Optimal Integration of Distribution Network Reconfiguration and Conductor Selection in Power Distribution Systems via MILP

Abstract

Power distribution systems (PDS) comprise essential electrical components and infrastructure that facilitate the delivery of electrical energy from a power transmission system to end users. Typically, the topology of distribution systems is radial, so that power goes from the substations to end users through main lines or feeders. However, the expansion of new feeders to accommodate new users and ever-growing energy demand have led to higher energy losses and deterioration of the voltage profile. To address these challenges, several solutions have been proposed, including the selection of optimal conductors, allocation of voltage regulators, utilization of capacitor banks, implementation of distributed generation, and optimal reconfiguration. Although reconfiguring the network is the most cost-effective approach, this solution might not be sufficient to completely minimize technical losses and improve system performance. This paper presents a novel approach that combines optimal distribution network reconfiguration (ODNR) with optimal conductor selection (OCS) to minimize power losses and enhance the voltage profiles of PDS. The key contribution lies in the integration of the ODNR and OCS into a single MILP problem, ensuring the attainment of globally optimal solutions. The proposed model was tested with benchmark 33-, 69-, and 85-bus test systems. The results allowed us to conclude that the combined effect of ODNR and OCS presents better results than when any of these approaches are applied either separately or sequentially.

1. Introduction

In today’s electric power industry, the efficient and reliable operation of PDS has become a predominant concern. The increasing demand for electricity, coupled with the integration of renewable energy sources and the advent of smart grid technologies, increases the complexity of PDS planning and operation. To address the challenges posed by this complexity, researchers are turning their attention to advanced optimization techniques. This paper explores a critical aspect of PDS optimization—the seamless integration of ODNR and OCS through the application of mixed-integer linear programming (MILP). By harnessing the potential of MILP, this study aims to improve voltage regulation and mitigating power losses in PDS.

OCS plays an important role in ensuring the reliable and efficient operation of PDS. Conductors distribute electrical energy from distribution substations to final customers through principal feeders and laterals. With the increasing demand for electricity and the integration of renewable energy sources, the need for OCS becomes crucial. The choice of conductors impacts various aspects of PDS, including power losses, voltage regulation, and overall system reliability. An inefficient conductor selection may lead to elevated energy losses, voltage drop issues, and increased operational costs. Conversely, OCS results in reduced power losses, improved voltage profiles, and enhanced system longevity.

The primary objective of OCS is to replace the current conductors in a feeder with different types of conductors. This replacement aims to decrease active power losses, enhance the capacity of circuits, and raise voltage levels. The OCS is a highly complex problem, which can be represented as a mixed integer nonlinear programming (MINLP) problem. Researchers have employed exact techniques as well as heuristic and metaheuristic approaches to tackle this optimization problem.

Heuristics are search techniques that prioritize speed over optimality and are often used when dealing with complex or computationally challenging tasks [1,2]. Although they may not always guarantee the global optimal solution, heuristic and metaheuristic techniques are valuable tools in non-convex optimization and decision-making processes. In the context of OCS, researchers have proposed various heuristic-based approaches. For instance, in [3] the authors presented a methodology combining an economically driven current density-based approach with a heuristic approach for optimizing the conductor selection in radial PDS. In this case, a branching feeder approach without uniform load distribution was used to approximate the real conditions of most PDS. Another study [4] proposed a branch-wise minimization technique for selecting the optimum size of conductors in radial PDS. The conductors selected by the proposed approach maximized the total savings in costs, including conductor material and energy losses while maintaining acceptable voltage profiles. In [5], the authors considered financial and engineering factors as key aspects of OCS. In this case, operating and capital costs were considered bearing in mind a set of conductors with the most economic cost characteristics and enough thermal capacity to meet high-demand scenarios. In [6], a general methodology for optimal conductor size selection in PDS was presented aiming to minimize the total conductor and power loss cost. The model includes diversity in load peaks, load factors, cost of power, load increments, and cost of energy in the decision-making process.

A comparative study between an analytical method and a genetic algorithm (GA) to solve the OCS problem was carried out in [7]. In this case, the analytical approach was based on consecutive load flows. A two-phase methodology employing the branch-wise minimization technique was proposed in [8] to solve OCS in radial PDS. In [9], the authors presented an analytical approach comparing power flow results for distinct ACSR conductors. It is important to note that heuristic optimization techniques cannot ensure a solution that is globally optimal; instead, they offer a proper estimation. These methods may encounter locally optimal solutions and can be computationally intensive, especially when dealing with large-scale problems involving numerous variables.

Metaheuristics have also gained significant popularity in solving the OCS problem. These approaches are inspired by natural and social phenomena such as evolutionary processes or swarm intelligence. In evolutionary and genetic algorithms, a population of potential solutions is evolved over generations through processes such as selection, crossover, and mutation. OCS has been solved through genetic algorithms (GAs) [7,10], adaptive genetic algorithm (AGA) [11], evolutionary strategies (ES) [12], differential evolution algorithm (DEA) [13], and discrete genetic algorithm (DGA) [14].

Particle swarm optimization (PSO) is a metaheuristic approach inspired by the social behavior of some organisms such as schools of fish and flocks of birds. The OCS has also been solved through PSO [15,16], selective particle swarm optimization (SPSO) [17], discrete particle swarm optimization (DPSO) [18] and salp swarm optimization (SSO) [19].

In harmony search algorithms (HSAs), a set of candidate solutions, labeled as harmonies, represent potential solutions to the optimization problem. The OCS problem was solved using HSA with a differential operator (HSDE) in [20]. Other metaheuristic approaches adapted for solving the OCS problem in PDS include crow search algorithm (CSA) [21], sine–cosine optimization algorithm (SCA) [22], Tabu search (TS) [23] and Newton’s metaheuristic algorithm (NMA) [24].

Even though metaheuristic approaches are suitable for solving complex optimization problems, they may not consistently reach the absolute global solution. Moreover, they often need significant adjustments to achieve acceptable results, and this can consume a significant amount of time and necessitate specialized knowledge.

Exact techniques have been employed to a lesser extent in solving the OCS problem. These techniques ensure convergence by utilizing existing optimization software. In the literature review, few research studies were found where exact techniques were employed for solving the OCS. Among these papers, one of them utilized a linear model. In the study conducted by the authors of [25], a MILP problem was proposed, accompanied by a heuristic approach to derive the Pareto front for the problem of optimal conductor sizing. The authors of [26] presented a MINLP model for the OCS problem, which was resolved utilizing the general algebraic modeling system (GAMS) with the aid of the DICOPT optimization solver. In [27], the authors developed an exact nonlinear model for the conductor selection, utilizing available MINLP solvers. Lastly, the authors of [28] proposed a MINLP formulation for OCS in DC radial PDS.

Planning strategies, such as reconfiguration, conductor selection, capacitor placement, and DG placement, are commonly studied separately. Nonetheless, the combination of two or more of these techniques may lead to a better-planned system. The problem of OCS has been integrated with capacitor placement in numerous research investigations, where the researchers employed metaheuristic approaches to direct the exploration procedure. However, from the review of existing literature, OCS has not been discussed in simultaneity with optimal distribution network reconfiguration (ODNR) so far.

ODNR is carried out by altering the topology of the distribution network, considering objectives such as minimizing power losses, improving voltage profile, and enhancing network reliability. ODNR is executed by opening and closing tie and sectionalizing switches, respectively, [29]. Due to the nature of its decision variables and constraints, ODNR can be classified as a mixed-integer nonlinear (MINL) optimization problem, usually requiring the aid of metaheuristic techniques for its solution. Early reconfiguration studies were limited to small-sized PDS [30]. This is due to the fact that ODNR is a complex optimization problem that involves both discrete and continuous decision variables. Furthermore, ensuring a radial topology is not a trivial task [31]. Basically, two optimization paradigms are applied to solve the ODNR problem: mathematical programming methods and metaheuristic techniques.

