Topology Design and Operation of Distribution Network Based on Multi-Objective Framework and Heuristic Strategies

Abstract

This work elucidates a methodological approach employed in the process of planning the expansion of distribution network (DN) lines, wherein the amalgamation of system reconfiguration capabilities with operational equilibrium and reliability is paramount. The expansion proposals for the DN and the radial operation schemes post-expansion are encompassed. We formulate a bi-objective DN planning optimization model that concurrently addresses resource optimization configuration and operational optimization. Subsequently, the NSGA-II algorithm is employed to solve the optimization model, providing a coordinated presentation of multiple alternative solutions. However, this problem diverges from conventional bi-objective optimization problems due to its nature as a bi-objective optimization problem with embedded sub-optimization problem, consequently imposing a substantial computational burden. To address this issue, heuristic algorithms are designed to optimize system operational configuration, which is regarded as a sub-optimization problem. The proposed metric, model, and algorithms are validated on two case studies using the IEEE 33-bus and 70-bus test systems. Notably, the proposed method achieves solution efficiency by over 200 times compared to existing methods.

1. Introduction

The power system classified as a “Critical Infrastructure” (CI) [1] is primarily composed of three integral components: generation, transmission, and DN. It is noteworthy that the DNs serving as the “last mile” of the power system hold a pivotal position. However, ensuring the stable operation of the DN remains a formidable challenge, given the occurrence of random failures, natural disasters [2], and the stochastic nature of renewable energy sources. The network reconfiguration [3] emerges as a potent strategy to adapt to dynamic factors, including varying load demands, integration of new energy sources, and faults, thereby enhancing the flexibility and operability of the DN.

Conventional studies in the realm of DN planning have predominantly focused on objectives such as reducing network losses [4] and economic cost [5], achieving load balancing [6], and neglecting the influence of network reconfiguration capabilities [7]. The stronger the reconfiguration capability of the DN, the more formidable its resilience against potential failures becomes. Therefore, taking into comprehensive account the reconfiguration capabilities during the establishment of DNs can significantly enhance the probability of successful subsequent reconfigurations, thereby enhancing the overall efficiency of the system to adapt to various risks and uncertainties. Existing studies typically analyze the vulnerability of power systems based on complex networks [8], introducing various metrics, such as degree centrality [9] and natural connectivity [10]. Nevertheless, these metrics are predominantly employed for identifying critical nodes or vulnerable lines within the power system, posing a challenge in their direct translation to the intuitive mapping of the system’s reconfiguration capabilities. In contrast, the “spanning tree metric” proposed based on complex networks exhibits heightened interpretability. It efficiently computes through graph theory methods, presenting distinct advantages, as initially proposed in our seminal work [11]. Nevertheless, the spanning tree metric solely focuses on the number of alternative paths in the system, disregarding variances between source and load buses and overlooking electrical factors. In addition, an initial radial operation scheme following the proposed design scheme has not been provided. Hence, there is a pressing need to incorporate electrical parameter considerations for better alignment with real-world scenarios, thereby proposing solid initial radial operation schemes based on the “source-to-load shortest distance metric” proposed while concurrently achieving robust resilience under meshed planning frameworks, termed the Distribution Network Topology Planning and Design Problem (DNTPDP). The objective of DNTPDP is to optimize the design of DN topology under specified constraints, harmonizing network reconfiguration capabilities, operation equilibrium, and reliability (see more details in Section 3). Consequently, the optimization of DNTPDP holds paramount significance in ensuring the safety, reliability, and economically efficient operation of the entire power system.

The DNTPDP poses a complex mixed-integer nonlinear programming challenge [12]. Prior research has illuminated that evolutionary algorithms (EAs), renowned for their robustness and formidable adaptability, have proven to be effective solutions for addressing complex engineering problems [13]. As presented in Ref. [14], researchers successfully employed a differential evolution algorithm based on large-scale binary matrices to address the problem of multi-microgrid network structure design. It is noteworthy that existing studies have predominantly treated the optimization of meshed design schemes and radial operation schemes as independent endeavors. However, the initial radial operation schemes and meshed design schemes of the system are not inherently autonomous; they exhibit inheritances and interdependencies. Given the inherently bi-objective optimization with the sub-optimization problem nature of the DNTPDP, the utilization of conventional multi-objective frameworks with EAs introduces challenges. The stochastic nature of EA optimizers may lead to varying evaluations for the source-to-load shortest distance metric within a indetical meshed topology (the same solution) during the evaluation of the sub-problem objective. This instability, manifested as random oscillations, may compromise the accuracy of the obtained Pareto front. Furthermore, employing EAs for this bi-objective problem is time-consuming, as it entails a cumulative multiplication rather than addition of computational time, resulting in a substantial computational burden. In order to address this issue, we propose the application of heuristic algorithms for the evaluation of the sub-problem objective. This approach not only proves efficient but also ensures stability (see more details in Section 4).

This paper aims to advance the assessment of the planning and design of DN topology by incorporating considerations of equilibrium and reliability based on a reconfiguration capability metric. The objective is to propose optimized radial operation schemes and devise planning strategies endowed with robust reconfiguration capabilities. The main contributions can be summarized as follows:

• This work introduces a novel metric termed the source-to-load shortest distance which is based on the shortest distance between source and load buses. This metric quantifies the equilibrium and reliability of DN during operation, capturing electrical parameters.
• To concurrently enhance the reconfiguration capability, operation equilibrium, and reliability of DNs, a bi-objective optimization model is established which can facilitate the optimization of expansion planning and operation schemes, with a specific focus on the optimized deployment of source buses.
• To efficiently and effectively solve the optimization model, heuristic algorithms are designed to significantly improve the efficiency of the solution process without compromizing solution quality.
• The effectiveness and competitiveness of the proposed metric, model, and algorithms are verified on IEEE 33-bus and 70-bus test systems, demonstrating their advantages over existing methods.

The rest of this paper is organized as follows: Section 2 introduces metrics serving as the two objective functions for the DNTPDP. These metrics evaluate DN reconfiguration capability based on the number of spanning trees and assess system operational equilibrium using the source-to-load shortest distance. Section 3 details the proposed bi-objective optimization mathematical model for the DNTPDP. In Section 4, the specific method for solving the proposed model is presented, incorporating two heuristic strategies designed to expedite the assessment of system operational equilibrium. The experiment description and results are provided in Section 5. Finally, Section 6 concludes this work and outlines potential research directions.

