A MILP Model for Optimal Conductor Selection and Capacitor Banks Placement in Primary Distribution Systems

Abstract

Power distribution systems (PDS) are the infrastructure and equipment used to distribute electricity from the transmission system to end-users, such as homes and businesses. PDS are usually designed to operate in a radial mode, where power flows from one substation to the end user through a series of feeders. The extension of distribution lines to attend new customers along with the growing demand for electricity result in increased energy losses and voltage reductions. Various solutions have been proposed to solve these issues, such as selecting the optimal set of conductors, optimizing the placement of voltage regulators, using capacitor banks, reconfiguring the distribution system, and implementing distributed generation. A well-known approach for reducing energy losses and enhancing voltage profile is the optimal conductor selection (OCS). While this can be beneficial, it may not be sufficient to fully reduce technical losses and improve the system voltage profile; therefore, it must be combined with other strategies. This paper presents a new approach that combines the OCS with the optimal placement of capacitor banks (OPCB) to minimize technical losses and improve the voltage profile in PDS. The main contribution of this paper is the integration of these two problems into a single mixed integer linear programming (MILP) model, therefore guaranteeing the achievement of globally optimal solutions. Three test systems of 27, 69, and 85 buses were used to illustrate the effectiveness of the proposed modeling approach. The results indicate that the combination of OCS and OPCB effectively minimizes energy losses and enhances the voltage profile. In all cases, the solutions obtained by the proposed MILP approach were better than those previously reported through metaheuristics for the combined OCS and OPCB problem.

1. Introduction

Power distribution systems (PDS) are the largest portion of power systems. They distribute electrical energy to end-users through substations, distribution feeders, distribution transformers, secondary conductors, and service mains. These grids are designed to be radial and operate at low voltage levels compared to transmission power systems. Technical losses and poor voltage regulation are common issues in PDS; therefore, several planning techniques, such as reconfiguration, conductor selection, capacitor placement, and distributed generation placement, have been employed to reduce technical losses and improve voltage levels. Each of these strategies has been the subject of numerous publications.

Most PDS were built several decades ago to meet the demand of that time. Some of these networks have not been updated and, therefore, their conductors may be undersized due to the gradual increase in demand. In many cases, this has led to an increase in power losses and excessive voltage drops, especially in the nodes farthest from the substation. Moreover, accessing new corridors to expand the current network can be costly due to the processing of environmental permits, as well as physical and geographic restrictions, etc. In this sense, the optimal conductor selection (OCS) becomes an attractive alternative to an upgrade of the network in order to improve technical and economic aspects.

The OCS has the objective of changing the existing circuit conductors to other conductor types (from an available set) to reduce electrical energy losses, increase the current capacity of existing circuits, and increment voltage magnitudes when they are below acceptable limits [1]. The OCS in PDS has been the focus of several studies since it constitutes a complex optimization problem; due to its nature, the OCS is commonly modeled as a mixed integer nonlinear programming (MINLP) problem. Heuristic methods, metaheuristics, and exact techniques have been used to address this optimization problem.

Heuristic techniques are problem-solving methods that are based on practical experience and knowledge. They are robust, easily applied, and normally converge to a local optimum solution [2]. The authors of [3] developed a practical approach that combines an economical current density-based method with a heuristic index-directed method for solving the OCS problem. In [4], a branch-wise minimization technique that employs the load flow solution of the system was proposed to solve the OCS. A systematic approach that considers various financial and engineering factors was proposed by the authors in [5]. In [6], the authors presented a voltage deviation index as well as a simple computer algorithm for the OCS. In [7,8], a sequential algorithm that considered the cost of conducting material, the cost of energy losses, the bus voltage profile, and the current carrying capacity of conductors was proposed. A method in which the search space was enumerated systematically and logical decisions was used to reduce the search space was developed by the authors of [9]. An analytical method based on consecutive power flows was presented in [10] to solve the OCS problem. In [11], an algorithm based on the cost of energy loss and interest and depreciation on capital investments was implemented to solve the OCS problem. The authors of [12] proposed a two-phase method based on the branch-wise minimization technique. Lastly, an analytical approach that compared the results of power flow for different ACSR conductors was proposed by the authors of [13]. Heuristics optimization methods do not guarantee finding the global optimal solution, only a good approximation of it. Moreover, they may get stuck in a local optimum and can be computationally expensive, especially when the number of variables or the size of the problem is large.