In [32], the authors proposed an optimal power flow and sensitivity analysis approach to solve the ODNR problem, aiming to minimize active power losses. A heuristic approach was implemented by closing all sectionalizing switches and then determining the ones to be reopened to avoid loops in the system. PSO was implemented in [33,34,35] to tackle the ODNR problem to minimize power losses. In [36,37,38,39,40,41], several variants of GAs were tested to solve the ODNR problem. In this case, network topologies were represented by binary strings representing the open or closed states of the switches. Then, other topologies were created through both the selection and mutation stages of the GA. In each iteration, the radial condition of the new solutions was verified. In [42,43], the authors solved the ODNR problem through firefly optimization (FO). In this case, ref. [42] considered both ODNR and optimal DG sizing, whereas [43] did not optimize the size of DG in the network; nonetheless, a search space-reducing strategy is implemented to accelerate convergence. The authors of [44,45] proposed an HSA to solve the ODNR problem for minimizing power losses. The proposed approach in [45] also includes island detection to enforce radiality. The TS metaheuristic technique was also applied in [46,47] to solve the ODNR problem. In [46], the authors considered a mutation mechanism to escape from local optima, whereas [47] implemented a random mechanism with the same purpose.

In [48], the authors developed a hybrid data-driven and model-based distribution network reconfiguration approach. A hierarchical network recovery process was implemented to speed up the process. ODNR can also be implemented to enhance network reliability. In [49], a column and constraint generation algorithm was proposed to minimize load curtailments under failures of lines or generators. In [50], the authors presented a reinforcement learning approach that resorts to ODNR to minimize load curtailment. The distribution system is modeled as a graph and the ODNR is determined by searching for a spanning tree that presents minimum curtailed power.

Multi-objective approaches have also been implemented in the reconfiguration problem. In [51], the authors developed an NSGAII approach that minimizes both active power losses and voltage offset of distribution networks. In [52], a multi-objective PSO was proposed to minimize total active power losses and maximize the absorption of renewable DG through a time-varying ODNR. In [53], the authors carried out ODNR with four objectives, namely, power loss minimization, voltage profile improvement, network reliability improvement, and operation costs minimization. The literature on metaheuristics applied to solve the ODNR is wide and varied, and a comparative study on this subject can be consulted in [54].

Apart from heuristic and metaheuristic techniques, some mathematical approaches have also been explored to tackle the ODNR problem. In [55], the authors presented a mathematical model of path connectivity for ODNR. This model is based on the closed-loop design and open-loop operation of DPS. In [56], the authors solve the ODNR problem bearing in mind the power loss minimization and the improvement of reliability. The epsilon-constrained method is used, and the proposed mathematical model is then solved through the algebraic modeling systems (GAMS) software. In [57], a mixed-integer two-stage formulation is proposed to solve the ODNR for minimizing power losses. The master–slave methodology was modeled through a decomposition algorithm in AMPL and subsequently resolved with the utilization of CPLEX.

In DPSs that feature fairly loaded feeders and poor voltage profiles, ODNR alone may not be enough to minimize power losses. Furthermore, as the size and type of conductor for each feeder segment are chosen based on the current carrying capacity of the feeder configuration, and ODNR affects the system operational conditions, simultaneous ODNR and OCS could lead to low-cost planning of PDS; nonetheless, this approach has not been reported in the specialized literature Therefore, the main contribution of this paper lies in the simultaneous formulation and resolution of ODNR and OCS in PDS; furthermore, the proposed MILP model guarantees the globally optimal solution. Finally, the proposed model is suitable for applications in real-size distribution systems through commercially available software. Although a specific table of conductors was used for the test and results, the model allows the use of any set of candidate conductors for OCS.

The remainder of this paper is structured as follows: In Section 5, a nonlinear mathematical framework is presented to tackle the combined ODNR and OCS. Elaboration on the linearization procedures employed to transform the initial model into a MILP problem is provided in Section 3. The outcomes of implementing the suggested model on various benchmark test systems are shown in Section 4. Lastly, the conclusions drawn from this study are presented in Section 5.

2. Nonlinear Mathematical Model for the Integrated ODNR and OCS

This section presents the initial nonlinear mathematical model for the integration of ODNR and OCS in PDS. This initial model presents integer and continuous variables; nonetheless, there are some variable multiplications that will be later linearized to recast this initial approach into a MILP model.

In [58], starting from a nonlinear model of the power flow in PDS, the authors propose and validate a linear model of it. In this case, a comparative analysis was conducted between a linear version of the power flow and the nonlinear power flow applied to seven distribution test systems. The results revealed that the linear model had a maximum error of 0.16%, demonstrating that this model accurately solves the load flow. The linear model proposed by the authors in [58] is then adapted to solve the ODNR problem.

On the other hand, in [59] a MILP for the OCS and the optimal capacitor placement in PDS is presented by the same authors. In this case, the linear power flow proposed in [58] is used for the optimal conductor selection. The models presented by the authors in [58,59] are modified and adapted in this paper to have a single MILP model that solves the OCS and ODNR problems jointly, separately or sequentially. The mathematical optimization model implemented in this paper is designed for radial EDNs and considers the following hypotheses: (i) the EDN is represented by a monophasic equivalent; (ii) Loads are represented as constant power injections; (iii) only an electric source (substation) is considered; (iv) active and reactive power losses in distribution lines are concentrated in their sending bus; and (v) the capacitive reactance of distribution lines is neglected.

2.1. Objective Function

The proposed objective function is formulated through Equation (1), comprising two components. The initial component aims to minimize the cost of annual energy loss. Here, K_p, K_e, T, and LF correspondingly stand for the annual demand cost averaged over time, energy expenses, the annual time in hours, and the system loss factor. Ω_l signifies the collection of branches, whereas Ω_c denotes the collection of available conductor types. It is important to note that the nonlinearity in the first term of the objective function arises from the multiplication of decision variables $W_{ij,c}$ and $I_{ij}^{sqr}$, representing the selected conductor type c for installation in branch ij (a binary variable) and the square of the current magnitude in the same branch. The linearization of this product is detailed in a subsequent section of this paper. Furthermore, $L_{ij}$ signifies the length of the conductor linked with branch ij, and $R_c$ represents the resistance of conductor type c (measured in kΩ/km).

