An Innovative Approach to Radiality Representation in Electrical Distribution System Reconfiguration: Enhanced Efficiency and Computational Performance

Abstract

The reconfiguration problem (DPSR) in electrical distribution systems is a critical area of research, aimed at optimizing the operational efficiency of these networks. Historically, this problem has been approached through a variety of optimization methods. Regarding mathematical models, a key challenge identified in these models is the formulation of equations that ensure the radial operation of the system, along with the nonlinear equations representing Kirchhoff’s laws, the last often necessitating complex relaxations for practical application. This paper introduces an alternative representation of system radiality, which potentially surpasses or matches the existing methods in the literature. Our approach utilizes a more intuitive and compact set of equations, simplifying the representation process. Additionally, we propose a linearization of the current calculation in the power flow model typically used to solve DPSR. This linearization significantly accelerates the process of obtaining feasible solutions and optimal reconfiguration profiles. To validate our approach, we conducted rigorous computational comparisons with the results reported in the existing literature, using a variety of test cases to ensure robustness. Our computational results demonstrate a considerable improvement in computational time. The objective functions used are competitive and, in many instances, outperform the best reported results in the literature. In some cases, our method even identifies superior solutions.

1. Introduction

In contemporary power distribution systems, a significant evolution is essential to accommodate the rising energy requirements and the increasing focus on sustainability and reliability. Central to this evolution is the reconfiguration challenge, which is crucial for optimizing network efficacy and augmenting operational efficiency [1]. Reconfiguration entails modifications in the topology of the power distribution network, primarily aimed at minimizing losses, enhancing load balancing, and improving fault response capabilities. This strategy is not merely vital for the effective operation of power grids but is also integral to incorporating renewable energy sources and adjusting to the dynamic characteristics of contemporary energy demands [2].

The inherent complexity of the DPSR in power distribution systems largely stems from the necessity to optimize network configurations while strictly maintaining a radial topology [3]. A radial topology is fundamentally a tree-like structure where each consumer or node is connected to a single source through a unique path. This simplicity is advantageous for isolating faults and simplifying maintenance but poses significant challenges during reconfiguration.

The primary complexity in reconfiguration arises because every potential change to the network’s topology must ensure that the network remains radial. Any deviation that inadvertently creates loops (or a mesh structure) can lead to multiple paths for power flow, which complicates fault detection and protection schemes significantly [3]. Thus, each potential reconfiguration must be validated not just for its efficiency or cost-effectiveness but also for its adherence to the radial structure.

Furthermore, the nonlinear nature of the power flow equations adds another layer of complexity. These equations, which describe how voltage, current, and power behave and interact across the network, are influenced by the physical layout of the network [4]. When reconfiguring the network, each change in topology can alter these parameters in complex ways. Efficiently solving these nonlinear equations to find an optimal configuration that adheres to all operational, safety, and regulatory constraints under varying load conditions becomes a computationally intensive task.

Therefore, the complexity of the DPSR in distribution systems is twofold: ensuring the network remains strictly radial while dynamically optimizing its performance in accordance with deeply intertwined and nonlinear electrical properties. This makes reconfiguration a critical yet intricate endeavor requiring sophisticated tools and analytical techniques to navigate effectively.

The current landscape of research is marked by an extensive array of mathematical models designed to tackle the complexities of reconfiguration in power distribution systems. On the other hand, these models often vary significantly in their methodologies, particularly in how they represent radiality and handle the nonlinear equations involved. Some models opt for computational simplicity, sacrificing accuracy, while others strive for high fidelity, only to encounter issues with computational intractability [5,6,7]. This variation has resulted in a dynamic yet fragmented research field, with no universally accepted method for addressing the DPSR effectively.

In response to these challenges, our study introduces a novel approach to the reconfiguration issue. We propose an alternative representation of radiality that is more intuitive and requires fewer equations than existing models. This method not only simplifies the mathematical formulation but also enhances the computational efficiency of the reconfiguration process. Moreover, we adopt a linearization technique for current calculations in the power flow model, originally developed by Baran and Wu, to facilitate quicker derivation of feasible solutions.