Metaheuristics are techniques that are easy to design and implement. They guide the search process to efficiently examine the solution space and find good-enough solutions for an optimization problem in a reasonable time frame. Metaheuristics have been widely used to solve the OCS. The techniques reported in the specialized literature are: Evolutionary Strategies (ES) [14], Particle Swarm Optimization (PSO) [15,16,17,18], Differential Evolution Algorithm (DEA) [19,20], Genetic algorithm (GA) [10,21,22,23,24,25], Adaptive Genetic Algorithm (AGA) [26], Harmony Search Algorithm (HSA) [27], Harmony Search Algorithm with a Differential Operator (HSDE) [28], Selective Particle Swarm Optimization (SPSO) [29], Discrete Genetic Algorithm (DGA) [30], Colonial Selection Algorithm (CSA) [31], Modified Differential Evolution Algorithm (MDEA) [32], Imperialism Competitive Algorithm (ICA) [17], Bacterial Foraging Algorithm (BFA) [33,34], Discrete Particle Swarm Optimization (DPSO) [35], Crow Search Algorithm (CSA) [36], Sine-Cosine Optimization Algorithm (SCA) [37], Tabu Search (TS) [38], Salp Swarm Optimization Algorithm (SSO) [39], Whale Optimization Algorithm (WOA) [40], Discrete Version of the Vortex Search Algorithm (DVSA) [41], and Newton’s Metaheuristic Algorithm (NMA) [42]. Although these techniques are more sophisticated than heuristic approaches, they usually find approximate solutions and may not always find the true global optimum; besides, they require a significant amount of fine-tuning to achieve good performance, which can be time-consuming and require a great deal of expertise. Finally, metaheuristics can be computationally expensive, especially for large-scale problems.

Exact techniques have also been used to solve the OCS problem, but to a lesser degree. The use of these techniques guarantees convergence to optimality using existing optimization software. In the review of the specialized literature, only four papers were found that used exact techniques for OCS, one of which used a linear model. The authors of [43] proposed a Mixed-Integer Linear Programming Model and a heuristic to obtain the Pareto front of the conductor size selection problem. In [44], a MINLP formulation for the OCS, which was solved using the General Algebraic Modeling System (GAMS) and DICOPT solver, was presented. An exact nonlinear model that was solved by available MINLP solvers was developed by the authors of [45] to select the optimal conductors. Finally, a MINLP model for OCS in DC radial PDS was proposed by the authors of [46].

The optimal placement of capacitor banks (OPCB) in PDS is crucial to minimize voltage drops and reduce line losses. It refers basically to the installation of capacitors at strategic locations in the network to compensate for the reactive power demand and improve technical features. The OPCB also ensures the proper distribution of reactive power in the system, which helps to reduce overall power consumption. Additionally, it may reduce the cost of electricity, as there are fewer losses in the system.

A bunch of methodologies have been proposed in the specialized literature to solve the OPCB. Early studies used discrete programming, optimal power flow (OPF) and sensitivity factor techniques such as [47,48,49], respectively. Nonetheless, due to the combinatorial nature of the problem, heuristic and metaheuristic techniques have been mostly explored. In [50], a heuristic methodology based on graph search was proposed to optimally size and allocate reactive compensation in PDS. Deterministic and GAs were used in [51] to solve the OPCB. In [52], the authors implemented a GA for reactive compensation in PDS having customers with different load patterns. A GA was also implemented in [53] for the OPCB. In this case, the authors limit the maximum number of operations in switched capacitive banks to consider the ageing of the equipment. In [54] a GA was used for the OPCB using probabilistic generation models with correlations.

A hybrid Tabu Search approach, extended with features taken from GA and simulated annealing was proposed in [55]. In [56], the concept of Shannon’s Entropy was used for the OPCB in PDS to consider multiple criteria. In [57], the authors considered unbalanced networks and implemented a micro-GA to solve the OPCB in radial distribution networks. In [58], a PSO was proposed for the optimal placement and sizing of capacitors considering the effect of harmonics, also in unbalanced networks. The same technique was implemented in [59] for the OCPB in microgrids and [60] considering the effect of switchable capacitive banks. In [61], the authors proposed a multiverse optimizer approach based on a partial modification of conventional loss sensitivity factors.

Multi-stage approaches have also been considered to solve the OPCB. A two-stage method is proposed in [62] that also implements a loss sensitivity technique to select the candidate locations for the capacitor placement. In [63], a three-stage optimization approach was carried out for the OPCB considering unbalanced PDN. In the first stage, a multi-phase OPF was solved considering both heavy and light load conditions. In this step, the reactive power as well as the number of fixed and switched banks was determined. In the second stage, the position of the capacitors was found by a GA. Finally, the third level determined the states of the switched capacitors, using binary optimization.

The OPCB has also been combined with the optimal allocation of DG in [64] using the water cycle algorithm and in [65] using GAs. In [66,67] the OPCB was implemented with the optimal reconfiguration problem using a MILP model and a thief police algorithm, respectively. In [68] the OPCB was implemented along with voltage regulators. A comprehensive review regarding OPCB techniques for power loss reduction, voltage stability improvement, and voltage profile enhancement can be consulted in [69].

Planning techniques such as optimal conductor selection, capacitor placement, reconfiguration, placement of new substations, and distributed generation (DG) placement are commonly studied separately. However, in PSD that have heavily loaded feeders and poor voltage profiles, implementing just one technique alone may not be enough to minimize power losses and improve voltage profiles. Therefore, combining these techniques may result in a more effectively planned system. The OCS problem has been combined with OPCB in several studies [16,20,21,22,23,27,29,30,33,45], where the authors used metaheuristics to guide the search process. According to the literature survey, the integration of OCS and OPCB into a single mixed integer linear programming (MILP) problem, and its solution by using exact techniques and commercial software, has not been proposed yet.