The second term of the objective function aims at minimizing the annual conductor selection costs. In this case, CRF is the capital recovery factor related to the selection of a new conductor, and $C_c$ is the conductor cost of conductor type c.

$$\begin{array}{c}\hfill \mathrm{Minimize}\mathit{f}=\left[K_p+\left(K_e·T·LF\right)\right]\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}{L}_{ij}·R_c·W_{ij,c}·I_{ij}^{sqr}+CRF\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}{L}_{ij}·C_c·W_{ij,c}\end{array}$$

(1)

2.2. Power Balance Constraints

Equation (2) expresses the balance of active power within every bus of the PDS. $P_{ki}$ and $P_{ij}$ denote the active power transmission in branches ki and ij, respectively. $P_i^s$ signifies the active power provided from the substation at bus i. $P_i^d$ represents the active power requirement at bus i. Lastly, Ω_b denotes the collection of buses within the PDS.

$$\begin{array}{c}\hfill \sum _{\forall ki\in {\Omega }_l}{P}_{ki}-\sum _{\forall ij\in {\Omega }_l}{P}_{ij}-\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}\left(L_{ij}·R_c·W_{ij,c}·I_{ij}^{sqr}\right)+P_i^s=P_i^d;\forall i\in {\Omega }_b\end{array}$$

(2)

Equation (3) signifies the equilibrium of reactive power within each bus of the PDS. $Q_{ki}$ and $Q_{ij}$ represent the reactive power flow in branches ki and ij, respectively. $X_{ij,c}$ denotes the inductive reactance of branch ij related to conductor type c. $Q_i^s$ is the reactive power provided from the substation at bus i. $Q_i^d$ stands for the reactive power demand at bus i.

$$\begin{array}{c}\hfill \sum _{\forall ki\in {\Omega }_l}{Q}_{ki}-\sum _{\forall ij\in {\Omega }_l}{Q}_{ij}-\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}\left(L_{ij}·X_c·W_{ij,c}·I_{ij}^{sqr}\right)+Q_i^s=Q_i^d;\forall i\in {\Omega }_b\end{array}$$

(3)

Note that Equations (2) and (3) are nonlinear due to the multiplication of the decision variables $W_{ij,c}$ and $I_{ij}^{sqr}$. The linearization of this product is presented in a later section of this paper.

2.3. Voltage Drop in the Branches of the PDS

Equation (4) illustrates the voltage reduction within each branch of the PDS. In this context, voltage magnitudes are given in relation to the electrical characteristics of the branches and their load flow. Variables $V_i^{sqr}$ and $V_j^{sqr}$ represent the squared voltage magnitudes at buses i and j, respectively. $Z_c^2=R_c^2+X_c^2$ is the square of the impedance of conductor type c, and $b_{ij}$ is a continuous auxiliary variable used to enforce Equation (4); it may take different values depending on whether the circuit ij is open or closed.

$$\begin{array}{c}\hfill V_i^{sqr}-V_j^{sqr}=2·L_{ij}\sum _{\forall c\in {\Omega }_c}\left(R_c·W_{ij,c}·P_{ij}+X_c·W_{ij,c}·Q_{ij}\right)+L_{ij}^2·Z_c^2·W_{ij,c}·I_{ij}^{sqr}+b_{ij};\forall ij\in {\Omega }_l\end{array}$$

(4)

2.4. Voltage and Current Limits

Equation (5) indicates the voltage magnitude limit at bus i. Note that ${\overline{V}}_i^2$ and ${\underline{V}}_i^2$ represent the upper and lower squared voltage magnitude limits at bus i, respectively.

$$\begin{array}{cc}\hfill & {\underline{V}}_i^2\le V_i^{sqr}\le {\overline{V}}_i^2;\forall i\in {\Omega }_b\hfill \end{array}$$

(5)

Equation (6) represents the current magnitude limit of the branch ij, where ${\overline{I}}_{ij,c}^2$ is the upper limit of the square current in branch ij of the conductor type c; $y_{ij}^{+}$ and $y_{ij}^{-}$ are binary variables related to the power flow direction of the branch ij and are used in the ODNR problem. If any one of these variables is equal to one, the switch in the respective branch is closed; if both variables are zero, the circuit is open. $W_{ij,c}$ is used to select the conductor type.

$$\begin{array}{cc}\hfill & 0\le I_{ij}^{sqr}\le \sum _{c\in {\Omega }_c}({\overline{I}}_{ij,c}^2·W_{ij,c})·(y_{ij}^{+}+y_{ij}^{-});\forall ij\in {\Omega }_l\hfill \end{array}$$

(6)

2.5. Apparent Power Constraint

The left side of Equation (7) is the linearization of $V_j^{sqr}$ times $I_{ij}^{sqr}$; furthermore, the right side of this equation is the linearization of $P_{ij}^2$ plus $Q_{ij}^2$ which are carried out as indicated in [58].

$$\begin{array}{c}\hfill \left({\underline{V}}^2+\frac{1}{2}\Delta V\right)·I_{ij}^{sqr}+\sum _{s=1}^S{P}_{j,s}^c=\sum _{y=1}^Y{m}_{ij,y}^s·\Delta P_{ij,y}+\sum _{y=1}^Y{m}_{ij,y}^s·\Delta Q_{ij,y};\forall ij\in {\Omega }_l\end{array}$$

(7)

In Equation (7), Y is the number of linearization blocks, $m_{ij,y}^s$ is the slope of the $y_{th}$ block of the power flow at circuit ij; $\Delta P_{ij,y}$ and $\Delta Q_{ij,y}$ represent the values of the $y_{th}$ block of $P_{ij}$ and $Q_{ij}$, respectively.

Equations (8)–(12) are complementary expressions used in the linearization of the left-hand side of Equation (7). Equation (8) indicates the minimum and maximum limits of $V_j^{sqr}$ taking into account the discretization steps. In this case, $x_{j,s}$ is a binary variable used in the discretization of $V_j^{sqr}$, ΔV is the discretization step, and S is the number of discretizations. Equation (9) represents the limit of the power discretization steps, where $P_{j,s}^c$ is the correction used in $V^{sqr}$ times $I_{sqr}$. Equation (10) indicates the limits of $P_{j,s}^c$, and Equations (11) and (12) represent the limits of the binary variable $x_{j,s}$ and its nature, respectively.

$$\begin{array}{cc}\hfill & {\underline{V}}^2+\sum _{s=1}^S\left(\Delta V·x_{j,s}\right)\le V_j^{sqr}\le {\underline{V}}^2+\sum _{s=1}^S\left(\Delta V·x_{j,s}\right)+\Delta V;\forall j\in {\Omega }_b\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill & 0\le \Delta V·I_{ij}^{sqr}-P_{j,s}^c\le \Delta V·\sum _{c\in {\Omega }_c}{\overline{I}}_c^2·W_{ij,c}·\left(1-x_{j,s}\right);\forall ij\in {\Omega }_l,\forall s\in 1..S\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & 0\le P_{j,s}^c\le \Delta V·\sum _{c\in {\Omega }_c}{\overline{I}}_c^2·W_{ij,c}·x_{j,s};\forall ij\in {\Omega }_l\hfill \end{array}$$

(10)

$$\begin{array}{cc}\hfill & x_{j,s}\le x_{j,s-1};\forall j\in {\Omega }_b;\forall s\in 2..S\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill & x_{j,s}\in \left\{0,1\right\};\forall j\in {\Omega }_b;\forall s\in 1..S\hfill \end{array}$$

(12)

Equations (13)–(19) serve the purpose of linearizing the right-hand side of Equation (7). In this instance, $P_{ij}^{+}$ and $P_{ij}^{-}$ are applied to determine $\left|P_{ij}\right|$, and $Q_{ij}^{+}$ and $Q_{ij}^{-}$ are employed to determine $\left|Q_{ij}\right|$, with $\overline{\Delta S}ij$ denoting the upper limit for each load flow block on branch ij. Equation (19) enforces the non-negativity of auxiliary variables $P{ij}^{+}$, $P_{ij}^{-}$, $Q_{ij}^{+}$, and $Q_{ij}^{-}$.

$$\begin{array}{cc}\hfill & P_{ij}^{+}-P_{ij}^{-}=P_{ij};\forall ij\in {\Omega }_l\hfill \end{array}$$