The primary goal of our research is to reconcile the need for accurate radial network representation with the computational demands of the DPSR. Our findings indicate significant improvements in both processing times and the quality of solutions, suggesting a promising direction for future research in the optimization of power distribution systems [2,4].

2. Literature Review

The reconfiguration of electric power distribution systems (DPSR) has been extensively studied due to its significant impact on enhancing the efficiency and reliability of networks. This review examines pivotal advancements in the sector, particularly focusing on the roles of mathematical modeling and algorithmic solutions in addressing DPSR challenges [5].

Mathematical optimization techniques are adept at resolving linear optimization challenges, consistently achieving the global optimum [8]. Nevertheless, their application to combinatorial optimization issues, characterized by extensive search spaces, can be computationally demanding and sometimes impractical for real-world applications. Despite these constraints, the continuous improvements in computational technology and the emergence of quantum computing continue to drive research interest in these methodologies [9].

Additionally, alternative strategies such as heuristics, metaheuristics, and machine learning have become prevalent in tackling DPSR, primarily due to their reduced computational demands. Table 1 highlights key studies from the literature that utilize various techniques to solve DPSR. For a more extensive review, readers are referred to references [5,7]. This comprehensive approach ensures a nuanced understanding of both the historical context and the current state of the art in DPSR problem-solving techniques.

The first attempt to solve DPSR was carried out in 1975 with the seminal study by A. Merlin and H. Back [5], who modeled network losses using the Joule effect and devised a solution scheme, iteratively reconfiguring the network by opening circuit breakers in feeders with the least power flow. Though convergence was ensured, the iterative process was sluggish. Since then, various approaches have been proposed, from mathematical programming approaches to artificial intelligence and metaheuristics. In more recent years, techniques utilizing mixed-integer programming have been put forth, employing different approaches such as quadratic and second-order programming, as well as decomposition processes. The integration of microgrids and distributed generation has shaped current trends, with research endeavors aiming to maximize renewable source generation. Recent solutions have managed to reduce computational time and enhance solution accuracy, even considering the presence and strategic placement of distributed generation in the network. For a more detailed review of the historical evolution of techniques to solve DPSR you can refer to [8,10,11].

In the field of exact optimization, two problems have been extensively studied by researchers in recent years: radiality conditions and techniques to relax the DPSR, rendering it convex, quadratic, linear, etc. Regarding radiality conditions, Merlin et al. [5] initiated from an initial network configuration, and through a series of iterations, the algorithm progressively opened different branches of the circuit until achieving a radial configuration, verified when the number of active branches equaled the number of nodes minus one. Subsequently, Liu et al. considered the distribution system loads as current sinks. With this model, they transformed the network DPSR into a quadratic programming problem solved by linearizing the objective function [12]. To ensure the radial topology of the solution, they solved the uncapacitated transshipment problem, a linear programming problem which assumes that all sinks and sources can both receive and distribute shipments simultaneously.

Ramos et al. [13] introduced the concept of paths to define radiality conditions in the DPSR. They defined a path as a set of branches connecting a particular node with the substation. Under this definition, radiality is guaranteed if every node has at most one path. Jabr et al. [7] proposed spanning tree constraints, based on the relationship between parent and descendant nodes; with this approach, radiality is guaranteed if one node has only one parent.

Lavorato et al. [14] generalized the radiality conditions used by Merlin et al., considering cases where the network has more than one substation or nodes with zero demand. They also proposed another set of constraints, the so-called single commodity flow, to guarantee radiality and connectivity of the network.

Borges et al. [4] adopted a similar approach to Lavorato et al. in expressing radiality conditions as a single equality between the number of active branches and the number of nodes minus the number of substations, with an additional constraint to ensure current flows in only one direction. This final set of radiality constraints has been widely accepted by researchers in the field, as demonstrated by Pareja et al. [2] who employed a mathematical model closely resembling that described by Borges et al., incorporating the presence of distributed generation in the network along with the optimization problem of strategically placing such distributed generation.

Concerning the relaxation of the original problem, Merlin et al. [5] assumed a purely resistive distribution network, making power flow calculations to estimate losses in a DC load flow. This approach has significant limitations, as distribution networks typically do not exhibit a significantly higher resistive component than inductive [15]. The same assumption was made by Broadwatert et al. [16], by assuming a power factor near one throughout the distribution network.