2. Evaluation Metrics of DN

2.1. Spanning Tree Metric

Given the widespread adoption of closed-loop design and radial operations within DNs, the significance of radial operations extends to the majority of optimization challenges associated with DNs [15]. Deliberating upon radial operations is imperative due to their advantages in facilitating coordination, enhancing protection, and reducing short-circuit current [16]. Therefore, the operation modes of meshed grids surpass the scope of investigation within this study. An effective topology structure for the operation configuration of DNs should exhibit a radial pattern, mitigating the presence of isolated grids or loops. Drawing on the intrinsic connection between spanning trees and the normal operation topology of DNs, as presented in Ref. [11], the proposal advocates for the utilization of the quantity of spanning trees as a metric for gauging the system’s reconfiguration capability. This metric adeptly captures the diversity and flexibility of the DN’s operation states. Consequently, we adopt the number of spanning trees as a metric for evaluating the system’s reconfiguration capability, and its efficient computation is facilitated by Kirchhoff theorem [17], as follows:

$$L\left(X\right)=D\left(X\right)-X$$

(1)

$$T\left(X\right)=detL\left(X\right)\left(\begin{array}{c}1,2,\cdots ,i-1,i+1,\cdots ,N\hfill \\ 1,2,\cdots ,i-1,i+1,\cdots ,N\hfill \end{array}\right)$$

(2)

where $D\left(X\right)$ and X represent the degree matrix and the adjacency matrix, respectively, of graph G with N nodes,  $T\left(X\right)$ represents the reconfiguration capability of G, which can be obtained by calculating the determinant of $L\left(X\right)\left(\begin{array}{c}1,2,\cdots ,i-1,i+1,\cdots ,N\hfill \\ 1,2,\cdots ,i-1,i+1,\cdots ,N\hfill \end{array}\right)$, and $L\left(X\right)\left(\begin{array}{c}1,2,\cdots ,i-1,i+1,\cdots ,N\hfill \\ 1,2,\cdots ,i-1,i+1,\cdots ,N\hfill \end{array}\right)$ is the submatrix obtained from matrix $L\left(X\right)$ by deleting row i and column i, $i\in [1,N]$. An increase in the quantity of spanning trees implies a greater number of alternative operational configurations available to the system in the event of faults [18]. Obviously, the larger the value of $T\left(X\right)$, the stronger the reconfiguration capability of network G corresponding to X.

The aforementioned metric provides an evaluation of the number of alternative paths in the system from the perspective of complex networks. However, they fall short in distinguishing between source buses and load buses and neglect the impact of electrical factors. Consequently, it becomes necessary to consider the other objectives along with the DN reconfiguration capability.

2.2. Source-to-Load Shortest Distance Metric

From the perspective of equilibrium and reliability during the operational phase of DNs, it can be reasonably concluded that a shorter distance between each load bus and the source bus corresponds to a more equilibrious system. This, in turn, diminishes the likelihood of bus isolation, thereby enhancing system reliability. Simultaneously, an increased distance between the source and load buses leads to substantial network losses, potentially causing voltage drops that may result in non-compliance with electrical quality standards. Consequently, we propose the source-to-load shortest distance metric, emphasizing the requirement for minimizing the distance from the source bus to each load bus.

The shorter the distance between a load bus and the source bus, the more secure the operational assurance of the bus. Given the potential occurrence of faults in transmission lines, a reduced distance from the source bus to a load bus correlates with a diminished likelihood of the line supplying energy to the load bus experiencing a fault. Considering the radial operational configuration of DNs, any fault in a transmission line results in the incapacitation of the load bus’s ability to receive energy supply from the source bus, leading to system failure. We let $\lambda$ represent the probability of a fault or attack occurring on a transmission line. Thus, $(1-\lambda )$ signifies the probability of the line operating normally. If the distance from load bus i to source bus j is denoted as c, with the unit step length, in the normal operation state of the DN, the reliability of load bus i receiving energy support from source bus j can be expressed as follows:

$${(1-\lambda )}^c$$

(3)

As illustrated in Table 1, it is evident that an increased distance between the source and load buses corresponds to a diminished level of assurance in the energy supply to the load bus from the source bus. This effect is particularly pronounced when $\lambda =0.1$, signifying that an augmentation in the distance between the source and load buses substantially weakens the assurance of normal system operation.

Table 1. Reliability of the load bus receiving energy support from the source bus, where $\lambda$ is the probability of line failure.

Step Length	$\mathit{\lambda }=0.01$	$\mathit{\lambda }=0.05$	$\mathit{\lambda }=0.1$
1	0.99	0.95	0.9
2	0.9801	0.9025	0.81
3	0.9703	0.8574	0.729
4	0.9606	0.8145	0.6561
5	0.9510	0.7738	0.5905

Figure 1 illustrates a schematic diagram of a DN topology with two source buses and five load buses, serving to elucidate the definition of the source-to-load shortest distance metric. For clear depiction, blue circles and red triangles symbolize load and source buses, respectively. Dashed lines represent the pre-established topology, while solid lines depict the actual operational topology of the system.

Figure 1. A seven-bus illustrative example.