To summarize, the main contribution of this paper is the simultaneous modeling of the OCS and OPCB in PDS through an MILP approach. Such type of modeling has not been previously reported in the specialized literature. The proposed MILP model presents several advantages: (i) it is adaptable to solve either jointly or separately the OCS and OPCB; (ii) globally optimal solutions are found due to the nature of the proposed model; and (iii) the proposed model is suitable to be implemented in commercially available solvers.

To demonstrate the applicability and effectiveness of the proposed model, several tests have been conducted on PDS with 27, 69, and 85 buses. In terms of OCS the proposed approach successfully replicates, and sometimes outperforms, the results reported in the specialized literature. Regarding the simultaneous OCS and OPCB, the proposed model identifies better solutions than those reported in the reviewed literature.

The rest of this document is organized as follows: Section 2 describes the proposed MILP model to solve the OCS and OPCB in PDS that can be carried out either jointly or separately. Section 3 presents the results of the proposed model applied to several test systems. Finally, Section 4 presents the research conclusions.

2. Mathematical Model

This section presents the proposed mathematical model for OCS and OPCB in distribution networks. Initially, a nonlinear model is presented; then, some linearizations are proposed and explained to finally recast the initial model into a MILP problem.

2.1. Nonlinear Model for OCS and OPCB in Radial Distribution Networks

The mathematical model for the simultaneous OCS and OPCB is given by Equations (1)–(28). As proposed in [43,70], a change of variables can be made to reduce the nonlinearity of the power flow equations. In this case, the original variables $V_i^2$, $V_j^2$ and $I_{ij}^2$ are replaced by an new set of variables labeled as $V_i^{sqr}$, $V_j^{sqr}$, and $I_{ij}^{sqr}$. These last variables do not represent the square of the corresponding quantities; instead, they are just the labels of new variables. Once the power flow is solved, the final values of currents and voltages are available by computing the square roots of the obtained variables. Furthermore, the mathematical models corresponding to the power flow and OPCB are based on previous research works carried out by the authors in [66,71,72].

The proposed objective function, given by Equation (1), aims at minimizing three components: the annual energy losses cost ($f_1$), the annual conductor selection cost ($f_2$), and the annual capacitor bank placement cost ($f_3$).

$$\begin{array}{cc}\hfill & \mathrm{Minimize}\mathit{f}=f_1+f_2+f_3\hfill \end{array}$$

(1)

The function $f_1$ represents the annual energy loss cost, and it is calculated as indicated by Equation (2).

$$\begin{array}{c}\hfill f_1=\left[K_p+\left(K_e\times T\times LSF\right)\right]\times \sum _{\forall ij\in {\mathsf{\Omega }}_l}\sum _{\forall c\in {\mathsf{\Omega }}_c}{l}_{ij}·R_c·W_{ij,c}·I_{ij}^{sqr}\end{array}$$

(2)

In this case, $K_p$ and $K_e$ represent the levelized annual demand cost and the energy costs, respectively. For comparative purposes, these values are given either in ($/kW) or (Rs/kW), according to the test system. T is number of hours per year; $LSF$ is the system loss factor, ${\mathsf{\Omega }}_l$ is the set of branches, ${\mathsf{\Omega }}_c$ is the set of available conductor types, $l_{ij}$ is the length of the conductor associated with branch $ij$, $R_c$ is the resistance (k$\mathsf{\Omega }$/km) of conductor type c, $W_{ij,c}$ is a binary decision variable that indicates the conductor type c selected to be installed at branch ij; finally, $I_{ij}^{sqr}$ represents the square of the current magnitude at branch $ij$.

The function $f_2$ represents the annual conductor selection cost, which is calculated as indicated in Equation (3).

$$\begin{array}{c}\hfill f_2=\sum _{\forall ij\in {\mathsf{\Omega }}_l}\sum _{\forall c\in {\mathsf{\Omega }}_c}IDF·S_c·C_c·l_{ij}·W_{ij,c}\end{array}$$

(3)

where $IDF$ is the interest factor associated with the conductor selection, $S_c$ is the conductor area (in ${\mathrm{mm}}^2$) of conductor type c; $C_c$ is the conductor cost expressed in function of its area and length (given in $\$/\left({\mathrm{mm}}^2\mathrm{km}\right)$ or $Rs/\left({\mathrm{mm}}^2\mathrm{km}\right)$ according to the test system); finally, $W_{ij,c}$ is a binary decision variable indicating the type of conductor selected.