Table 1. Relevant works developed in the literature divided into the main categories: optimization models, heuristics, metaheuristics, and artificial intelligence.

Ref.	Year	Model Description	Radiality Conditions	Apparent Power
[5]	1975	Quadratic Programming Model—Linearization of the Objective Function	The solution is updated iteratively by opening branches until the number of active branches equals the number of nodes minus one.	Network is assumed to be purely resistive.
[16]	1990	Quadratic loss function. DC power flow.	The radial topology of the circuit is described using circuit component trace.	Distribution system power factor near to 1. DC power flow model.
[12]	1991	Algorithm based on the linear transportation problem. Efficient for small networks.	It solves the linear uncapacitated transshipment problem to ensure radial topology.	Power losses are estimated using a radial power flow method (i.e., a backward and forward trace procedure).
[13]	2010	Mixed-integer quadratic programming model. Voltage drops approximated through Blondel equation.	Every node of the final network must have at most one active path. They expressed radiality conditions as $\sum paths\_to\_i=1$; in this study, a path is defined as a set of branches connecting bus bar I to the substation.	It is expressed as the inequality:       $P_{ij}^2+Q_{ij}^2\le S\_max$
[7]	2012	Mixed-integer conic linear programming model—Mixed-integer linear programming model (polyhedral representation of conic constraints).	Spanning tree constraints of the form:                $b_{ij}+b_{ji}=a_l$                ${\displaystyle {\displaystyle \sum }_j}{b}_{ij}=1$	The authors proposed a convex relaxation for the equation:       $V_j^2{I}_{ij}^2=P_{ij}^2+Q_{ij}^2$ In the form of       $V_j^2{I}_{ij}^2\ge P_{ij}^2+Q_{ij}^2$
[14]	2012	Nonlinear programming model with generalized radiality conditions.	Generalization of radiality conditions to equation              ${\displaystyle {\displaystyle \sum }_{\left(i,j\right)ϵ\mathsf{\Omega }}}{y}_{ij}=\left|N\right|-N_{substations}$ It also introduces single commodity flow constraints to guarantee the network is connected.	Active and reactive power flows are expressed in terms of voltages and phase angles between bus bars, leading to a high nonlinear model.
[4]	2014	Method based on MILP, involving the linearization of the equation relating currents, voltages, and active and reactive powers.	The radiality is guaranteed with the following set of equations:                ${{\displaystyle \sum }}_{\left(i,j\right)\in \mathsf{\Omega }}{y}_{ij}=N-1$                  $y_{ij}+y_{ji}\le 1$	Linearization of $P_{ij}^2$ y $Q_{ij}^2$ through piecewise linear approximations, and $V_j^2{I}_{ij}^2$by linearizing $V_j^2.$
[2]	2022	Method based on mixed-integer linear programming with distributed generation in the network.	The radiality is guaranteed with the following set of equations:             ${{\displaystyle \sum }}_{\left(i,j\right)\in \mathsf{\Omega }}{y}_{ij}=N-1$, $y_{ij}+y_{ji}\le 1$	The authors used the same approach described in [4].

Table 1. Relevant works developed in the literature divided into the main categories: optimization models, heuristics, metaheuristics, and artificial intelligence.