In a given meshed planning scenario, as depicted in Figure 1a, for Load bus 5, the shortest distance from Source bus 1 is three (path: 5-4-2-1), and from Source bus 6, it is one (path: 5-6). Therefore, the shortest distance from Load bus 5 to the source bus is one, specifically the distance from Load bus 5 to Source bus 6, denoted as the “source-to-load shortest distance”. By this definition, the shortest distance from Load bus 2 to the source is one (path: 2-1). It is imperative to note that the aforementioned computation of the source-to-load shortest distance metric is contingent upon a specific planning scheme, namely determining the closest proximity of each respective bus to the source bus within a meshed topology. However, during the actual operation of a DN, the requirement is typically radial. Consequently, under the prescribed operation conditions within the meshed topology, Load buses 5 and 2 may find it challenging to concurrently meet the criterion of the source-to-load shortest distance in the mesh grid. As illustrated in Figure 1b, only Load bus 2 satisfies the condition of being the shortest to the source bus, with the metric of one (path: 2-1), consistent with the result in the mesh grid. In this operation condition, the source-to-load shortest distance for Load bus 5 becomes two (path: 5-4-6), deviating from the result in the mesh grid. In the operation state depicted in Figure 1c, Load bus 5 exhibits a shortest path consistent with the meshed planning phase (path: 5-6), with a metric of one. Meanwhile, the metric for Bus 2 becomes two (path: 2-4-6). When operating under the radial topology depicted in Figure 1d, both Load buses 5 and 2 simultaneously meet the requirements for the source-to-load shortest distance in the mesh grid, i.e., the source-to-load shortest distance and path under this configuration align with the results obtained in the mesh grid illustrated in Figure 1a. Consequently, it is imperative to compute the source-to-load shortest distance metric within a given radial topology.

For a more precise formalization, we let V and U denote the sets of load and source buses, respectively, where $V=\{v_1,v_2,v_3,\dots ,v_n\}$, $U=\{u_1,u_2,u_3,\dots ,u_s\}$, $n=\left|V\right|$ is the number of load buses, and $s=\left|U\right|$ is the number of source buses, and $N=n+s$ represents the total number of buses. The distance between Load bus $v_i$ and Source bus $u_j$ is denoted as $d(v_i,u_j)$. It is important to note that the acquirement of $d(v_i,u_j)$ is performed in a radial topology, not in a mesh topology. Then, the source-to-load shortest distance between Load bus $v_i$ and the source bus can be obtained as follows:

$$min\left\{d(v_i,u_j)|j\in \{1\dots s\}\right\}$$

(4)

For the DN system, the total source-to-load shortest distance can be formulated as follows:

$$minP=\sum _{i=1}^{n}min\left\{d(v_i,u_j)\right|j\in \{1\dots s\}\}$$

(5)

where $minP$ represents the equilibrium and reliability of the radial operation scheme of the system.

3. Problem Formulation of DNTPDP

In this work, we concurrently incorporate the metric of the number of spanning trees during topology design and the objective of the source-to-load shortest distance during radial operations. Traditionally, solving this problem involves a two-step approach. First, it entails providing the optimal topology design scheme for adding K lines based on the given initial radial topology according to the spanning tree metric. Subsequently, it involves optimizing the radial operation scheme based on the optimized topology design with respect to the source-to-load shortest distance metric. Evidently, this approach exhibits a notable drawback. Specifically, during the optimization of the operation scheme, it focuses on optimizing the radial operation scheme under a given topology design scheme, ensuring that the optimization result achieves a superior solution in terms of system reconfiguration capability. However, it falls short of further enhancing the equilibrium and reliability of the system operation. Moreover, it provides limited decision alternatives, offering only a singular solution.

Motivated by the insights gained from the aforementioned analysis, we advocate for a coordinated optimization approach that concurrently considers both planning and operation schemes. Employing a multi-objective framework for optimization, we aim to address the limitations inherent in the two-step approach, which ignores the trade-offs and conflicts inherent in bi-objective optimization. By considering the optimization of the system’s topology design and operation scheme within a multi-objective framework, we can effectively explore a range of decision alternatives, rather than focusing on a single “optimal” solution. Therefore, it becomes necessary to formulate a bi-objective expression that accounts for the equilibrium of radial operation, reliability, and DN reconfiguration capability. The expansion of lines entails increased costs and complexity and therefore consideration of the addition of a certain number of network connections. To be specific, line expansion in an existing network to maximize reconfiguration capability can be measured by the number of spanning trees, as well as improving equilibrium when the network is operating, which can be measured by the source-to-load shortest distance. To this end, we establish the bi-objective optimization model as follows:

$$minF\left(X\right)=\{f_1\left(X\right)=-T\left(X\right),min{f}_2\left(\tilde{X}\right)=minP\}$$

(6)

$$\begin{array}{ccc}s.t.& & \sum _{i=1}^{N-1}\sum _{j=i+1}^N{X}_{ij}\le N-1+K\end{array}$$

(7)

$$\begin{array}{cccc}& & & \sum _{i=1}^{N-1}\sum _{j=i+1}^N{a}_{ij}{X}_{ij}=N-1\end{array}$$

(8)

$$\begin{array}{cccc}& & & X_{ij}=X_{ji},i,j=1·····N\end{array}$$

(9)

$$\begin{array}{cccc}& & & X_{ij}\in \{0,1\},i,j=1·····N\end{array}$$

(10)

$$\begin{array}{cccc}& & & \sum _{i=1}^{N-1}\sum _{j=i+1}^N{\tilde{X}}_{ij}=N-1\end{array}$$

(11)

$$\begin{array}{cccc}& & & \sum _{i=1}^{|S^{\prime }|-1}\sum _{j=i+1}^{|S^{\prime }|}{\tilde{X}}_{ij}\le \left|S^{\prime }\right|-1,\forall S^{\prime }\in S\end{array}$$

(12)

$$\begin{array}{cccc}& & & \sum _{i=1}^{N-1}\sum _{j=i+1}^N{\tilde{X}}_{ij}{X}_{ij}=N-1\end{array}$$

(13)

$$\begin{array}{cccc}& & & {\tilde{X}}_{ij}={\tilde{X}}_{ji},i,j=1·····N\end{array}$$

(14)

$$\begin{array}{cccc}& & & {\tilde{X}}_{ij}\in \{0,1\},i,j=1......N\end{array}$$

(15)

The above mathematical model consists of the optimization objective function and constraints of the DNTPDP, where $A={\left(a_{ij}\right)}_{N\times N}$ represents the adjacency matrix of the existing system, which is radial, $S=\{v_1^{\prime },\dots ,v_t^{\prime },u_1^{\prime },\dots u_h^{\prime }|v_t^{\prime }\in V,u_h^{\prime }\in U\}$. Objective function (6) encapsulates the maximization of the DN’s reconfiguration capability while simultaneously aiming to enhance the equilibrium and reliability of system operation to the greatest extent possible. Given the adoption of the standard form of the bi-objective formulation involving minimization ($min$), $min-T\left(X\right)$ is equivalent to $maxT\left(X\right)$. Constraint (7) restricts the number of newly added lines to K, which serves as a constant constraint on the maximum number of expansion lines. Constraint (8) ensures that the initially given radial topology remains unchanged during the line expansion process. Constraints (11) and (12) are used to ensure that the radial operation scheme of the system adheres to a tree-like structure. Constraint (13) imposes the condition that the adjacency matrix $\tilde{X}$ corresponds to a subgraph generated from the graph represented by the adjacency matrix X. Equations (9) and (10) and Equations (14) and (15), respectively, indicate that X and $\tilde{X}$ are an $N\times N$-dimensional symmetric binary variable, where $X_{ij}=1$ or ${\tilde{X}}_{ij}=1$ represent that there is a line between Bus i and Bus j. In fact, $\tilde{X}$ emerges as an incidental byproduct when evaluating the second objective function ($f_2$) of X.