The function $f_3$ represents the annual capacitor banks placement cost, which is calculated according to Equation (4).

$$\begin{array}{c}\hfill f_3=\sum _{\forall i\in {\mathsf{\Omega }}_b}\sum _{\forall ca\in {\mathsf{\Omega }}_{ca}}\left({\alpha }_c·K_f+C_{ca}·Q_{i,ca}\right)\end{array}$$

(4)

where ${\mathsf{\Omega }}_b$ is the set of system buses, ${\mathsf{\Omega }}_{ca}$ is the set of available capacitor bank types; ${\alpha }_c$ is the depreciation factor associated with the capacitor bank selection, $k_f$ is the cost of capacitor bank in $Rs$ or $; $C_{ca}$ is the annual installation cost of capacitor bank in Rs/kVAr or $\$/\mathrm{kVAr}$; finally, $Q_{i,ca}$ is the injected reactive power at bus i by capacitor type $ca$.

The aforementioned objective function is subject to constraints (5)–(27).

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

(5)

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

(6)

$$\begin{array}{cc}\hfill & V_i^{sqr}-2·l_{ij}·\sum _{\forall c\in {\mathsf{\Omega }}_c}\left(R_c·W_{ij,c}·P_{ij}+X_c·W_{ij,c}·Q_{ij}\right)-l_{ij}^2·Z_c^2·W_{ij,c}·I_{ij}^{sqr}-V_j^{sqr}=0;\forall ij\in {\mathsf{\Omega }}_l\hfill \end{array}$$

(7)

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

(8)

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

(9)

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

(10)

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

(11)

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

(12)

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

(13)

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

(14)

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

(15)

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

(16)

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

(17)

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

(18)

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

(19)

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

(20)

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

(21)

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

(22)

$$\begin{array}{cc}\hfill & \sum _{\forall c\in {\mathsf{\Omega }}_c}{W}_{ij,c}\le 1\forall ij\in {\mathsf{\Omega }}_l\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill & W_{ij,c}\in \left\{0,1\right\};\forall ij\in {\mathsf{\Omega }}_l,\forall c\in {\mathsf{\Omega }}_c\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill & Q_i^{}=\sum _{\forall ca\in {\mathsf{\Omega }}_{ca}}{Q}_{ca}·W_{i,ca};\forall i\in {\mathsf{\Omega }}_b\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & \sum _{\forall ca\in {\mathsf{\Omega }}_{ca}}{W}_{i,ca}\le 1;\forall i\in {\mathsf{\Omega }}_i\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & \sum _{\forall i\in {\mathsf{\Omega }}_b}\sum _{\forall ca\in {\mathsf{\Omega }}_{ca}}{W}_{i,ca}={\overline{N}}^{ca}\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & W_{i,ca}\in \left\{0,1\right\};\forall i\in {\mathsf{\Omega }}_b,\forall ca\in {\mathsf{\Omega }}_{ca}\hfill \end{array}$$

(28)

Equations (5) and (6) indicate the balance of active and reactive power in each bus of the system, respectively. $P_{ki}$ and $P_{ij}$ are the active power flow in branches ki and ij; $Q_{ki}$ and $Q_{ij}$ represent the reactive power flow in branches ki and ij, respectively. $P_i^s$ and $Q_i^s$ stand for the active and reactive power provided through the substation at bus i. $P_i^d$ and $Q_i^d$ represent the active and reactive power demands at bus i; $X_{ij,c}$ is the reactance of branch ij associated to conductor type c.

Equation (7) represents the voltage drop in every branch of the system. In this case, the voltage magnitudes are given terms of the electrical parameters of the branches and their power flows. $V_i^{sqr}$ and $V_j^{sqr}$ are variables that represent the square of the voltage magnitudes at buses i and j, respectively. $Z_c^2$ is the square of the impedance of conductor type c, and is calculated as $Z_c^2=R_c^2+X_c^2$.

The left-hand side of Equation (8) is a linear approximation of the square of the apparent power flow in branch ij given with the expression $V_j^{sqr}·I_{ij}^{sqr}$. It results from using the middle point of the first interval of the discretization of the square voltage magnitude multiplied by the square current flow magnitude, plus the successive power corrections. In this case, Equations (11)–(15) are used to carry out such linearization as indicated in [66,71,72]. The following factors are taken into consideration in this linearization: $\Delta V$ is the discretization step, S is the number of discretizations, $P_{j,s}^c$ is the power correction factor, and $x_{j,s}$ is a binary variable related to the number of blocks s.

The right-hand side of Equation (8) is a linear approximation of the expression $P_{ij}^2+Q_{ij}^2$. Equations (16)–(22) are used to carry out such linearization as detailed in [66,71,72]. In this case, $\Delta P_{ij,y}$ and $\Delta Q_{ij,y}$ are the values of the y block of $\left|P_{ij}\right|$ and $\left|Q_{ij}\right|$, respectively; $P_{ij}^{+}$ and $P_{ij}^{-}$ are used to obtain $\left|P_{ij}\right|$; and $Q_{ij}^{+}$ and $Q_{ij}^{-}$ are used to obtain $\left|Q_{ij}\right|$. ${\overline{\Delta S}}_{ij}$ is the maximum limit of every block of the load flow at branch ij. The following factors are taken into consideration in this linearization: Y is the number of blocks of the piecewise linearization; $m_{ij,y}^s$ is the slope of block $y_{th}$ of the load flow at branch ij; finally, $\Delta P_{ij,y}$ and $\Delta Q_{ij,y}$ represent the values of the $y_{th}$ block of $\left|P_{ij}\right|$ and $\left|Q_{ij}\right|$, respectively.

