy after reconfiguration, the real power loss of each branch and the relative NID of each node are discussed. The weight coefficients are set to 0.5 because the two evaluation indices are equally important in the DNR. Figures 9 and 10 represent the reconfiguration results of the HEPSO which could have achieved the optimal configurations of both systems.

In the 33-bus system, the obtained best opened switches combination is [7 14 10 32 28]. A comparison between each branch’s active power loss of the best configuration and the initial network is provided in Figure 9(a). The total power loss after reconfiguration is 140.5971 kW which is approximately 30.4% reduction of initial loss from Table 5. Figure 9(b) compares the importance degree of each node for both networks. It can be realized that each node importance degree of the best configuration is improved. As it can be gathered from the figure, the sum of nodes importance degree is greatly increasing than the initial network.

Figure 9 Reconfiguration results of 33-bus system:

Figure 10 Reconfiguration results of 69-bus system:

The initial network topology of the 69-bus system is modified by closing normally open switches 69, 70, 71, 72, and 73 and opening closed switches 63, 17, 14, 52, and 47 to represent final topology. The system power loss profiles before and after reconfiguration are shown in Figure 10(a). This amounts to a reduction of 46.73% in total power loss. And the relative NID of nodes is depicted in Figure 10(b). To enhance the invulnerability of the system, the nodes with high importance degree should be protected. The minimum NID is increased to 0.8750.

In order to investigate the relation of the objectives, Tables 5 and 6 give the results of different weight coefficients. If the weight value w₁ is greater than w₂, that is, the objective of power loss takes priority over the NID. Otherwise, the NID is considered as dominated optimization objective. It is possible to say that reduction of the value of power loss could reduce the importance degree of nodes. It can be seen that the power loss and NID are in conflict, but not a simple inversely correlated.

From Table 5, it can be observed that different weight coefficients would lead to the same reconfiguration results. In the 33-bus system, the scenarios of weight-vectors adopted w=(0.9, 0.1) and (0.8, 0.2) lead to the same performance. It illustrates that the reconfiguration results are less sensitive to the weight coefficients. Under the circumstances, we obtain reconfiguration results by selecting randomly weight coefficients. The obtained results are shown in Table 8. Table 6 summarizes the results of the 69-bus system. These solutions bring about an extra flexibility for the decision-maker. They can choose an appropriate plan in accordance with the operation condition and different system requirements. In the case of no special operating requirements, we use the geometric mean method to determine the weight vector, that is, w₁=w₂=0.5. In both systems, the best configurations are [7 14 10 32 28] and [63 17 14 52 47].

Table 8 Final reconfiguration results for 33-bus system

6.3 Comparative tests

In this experiment, the results of our algorithm are compared to the results by other methods in Refs. [23, 26, 31]: the genetic algorithm (GA), the modified particle swarm optimization (MPSO) and the standard quantum particle swarm optimization (QPSO). In order to provide a fair comparison, all the optimization methods are repeatedly 100 runs and the weight coefficients are set to 0.5. The best and worst values among the best solutions are listed in Table 9.

1) Algorithm robustness

This work uses the best solution to assess the solution’s quality. The mismatch between the best and the worst solutions enables to measure the robustness of the algorithm. The open switches of initial 33-bus system are 33, 34, 35, 36, and 37. The best, worst, and average cases for this system in 100 runs are shown in Table 9. All the methods can be obtained the best solution which has the fitness value is 0.8001. Computational results are shown that the robustness of the HEPSO is better than GA, MPSO, and QPSO. It can be observed that 0.8001 of worst fitness value is achieved by HEPSO comparing with 0.8030 by the MPSO, 0.8096 by the GA, and 0.8030 by the QPSO.

Similarly, the proposed approach is also applied to 69-bus system. It is found that the fitness value has been reduced. Although all the methods can obtain the best configuration, only the proposed method between the best and worst cases is matched. The average fitness value of the HEPSO is 0.6618 which is less than other conventional methods of MPSO, GA, and QPSO. Simulation results demonstrate that the HEPSO has strong robustness by adopting the loop switches hierarchical encoded scheme to eliminate infeasible solutions during optimization.

