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.
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:
The above mathematical model consists of the optimization objective function and constraints of the DNTPDP, where
represents the adjacency matrix of the existing system, which is radial,
. 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 (
),
is equivalent to
. 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
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
are an
-dimensional symmetric binary variable, where
or
represent that there is a line between Bus
i and Bus
j. In fact,
emerges as an incidental byproduct when evaluating the second objective function (
) of
X.
It is essential to note that in contrast to traditional bi-objective optimization, the evaluation of the operation scheme (
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,
. However, in the context of the bi-objective optimization problem proposed in this study, assessing the first objective function value
for a given
X is straightforward and also operates within
complexity. Conversely, evaluating the second objective function value
necessitates revisiting the optimization process, constituting a re-optimization rather than a mere calculation. Thus, determining the second objective function value
associated with
X no longer adheres to
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.
To further elucidate the evaluation of planning scheme
X corresponding to its
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
value for
X. However, determining which open-loop operational strategy yields the optimal system equilibrium necessitates invoking the optimizer once again for resolution.
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
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
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.
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:
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:
, set the weight matrix W to X; - 3:
; - 4:
while
do - 5:
Obtain the ith path and obtain the number of buses included in , ; - 6:
; - 7:
while do - 8:
W, according to the jth and th buses of , increment the element at the corresponding position by 1; - 9:
; - 10:
end while - 11:
; - 12:
end while - 13:
Apply the minimum spanning tree algorithm to to obtain the high-quality initial radial operation scheme;
|
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.
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 , the high-quality source bus location scheme;
- 1:
Let 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 ; - 4:
; - 5:
while
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 ; - 8:
; - 9:
end while - 10:
Output the high-quality source bus location scheme, ;
|