Ref.	Year	Model Description	Radiality Conditions	Apparent Power
[5]	1975	Quadratic Programming Model—Linearization of the Objective Function	The solution is updated iteratively by opening branches until the number of active branches equals the number of nodes minus one.	Network is assumed to be purely resistive.
[16]	1990	Quadratic loss function. DC power flow.	The radial topology of the circuit is described using circuit component trace.	Distribution system power factor near to 1. DC power flow model.
[12]	1991	Algorithm based on the linear transportation problem. Efficient for small networks.	It solves the linear uncapacitated transshipment problem to ensure radial topology.	Power losses are estimated using a radial power flow method (i.e., a backward and forward trace procedure).
[13]	2010	Mixed-integer quadratic programming model. Voltage drops approximated through Blondel equation.	Every node of the final network must have at most one active path. They expressed radiality conditions as $\sum paths\_to\_i=1$; in this study, a path is defined as a set of branches connecting bus bar I to the substation.	It is expressed as the inequality:       $P_{ij}^2+Q_{ij}^2\le S\_max$
[7]	2012	Mixed-integer conic linear programming model—Mixed-integer linear programming model (polyhedral representation of conic constraints).	Spanning tree constraints of the form:                $b_{ij}+b_{ji}=a_l$                ${\displaystyle {\displaystyle \sum }_j}{b}_{ij}=1$	The authors proposed a convex relaxation for the equation:       $V_j^2{I}_{ij}^2=P_{ij}^2+Q_{ij}^2$ In the form of       $V_j^2{I}_{ij}^2\ge P_{ij}^2+Q_{ij}^2$
[14]	2012	Nonlinear programming model with generalized radiality conditions.	Generalization of radiality conditions to equation              ${\displaystyle {\displaystyle \sum }_{\left(i,j\right)ϵ\mathsf{\Omega }}}{y}_{ij}=\left|N\right|-N_{substations}$ It also introduces single commodity flow constraints to guarantee the network is connected.	Active and reactive power flows are expressed in terms of voltages and phase angles between bus bars, leading to a high nonlinear model.
[4]	2014	Method based on MILP, involving the linearization of the equation relating currents, voltages, and active and reactive powers.	The radiality is guaranteed with the following set of equations:                ${{\displaystyle \sum }}_{\left(i,j\right)\in \mathsf{\Omega }}{y}_{ij}=N-1$                  $y_{ij}+y_{ji}\le 1$	Linearization of $P_{ij}^2$ y $Q_{ij}^2$ through piecewise linear approximations, and $V_j^2{I}_{ij}^2$by linearizing $V_j^2.$
[2]	2022	Method based on mixed-integer linear programming with distributed generation in the network.	The radiality is guaranteed with the following set of equations:             ${{\displaystyle \sum }}_{\left(i,j\right)\in \mathsf{\Omega }}{y}_{ij}=N-1$, $y_{ij}+y_{ji}\le 1$	The authors used the same approach described in [4].

Liu et al. [12] employed an iterative backward–forward trace power flow to estimate electrical losses for a particular network configuration. These approaches leverage assumptions or iterative procedures. However, the first to adapt the DPSR by an off-the-shelf optimizer was Jabr et al. [7], who proposed a convex relaxation for apparent power.

Since then, many efforts have focused on linearizing the relaxed equation proposed by Jabr et al. to reduce computational complexity. For instance, Borges et al. [4] utilized piecewise linear approximations to linearize quadratic active and reactive power, as well as quadratic voltage. This approach has been widely accepted, with small variations by Pareja et al. [2], and has proven to be an effective way to reduce computational effort.

As has been shown, in recent years the tendency in most of the up-to-date research regarding DPSR has been to use the criterion proposed by Merlin et al. and generalized by Lavorato et al. At the same time, regarding the convexification and linearization of the DPSR, various linearization techniques for quadratic constraints have been proposed in the scientific literature to enhance computational efficiency. Considering this, the contributions of the current research to the field of DPSR can be summarized as follows:

• Introducing and evaluating the impact of alternative radiality conditions, by expressing the spanning tree constraints in a more compact way.
• Proposing a simplified linearization approach for calculating currents, based on the power flow model originally proposed by [1].
• Assessing the methodology of linearization along with its advantages and limitations.

3. Proposed Methodology

The reconfiguration of electric power distribution systems is an integral task in the pursuit of network optimization, addressing demands for improved efficiency and enhanced reliability. Central to this endeavor is the development and refinement of mathematical models that navigate the complexities of power distribution—managing the nuances of voltage, load distribution, and the intrinsic properties of the network’s infrastructure. This paper presents an in-depth review of these modeling efforts, particularly emphasizing the nonlinear basis model that underpins the reconfiguration process.

Within the framework of this model as shown in Figure 1, two primary constraints are considered: the Dist-Flow Constraint and the Radiality Constraint. The Dist-Flow Constraint encompasses the essential considerations of power balance, voltage drop, and the calculation of current magnitude, along with the adherence to voltage and current limitations. On the other hand, the Radiality Constraint ensures the maintenance of a tree-shaped, loop-free network topology, a critical requirement for the practical applicability of the reconfiguration process.