2) Convergence rate

The convergence rate can be used as an acceptable index to measure the search algorithms. It shows that the algorithm is reliable to reach optimum solutions. The number of iteration is recorded when the optimal result is achieved. In order to show the HEPSO convergence, Figure 11 illustrates the best fitness in 100 runs of all the methods for the two test systems. In Figure 11(a), the proposed algorithm can find the global optimum point of the 33-bus system in the 11th iteration which is less than the methods in Refs. [23], [26], and [31]. From Table 9, the average iteration number needed to obtain the optimum solution for all the methods are 174, 109, 67, and 17, respectively. Moreover, Figure 11(b) illustrates that the proposed algorithm has the ability to get the optimal solution in iteration 20 for the 69-bus system. Comparing the average iteration number, the proposed HEPSO is faster than other methods in finding the optimum solution, i.e., 30 iterations against 142 in MPSO, 33 iterations compared to 208 in GA, and 33 iterations instead of 116 in QPSO. It is observed that the HEPSO algorithm can obtain find the best solution in fewer iterations and more efficient than other conventional methods.

Table 9 Configuration results of 33-bus system

Figure 11 Convergence characteristics of all methods for tested systems:

3) Time consuming

Resuming the comparison in terms of complexity and rapidity, Figure 12 illustrates the best fitness found by each algorithm with respect to running time. And Table 9 gives the average time consuming (TC) of all the methods for the tested systems. For the 33-bus system, the average time consuming for HEPSO to obtain the optimal solution is 4.5543 s which remains at the forefront; QPSO is second, with 9.1317 s; MPSO is third, with 11.1179 s; and only then, GA, with 15.9566 s. The proposed HEPSO with avoiding infeasible solution present advantage, the time consuming taken by the processor to carry out the simulations for 100 runs is less than other methods.

Figure 12 Time consuming curve of all methods for tested systems:

7 Conclusions

Based on the loop switches hierarchical encoding scheme, a novel quantum particle swarm optimization approach is proposed for the compound objective network reconfiguration. In order to evaluate the economy and important nodes of reconstructed network, the compounding objective reconfiguration with considering power loss and NID for distribution systems is studied. The quantitative NID based on the complex theory can be obtained by node merging arithmetic. Consequently, this paper extends and illustrates a procedure for optimal distribution reconfiguration calculating the compounding objective.

In this reconfiguration strategy, the hierarchical encoded scheme is applied to generating population swarm particles which guarantee the diversity of groups. Moreover, the trial solutions generated represent the radial networks without verifying the feasibility of the solution.

Simulation results and the performance assessment analysis illustrate that the effectiveness of the compound evaluation model and the strategy intends to obtain a set of solutions by adjusting the weight coefficients. Using this strategy, we will get a set of solutions bringing about an extra flexibility. Furthermore, the proposed scheme can be utilized to solve the conflicts among optimization objectives of the system. The decision-maker can choose a plan which is appropriate in accordance with the operation condition and different requirements of the system. The reconfiguration strategy will benefit the guidance of dispatching operation.

References

[1] Paterakis G N,Mazza A,SANTOS F S, ERDINC O. Multi-objectivereconfigurationof radial distribution systems using reliability indices [J]. IEEE Transactions on Power Systems, 2016, 31(2): 1288–1298. DOI: 10.1109/TPWRS. 2015.2425801.

[2] RAVADANEGH S N, OSKUEE M R J,KARIMI M. Multi- objective planning model for simultaneous reconfiguration of power distribution network and allocation of renewable energy resources and capacitors with considering uncertainties [J]. Journal of Central South University, 2017,24(8): 1837–1849. DOI: 10.1007/s11771-017-3592-8.

[3] Jazebi S, Hadji M M, Naghizadeh R A. Distribution network reconfiguration in the presence of harmonic loads: Optimization techniques and analysis [J]. IEEE Transactions Smart Grid, 2014, 5(4): 1929–1937. DOI: 10.1109/TSG.2014. 2314124.

[4] Gupta N, Swarnkar A, Niazi K R. Distribution networker configuration for power quality and reliability improvement using genetic algorithms [J]. International Journal of Electrical Power & Energy Systems, 2014, 54(54): 664–671. DOI: 10.1016/j.ijepes. 2013.08.016.

[5] Spitsa V, Ran X, Salcedo R, MARTINEZ J F. On the transient behavior of large-scale distribution networks during automatic feeder reconfiguration [J]. IEEE Transactions Smart Grid, 2012, 3(2): 887–896. DOI: 10.1109/TSG.2012. 2186319.