Equation (9) represents the voltage limit at bus i. Note that ${\overline{V}}_i$ and ${\underline{V}}_i$ are the upper and lower voltage limits at bus i, respectively. Equation (10) represents the current limit at branch ij, where ${\overline{I}}_c$ is the maximum current of conductor type c.

Equation (23) indicates that only one type of conductor can be chosen for each branch ij. In this case, the selected conductor corresponds to a value of 1 for $W_{ij,c}$ and zero for the other type of conductors. Equation (24) defines the binary nature of the variable $W_{ij,c}$. Equations (25)–(28) are used to model the OPCB problem, where $Q_i$ represents the reactive power injected at bus i. $W_{i,ca}$ is a binary decision variable that indicates the capacitor bank type selected to be installed at bus i. Finally, ${\overline{N}}_{ca}$ is the maximum number of capacitor banks to be allocated in the distribution system.

Equations (29) and (30) are used to calculate the values of ${\overline{\Delta S}}_{ij}$, and $m_{ij,y}^s$.

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

(29)

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

(30)

2.2. Linearization

Note that the model given by Equations (1)–(27) presents nonlinearities associated with the multiplication of binary variables as well as the multiplication of binary variables with continuous variables. This section explains the linearization carried out to recast the model given by (1)–(27) into a MILP problem. The main advantage is that the later formulation guarantees obtaining global optimal solutions to the OCS and OPCB.

Note that the first term of the objective function ($f_1$), as well as Equation (7), contain a multiplication of a binary variable with a continuous variable given by $W_{ij,c}·I_{ij}^{sqr}$. A new variable ${\varphi }_{ij,c}$ is introduced as the result of this multiplication. Then, the linearization of this expression is carried out using the big-M method given by Equations (31) and (32), where $M_1$ is a sufficiently large number that guaranties the enforcement to both constraints.

$$\begin{array}{cc}\hfill & 0\le -{\varphi }_{ij,c}+I_{ij}^{sqr}\le M_1·\left(1-W_{ij,c}\right)\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill & 0\le {\varphi }_{ij,c}\le M_1·W_{ij,c}\hfill \end{array}$$

(32)

Equation (7) contains two nonlinearities given by $W_{ij,c}·P_{ij}$ and $W_{ij,c}·Q_{ij}$. Both expressions can be linearized using the big-M method and introducing two new variables ${\beta }_{ij,c}$ and ${\delta }_{ij,c}$ as indicated in Equations (33)–(36), where $M_2$ and $M_3$ are sufficiently large numbers to enforce the constraints.

$$\begin{array}{cc}\hfill & 0\le -{\beta }_{ij,c}+P_{ij}^{sqr}\le M_2·\left(1-W_{ij,c}\right)\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & 0\le {\beta }_{ij,c}\le M_2·W_{ij,c}\hfill \end{array}$$

(34)

$$\begin{array}{cc}\hfill & 0\le -{\delta }_{ij,c}+Q_{ij}^{sqr}\le M_3·\left(1-W_{ij,c}\right)\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & 0\le {\delta }_{ij,c}\le M_3·W_{ij,c}\hfill \end{array}$$

(36)

It is worth mentioning that when the big-M method is implemented, each parameter ($M_1$, $M_2$, and $M_3$) must be fitted in the model. Small values of M might lead to conflicts with the original constraints of the model, while excessively high values of M might lead to convergence problems. Therefore, these parameters must be found by trial and error.

Equations (12) and (13) involve the multiplication of two binary variables given by $W_{ij,c}·x_{j,s}$. In this case, a new binary variable $A_{ij,c,s}$ is the result of this multiplication. Then the linearization of this expression is carried out as indicated in Equations (37)–(39).

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

(37)

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

(38)

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

(39)

Following the linearization strategy, $f_1$ can be rewritten as indicated in Equation (40), relabeled as $f_1^{{}^{\prime }}$ to indicate its linear expression.

$$\begin{array}{c}\hfill f_1^{{}^{\prime }}=\left[K_p+\left(K_e\times T\times LSF\right)\right]\times \sum _{\forall ij\in {\mathsf{\Omega }}_l}\sum _{\forall c\in {\mathsf{\Omega }}_c}{l}_{ij}·R_c·{\varphi }_{ij,c}\end{array}$$

(40)

The balance of active and reactive power given by Equations (5) and (6) are modified as follows:

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

(41)

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

(42)