To advance beyond the limitations inherent to the nonlinearities of these constraints, this paper proposes an enhanced model that introduces a linearization approach for the nonlinear equations related to current magnitude calculations. Moreover, we present refinements to the Radiality Constraints, aiming to yield a more intuitive and computationally efficient reconfiguration process.

3.1. Nonlinear Non-Convex Integer Basis Model

This section presents the mixed-integer nonlinear model for the PDSR problem using the set of Constraints (1)–(9). Furthermore, the application of the linearization schemes presented in Section 3.2 yields a mixed-integer quadratic model. As a result, the global optimum can be guarantee using properly off-the-shelf software (Gurobi 11.0).

Constraints (1)–(9) represent the objective function and the AC power flow model for a radially operated distribution network, i.e., Dist-Flow Constraints, based on [1], which is represented in Figure 2. Jointly with (9), they hold the tree-shape form of the network.

$$\min {\displaystyle \sum }_{\left(k,i\right)ϵ\mathsf{\Omega }}{R}_{k,i}\cdot I_{k,i}^2$$

(1)

$${\displaystyle \sum }_{\left(k,i\right)ϵ\mathsf{\Omega }}{P}_{k,i}+P_i^S=P_i^D+{\displaystyle \sum }_{\left(i,j\right)ϵ\mathsf{\Omega }}(P_{i,j}+R_{i,j}*I_{i,j}^2) \forall i \in \mathsf{{\rm N}}$$

(2)

$${\displaystyle \sum }_{\left(k,i\right)ϵ\mathsf{\Omega }}{Q}_{k,i}+Q_i^S=Q_i^D+{\displaystyle \sum }_{\left(i,j\right)ϵ\mathsf{\Omega }}(Q_{i,j}+X_{i,j}*I_{i,j}^2) \forall i \in \mathsf{{\rm N}}$$

(3)

$$V_k^2-2\left(R_{k,i}{P}_{k,i}+X_{k,i}{Q}_{k,i}\right)-Z_{k,i}^2{I}_{k,i}^2=V_i^2+{\Delta }_{k,i}^v \forall \left(k,i\right)\in \mathsf{\Omega }$$

(4)

$$\left|{\Delta }_{k,i}^v\right|\le \left(\Delta V_{k,i}^{{ub}^2}-\Delta V_{k,i}^{{ub}^2}\right)\cdot \left(1-y_{k,i}\right) \forall \left(k,i\right)\in \mathsf{\Omega }$$

(5)

$$V_i^2{I}_{k,i}^2=P_{ki}^2+Q_{k,i}^2 \forall \left(k,i\right)\in \mathsf{\Omega }$$

(6)

$$V_i^{lb}\le V_i\le V_i^{ub} \forall i \in \mathsf{{\rm N}}$$

(7)

$$I_{k,i}^{lb}\le I_{k,i}\le I_{k,i}^{ub}{y}_{k,i} \forall \left(k,i\right)\in \mathsf{\Omega }$$

(8)

$${\displaystyle \sum }_{\left(i,j\right)ϵ\mathsf{\Omega }}{y}_{ij}=\left|N\right|-N_{substations}$$

(9)

Equations (2) and (3) guarantee the maintenance of active and reactive power balance at each node, which effectively constitutes the application of Kirchhoff’s first law, while also considering the losses within the network. Equations (4) and (5) pertain to Kirchhoff’s second law.

Equation (4) introduces auxiliary variables ${\Delta }_{i,j}^v$ to model the voltage drops across branches, and these variables are constrained as defined in (5). According to (5), ${\Delta }_{i,j}^v$ is set to zero for active branches $ij$, where $y_{i,j}$ equals one, nullifying the effect of ${\Delta }_{i,j}^v$ in Equation (4) as intended. Conversely, for inactive branches $ij$, where $y_{i,j}$ is zero, ${\Delta }_{i,j}^v$ can vary within the range (5).