It is essential to note that in contrast to traditional bi-objective optimization, the evaluation of the operation scheme ($min{f}_2\left(\tilde{X}\right))$ in the above context is framed as an optimization problem rather than a computational one. In the realm of traditional bi-objective optimization, evaluating any solution X involves directly computing its corresponding objective function values, a task achievable in constant time, $O\left(1\right)$. However, in the context of the bi-objective optimization problem proposed in this study, assessing the first objective function value $f_1$ for a given X is straightforward and also operates within $O\left(1\right)$ complexity. Conversely, evaluating the second objective function value $f_2$ necessitates revisiting the optimization process, constituting a re-optimization rather than a mere calculation. Thus, determining the second objective function value $f_2$ associated with X no longer adheres to $O\left(1\right)$ complexity. In other words, this problem simultaneously represents a bi-objective optimization with a sub-optimization problem challenge, which can be also called a bi-level challenge. Figure 2 illustrates the conceptual framework of the DNTPDP, which has the bi-level nature. To solve the DNTPDP, as shown in Figure 2, first, upper-level optimization involves obtaining an optimal topology design (mesh grid) through the optimization of line additions within a given radial topology. Subsequently, the sub-problem optimization in the lower level engages in radial operation scheme based on each designed topology (mesh grid) where the radial network corresponds to the subgraph generated by the design scheme. Given the diverse radial strategies available, the evaluation of the operation scheme becomes a sub-optimization problem. Therefore, when evaluating a new solution, it necessitates a re-optimization process. This entails invoking the optimizer again to solve the sub-optimization problem for the evaluation of the operation scheme.

Figure 2. The bi-level nature of DNTPDP.

To further elucidate the evaluation of planning scheme X corresponding to its $f_2$ value as an optimization rather than a computational issue, Figure 3 provides an illustrative example involving a four-bus test system. The dashed black lines and solid black lines denote the topological configuration derived from the planning stage, and the topology formed during actual operation, respectively. The optimized topology corresponding to X, as depicted in the upper layer of Figure 3, reveals various scenarios during normal operation, as shown in the lower layer. Among these four potential operational scenarios, we designate the source-to-load shortest metric of the system with the best equilibrium as the evaluated $f_2$ value for X. However, determining which open-loop operational strategy yields the optimal system equilibrium necessitates invoking the optimizer once again for resolution.

Figure 3. Illustrative representation of DN operation equilibrium optimization, where the blue dot represents the bus and the number represents the bus number.

4. Solution Methodology

In order to verify the proposed idea, a popular algorithm, NSGA-II, is selected as the foundational algorithm framework for the bi-objective DN optimization model proposed in Section 3. Figure 4 depicts the comprehensive flowchart of the whole methodology employed to solve the DNTPDP. As shown in Figure 4, the value of $f_1$ can be calculated directly from the corresponding solution of the individual to evaluate the reconfiguration capability of the system. The position highlighted in red boxes denotes where heuristic algorithms are integrated into the NSGA-II algorithm framework, aimed at solving sub-optimization problems and evaluating the $f_2$ metric based on optimized results of the source-to-load shortest distance metric. Next, we elaborate on the implementation specifics of NSGA-II and the proposed heuristic algorithms.

Figure 4. Flowchart of the solving process for the DNTPDP.

4.1. NSGA-II

NSGA-II, renowned for its robustness, has found widespread application in DN planning problems [19]. Additionally, this method has a wide range of applications in various fields, such as transportation [20], portfolio optimization [21], supply chain [22], and biomedical engineering [23]. This provides a solid theoretical and practical support for us to choose NSGA-II for optimization.

The topology of a DN can be effectively represented by its adjacency matrix. While matrix encoding can fully capture the topological connectivity status for DNTPDP, it is prone to generating invalid solutions. Therefore, we adopt integer encoding. We assign numbers to all feasible newly constructed lines in the network, namely the positions in the adjacency matrix corresponding to the elements of zero. The length of the solution encoding corresponds to the number of lines to be expanded. The values of the genes represent the lines that need to be expanded. For crossover and mutation operations, we employ a multi-point crossover and multi-point mutation strategy.

4.2. Weighted Radial Operation Heuristic Algorithm

When provided with the topology and source bus locations of a DN system, assessing its optimal operational configuration becomes imperative. Approaching from the perspective of complex networks and leveraging the concept of preserving critical pathways, we introduce the weighted radial operation heuristic (WROH) algorithm to determine the optimal radial operational scheme for the network to speed up the solution process. The WROH algorithm is primarily focused on the path corresponding to the total source-to-load shortest distance. The importance of a specific line is contingent upon the frequency of its repeated traversal. Consequently, in the selection of radial operation schemes, greater emphasis should be placed on retaining paths that exhibit a higher recurrence of traversal, signifying their increased significance in the system. In radial operation schemes, priority should be given to retaining such lines.

To be specific, the approach involves computing the source-to-load shortest distances and corresponding paths from each load bus to source bus. Subsequently, for each line traversed by each shortest path, the corresponding position in the adjacency matrix is subjected to a weighted treatment. The resulting weighted matrix is then employed to assess the importance of each line, and a tree-like radial operation topology, preserving crucial lines, is computed using the minimum spanning tree (MST) algorithm [24]. It is important to note that as the MST algorithm is applied here, where smaller weights indicate greater importance, the reciprocals of the weights in the weighted matrix are used as input for the MST algorithm. Figure 5 presents the workflow of the WROH algorithm. The pseudocode of WROH in Algorithm 1 is presented. The algorithm’s input and output mainly consist of a mesh-designed topological structure and a radial operational scheme, each represented by a symmetric adjacency matrix.