[6] Xu Yan, ZHAO Yang-dong, WONG K P, LIU Evan, YUE B. Optimal capacitor placement to distribution transformers for power loss reduction in radial distribution systems [J]. IEEE Transactions on Power Systems, 2013,28(4): 4072–4079. DOI: 10.1109/TPWRS.2013.2273502.

[7] MENDONZA J E, LOPEZ M E, COELLO C A C, LOPEZ E A. Microgenetic multiobjective reconfiguration algorithm considering power losses and reliability indices for medium voltage distribution networks [J]. IET Generation Transmission & Distribution, 2009, 3(9): 825–840. DOI: 10.1049/iet-gtd.2009.0009.

[8] ALONSO F R, OLIVEIRA D O, SOUZA A C. Artificial immune systems optimization approach for multiobjective distribution system reconfiguration [J]. IEEE Transactions on PowerSystems, 2015, 30(2): 840–847. DOI: 10.1109/ TPWRS.2014.2330628.

[9] TAPIAJUREZ R, ESPINOSAJUREZ E, GRAFF M. Optimal reconfiguration of radial distribution networks for reducing voltage sags [C]// IEEE Conference on CCE. Mexico, 2013: 280–285. DOI: 10.1109/ICEEE. 2013. 6676027.

[10] Asrari A, Lotfifard S, Payam M S. Pareto dominance-based multiobjective optimization method for distribution network reconfiguration [J]. IEEE Transactions on Smart Grid, 2016, 7(3): 1401–1410. DOI: 10.1109/TSG. 2015.2468683.

[11] Rao R S, RAVINDRA K, SATISH K, NARASIMHAM S V L. Power loss minimization in distribution system using network reconfiguration in the presence of distributed generation [J]. IEEE Transactions on Power Systems, 2013, 28(1): 317–325. DOI: 10.1109/TPWRS.2012.2197227.

[12] YUAN Rong-kun, MENG Xiang-ru, LI Ming-xun, WEN Xiang-xi. Evaluation method for network invulnerability based on node importance [J]. Fire Control＆Command Control, 2012, 37(10): 40–42. (in Chinese)

[13] Arianos S, Bompard E, CARBONE A, XUE F. Power grid vulnerability: A complex network approach [J]. Chaos, 2009, 19(1): l–6. DOI: 10.1063/1.3077229.

[14] LI Fu-huang, WEN Jie, XIAO Sheng, LI Yuan, GUO Shi-fan. Vulnerability assessment to small-world power grid based on weight topological model [C]// IEEE Conference on Powerand Energy Engineering. Chengdu, 2010: 1–4. DOI: 10.1109/APPEEC.2010. 5448813.

[15] Deng Chang-hong, HU Nan-nan, Xie Qiong-yao, DONG Chao. Evaluation of the importance of network nodes based on weighted network model [C]// IEEE Conference on Power and Energy Engineering. Wuhan, 2009: 27–31. DOI: 10.1109/APPEEC.2009.4918345.

[16] LI Can-bing,LIU Wen-can,CAO Yi-jia, CHEN Hao, FANG Ba-ling, ZHANG Wei, SHI Hai-qing. Method for evaluating the importance of power grid nodes based on PageRank algorithm [J]. IET Generation Transmission & Distribution, 2014, 8(11): 1843–1847. DOI: 10.1049/iet-gtd.2014.0051.

[17] Crucitti P, Latora V, Marchiori M. Model for cascading failures in complex networks [J]. Physical Review E: Statistical Nonlinear & Soft Matter Physics, 2004, 69(4): 1–4. DOI: 10.1103/PhysRevE.69.045104.

[18] LIU Yan, GU Xue-ping. Skeleton-network reconfiguration based on topological characteristics of scale-free networks and discrete particle swarm optimization [J]. IEEE Transactions on Power Systems, 2007, 22(3): 1267–1274. DOI: 10.1109/TPWRS.2007. 901486.

[19] Huang Jin-kai,Du Liang, Zhang Guo-song. Skeleton- network reconfiguration based on node importance and line optimization [C]// IEEE Conference on Power and Energy Engineering. Shanghai: IEEE, 2012: 1–4. DOI: 10.1109/APPEEC. 2012. 6307449.

[20] Zhang Can, LIN Zhen-zhi, WEN Fu-shuan, LEDWICH G, XUE Yu-sheng.Two-stage power network reconfiguration strategy considering node importance and restored generation capacity [J]. IET Generation Transmission & Distribution, 2014, 8(1): 91–103. DOI: 10.1049/iet-gtd. 2013.0065 .