(13)

$$\begin{array}{cc}\hfill & Q_{ij}^{+}-Q_{ij}^{-}=Q_{ij};\forall ij\in {\Omega }_l\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & P_{ij}^{+}+P_{ij}^{-}=\sum _{y=1}^Y\Delta P_{ij,y};\forall ij\in {\Omega }_l\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & Q_{ij}^{+}+Q_{ij}^{-}=\sum _{y=1}^Y\Delta Q_{ij,y};\forall ij\in {\Omega }_l\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & 0\le \Delta P_{ij,y}\le {\overline{△S}}_{ij};\forall ij\in {\Omega }_l,\forall y\in 1..Y\hfill \end{array}$$

(17)

$$\begin{array}{cc}\hfill & 0\le \Delta Q_{ij,y}\le {\overline{△S}}_{ij};\forall ij\in {\Omega }_l,\forall y\in 1..Y\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill & P_{ij}^{+},P_{ij}^{-},Q_{ij}^{+},Q_{ij}^{-}\ge 0;\forall ij\in {\Omega }_l\hfill \end{array}$$

(19)

2.6. Constraints Associated with the ODNR

Equations (20) to (26) model the ODNR problem taking into account the OCS. The upper limits of auxiliary variables $P_{ij}^{+}$ and $P_{ij}^{-}$ are given by Equations (20) and (21), respectively. The reactive power flow limit in branch ij is given by Equation (22). Equation (23) indicates the limits of $b_{ij}$; which is zero if the circuit ij is closed; otherwise, the value of $b_{ij}$ is defined by Equation (23). Equation (24) defines the condition of radiality of the PDS, and N is the number of buses of the PDS. Equation (25) indicates that if there is power flow in branch ij, it must have only one direction. Therefore, if $y_{ij}^{+}=1$, it follows that $y_{ij}^{-}=0$ and vice versa; otherwise, when $y_{ij}^{+}=0$ and $y_{ij}^{-}=0$, the circuit is open (there is no power flow). Equation (26) denotes the binary nature of $y_{ij}^{+}$ and $+y_{ij}^{-}$.

$$\begin{array}{cc}\hfill & P_{ij}^{+}\le \overline{V}·{\overline{I}}_{ij,c}·W_{ij,c}·y_{ij}^{+};\forall ij\in {\Omega }_l;\forall ij\in {\Omega }_l\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill & P_{ij}^{-}\le \overline{V}·{\overline{I}}_{ij,c}·W_{ij,c}·y_{ij}^{-};\forall ij\in {\Omega }_l;\forall ij\in {\Omega }_l\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & |Q_{ij}|\le \overline{V}·{\overline{I}}_{ij,c}·W_{ij,c}·\left(y_{ij}^{+}+y_{ij}^{-}\right);\forall ij\in {\Omega }_l\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill & |b_{ij}|\le \left({\overline{V}}^2-{\underline{V}}^2\right)\left(1-\left(y_{ij}^{+}+y_{ij}^{-}\right)\right);\forall ij\in {\Omega }_l\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & \sum _{\forall ij\in {\Omega }_l}\left(y_{ij}^{+}+y_{ij}^{-}\right)=N-1;\forall ij\in {\Omega }_l\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & \left(y_{ij}^{+}+y_{ij}^{-}\right)\le 1;\forall ij\in {\Omega }_l\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & y_{ij}^{+},y_{ij}^{-}\in \left\{0,1\right\};\forall ij\in {\Omega }_l\hfill \end{array}$$

(26)

2.7. Constraints Associated with the OCS

Equations (27) and (28) are used in the OCS problem, and define, respectively, the possible values of the variable $W_{ij,c}$ and its binary nature.

$$\begin{array}{cc}\hfill & \sum _{c\in {\Omega }_c}{W}_{ij,c}\le \left(y_{ij}^{+}+y_{ij}^{-}\right);\forall ij\in {\Omega }_l\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & W_{ij,c}\mathrm{binary};\forall ij\in {\Omega }_l,\forall c\in {\Omega }_c\hfill \end{array}$$

(28)

The expressions given by Equations (29) and (30) are used to determine the input parameters of the objective function. In this case, the constant parameters presented in Equation (29) are taken from [8]. The system loss factor LF is calculated with the demand factor (DF), as indicated in Equation (29), whereas CRF is the capital recovery factor, which is calculated considering the interest rate (i) and the number of years (n) envisaged in the distribution planning, as indicated in Equation (30).

$$\begin{array}{cc}\hfill & LF=0.16·DF+0.84·D{F}^2\hfill \end{array}$$

(29)

$$\begin{array}{cc}\hfill & CRF=\frac{i{\left(i+1\right)}^n}{{\left(i+1\right)}^n-1}\hfill \end{array}$$

(30)

Constraints (31) and (32) define the upper and lower limits of each block’s contribution to $|P_{ij}|$ and $|Q_{ij}|$, respectively, where $\Delta S_{ij}$ is the upper limit of each block of the power flow at circuit ij.

$$\begin{array}{cc}\hfill & m_{ij,y}^s=\left(2y-1\right)\Delta S_{ij}\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & \Delta S_{ij}=\overline{V}·{\overline{I}}_{ij}/Y\hfill \end{array}$$

(32)

3. Integration of ODNR and OCS via Mixed Integer Linear Programming Model

Equations (1) to (28) represent a MINLP problem. The linearizations that allow recasting this model into a MILP problem are presented in this section. Equations (1) to (4) are nonlinear due to the multiplication of the variables $W_{ij,c}·I_{ij}^{sqr}$. This product of a binary and continuous variable is labeled as ${\varphi }_{ij,c}$. The process of linearizing this expression can be accomplished through the utilization of the big-M method (where M represents a suitably large value), exemplified in Equations (33) and (34).

$$\begin{array}{cc}\hfill & 0\le -{\varphi }_{ij,c}+I_{ij}^{sqr}\le M·\left(1-W_{ij,c}\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & 0\le {\varphi }_{ij,c}\le M·W_{ij,c}\hfill \end{array}$$

(34)

The voltage drop in the branches of the PDS defined by Equation (4) is nonlinear due to the multiplications of $W_{ij,c}·P_{ij}$ and $W_{ij,c}·Q_{ij}$. These nonlinearities can be converted into linear forms using the big-M approach, as illustrated in Equations (35) through (38). In this case, the products $W_{ij,c}·P_{ij}$ and $W_{ij,c}·Q_{ij}$ are labeled as new continuous variables ${\beta }_{ij,c}$ and ${\delta }_{ij,c}$, respectively.

$$\begin{array}{cc}\hfill & 0\le -{\beta }_{ij,c}+P_{ij}\le M·\left(1-W_{ij,c}\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & 0\le {\beta }_{ij,c}\le M·W_{ij,c}\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & 0\le -{\delta }_{ij,c}+Q_{ij}^{sqr}\le M·\left(1-W_{ij,c}\right)\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill & 0\le {\delta }_{ij,c}\le M·W_{ij,c}\hfill \end{array}$$

(38)

Equations (6) and (20) to (22) are nonlinear due to the multiplication of binary variables $W_{ij,c}·y_{ij}^{+}$ and $W_{ij,c}·y_{ij}^{-}$. These multiplications are renamed using new binary variables labeled as $U_{ij,c}^{+}$ and $U_{ij,c}^{-}$, respectively. The linearization of these expressions is indicated by Equations (39) to (44).

$$\begin{array}{cc}\hfill & 0\le U_{ij,c}^{+}\le W_{ij,c}\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill & 0\le U_{ij,c}^{+}\le y_{ij}^{+}\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & W_{ij,c}+y_{ij}^{+}-1\le U_{ij,c}^{+}\le 1\hfill \\ \hfill & \hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & 0\le U_{ij,c}^{-}\le W_{ij,c}\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & 0\le U_{ij,c}^{-}\le y_{ij}^{-}\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & W_{ij,c}+y_{ij}^{-}-1\le U_{ij,c}^{-}\le 1\hfill \end{array}$$