Algorithm 1: Weighted Radial Operation Heuristic Algorithm

Input:  X, the mesh topology design plan obtained by the DN reconfiguration capability optimization; n, the number of buses; Output:  $\tilde{X}$ the high-quality initial radial operation scheme; 1: Find the source-load shortest path from each load bus to source bus, and all shortest paths are grouped as set P; 2: $W=X$, set the weight matrix W to X; 3: $i=1$; 4: while  $i\le n$do 5:    Obtain the ith path $P_i$ and obtain the number of buses included in $P_i$, $len$; 6:    $j=1$; 7:    while $j\le len-1$ do 8:      W, according to the jth and $(j+1)$th buses of $P_i$, increment the element at the corresponding position by 1; 9:      $j=j+1$; 10:    end while 11:    $i=i+1$; 12: end while 13: Apply the minimum spanning tree algorithm to $\frac{1}{W}$ to obtain the high-quality initial radial operation scheme;

Figure 5. Flowchart of a weighted radial operation heuristic algorithm.

4.3. Source Bus Determination Heuristic Algorithm

The implicit assumption of the WROH algorithm proposed in Section 4.2 is the given position of the source bus. The source bus determination heuristic algorithm proposed is applied to a scenario where the topological structure is planned and designed, but the exact locations of the source buses remain uncertain. Further optimization of the deployment of the source bus can be conducted to enhance the system’s equilibrium and reliability. Buses of greater importance should exhibit higher degrees, indicating stronger connections with other buses. Additionally, considering the distance between the bus and others is essential for the overall equilibrium of the system. Figure 6 illustrates the workflow of the algorithm for determining the source bus position, termed as the degree-distance heuristic (DDH) algorithm. This algorithm aims to optimize the deployment of the source bus by considering both bus importance, reflected in their degrees, and their spatial relationships with other buses.

Figure 6. Flowchart of degree-distance heuristic algorithm.

As depicted in Figure 6, when optimizing the deployment of a single source bus, the selection criteria should prioritize buses with a high degree and the shortest cumulative distance to other buses. A greater degree signifies the bus’s increased importance, while a shorter total shortest distance to other buses indicates a more balanced distribution of load buses around the source bus. This aligns with the proposed metric of source-to-load shortest distance, which measures the system’s equilibrium. Upon this foundational premise, the continued augmentation of source buses necessitates a concomitant consideration of the significance of each bus, as denoted by its degree. Subsequently, the evaluation should extend to the distance between the newly contemplated source bus and the source buses already added. Ideally, this distance should be maximized. Proximity among distinct source buses corresponds to a concentrated spatial distribution of these buses. In view of the equilibrium governing system operations, a greater distance between the prospective source bus and those already integrated implies a superior dispersion of sources within the network, thereby fostering a more balanced spatial distribution. The pseudocode of DDH in Algorithm 2 is presented. The algorithm receives as input a mesh-designed topological structure, represented by a symmetric adjacency matrix, alongside a constant constraint specifying the number of source buses to be added. The output comprises a set of schemes detailing the positions of these source buses.

Algorithm 2: Degree-distance heuristic algorithm.

Input:  X, the mesh topology design plan obtained by the DN reconfiguration capability optimization; s, the number of source buses; Output:  Set $P_o$, the high-quality source bus location scheme; 1: Let $P_o$ to an empty set; 2: Calculate the degree of each bus and the shortest distance from each bus to the other buses; 3: Select the bus with highest degree and the shortest distance to other buses as the newly added position of the source bus, and add it to set $P_o$; 4: $i=1$; 5: while  $i<s$do 6:    Calculate the degree of each bus and the total shortest distance from the already added source buses; 7:    Select the bus with highest degree and the farthest distance to the already added source buses, and add it to set $P_o$; 8:    $i=i+1$; 9: end while 10: Output the high-quality source bus location scheme, $P_o$;

5. Numerical Experiments

5.1. Validation of Heuristic Algorithms

To further illustrate the effectiveness of the heuristic algorithms proposed in Section 4, we selected 3 advanced algorithms to be used as competitor algorithms for the heuristic and tested them extensively in 500 randomly generated test cases.

5.1.1. Comparison of Different Algorithms for Radial Operation Schemes

To further elucidate the effectiveness and applicability of the heuristic algorithm proposed for the sub-optimization problem radial operation, we employed additional EAs as comparative benchmarks in this section. We opted for three advanced algorithms: GA [25], differential evolution (DE) [26], and particle swarm optimization (PSO) [27], each relying on distinct operators. GA utilizes simulated binary crossover and polynomial mutation operators, which has been proven effective in overcoming challenges posed by sparse individual spaces and real-valued encoding.

In accordance with the characteristics of the system’s radial operation, for all EAs, we adopted real-valued encoding with a code length equivalent to the total number of lines in the system. Specifically, each real number is mapped to the weight of each edge, and the MST algorithm is subsequently applied for decoding, yielding the radial operation scheme. The population size was set as $Popsize=2\ast N$, where N represents the bus count, and the iteration termination criterion was established as no updates for 50 consecutive generations, indicating algorithm convergence. To ensure a fair comparison, default parameters were employed, which have been proven effective for benchmark problems. We randomly generated 500 test cases, with the number of bus $N\in [16,70]$ and line numbers within range $[N,N+9]$. This encompasses adding 1 to 10 lines as expansion lines to a given tree-like network. The quantity of source buses was $s\in [1,3]$ in these instances, serving to validate the effectiveness of the proposed WROH algorithm.

Table 2 presents the ranking and total runtime of the GA, DE, PSO, and the WROH algorithm based on source-to-load shortest distance metric across 500 test cases. For each case, the source-to-load shortest distance metric optimized by the four algorithms was arranged in ascending order, resulting in rankings from 1 to 4. In other words, given a test case, “Rank 1” signifies that the algorithm’s solution quality on this instance is no less than that of other algorithms; “Rank 2” indicates that its solution on this instance is inferior to that of one other algorithm; “Rank 3” denotes that its solution on this instance is inferior to that of three other algorithms; and “Rank 4” indicates that its solution on this instance does not perform as well as that of the other three algorithms.