The voltage profile in every branch given by Equation (7) is modified as follows:

$$\begin{array}{cc}\hfill & V_i^{sqr}-2·l_{ij}·\left(\sum _{\forall c\in {\mathsf{\Omega }}_c}{R}_c·{\beta }_{ij,c}+\sum _{\forall c\in {\mathsf{\Omega }}_c}{X}_c·{\delta }_{ij,c}\right)-l_{ij}^2·Z_c^2·{\varphi }_{ij,c}-V_j^{sqr}=0;\forall ij\in {\mathsf{\Omega }}_l\hfill \end{array}$$

(43)

2.3. MILP Model for OCS and OPCB

Following the aforementioned linearization approach, the mathematical model given by Equations (1)–(27) is recast as a MILP equivalent as indicated below:

$$\begin{array}{cc}\hfill & \mathrm{Minimize}\left(39\right)+\left(2\right)+\left(3\right)\hfill \end{array}$$

(44)

$$\begin{array}{cc}\hfill & \mathrm{Subject}\mathrm{to}:\left(40\right)-\left(42\right),\left(8\right)-\left(26\right),\left(30\right)-\left(38\right)\hfill \end{array}$$

(45)

3. Validation of the Proposed Model

The mathematical model for the OCS and OPCB proposed in this paper was implemented in AMPL and solved using CPLEX. Three different test systems were used to show the performance of the proposed model. The following cases were considered for each distribution test system:

• Initial base case.
• Only OCS.
• Only OPCB.
• Simultaneous OCS and OPCB.

Parameter $M_1$ used in Equations (32) and (33) was $1\times {10}^{14}$, $M_2$ and $M_3$ used in Equations (34)–(37) were adjusted to $1\times {10}^7$. Furthermore, the number of piece-wise linearizations (Y) was equal to 50, and S was set to 5 for all test systems.

A comparison of the results of the proposed approach with previously reported research was carried out for each test system. Although both OCS and OPCB have been widely studied independently, there are only a few papers that have integrated both problems, all of them using metaheuristic techniques.

To better illustrate the conductor selection, in all single-line diagrams of the test system, the intensity of the colors was proportional to the current-carrying capacity of the conductors.

The information regarding capacitor banks for OPCB is presented in

Table 1. Capacitor bank data for the simulations.

Size (kVAr)	150	300	450	600	750	900	1050	1200	1350	1500	1650	1800	1950	2100
Cost ($/kVAr)	0.5	0.35	0.253	0.22	0.276	0.183	0.228	0.17	0.207	0.201	0.193	0.187	0.211	0.176
Size (kVAr)	2250	2400	2550	2700	2850	3000	3150	3300	3450	3600	3750	3900	4050
Cost ($/kVAr)	0.197	0.17	0.189	0.187	0.183	0.18	0.195	0.174	0.188	0.17	0.183	0.182	0.179

, according to [73].

3.1. 27-Bus Test System

The 27-bus test system operated at a nominal voltage of 23 kV and had a total demand of $(16,541.66+j9932.42)$ kVA. Its electrical data can be found in [21,22]. The active power losses for the initial base case was 1574.61 kW, and the voltage at bus 10 reached a minimum of 0.7936 p.u. The first two branches of this system, namely 0-1 and 1-2, had the following parameters: $R_{0-1}=0.1233[\mathsf{\Omega }/\mathrm{km}]$, $X_{0-1}=0.4127[\mathsf{\Omega }/\mathrm{km}]$, $R_{1-2}=0.014[\mathsf{\Omega }/\mathrm{km}]$ and $X_{1-2}=0.6451[\mathsf{\Omega }/\mathrm{km}]$. For comparative purposes, these branches are not considered for OCS; moreover, the minimum and maximum voltage limits were 0.95 p.u. and 1.00 p.u., respectively. The total annual cost of active power losses $\left(K_p+\left(K_e\times T\times LSF\right)\right)$ was 168 US$/kW-year.

Table 2. Conductors data for the 27-bus test system.

Conductor	Reference	R	X	Imax	Cost
Type	Color	[Ω/km]	[Ω/km]	[A]	[US$/km]
1		0.1576	0.2277	520	151
2		0.2712	0.2464	310	81
3		0.2733	0.2506	288	70
4		0.4545	0.2664	212	48
5		0.7822	0.2835	150	31

lists the types of conductors used in this test system along with their corresponding electrical parameters, as described in [21,22]. The second column provides a reference color for each conductor to better visualize the proposed solution (Figure 1). The third to fifth columns of the table show the electrical parameters of the conductors. The last column indicates the cost of each conductor, which results from multiplying its cross-sectional area ($S_c$) and the cost per unit area ($C_c$). The $IDF$ is equal to 10%.

To compare the costs of the OPCB in this test system, the value of ${\alpha }_c$ was set to zero. Therefore, the cost of placing capacitor banks is simply given by the product of the cost per unit of capacitance ($C_{ca}$) and the total amount of reactive power ($Q_{i,ca}$) required, is denoted as $f_3=C_{ca}·Q_{i,ca}$.

Table 3. Conductors selected for the 27-bus test system.