(44)

Equations (9) and (10) are nonlinear due to the multiplication of binary variables $W_{ij,c}·x_{j,s}$. This multiplication is renamed using the binary variable $A_{ij,c,s}$, and its linearization is performed as shown in Equations (45) to (47).

$$\begin{array}{cc}\hfill & 0\le A_{ij,c,s}\le W_{ij,c}\hfill \end{array}$$

(45)

$$\begin{array}{cc}\hfill & 0\le A_{ij,c,s}\le x_{j,s}\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill & W_{ij,c}+x_{j,s}-1\le A_{ij,c,s}\le 1\hfill \end{array}$$

(47)

Following the linearizations presented above, the objective function (see Equation (1)) is reformulated as shown in Equation (48).

$$\begin{array}{cc}\hfill & \mathrm{Minimize}\mathit{f}=k_c\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}{L}_{ij}·R_c·{\varphi }_{ij,c}+CRF\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}{L}_{ij}·C_c·W_{ij,c}\hfill \end{array}$$

(48)

The power balance constraints (Equations (2) and (3)) are also modified as follows:

$$\begin{array}{cc}\hfill & \sum _{\forall ki\in {\Omega }_l}{P}_{ki}-\sum _{\forall ij\in {\Omega }_l}{P}_{ij}-\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}\left(L_{ij}·R_c·{\varphi }_{ij,c}\right)+P_i^s=P_i^d;\forall i\in {\Omega }_b\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill & \sum _{\forall ki\in {\Omega }_l}{Q}_{ki}-\sum _{\forall ij\in {\Omega }_l}{Q}_{ij}-\sum _{\forall ij\in {\Omega }_l}\sum _{\forall c\in {\Omega }_c}\left(L_{ij}·X_c·{\varphi }_{ij}^{sqr}\right)+Q_i^s=Q_i^d;\forall i\in {\Omega }_b\hfill \end{array}$$

(50)

The voltage drops in branches (Equation (4)) are rewritten as follows:

$$\begin{array}{cc}\hfill & V_i^{sqr}-V_j^{sqr}=2·L_{ij}\sum _{\forall c\in {\Omega }_c}\left(R_c·{\beta }_{ij,c}+X_c·{\delta }_{ij,c}\right)+L_{ij}^2·Z_c^2·{\varphi }_{ij,c}+b_{ij};\forall ij\in {\Omega }_l\hfill \end{array}$$

(51)

The current limit given by Equation (6) is rewritten as indicated in Equation (52)

$$\begin{array}{cc}\hfill & 0\le I_{ij}^{sqr}\le \sum _{c\in {\Omega }_c}{\overline{I}}_{ij,c}^2\left(U_{ij,c}^{+}+U_{ij,c}^{-}\right);\forall ij\in {\Omega }_l\hfill \end{array}$$

(52)

Equations (9) and (10) are rewritten as indicated in Equations (53) and (54).

$$\begin{array}{cc}\hfill & 0\le \Delta V·I_{ij}^{sqr}-P_{j,s}^c\le \Delta V·\sum _{c\in {\Omega }_c}\left({\overline{I}}_c^2·W_{ij,c}-A_{ij,c}\right);\forall ij\in {\Omega }_l,\forall s\in 1..S\hfill \end{array}$$

(53)

$$\begin{array}{cc}\hfill & 0\le P_{j,s}^c\le \Delta V·\sum _{c\in {\Omega }_c}{\overline{I}}_c^2·A_{ij,c};\forall ij\in {\Omega }_l\hfill \end{array}$$

(54)

Equations (55) to (57) indicate the linearization of Equations (6), and (20) to (22), respectively.

$$\begin{array}{cc}\hfill & P_{ij}^{+}\le \overline{V}·{\overline{I}}_{ij,c}·U_{ij,c}^{+};\forall ij\in {\Omega }_l;\forall ij\in {\Omega }_l\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill & P_{ij}^{-}\le \overline{V}·{\overline{I}}_{ij,c}·U_{ij,c}^{-};\forall ij\in {\Omega }_l;\forall ij\in {\Omega }_l\hfill \end{array}$$

(56)

$$\begin{array}{cc}\hfill & |Q_{ij}|\le \overline{V}·{\overline{I}}_{ij,c}·\left(U_{ij,c}^{+}+U_{ij,c}^{-}\right);\forall ij\in {\Omega }_l\hfill \end{array}$$

(57)

Finally, using the proposed linearization approach, the initial MINLP model given by Equations (1) to (28) can be represented as a MILP model as indicated below:

Minimize (48)

(58)

Subject to: (49), (50), (51), (5), (7), (8), (11)–(19),
           (23)–(28), (33)–(47), (52)–(57)

(59)

4. Test and Results

The simultaneous implementation of ODNR and OCS was simulated using AMPL 4.23 and solved using the standard settings in CPLEX 22.1.1.0. The effectiveness of the proposed model is illustrated using 32-, 69-, and 83-bus test systems. Each of these test systems is evaluated under six distinct scenarios:

• Initial scenario (base case).
• Optimal conductor selection (only OCS).
• Optimal distribution network reconfiguration (only ODNR).
• ODNR and then OCS (sequential approach).
• OCS and then ODNR (sequential approach).
• Simultaneous OCS and ODNR.

Table 1. Available conductor types for all test systems.

Conductor	Name	Area	R	X	Imax	Cost
Type		mm2	[Ω/km]	[Ω/km]	[A]	[US$/km]
1	Mole	6.5	2.7180	0.374	70	90
2	Squirrel	13	1.3740	0.355	120	170
3	Gopher	16	1.0980	0.349	130	210
4	Weasel	20	0.9116	0.345	150	260
5	Ferret	25	0.6795	0.339	175	340
6	Rabbit	30	0.5449	0.335	200	420
7	Mink	40	0.4565	0.353	250	500
8	Horse	42	0.3977	0.327	270	540
9	Beaver	45	0.3841	0.327	257	590
10	Raccoon	48	0.3656	0.329	260	630
11	Otter	50	0.3434	0.328	270	770
12	Cat	55	0.3020	0.327	290	760
13	Dog	65	0.2745	0.315	305	820
14	Leopard	80	0.2193	0.282	395	1010
15	Coyote	80	0.2214	0.268	380	1040
16	Tiger	80	0.2221	0.271	385	1130
17	Wolf	95	0.1844	0.266	425	1370
18	Lynx	110	0.1589	0.261	470	1590
19	Panther	130	0.1375	0.256	510	1840
20	Lion	140	0.1223	0.252	560	2060

presents the conductor types used for all test systems, taken from [22], and

Table 2. Objective function parameters for all simulations.

Parameter	Value
$k_p$ [U$/kW]	1.04
$K_e$ [U$/kWh]	0.012
T (hours)	8760
DF	0.4019
i (%)	8
n (year)	20

indicates the parameters adopted for the objective function, which are based on [8].

There are several benchmark test systems used to evaluate the effectiveness of the ODNR problem. However, due to the nature of ODNR, these systems only provide data on the resistance and reactance of conductors. Crucial information such as the length of feeder sections and conductor costs is not specified, which is necessary to solve the OCS problem. To combine the ODNR and OCS problems, we use the information in Table 1, reported in [22]; although other conductor-type tables that exist in the specialized literature may also be used. The original conductors of the test systems were substituted with their counterparts from Table 1, while adjusting the distances between nodes to ensure similar results compared to those of the original systems. Following the conductor replacement, a power flow analysis was carried out to assess the active power losses and minimum voltage in the test systems.