Table 2. Performance comparison of different algorithms on 500 test cases for optimization of radial operation scenarios.

Algorithm	# Rank 1	# Rank 2	# Rank 3	# Rank 4	Time(s)
DE	389	0	28	83	2649.7349
GA	499	0	0	1	2080.2186
PSO	417	0	38	45	2092.8237
WROH	500	0	0	0	1.4536

As shown in Table 2, # Rank 1, # Rank 2, # Rank 3, # Rank 4 represent the number of cases in which the algorithm ranked Rank 1, Rank 2, Rank 3, and Rank 4 out of 500 test cases, respectively. It is evident that the proposed heuristic algorithm consistently outperforms the radial operation schemes obtained from the other three EAs across 500 random test cases, achieving encouraging results. Additionally, the GA exhibits commendable performance, securing the optimal solution in 499 test cases and ranking fourth in the remaining 1 case. The PSO algorithm attains the optimal radial operation solution in 417 out of 500 cases, ranked third in solution quality in 38 test cases, and fourth in 45 cases, showcasing robust performance. In contrast, the DE algorithm performs less satisfactorily, achieving the optimal radial operation solution in only 389 cases, ranking third in solution quality in 28 test cases and fourth in 83 cases.

Notably, while there is no significant difference in the quality of radial operation solutions between GA and the WROH algorithm, there is a substantial disparity in computational efficiency. GA requires 2080.2186 s to optimize 500 test cases, while the proposed WROH algorithm accomplishes the same task in a mere 1.4536s. The utilization of the WROH algorithm, in contrast to the GA, results in an efficiency enhancement exceeding one thousandfold. As illustrated above, the WROH algorithm for radial operation schemes, considering given source bus positions and network expansion planning topology, exhibits exceptional solving performance with minimal time consumption. The efficiency advantage is conspicuous. Therefore, for the the sub-optimization objective ($f_2\left(\tilde{X}\right)$), we opt for the WROH algorithm due to its simplicity, efficiency, and stability.

5.1.2. Comparison of Different Algorithms for Source Bus Determination

In Section 5.1.1, we demonstrated the efficacy of the WROH algorithm for radial operations given fixed source bus position. Building upon this foundation, we proceed to optimize the deployment of source bus position. Utilizing 500 randomly generated test cases, with rules similar to those outlined in Section 5.1.1, the distinction lies in the fact that source bus positions are no longer fixed but require optimization. To validate the effectiveness of the proposed heuristic algorithm for determining source bus position, we also select GA, PSO, and DE algorithms as competitor algorithms. Consistent with Section 5.1.1, we employ real-valued encoding for determining the final radial scheme and source bus deployment positions.

It is essential to note that when the number of lines in the DN is ($N-1+K$) and the desired number of source buses is s, the last s bits are mapped through real number encoding to integer values in range $[1,N]$. These integers serve as indicators for the positions of the source buses. The first ($N-1+K$) bits are subjected to the MST algorithm, decoded into an radial operation scheme. To substantiate the efficacy of the DDH algorithm employed in determining the positions of source buses, we initially apply this strategy to ascertain the positions of source buses. Then, the proven effective WROH algorithm is applied, culminating in the determination of the source-to-load shortest distance metric.

Table 3 presents the ranking of GA, DE, PSO, and the DDH algorithm concerning source bus optimization deployment and radial schemes across 500 test cases, with the source-to-load shortest distance as the metric. GA exhibits exceptional performance, securing optimal results in 234 cases, a significantly higher likelihood than what other algorithms can achieve. It ranks second in solution quality in 217 test cases, third in 38 cases, and fourth in 11 cases. The DDH algorithm for determining source bus positions follows suit, obtaining optimal results in 196 cases, ranking second in 7 test cases, third in 13 cases, and fourth in 284 cases. Notably, the DDH algorithm outperforms GA significantly in terms of computational efficiency, requiring 4.099 s compared to GA’s 2436.0307 s, resulting in an efficiency augmentation of several hundredfold. In contrast, DE and PSO algorithms exhibit less satisfactory performance. The DE algorithm achieves first, second, third, and fourth rankings in 84, 120, 194, and 102 cases, respectively, while the PSO algorithm achieves these rankings in 119, 135, 159, and 187 cases, respectively.

Table 3. Performance comparison of different algorithms for optimization of determining source bus position on 500 test cases.

Algorithm	# Rank 1	# Rank 2	# Rank 3	# Rank 4	Time(s)
DE	84	120	194	102	3213.8257
GA	234	217	38	11	2436.0307
PSO	119	135	159	87	2644.5641
DDH	196	7	13	284	4.099

While the DDH algorithm employed for determining the positions of source buses sacrifices a degree of optimality in comparison to GA, its noteworthy enhancement in computational efficiency deserves acknowledgment. It has promising applications in large-scale systems and is worthy of further refinement, optimization, and adjustment.

Notably, the proposed heuristic algorithms in Section 4.2 and Section 4.3 are equally applicable to any other network that is suitable for the source-to-load shortest distance metric measurement.

5.2. Bi-Objective Optimization of DNTPDP

To demonstrate the efficacy and competitiveness of the proposed model and algorithms, a series of comparative studies were conducted, involving validation experiments conducted on the IEEE 33-bus [28] and 70-bus [29] test systems. To be specific, the top N-1 lines were selected as the initial radial topology of the preset, and the number of remaining lines were chosen as the limit for newly added lines.

Given the bi-objective optimization with the sub-optimization nature of this problem, EAs for the resolution of the sub-optimization objective proves to be excessively time-consuming. To address this issue, we integrated the proposed heuristic algorithm into a multi-objective optimization algorithm (NSGA-II) framework. To substantiate the efficacy of the proposed algorithm, we opted solely for the integration of the proficient GA as a benchmark algorithm within the framework of NSGA-II, facilitating a comparative analysis. For GA, we let population size be $min\{2\ast N,100\}$ while setting the termination criterion for iterations as the absence of updates in the optimal fitness over a consecutive span of 10 generations. For the NSGA-II, the parameter settings listed in Table 4 were used.