	Conductor Type			Conductor Type
Branch Number	Base	Only	Branch Number	Base	Simultaneous
	Case	OCS		Case	OCS and OPCB
3–6	1	1	3–6	1	1
7–11; 17; 18	2	1	7–11; 17–18	2	1
12–16; 19–21; 23–25	3	1	12–15; 19; 20	3	1
22	3	3	16; 21; 23–25	3	3
26	4	1	22	3	5
27	4	4	26	4	3
			27	4	5

presents the conductor selection for both OCS and simultaneous OCS and OPCB scenarios. The first and fourth columns of the table specify the branch numbers included in each selection. For instance, in the first line of the table, the notation 3–6 indicates the branches from 3 to 6. Similarly, the second line lists branches 7 to 11, 17, and 18, and so on. It should be noted that in some branches where simultaneous OCS and OPCB were employed, the selected conductors had a lower current capacity compared to the base case with only OCS. This reduction in capacity was due to the effect of the capacitor banks.

Table 4. Optimal solution for only OPCB and simultaneous OCS and OPCB (27-bus test system).

Bus	Capacitor Size for Only OPCB [kVAr]	Bus	Capacitor Size for OCS and OPCB [kVAr]
6	3300	2; 4	1350
8	1200	3	2400
10	1500	6	900
14; 15	600	8; 14	1200
16	1350	10; 13; 16	750
21; 26	900	26	600
23	450
27	300
Total reactive power [kVAr]	11,100		10,050

shows the reactive power injected by the capacitor banks for both OPCB and simultaneous OCS and OPCB scenarios. The amount of reactive power for the simultaneous case was lower than for the OPCB alone. This reduction in reactive power (9.45%) was due to the effect of the conductor selection employed in the simultaneous scenario.

Table 5. Summary of results and comparison for the 27-bus test system.

Description	Base Case	Proposed Solution			Proposed Solution for the OCS and OPCB
		OPCB	OCS	OCS and OPCB	[29]	[21]	[22]
Minimum voltage [p.u]	0.7936	0.9500	0.8122	0.9503	0.9520	0.9360	0.9323
Active power losses [kW]	1574.61	1025.42	930.83	446.31	835.80	868.17	842.2
Active power losses cost [$]	264,534.48	172,270.56	156,379.44	74,980.08	140,415.60	145,855	141,490
Conductor cost [$/year]	32,338.18	0.00	13,440.00	17,878.05	23,000.00	11,930	13,442
Capacitor cost [$/year]	0.00	2171.40	0.00	2292.60	965.40	3421.9	3470
Total cost [$/year]	296,872.66	174,441.96	169,819.44	95,150.73	164,381	161,207	158,402
Total economic benefit [$/year]		122,430.70	127,053.22	201,721.93	132,491.66	135,665.66	138,470.66

summarizes the results obtained for each simulation and provides a comparison with [21,22,29], all of which consider the simultaneous OCS and OPCB. The results showed that the simultaneous OCS and OPCB improved the minimum voltage magnitude by 19.74%, reduced power losses up to 71.65%, and total investment cost up to 67.95% compared to the base case. Notably, the proposed model achieved a 46.60% reduction in active power losses compared to the results reported in [29], which presented the lowest power losses. Moreover, there was a 39.93% reduction in the total cost of the solution compared to [22]. These results demonstrate the superiority of the proposed approach over the metaheuristics implemented in [21,22,29].

The conductor selection and location of capacitor banks for the simultaneous OCS and OPCB are illustrated in Figure 1. As expected, higher capacity conductors are used for the branches closer to the substation, and capacitor banks are mainly located towards the end of heavily loaded feeders.

Figure 2 shows the voltage profile of the 27-bus test system for each case studied. It is worth noting that the OCS alone does not provide significant improvements in the voltage profile. However, the voltage profile was noticeably improved with the OPCB, and even more so when OCS was combined with OPCB. In particular, nodes 11 and 27 exhibited the highest improvement in voltage magnitudes.

3.2. 69-Bus Test System

The 69-bus test system operated at a nominal voltage of 12.66 kV and had a total demand of $(3802.19+j2694.6)$ kVA. Its electrical data can be found in [74]. The active power losses for the initial base case were 224.9931 kW, and the voltage at bus 65 reached a minimum of 0.9092 p.u. For comparative purposes, the minimum and maximum voltage limits were set at 0.95 p.u. and 1.00 p.u., respectively.

Table 6. Objective function parameters for the 69-bus test system.

Parameter	Value	Parameter	Value
$k_p$ [$/kW]	1.04	$K_e$ [$/kWh]	0.012
i [% ]	8	n [year]	20
LSF	0.2	T [hour]	8760
IDF	0.1

presents the parameters used in [12] to compute the objective function. These parameters were also adopted in the present study to calculate objective functions $f_1$ and $f_3$. In this case, the value of ${\alpha }_c$ was set to zero. Therefore, the cost of placing capacitor banks was simply given by the product of the cost per unit of capacitance ($C_{ca}$) and the total amount of reactive power ($Q_{i,ca}$) required, denoted as $f_3=C_{ca}·Q_{i,ca}$.

Table 7. Conductors data for the 69-bus test system.