Equation (6) describes the interrelation between the flows of active and reactive power, represented by $P_{k,i}$and $Q_{k,i}$, respectively, and square current magnitudes $I_{k,i}^2$, as well as the squared voltage magnitudes $V_i^2$. Following this, Equations (7) through (8) define the permissible operational boundaries for the system’s variables, setting the acceptable limits for voltage and current values in the network.

Finally, the radiality conditions that ensure a radial topology in the distribution system are expressed in Constraint (9) as set in [14].

3.2. Enhanced Model

Note that (6) contains nonlinear terms [4,8], and is also a non-convex constraint that can be convexified as described in (10). Based on the limited range within which nodal voltage magnitudes lie in practice, the product $V_i^2{I}_{k,i }^2$in (10) can be linearized as shown in (11). This approach is simpler than the one commonly used to linearize the term $V_i^2{I}_{k,i}^2$ [2,4], and as shown in the results section, it is very accurate, allowing us to reduce computational effort. As far as we are concerned, this approach has never been used in DPSR.

$$V_i^2{I}_{k,i}^2\ge P_{ki}^2+Q_{k,i}^2 \forall \left(k,i\right)\in \mathsf{\Omega }$$

(10)

$$I_{k,i}^2\ge P_{ki}^2+Q_{k,i}^2 \forall \left(k,i\right)\in \mathsf{\Omega }$$

(11)

On the other hand, the radiality-related Constraint (9) can be reformulated as a set of spanning tree constraints as in (14)–(16). The idea behind this set of constraints is that a bus bar can only be fed by one conductor to ensure radial topology, while it also ensures there are no current flows towards the substations.

The above replacement process has the potential to enhance solution speed, as certain bus bars are just one path to be fed and are fixed during the pre-solve step, since, in some networks, certain nodes only have one connection. The proposed set of constraints also effectively addresses bus bars with zero demand, preventing the generation of non-connected or non-radial topologies. Constraint (12) ensures that each bus bar in the network with a demand other than zero is supplied with power from only one other bus bar. Constraint (13) ensures that the substation is not supplied by any bus bar. To the best of our knowledge, this representation of spanning tree constraints is the first time it has been employed in the DPSR.

$${\displaystyle \sum }_{j ϵ N}{y}_{ij}\le 1 \forall i \in N ⋏j\notin N_{substations}$$

(12)

$${\displaystyle \sum }_{i ϵ N}{y}_{ij}=0 \forall j \in N_{substations}$$

(13)

4. Numerical Results

In this section, we offer a detailed overview of the five instances selected for evaluation, which served to determine the effectiveness and performance of the models outlined in Section 3. These instances were chosen to represent a range of scenarios and complexities to provide a comprehensive assessment of the models’ capabilities.

The analysis of the numerical results obtained from these test instances is comprehensive, involving an in-depth examination of performance metrics and comparative evaluations. This analytical approach facilitates a robust understanding of how each model performs under different conditions and configurations.

After presenting the results, the discussion shifts to exploring the implications, insights, and potential applications of the findings. This discourse aims to interpret the practical significance of the results and how they might influence future research or practical applications in the field.

The computational equipment used for the simulations was an AMD Ryzen 5 3550H processor with a clock speed of 2100 MHz and four cores, providing a baseline for understanding the computational demands and performance limitations of the tested models. The simulations were conducted within a fixed time frame, with a simulation time limit set at 3600 s, ensuring consistency in the evaluation of each model’s efficiency.

To enhance the clarity and comprehension of the models under review, we integrated the spanning tree constraints with the linearization strategies developed in Section 3.2. Each model selected for investigation is briefly described below to provide context and facilitate a better understanding of their structure and intended applications:

• Nonlinear non–convex integer basis model [4]: the base model consists of Constraints (2)–(9)
• Model with modified Radiality Constraints: the base model can be modified by including the modified Radiality Constraints; in this case, the incumbent constraints are (2)–(8), (14) and (15)
• Model with a simplification of the term $V_i^2{I}_{k,i}^2$: in this case, Constraint (11) is included instead of Constraint (6); the model then consists of Constraints (2)–(5), (7), (8) and (11)–(13)
• Model with a double power triangle constraint: in this case, the set of Constraints (10) and (11) is included instead of Constraint (6); the model then consists of Constraints (2)–(5), (7), (8) and (10)–(13)
• It would be useful to note that, except for the base model, all other models are modifications proposed within this research, as outlined in Section 2. These modifications represent novel contributions aimed at enhancing computational efficiency in the context of the DPSR.