Table 3. Base case results of the test systems considering conductors of Table 1.

Test System	Active Power Losses [kW]			Minimum Voltage [p.u.]
	Original	Proposed	Error [%]	Original	Proposed	Error [%]
33-bus	202.67	203.23	0.27	0.9131	0.9128	0.03
69-bus	224.99	233.04	3.58	0.9092	0.8919	1.90
83-bus	531.99	515.77	3.04	0.9285	0.9514	2.46

compares the original values of active power losses and minimum voltage magnitudes for each test system with respect to those obtained after updating the conductor types from Table 1. In both cases, the minimum voltage magnitudes were obtained at the same buses for the original test systems. The new conductor types used in the test systems are indicated in Appendix A. Notably, the errors with the new conductor data range from 0.03% to 3.58%, confirming the equivalence between the original and proposed system data.

4.1. OCS and ODNR for the 33-Bus Test System

The 33-bus test system comprises 37 branches, 32 normally closed tie switches, and 5 initially open interconnection switches. The system operates at a nominal voltage of 12.66 kV and has a total demand of 3715 + j 2300 kVA. A power flow was computed to determine the initial state of the network. In the initial base case, the active power losses amount to 203.23 kW, and the minimum voltage magnitude is 0.9128 p.u. at bus 18. For reference, the voltage limits range from a minimum of 0.92 p.u. to a maximum of 1.00 p.u.

Table 4. Proposed solutions for the 33-bus test system.

Description	Base Case	Only	Only	Sequential	Sequential	Simultaneous
		ODNR	OCS	ODNR → OCS	OCS → ODNR	OCS and ODNR
Minimum voltage [p.u]	0.9128	0.9294	0.9500	0.9600	0.9658	0.9689
Active power losses [kW]	203.23	161.72	79.44	65.37	70.79	63.85
Active power losses cost [$]	4483.72	3567.91	1752.68	1442.23	1561.79	1408.68
Conductor cost [$/year]	661.49	692.98	1441.16	1431.81	1435.65	1277.24
Total cost [$/year]	5145.21	4260.89	3193.84	2874.01	2997.44	2685.92
Economic benefit [%]	0.00	17.18	37.92	44.14	41.74	47.80

displays the information of the base case as well as the solutions obtained for only OCS, only ODNR, sequential ODNR and then OCS, sequential OCS and then ODNR and simultaneous OCS and ODNR.

Based on the findings presented in Table 4, it is evident that the simultaneous implementation of OCS and ODNR provides the highest economic benefit of 47.8%. Individually, the ODNR yields an economic benefit of 17.18%, whereas the OCS achieves a benefit of 37.92%. Furthermore, when sequential strategies of the ODNR and then OCS or vice versa are implemented, the economic benefits are 44.14% and 41.74%, respectively. Note that the improvement of these strategies is superior with respect to any single strategy; nonetheless, they are lower than the one obtained with the simultaneous strategy.

In terms of technical losses, the simultaneous implementation of OCS and ODNR, once again, yields the highest reduction rate at 68.58%. Individually, ODNR and OCS result in loss reductions of 20.43% and 60.9%, respectively. Regarding the voltage profile, the initial scenario, referred to as the base case, presents a minimum voltage of 0.9128 p.u. The most substantial enhancement in the voltage profile was attained by implementing the simultaneous OCS and ODNR, contrasting with the base case. In this situation, there was a 5.47% increase in the minimum voltage.

It was then demonstrated that the simultaneous combination of the two optimization approaches, OCS and ODNR, leads to a more optimized system (with minimal losses and investment costs) than when any of them is solved either individually or sequentially.

Table 5. Open switches for the 33-bus test system.

Case	Open Switches
Base case	33 to 37
Only OCS	33 to 37
Only ODNR	9, 14, 32, 33, 37
Sequential ODNR-OCS	9, 14, 32, 33, 37
Sequential OCS-ODNR	9, 14, 28, 32, 33
Simultaneous OCS and ODNR	14, 28, 33, 35, 36

indicates the open switches used in the analyzed cases. It is important to highlight that the open switches in the ODNR-only scenario are distinct from those employed in the simultaneous OCS and ODNR case; nonetheless, they coincide with the sequential scenario of ODNR and then OCS. The reason for this disparity lies in the impact of the OCS on the most efficient reconfiguration strategy aimed at minimizing losses.

Table 6. Selected conductors for all simulations carried out for the 33-bus test system.

Branch	A	B	C	D	Branch	A	B	C	D	Branch	A	B	C	D
1	15	20	20	20	14	2	2	–	–	27	2	8	8	1
2	15	20	20	20	15	2	2	4	2	28	2	8	8	–
3	7	18	18	14	16	2	2	3	1	29	2	8	8	8
4	7	18	14	13	17	2	1	3	1	30	2	4	5	4
5	7	14	14	13	18	1	3	8	3	31	2	2	3	2
6	2	8	8	8	19	1	2	6	2	32	2	1	–	1
7	2	8	8	8	20	1	1	5	1	33	2	2	–	–
8	2	8	5	6	21	1	1	4	1	34	2	2	5	2
9	2	6	–	3	22	2	8	8	14	35	2	2	3	–
10	2	5	1	3	23	2	8	8	14	36	2	2	1	–
11	2	5	1	2	24	2	4	4	14	37	2	2	–	8
12	2	5	2	2	25	2	13	13	1
13	2	3	1	1	26	2	13	8	1

A: base case; B: only OCS; C: sequential ODNR-OCS, and D: simultaneous OCS and ODNR.

presents the conductor types selected for the base case (A), only OCS (B), sequential ODNR and then OCS (C) and simultaneous OCS and ODNR (D). Note that the solutions differ significantly due to the impact of the ODNR. As the system reconfiguration is optimized, there are changes in the OCS, and this leads to a decrease in the investment conductor cost (see Table 4).

Figure 1 depicts the optimal solution for the simultaneous OCS and ODNR problem, where the conductor type of each branch is indicated in parenthesis in red, and the branch number is indicated in blue. Note that the solution involved the use of conductor types 20, 14, 13, 8, 6, 4, 3, 2, and 1. Larger capacity conductors were selected for branches located closer to the substation. Furthermore, Figure 1 also shows the specific switches that were opened to achieve the optimal reconfiguration of the system.

Figure 2 illustrates the voltage profile of the 33-bus test system for the different cases under study. Note that the simultaneous OCS and ODNR improves the voltage profile, especially at buses far away from the substation.

According to Figure 2, buses 13 to 18 and 30 to 33 exhibit low voltage values in the base case. However, this issue is effectively rectified through the implementation of simultaneous OCS and ODNR (indicated by the green line). With this solution, all voltage magnitudes are maintained above 0.96 p.u. Furthermore, the overall voltage profile is significantly improved with the simultaneous implementation of OCS and ODNR.

4.2. OCS and ODNR for the 69-Bus Test System

The 69-bus test system has 73 branches, 68 normally closed tie switches, and 5 initially open interconnection switches. The system operates at a nominal voltage of 12.66 kV with a total demand of 3802 + j 2694 kVA. A power flow was calculated to find the initial state of the network. In the initial state, active power losses are 230.78 kW, and the minimum voltage magnitude of the system is 0.8973 p.u. at bus 65. Voltage limits are considered between 0.95 and 1.00 p.u.

Table 7. Proposed solutions for the 69-bus test system.