Table 4. NSGA-II parameter settings.

Parameter	Value
Population size	$min\{N\ast K,300\}$
$maxGen$	300
Crossover probability	0.8
Mutation probability	0.2

In order to reduce the impact of the inherent randomness of the EAs, 5 independent algorithm runs were performed. The optimal Pareto front composed of 5 experiments was recorded.

Note that experiments were conducted under two scenarios: one where the source bus positions are predetermined, referred to as the DNTPDP with fixed source bus scenario, and another where only the number of source buses is given without specifying their positions, referred to as the DNTPDP with source bus to be optimized scenario.

5.2.1. DNTPDP with Fixed Source Bus

Embedding the WROH algorithm into the NSGA-II to solve the DNTPDP with fixed source bus scenario is denoted as NSGA-II (WROH), while incorporating the GA into the NSGA-II is represented as NSGA-II (GA). Figure 7 illustrates the Pareto front obtained through the employment of NSGA-II (WROH) and NSGA-II (GA), where the x-axis and the y-axis, respectively, depict the source-to-load shortest distance and the number of spanning trees. As depicted in Figure 7a, negligible disparities in the Pareto front are discerned when applied to the IEEE 33-bus test system using both approaches. Table 5 delineates the average computation time taken for 5 independent runs of NSGA-II (WROH) and NSGAII (GA). The results underscore that for each test system, NSGA-II (WROH) necessitates a lesser computation time in comparison to the NSGA-II (GA) algorithm. Specifically, for the 33-bus test system, NSGA-II (GA) demands 34,565 s, whereas NSGA-II (WROH) requires a mere 123 s, revealing a substantial disparity. NSGA-II (WROH) demonstrates an efficiency improvement of 281 times compared to NSGA-II (GA) on the IEEE 33-bus test system. Similar trends are evident on the IEEE 70-bus system, where NSGA-II (WROH) achieves a 455-fold efficiency gain compared to NSGA-II (GA), as illustrated in Figure 5b and Table 5. In the context of DNTPDP with a fixed source bus, utilizing NSGA-II (WROH) to solve this problem improves computational efficiency by several hundredfold compared to employing NSGA-II (GA).

Figure 7. In the context of DNTPDP with a fixed source bus, the Pareto front obtained by NSGA-II (WROH) and NSGA-II (GA) in the two test systems.

Table 5. In the context of DNTPDP with a fixed source bus, a comparison of computation time between NSGA-II (WROH) and NSGA-II (GA).

Test System	Method
	NSGA-II (WROH)	NSGA-II (GA)
33-bus	123 s	34,565 s
70-bus	435 s	198,123 s

5.2.2. DNTPDP with Source Bus to Be Optimized

Embedding the DDH algorithm into the NSGA-II to solve the DNTPDP with a source bus to be optimized scenario is denoted as NSGA-II (DDH), while incorporating the GA into the NSGA-II is represented as NSGA-II (GA).Figure 8 and Table 6 delineate outcomes similar to those expounded in Section 5.2.1. Specifically, NSGA-II (DDH) exhibits commensurate performance with NSGA-II (GA) in terms of solution efficacy and performance, yet it markedly enhances computational efficiency. Within the 33-bus test system, NSGA-II (GA) necessitates 43,274 s to accomplish the optimization tasks encompassing line expansion schemes, optimal deployment of source bus positions, and the schemes of initial radial operation configurations. Conversely, NSGA-II (DDH) achieves these optimization objectives within a mere 134 s. Therefore, in the IEEE 33-bus test system, NSGA-II (DDH) demonstrates a 323-fold improvement in computational efficiency compared to NSGA-II (GA). In the 70-bus test system, NSGA-II (DDH) exhibits a 362-time enhancement in computational efficiency compared to NSGA-II (GA). Specifically, NSGA-II (GA) demands 306,658 s, while NSGA-II (DDH) accomplishes the same tasks in mere 845 s. This revelation underscores the pronounced advantages and efficiency of the proposed heuristic algorithm.

Figure 8. In the context of DNTPDP with a source bus to be optimized, the Pareto front obtained by NSGA-II (DDH) and NSGA-II (GA) in the two test systems.

Table 6. In the context of DNTPDP with a source bus to be optimized, a comparison of computation time between NSGA-II (DDH) and NSGA-II (GA).

Test System	Method
	NSGA-II (DDH)	NSGA-II (GA)
33-bus	134 s	43,274 s
70-bus	845 s	306,658 s

In summary, this experimental evaluation establishes the efficacy of the proposed algorithm in terms of both efficiency and performance. This achievement can be attributed to the utilization of the proposed WROH and DDH algorithms.

5.2.3. The Effect of Optimizing the Source Bus on DNTPDP

We demonstrated the advantages inherent in the proposed NSGA (WROH) and NSGA (DDH) in addressing DNTPDP. In this section, an analysis is conducted on the impact of optimizing the position of the source buses on the overall system equilibrium. Figure 9 illustrates the results of solving the bi-objective optimization model under two scenarios: one where optimization is performed based on a given source bus position and the other where optimization is carried out considering the deployment of source buses.

Figure 9. The Pareto front obtained under different scenarios: (a) Using heuristic algorithms in the IEEE 33-bus test system; (b) Using GA in the IEEE 33-bus test system; (c) Using heuristic algorithms in the IEEE 70-bus test system; (d) Using GA in the IEEE 70-bus test system.

As shown in Figure 9, the blue points represent the Pareto front obtained through optimization based on a given source bus position, while the red points depict the results obtained through optimization considering the deployment of the source bus. Through IEEE 33-bus and 70-bus test cases, it becomes apparent that, whether incorporating heuristic algorithms into the NSGA-II algorithm for solution or embedding the GA algorithm into the NSGA-II algorithm for solution, the overall Pareto front in red, compared to the blue Pareto front, consistently shifts upward. This implies that optimizing the deployment of source buses enhances the system’s reconfiguration capabilities, promoting a more balanced system. Consequently, it leads to obtaining superior initial radial operation schemes, thereby further enhancing the equilibrium and reliability of the system during operation.