4.1. Instances’ Description

The previously mentioned improvement strategies were evaluated using five case studies sourced from [17]. Each case study is distinguished by the number of branches and corresponding information about the loads at each bus bar.

Table 2. Case studies.

# of Bus Bars	# of Switches	Ref. Node	Base Voltage (kV)	Base Apparent Power (MVA)
14	16	14	23	100
33	74	1	12.66	10
84	192	84	11.4	10
136	312	1	13.8	100
417	473	1	10	100

includes data on the number of switches, substation nodes, and base voltages and powers for each considered system.

Typically, the literature utilizes small instances [18] (13 and 33 nodes), medium instances [2] (84, 119, 136 and 208 nodes) and big instances [19] (417 barras). To facilitate comparison with the latest literature results, instances from Table 2 were carefully selected, as they are commonly utilized in this research field [2,4]. These instances serve as benchmarks for evaluating and comparing the performance of different optimization algorithms and approaches to DPSR. Moreover, the following metrics were used to compare the effectiveness of our proposal:

• Active power losses: These refer to the power losses due to Joule effect and are estimated using the objective function (1).
• Computation time: This corresponds to the time taken by the code to load the data, build the model in the Gurobi off-the-shelf optimizer, and solve the system. The total time refers to the sum of these three times.

In

Table 3. Results of the three considered models.

Ins	Base Model			Model with Radiality Modified			Model with a Simplification of the Term ${\mathit{V}}_{\mathit{i}}^2{\mathit{I}}_{\mathit{k},\mathit{i}}^2$					Double Power Triangle Constraint
	TT	PPA	GO	TT	PPA	GO	TT	PPA	GO	OF	%ER	TT	PPA	GO
14	0.509	605.9	0	0.8642	605.9	0	0.4471	605.7	0	577.7	4.62	0.4991	605.9	0
33	1.735	139.4	0	1.1662	139.4	0	1.0551	139.2	0	131.8	5.31	1.0024	139.4	0
84	8.0939	469.3	0	3.0809	469.3	0	2.7497	468.6	0	447.4	4.52	3.7231	469.3	0
133	3601.85 **	NSF	NSF	82.785	279.6	0	8.0224	287.9	0	265.8	6.7	32.702	279.6	0
417	3606.13 **	NSF	NSF	3606.6 **	1653 ***	68.5	3606.3 **	581.5 ***	2.55	565.0	2.8	3606.8	582.5	3.0

Ins: Instance; TT: Total time (s); PPA: Active power losses (kW); GO: Solver gap (%). OF: Objective function (kW); %ER: Percental difference among objective function and losses computed by Gauss Seidel power flow analysis. ** Wall time reached. *** Unable to find the optimum value. NSF: Unable to find a feasible solution.

, the overall results obtained for the test systems with 14, 33, 84, 133, and 417 buses are presented.

4.2. Discussion of Results

As demonstrated in Table 3, replacing the Radiality Constraint (6) with the set of Constraint (9) and Constraint(13), results in a significant reduction in computation time. This reduction is more pronounced in test systems with a larger number of nodes (84 and 136). Further reductions in computation time are achieved by simplifying Constraint (6), based on the assumption that voltage variation at the nodes is negligible, and by applying convex relaxation (11). The model that incorporates these simplifications approaches an optimality gap of 0.0%.

The simplification of Constraint (11), detailed in the Appendix A, yields topological decisions consistent with the best outcomes reported in the literature for the 14, 33, and 84 case studies. However, in larger systems, such as the 133 bus system, the error from assuming a constant voltage of 1 p.u. in the power triangle constraint leads to deviations in topological decisions. The error associated with using the set of Constraint (16) was quantified using Gauss–Seidel power flow analysis [15] and is detailed in Table 3 under the column “%ER”. In the 133 bus system, the model shows a 3% discrepancy in power loss compared to those reported in the literature, despite achieving an optimality gap of 0%. In a system with 417 nodes, the discrepancy in power loss results between this model and those reported in the literature is a mere 0.03%, albeit with an optimality gap of 2.55%.