Description	Base Case	Only	Only	Sequential	Sequential	Simultaneous
		ODNR	OCS	ODNR → OCS	OCS → ODNR	OCS and ODNR
Minimum voltage [p.u]	0.8919	0.9123	0.9500	0.9500	0.9546	0.9725
Active power losses [kW]	233.04	174.61	71.00	69.84	64.22	34.4
Active power losses cost [$]	5141.40	3852.30	1566.47	1540.97	1372.72	759.97
Conductor cost [$/year]	880.11	873.90	2594.36	1878.94	2574.92	687.67
Total cost [$/year]	6021.51	4726.20	4160.83	3419.91	3947.64	1447.64
Economic benefit [%]	0.00	21.51	30.90	43.21	34.44	75.96

presents the results obtained with the 69-bus test system.

The results reported in Table 7 show that the simultaneous implementation of OCS and ODNR provides the highest economic benefit of 75.96%. Individually, OCS yields an economic benefit of 30.9%, whereas ODNR achieves 21.51%. Furthermore, the sequential optimization of ODNR and then OCS and vice versa yield economic benefits of 43.21% and 34.44% respectively.

The simultaneous implementation of OCS and ODNR also presents the highest reduction in power loss. Note that the power losses of the base case amount to 233.04 kW, whereas the ones obtained with the simultaneous optimization are 34.4 kW, representing a reduction of 85.23%. Individually, ODNR and OCS result in loss reductions of 66.0% and 58.3%, respectively, whereas the sequential optimization resulted in power loss reductions of 70.03% and 72.44% for ODNR and then OCS, and vice versa, respectively. Regarding the voltage profile, the initial scenario, referred to as the base case, presents a minimum voltage of 0.8919 p.u. The most substantial enhancement in voltage profile was attained by simultaneously implementing OCS and ODNR, with a minimum voltage of 0.9725 p.u. In this situation, there was an 8.26% increase in the minimum voltage. Finally, the results for this test system show that the combination of the two optimization approaches, OCS and ODNR, leads to a more optimized system than when any of them are solved individually.

Table 8. Open switches for the 69-bus test system.

Case	Open Switches
Initial base case	69 to 73
Only OCS	69 to 73
Only ODNR	13, 20, 61, 69, 72
Sequential ODNR-OCS	13, 20, 61, 69, 72
Sequential OCS-ODNR	12, 64, 69, 70, 72
Simultaneous OCS and ODNR	20, 56, 69, 71, 73

indicates the open switches used in the analyzed cases. Note that the open switches in the ODNR-only scenario are the same as the ones of the sequential ODNR-OCS scenario, but different from those employed in the simultaneous OCS and ODNR scenario. This is due to the impact of the OCS on the most efficient reconfiguration strategy aimed at minimizing losses.

Table 9. Selected conductors for all simulations carried out for the 69-bus test system.

Branch	A	B	C	D	Branch	A	B	C	D	Branch	A	B	C	D
1	14	20	20	20	26	1	1	3	1	51	1	1	1	1
2	14	20	19	20	27	1	1	1	1	52	6	20	14	1
3	14	20	20	20	28	1	1	1	1	53	6	19	14	1
4	14	20	20	8	29	1	1	1	1	54	6	20	14	1
5	14	20	20	8	30	1	1	1	1	55	6	19	14	1
6	14	20	20	8	31	1	1	1	1	56	6	19	14	–
7	14	20	19	8	32	1	1	1	1	57	6	19	14	1
8	14	20	19	8	33	1	1	1	1	58	6	19	14	1
9	2	8	8	8	34	1	1	1	1	59	5	19	14	14
10	2	8	8	8	35	1	2	3	2	60	5	19	14	14
11	2	6	8	6	36	1	1	3	1	61	1	8	–	3
12	2	3	5	3	37	1	1	3	1	62	1	10	1	3
13	2	3	–	1	38	1	1	3	1	63	1	8	1	3
14	2	3	1	1	39	1	1	3	1	64	1	4	2	1
15	2	3	1	1	40	1	1	2	1	65	1	1	1	1
16	2	3	1	1	41	1	1	2	1	66	1	1	1	1
17	2	2	1	1	42	1	1	2	1	67	1	1	1	1
18	2	2	1	1	43	1	1	2	1	68	1	1	1	1
19	2	2	1	1	44	1	1	2	1	69	1	1	–	–
20	1	2	4	–	45	1	1	2	1	70	1	1	5	2
21	1	1	–	1	46	2	8	8	14	71	1	1	1	–
22	1	1	4	1	47	2	8	8	14	72	1	1	–	14
23	1	1	4	1	48	2	8	8	14	73	1	1	3	–
24	1	1	3	1	49	2	4	4	14
25	1	1	3	1	50	1	1	1	1

A: Base case; B: only OCS; C: sequential ODNR-OCS, and D: simultaneous OCS and ODNR.

presents the conductor types selected for the base case (A), only OCS (B), sequential ODNR and then OCS (C) and simultaneous OCS and ODNR (D). The solutions differ significantly due to the impact of the ODNR. As the system reconfiguration is optimized, there are changes in the OCS, leading to a decrease in the investment conductor cost. Consequently, integrating OCS and ODNR in an optimization problem results in a more efficient and economically viable distribution system planning.

Figure 3 depicts the optimal solution for the simultaneous OCS and ODNR problem, where the conductor type of each branch is indicated in parenthesis in red, and the branch number is indicated in blue. Note that the solution involves the use of conductor types 20, 14, 8, 6, 3, 2, and 1. It was observed that larger capacity conductors are selected for branches located closer to the substation. Furthermore, Figure 3 also shows the specific switches that were opened to achieve the optimal reconfiguration of the system.

Figure 4 illustrates the voltage profile of the 69-bus test system considering the cases described in Table 7. Note that in the base case buses 60 to 66 exhibit low voltage values. However, this issue is effectively rectified through the implementation of OCS and ODNR. An initial improvement in these voltages is obtained with only ODNR, as indicated in the yellow line, these voltages are further improved with only OCS and with the sequential implementation of both. Finally, it is observed that the best voltage profile is obtained with the simultaneous OCS and ODNR. In this case, all voltage magnitudes remain above 0.97 p.u.

4.3. OCS and ODNR for the 83-Bus Test System

The 83-bus test system comprises 96 branches, 83 normally closed tie switches, and 13 initially open interconnection switches. The system operates at a nominal voltage of 11.4 kV and has a total demand of 28,350.9 + j 20,700 kVA. A power flow was computed to determine the initial state of the network. In the initial base case, the active power losses are 531.91 kW, and the minimum voltage magnitude is 0.9378 p.u. In this case, voltage limits are considered from 0.95 to 1.00 p.u.

Table 10. Proposed solutions for the 83-bus test system.

Description	Base Case	Only	Only	Sequential	Sequential	Simultaneous
		ODNR	OCS	ODNR → OCS	OCS → ODNR	OCS and ODNR
Minimum voltage [p.u]	0.9514	0.9733	0.9605	0.9799	0.9817	0.9605
Active power losses [kW]	515.77	491.33	263.72	251.79	255.45	253.23
Active power losses cost [$]	11,379.07	10,839.87	5818.36	5555.06	5635.81	5586.81
Conductor cost [$/year]	2134.76	2245.92	4136.38	4338.42	4164.07	4100.23
Total cost [$/year]	13,513.83	13,085.79	9954.75	9893.58	9799.88	9687.04
Economic benefit [%]	0.00	3.17	26.34	26.78	27.48	28.31

presents the optimal solutions found with the 83-bus test system for the different cases under study.