It is worth mentioning that each point on the Pareto front represents an alternative solution. Trade-offs between reconfiguration capability and operation equilibrium can be considered to obtain corresponding expansion planning and initial radial operation schemes as per actual needs. In the context of DNTPDP with a fixed source bus, taking the optimization results obtained using the NSGA-II (WROH) algorithm in the IEEE 33-bus test system as an example, Figure 10 presents the Pareto front obtained from the optimization, the planning scheme corresponding to the knee point, and the initial radial operation scheme, where $T\left(X\right)$ = 79,754 and $minP=111$. Figure 11 shows the Pareto front obtained by the IEEE 70-bus test system in the DNTPDP with a fixed source bus scenario, the planning scheme and initial radial operation scheme corresponding to the knee point, where $T\left(X\right)$ = 136,080,746 and $minP=324$.

Figure 10. Display of alternative schemes in IEEE 33-bus test system, where the red dot denote source bus, while the blue dots represent load buses, blue solid lines indicate the initial radial topology, red solid lines signify newly added lines, and black solid lines depict actual operational network connections. (a) The Pareto front obtained by the NSGA-II (WROH); (b) expansion planning scheme; (c) initial radial operation scheme.

Figure 11. Display of alternative schemes in IEEE 70-bus test system, where the red dots denote source buses, while the blue dots represent load buses, blue solid lines indicate the initial radial topology, red solid lines signify newly added lines, and black solid lines depict actual operational network connections. (a) The Pareto front obtained by the NSGA-II (WROH); (b) expansion planning scheme; (c) initial radial operation scheme.

5.3. Discussion

We demonstrated the advantages of the proposed bi-objective optimization model and algorithms in solving the DNTPDP. This section analyzes the influence of using the conventional two-step method to tackle the DNTPDP. The two-step method involves first obtaining the optimal expansion planning scheme based on the reconfiguration capability metric. Subsequently, leveraging this optimized blueprint, the attainment of an optimal radial operation scheme is pursued in accordance with the system equilibrium metric. Multiple experiments were conducted employing a staged strategy to resolve the DNTPDP. Specifically, when the source buses were fixed, distinct experiments were conducted using GA for expansion planning optimization, followed by another application of GA for radial operation optimization, denoted as GA-GA (fixed). Furthermore, GA was applied for expansion planning optimization, succeeded by optimization of radial operation using the WROH algorithm, denoted as GA-WROH. In scenarios where the source buses were not constrained, analogous experiments were conducted, employing GA for expansion planning optimization, followed by another application of GA for radial operation optimization, expressed as GA-GA (unfixed). Additionally, GA was utilized for expansion planning optimization, followed by radial operation optimization utilizing the DDH algorithm, denoted as GA-DDH. For a fair comparison between the two-step and joint optimization method, all parameters were maintained consistent with those previously employed.

Table 7 delineates the experimental outcomes employing the two-step optimization strategy. The findings manifest a certain enhancement in the system’s reconfiguration capability relative to bi-objective optimization. Additionally, a reduction in computational time is discernible across all test systems (as demonstrated by the comparison to Table 7 with Table 5 and Table 6). For instance, in the case where the source bus is fixed in a 33-bus test system, the two-step method yields an optimal reconfiguration capability of 102,675, whereas the optimal reconfiguration capability obtained through bi-objective optimization stands at 99,656. Nevertheless, compared with bi-objective optimization, the two-step approach only affords a single solution. While it can furnish plans for line expansion with heightened reconfiguration capabilities, it falls short in further refining the equilibrium of system operations, as its equilibrium remains fixed at $minP=138$. In contrast, bi-objective optimization offers multiple alternative schemes, abstaining from single-mindedly pursuing a particular optimal objective. The resulting optimization scheme $T\left(X\right)$ fluctuates within the range of 10,483–99,656, with $minP$ varying within $[75,142]$, as shown in Figure 7, affording decision-makers the latitude to balance considerations based on practical situations. Similar conclusions are evident in other test cases.

Table 7. Results obtained by two-step methods.

Test System	DNTPDP	Method	Metrics
			T(X)	minP	Time
33-bus	fixed source bus	GA-WROH	$1.0268\times {10}^5$	138	4.5655 s
		GA-GA(fixed)	$1.0268\times {10}^5$	138	7.0145 s
	source bus to be optimized	GA-DDH	$1.0268\times {10}^5$	132	4.5672 s
		GA-GA(unfixed)	$1.0268\times {10}^5$	132	7.0719 s
70-bus	fixed source bus	GA-WROH	$1.7935\times {10}^8$	353	17.9789 s
		GA-GA(fixed)	$1.7935\times {10}^8$	353	23.4411 s
	source bus to be optimized	GA-DDH	$1.7935\times {10}^8$	342	17.9807 s
		GA-GA(unfixed)	$1.7935\times {10}^8$	335	23.5640 s

6. Conclusions

In this paper, a novel metric is introduced to quantify the equilibrium of DN operations. A bi-objective optimization model is established, with the objectives of spanning tree count and source-to-load shortest distance consideration. Additionally, heuristic algorithms are designed to embed them into the NSGA-II evolutionary multi-objective algorithm to effectively solve the optimization model. The following conclusions are obtained:

• The novel metric for assessing the equilibrium and reliability of DN captures electrical factors during system operation, addressing the limitations of focusing solely on system reconfiguration capability;
• By solving the established bi-objective optimization model, a range of expansion planning scenarios and initial radial operation scheme references are provided for decision-makers, rather than focusing solely on a single “optimal” solution;
• The approach proposed in this work achieves an over 200 times increase in computational efficiency compared to existing methods, maintaining solution quality. This validates the effectiveness and efficiency of the heuristic algorithms introduced;
• Optimization of the placement of source buses further enhances the equilibrium and reliability of network operations.

In future work, we plan to extend our metric and model to consider other factors, such as the integration of renewables and the penetration. We are also interested in taking into account the nature of generation sources, such as the variability and unpredictability associated with renewable energy generation. Since we use the default parameter settings and strategies for NSGA-II, developing advanced and efficient strategies is also encouraged to enhance its competitive performance.