[21] G. Fuzzy-based monte carlo simulation for harmonic load flow in distribution networks [J]. IET Generation Transmission & Distribution, 2015, 9(3): 267–275. DOI: 10.1049/iet-gtd.2014.0138.

[22] TSAI M S,HSU F Y. Application of grey correlation analysis in evolutionary programming for distribution system feeder reconfiguration [J]. IEEE Transactions on Power Systems, 2010, 25(2): 1126–1133. DOI: 10.1109/TPWRS.2009. 2032325.

[23] BRAZ H D, SOUZA B A D. Distribution network reconfiguration using genetic algorithms with sequential encoding: subtractive and additive approaches [J]. IEEE Transactions on Power Systems, 2011, 26(2): 582–593. DOI: 10.1109/TPWRS.2010. 2059051.

[24] LE ROUX P F, MUNDA D, HAMAM Y. Distribution network reconfiguration using genetic algorithm and load flow [C]// IEEE Conference on IPEC. Ho Chi Minh: IEEE, 2012: 616–620. DOI: 10.1109/ASSCC.2012. 6523339.

[25] ANDERVAZH M,OLAMAEI J,HAGHIFAM M. Adaptive multi-objective distribution network reconfiguration using particles swarm optimization algorithm and graph theory [J]. IET Generation Transmission & Distribution, 2013, 7(4): 1801–1810. DOI: 10.1049/iet-gtd.2012.0712.

[26] ABDELAZIZ A Y, MOHAMMED F M, MEKHAMER S F, BADR M A L. Distribution systems reconfiguration using a modified particle swarm optimization algorithm[J]. Elec Pow Syst Res, 2009, 79(12): 1521–1530.

[27] SCENNA F,ANAUT D, PASSONI L I, MESCHINO G J. Reconfiguration of electrical networks by an ant colony optimization [J]. IEEE Latin America Transactions, 2013, 11(1): 538–544. DOI: 10.1109/TLA. 2013.6502858.

[28] SIVANAGARAJU S, RAO J V, RAJU P S. Discrete particle swarm optimization to network reconfiguration for loss reduction and load balancing [J]. Electric Power Components & Systems, 2008, 36(5): 513–524. DOI: 10.1080/ 15325000701735389.

[29] RAO R S,NARASIMHAM S V L,RAJU M R, RAO S A. Optimal network reconfiguration of large-scale distribution system using harmony search algorithm [J]. IEEE Transactions on PowerSystems, 2011, 26(3): 1080–1088. DOI: 10.1109/TPWRS.2010.2076839.

[30] CHIEH C C, SHEN T M. Application of novel charged system search with real number string for distribution system loss minimization [J]. IEEE Transactions on Power Systems, 2013, 28(4): 3600–3609. DOI: 10.1109/ TPWRS.2013. 2264836.

[31] GUAN Wang-lin, Tan Yang-hong, ZHANG Hai-xia, SONG Jian-li. Distribution system feeder reconfiguration considering different model of DG sources [J]. International Journal of Electrical Power & Energy Systems, 2015, 68(1): 210–211. DOI: 10.1016/ j.ijepes.2014.12.023.

[32] GANGULY S, SAHOO N C, DAS D. Mono-and multi-objective planning of electrical distribution networks using particle swarm optimization [J]. Applied Soft Computing,2011, 11(2): 2391–2405. DOI:10.1016/j.asoc. 2010.09.002.

[33] ENACHEANU B,RAISON B,CAIRE R, DEVAUX O, BIENIA W, HADJSAID N. Radial network reconfiguration using genetic algorithm based on the matroid theory [J]. IEEE Transactions on Power Systems, 2008, 23(1): 186–195. DOI: 10.1109/ TPWRS.2007.913303.

[34] ALRASHIDI M R, ELHAWARY M. A survey of particle swarm optimization applications in power system operations [J]. Electric Power Components and Systems, 2009, 34(12): 1349–1357. DOI: 10.1080/ 15325000600748871.

[35] TAN Yue, TAN Guan-zheng, DENG Shu-guang. Hybrid particle swarm optimization with chaotic search for solving integer and mixed integer programming problems [J]. Journal of Central South University, 2014,21(7): 2731–2742. DOI: 10.1007/s11771-014-2235-6.