From Table 10, it is evident that the simultaneous implementation of OCS and ODNR provides the highest economic benefit of 28.31%. This benefit is closely followed by the one obtained with sequential OCS and then ODNR of 27.48%. It was also observed that the sequential ODNR and then OCS present similar economic benefits of only OCS with 26.78% and 26.34%, respectively. This means that for this test system, it is difficult to further reduce power losses once OCS has been carried out. Finally, the lowest economic benefit of only 3.17% was achieved with only ODNR.

As regards technical losses, the ODNR offers a reduction of only 4.73% (from 515.77 kW of the base case to 491.33 kW), whereas the rest of the cases under study manage to reduce power losses by nearly 50%. In this case, the highest power loss reduction was obtained with the sequential ODNR and then OCS approach (51.18%) followed by the simultaneous approach (50.9%). It is worth mentioning that despite the fact that this sequential approach presented a higher power loss reduction, the overall economic benefit of the simultaneous approach is higher.

Regarding the voltage profile, the initial scenario, referred to as the base case, presents a minimum voltage of 0.9514 p.u. The most substantial enhancement in the voltage profile was attained by the sequential OCS and then ODNR approach. In this situation, the minimum voltage is 0.9817 p.u.; on the other hand, the minimum voltage attained by the simultaneous approach was 0.9605 p.u.

Table 11. Open switches for 83-bus test system.

Case	Open Switches
Initial base case	84 to 96
Only OCS	84 to 96
Only ODNR	7, 13, 34, 39, 42, 63, 72, 83, 84, 86, 89, 90, 92
Sequential ODNR-OCS	7, 13, 34, 39, 42, 63, 72, 83, 84, 86, 89, 90, 92
Sequential OCS-ODNR	7, 32, 34, 35, 41, 64, 72, 83, 84, 86, 88, 89, 90
Simultaneous OCS and ODNR	32, 39, 42, 72, 83 84, 85, 86, 88, 89, 90, 94, 96

indicates the open switches used in the analyzed cases. It is important to highlight that the open switches in the ODNR-only scenario are distinct from those employed in the simultaneous OCS and ODNR scenarios but the same as the sequential ODNR and then OCS scenario. This is due to the impact of the OCS on the most efficient reconfiguration strategy aimed at minimizing power losses.

Table 12. Selected conductors for all simulations carried out for the 83-bus test system.

Branch	A	B	C	D	Branch	A	B	C	D	Branch	A	B	C	D
1	14	20	19	20	33	6	14	8	14	65	15	20	19	20
2	14	20	19	20	34	6	14	8	14	66	15	20	20	20
3	14	20	19	20	35	6	6	8	5	67	15	20	20	20
4	14	20	18	20	36	6	5	5	4	68	15	20	20	20
5	14	20	14	20	37	6	3	8	3	69	9	19	14	18
6	12	20	14	20	38	6	3	20	2	70	9	19	20	18
7	5	14	8	14	39	6	1	20	–	71	9	19	20	18
8	1	3	8	3	40	6	1	20	1	72	9	2	20	–
9	1	3	8	3	41	6	3	14	2	73	5	14	14	14
10	1	4	19	4	42	6	1	20	–	74	5	14	14	14
11	13	20	19	20	43	2	13	20	13	75	5	14	5	14
12	13	20	14	20	44	2	13	20	13	76	5	3	14	3
13	5	8	20	8	45	2	13	20	13	77	17	20	8	20
14	13	8	20	8	46	2	2	20	2	78	17	20	14	20
15	17	20	20	20	47	13	20	20	20	79	17	20	13	20
16	17	20	20	20	48	13	20	18	20	80	7	14	5	8
17	16	20	20	20	49	13	20	14	20	81	7	13	14	8
18	16	20	14	20	50	13	20	8	20	82	7	6	4	1
19	7	19	14	20	51	13	20	19	20	83	7	5	14	–
20	7	8	4	14	52	13	14	19	14	84	13	13	13	–
21	1	6	5	6	53	13	13	19	13	85	5	5	5	–
22	1	1	3	1	54	13	8	18	8	86	13	13	13	–
23	1	1	20	1	55	13	2	18	2	87	13	13	13	2
24	1	1	20	1	56	5	14	8	14	88	5	5	5	–
25	7	19	19	20	57	5	14	8	14	89	16	16	16	–
26	7	19	19	20	58	5	14	8	14	90	20	20	20	–
27	7	18	8	20	59	5	8	20	9	91	7	7	7	5
28	7	18	20	20	60	5	8	20	9	92	17	17	17	14
29	6	2	20	3	61	5	8	20	8	93	6	6	6	1
30	17	20	13	18	62	5	8	20	8	94	6	6	6	–
31	17	20	8	18	63	5	8	19	8	95	6	6	6	1
32	17	14	13	–	64	5	3	19	3	96	5	5	5	–

A: base case; B: only OCS; C: sequential ODNR-OCS, and D: simultaneous OCS and ODNR.

presents the conductor types selected for the base case (A), only OCS (B), sequential ODNR and then OCS (C) and simultaneous OCS and ODNR (D). Note that the solutions differ significantly due to the impact of the ODNR. As the system reconfiguration is optimized, there are changes in the OCS, and this leads to a decrease in the investment conductor cost.

Figure 5 depicts the optimal solution for the simultaneous OCS and ODNR problem, where the conductor type of each branch is indicated in parenthesis in red, and the branch number is indicated in blue. Note that the solution involved the use of conductor types 20, 19, 14, 13, 11, 8, 6, 5, 4, 3, 2, and 1. Furthermore, as with the other test systems, larger capacity conductors were selected for branches near the substation. Figure 5 also shows the specific switches that were opened to achieve the optimal reconfiguration.

Figure 6 depicts the voltage profile of the 83-bus test system. Note that for the base case, all voltage magnitudes are above 0.95 p.u.; nonetheless, the voltage profile is further improved in all cases under study.

5. Conclusions

Choosing new conductors as replacements for outdated or deteriorated cables can lead to substantial reductions in power losses and enhancement of voltage profiles. Nevertheless, OCS alone might not always be fully effective in achieving significant reductions in technical losses or improving voltage profiles. As a result, OCS can be complemented using other technical approaches. In this study, we integrate OCS with ODNR, which is a method that modifies the distribution network’s topology to minimize power losses and enhance voltage profiles.

Both OCS and ODNR were integrated into a unified MILP model, enabling to obtain globally optimal solutions. This marks the main innovation of the study because the integration of these two issues through a MILP approach has not been previously put forth in the specialized literature. Moreover, the model offers flexibility in decision making, allowing the problems to be addressed jointly or separately as needed.

The efficiency and suitability of the suggested model was assessed using three standard distribution test systems. In each investigated scenario, it was proven that employing simultaneously both optimization methods (OCS and ODNR) within the proposed MILP model results in a more optimized system, characterized by higher economic benefits, compared to solving either of them separately or sequentially.

In the 33-bus and 69-bus test systems, the combined application of OCS and ODNR yielded the greatest decrease in technical losses and the most notable enhancement in voltage profile. In the 83-bus test system the sequential ODNR and then OCS approach resulted in slightly higher power loss reduction than the simultaneous approach; nonetheless, the latter reported higher overall economic benefits. Additionally, across all test systems, implementing OCS and DNR simultaneously led to the choice of lower current capacity conductors compared to using only OCS. This capacity reduction was attributed to the effect of DNR and significantly influenced the overall investment cost. Future work may include a multi-period optimization approach to consider active and reactive power demand variations and the impact of renewable generation.