Moreover, the combination of Constraints (16) and (17) enables solving the optimization problem within a computation time that is intermediate between those required for Constraints (14) and (16). This compromise between precision and computational effort results in an exact model. Despite improvements in computational efficiency, this model cannot solve the DPSR for the 417 node instance within the 3600 s time limit. Although it outperforms the base model and the model with modified Radiality Constraints in terms of computational efficiency, it falls short of achieving optimality within the allotted time. Nevertheless, the 3% optimality gap represents a significant improvement over other models, suggesting that it could still provide valuable insights and solutions for large-scale DPSR instances without compromising the decisions suggested by the optimizer.

Section 3 includes a comparison of solutions obtained with those documented in the literature.

Table 4. Results from the literature.

Instance	Lit. Solution (kW)	Solution Time (s)	CPU	Reference
33	139.54	0.46	Intel i7-8850H	[2]
84	469.87	2.58	Intel i7-8850H	[2]
133	280.19	7.07	Intel i7-8850H	[2]
417	581.57	171425	Inter i7PC @1.87 GHz	[4]

presents the values of active power losses for the test systems with 33, 84, 136, and 417 nodes. Notable differences are observed only in the model that assumes a constant voltage of 1 p.u. for each node in the power triangle constraint; however, these differences are less significant when power losses are calculated using the Gauss–Seidel power flow analysis on the topology determined by the model (PPA column in Table 3).

Future research could explore improving the simplification of the power triangle constraint by seeking a more accurate estimate of the constant voltage than 1 p.u., potentially determining an optimal voltage value that better reflects the network characteristics. Another natural step to further improve computational times is the development of decomposition algorithms that can leverage the speed of modern power flow algorithms to generate an iterative reconfiguration strategy. By decomposing DPSR into smaller, more manageable subproblems and utilizing efficient power flow algorithms, it may be possible to achieve significant reductions in computation time while maintaining solution accuracy.

This approach could lead to the development of more scalable and efficient reconfiguration algorithms capable of handling larger and more complex distribution networks, without the necessity of expensive solvers.

5. Conclusions

The results presented in Section 4 underscore significant achievements in computational efficiency within the mixed-integer nonlinear model. Notably, by replacing the Radiality Constraint proposed by [11] with a larger set of constraints, each involving fewer variables (15), we observed a marked improvement in computation time. This strategic adjustment reflects the benefit of this approach to represent radiality conditions, instead of the traditional method used in the literature. This effect is even more pronounced in large instances, where the base model is almost incapable of finding a feasible solution in a reasonable amount of time. This highlights the importance of continuously seeking innovative methods to improve computational efficiency, especially as problem instances scale up. Even more, this type of Radiality Constraint ensures a feasible solution even in cases where some nodes have zero demand, as it could be the case if distributed generation would be considered.

Moreover, our research yielded another substantial breakthrough by introducing a novel and simpler approach to linearizing the apparent power in Constraint (6). While this adaptation led to enhanced computational efficiency, it also revealed a nuanced challenge: the model tended to underestimate power losses and occasionally generated solutions slightly deviating from those obtained using Constraint (6). To address this discrepancy and maintain solution integrity, we advocate for conducting a power flow analysis on the solutions derived from this model, verifying that solutions found with this approach are equivalent to those from the literature. This emphasizes the importance of validating computational models with established methods to ensure their reliability and accuracy in practical applications. As a valuable conclusion, the simplification of constant voltage in the power triangle constraint is valid for small instances (up to 84 bus bars in this study) and may serve as a useful guideline in large instances. However, it is important to note that relying solely on this simplification may not always yield optimal solutions. While it can provide computational efficiency benefits, decisions made using this method may not be optimal in terms of solution accuracy, especially in larger instances where the approximation error becomes more pronounced.

Additionally, our other proposed set of Constraints (16) and (17) represents a remarkable compromise between computational efficiency and solution accuracy. Crucially, this model achieves this balance without necessitating additional post-processing through power flow analysis, and can find the optimal decisions. This suggests the potential for wider adoption of this approach in practical DPSR, where computational efficiency and solution accuracy are both crucial considerations.