[36] SUN Jun, FANG Wei, WU Xiao-jun, PALADE V, XU Wen-bo. Quantum-behaved particle swarm optimization: Analysis of individual particle behavior and parameter selection [J]. Evolutionary Computation, 2012, 20(3): 349– 393. DOI: 10.1162/EVCO_a_00049.

[37] KHAN S, YANG Shi-you, WANG Lu-yu, LIU Lei. A modified particle swarm optimization algorithm for global optimizations of inverse problems [J]. IEEE Transactions on Magnetics, 2015, 6729(12): 1–5. DOI: 10.1109/TMAG.2015. 2487678.

[38] JAMALIPOUR M, GHARIB M, SAYAREH R, KHOSHAHVAL F. PWR power distribution flattening using quantum particle swarm intelligence [J]. Annals of Nuclear Energy, 2013, 56(2): 143–150. DOI: 10.1016/j.anucene. 2013.01.026.

[39] MENDOZA J, LPEZ R, MORALES D, LOPEZ E, SESSANTE P, MORAGA R. Minimal loss reconfiguration using genetic algorithms with restricted population and addressed operators: Real application [J]. IEEE Transactions Power Systems, 2006, 21(2): 948–954. DOI: 10.1109/ TPWRS.2006.873124.

[40] SHI Jia-chuan, WANG Chun-yi, AN Peng. Loop-based coding reactive tabu search for comprehensive optimization in distribution networks [C]// IEEE Conf. Asia-pacific Power & Energy Engineering. Wuhan: IEEE 2011: 1–4. DOI: 10.1109/APPEEC.2011.5749119.

[41] GOMES F V. A new distribution system reconfiguration approach using optimum power flow and sensitivity analysis for loss reduction [J]. IEEE Transactions Power Systems, 2006, 21(4): 1616–1623. DOI: 10.1109/TPWRS.2006. 879290.

[42] MOHD Z A A,FERDAVANI A K, KHAIRUDDIN A B, NAEINI M M. Reconfiguration of radial electrical distribution network through minimum-current circular- updating-mechanismmethod [J]. IEEE Transactions on Power Systems, 2012, 27(2): 968–974. DOI: 10.1109/ TPWRS.2011.2174258.

[43] GOMES F V,CARNEIRO S,PEREIRA J L R, VINAGRE M P, GARCIA P A N, ARAUJO L R. Anew heuristic reconfiguration algorithmfor large distribution system [J]. IEEE Transactions on Power Systems, 2005, 20(3): 1373–1378. DOI: 10.1109/TPWRS.2005.851937.

(Edited by HE Yun-bin)

中文导读

基于分层编码粒子群优化的配电网复合目标重构方法研究

摘要：网络重构技术是实现系统优化运行最重要的手段之一。配电网重构通过改变配电系统内开关状态来改变网络结构，以达到降低损耗、消除过载、提高电压质量等目的，较少涉及到网络的结构特征。针对这一问题，本文引入节点重要度作为识别网络关键节点的评价指标，建立降低配电网网络损耗、提高网络节点重要度的配电网重构模型。此模型中采用加权系数法将2个目标函数转化为单一的复合目标函数，解决了目标函数之间相冲突的矛盾。然后，将环网分层编码方案与粒子群算法结合，提出求解上述模型的优化方法。该方法利用分层编码方案随机生成代表配网结构的种群粒子，保证生成的粒子均是可行的拓扑解，避免了繁琐的拓扑解的验证过程，降低了粒子群算法的搜索空间，有效地提高了网络重构的速度和优化方法的搜索效率。针对IEEE 33 节点系统和PG&E 69节点系统，对所提出的模型和方法进行验证，从算法的搜索空间、鲁棒性、收敛特征和运行时间等方面验证了方法的有效性。同时，通过修改复合目标函数的权重，得到不同权重情况下的网络最优配置，管理者可根据不同的要求选择最恰当的一组，对网络的运行有一定的指导意义。

关键词：配网重构；节点重要度；复合目标函数；分层编码方案

Foundation item: Project(61102039) supported by the National Natural Science Foundation of China; Project(2014AA052600) supported by National Hi-tech Research and Development Plan, China

Received date: 2016-06-17; Accepted date: 2017-12-20

Corresponding author: TAN Yang-hong, PhD, Professor; Tel: +86–731–87202461; E-mail: 809677326@qq.com; ORCID: 0000-0002- 6713-5618