Conductor	Reference	R	X	Imax	Cost	Area
Type	Color	[Ω/km]	[Ω/km]	[A]	[US$/(${\mathbf{mm}}^2·\mathbf{km}$)]	[${\mathbf{mm}}^2$]
1		0.2128	0.30747	760	692.2315	400.00
2		0.2128	0.30747	546	415.3389	240.00
3		0.5424	0.78370	440	271.5282	156.90
4		0.6803	0.98290	380	216.4967	125.10
5		0.7220	1.06880	330	171.8636	99.31
6		0.7741	1.10700	245	108.0582	62.44
7		0.8570	1.16640	215	85.62990	49.48
8		0.9361	1.25330	135	42.76160	24.71
9		0.9746	1.33220	105	105.0000	15.52
10		0.9920	1.35000	81	81.00000	13.44
11		1.1270	1.51740	70	70.00000	10.48
12		1.2620	1.73340	48	25.04710	9.71
13		1.4336	2.52260	31	26.85860	8.52

lists the conductor types and their electrical specifications reported in [12], which were used for all simulations. Its worth noting that this system includes 13 types of conductors with varying characteristics. The second column provides a reference color for each conductor to better visualize the proposed solution (Figure 3).

Table 8. Optimal solution for only OCS and simultaneous OCS and OPCB (69-bus test system).

Branch Number	Conductor Type for Only OCS	Branch Number	Conductor Type for OCS and OPCB
1–3	4	1–3	5
4–8	5	4–8; 52–54	7
52–60	7	9; 10; 46–48; 55–60	8
9–10; 46–48	8	11	11
11	11	12–17; 49; 61–63	12
12–17; 49; 61–63	12	18–45; 50; 51; 64–68	13
18–45; 50; 51; 64–68	13

presents the conductor selection for both the OCS and simultaneous OCS and OPCB scenarios. The first and third columns of the table indicate the branches for which the conductors were modified. Note that different branches are selected for OCS in both scenarios. For example, when only OCS was considered, conductor type 4 was used to upgrade branches 1 to 3; however, for the simultaneous OCS and OPCB scenario, the same branches were upgraded with conductor type 5. This pattern was also observed for other branches, such as 4 to 8 and 55 to 60.

Table 9. Optimal solution for only OPCB and simultaneous OCS and OPCB (69-bus test system).

Bus	Capacitor Size for Only OPCB [kVAr]	Bus	Capacitor Size for OCS and OPCB [kVAr]
58; 65	150	9	900
61	1650	61	900
64	600
Total reactive power [kVAr]	2400		1800

presents reactive power injected using the capacitor banks for only OPCB and simultaneous OCS and OPCB. In this case, due to the effect of the conductor selection, the amount of reactive power for the simultaneous case is 600 kVAr lower than for the OPCB alone.

Table 10. Summary of results (69-bus test system).

Description	Base Case	Proposed Solution
		OCS and OPCB	OPCB	OCS
Minimum voltage [p.u]	0.9092	0.9733	0.9501	0.9596
Active power losses [kW]	224.99	58.30	205.32	76.59
Active power losses cost [$]	4963.86	1286.28	4530.63	1689.89
Conductor cost [$/year]	–	1098.95	–	2252.77
Capacitor cost [$/year]	–	329.40	600.45	–
Total cost [$/year]	4963.86	2714.63	5131.08	3942.67

summarizes the results obtained for each simulation. The base case exhibited a minimum voltage of 0.9092 p.u and active power losses of 224.99 kW, resulting in a total cost of 4963.86 [$/year]. It is noteworthy that the initial cost of conductors was not available for this system.

The highest improvement in the voltage profile was achieved when OCS and OPCB were carried out simultaneously, as compared to the base case. In this case, the minimum voltage improved by 7.05%. The lowest active power losses were also obtained for the same case (OCS and OPCB). A reduction of 74.08% in active power losses was achieved compared to the base case, which implies a significant reduction in total costs per year.

The simultaneous OCS and OPCB is illustrated in Figure 3. In this case, only conductor types 5, 7, 8, 11, 12, and 13 were selected in the optimal solution. Note that conductors with higher capacity were used for the branches closer to the substation. Moreover, capacitor banks were installed at buses 9 and 61, each one of 900 kVAr.

Figure 4 illustrates the voltage profile of the 69-bus test system for the different cases under study. Note the base case presents very low voltages at buses 58 to 65; nonetheless, this was corrected with the OPCB that allowed all voltage magnitudes to be over 0.95 p.u. Moreover, the voltage profile was further improved with the OCS and also with the simultaneous OCS and OPCB.

3.3. 85-Bus Test System

The 85-bus test system operated at a nominal voltage of 11 kV and had a total demand of $(2570.28+j2622.21)$ kVA. Its electrical data can be found in [75]. The active power losses for the initial base case were 315.71 kW, and the voltage at bus 54 reached a minimum of 0.8628 p.u. For comparative purposes, the minimum and maximum voltage limits were 0.95 p.u. and 1.00 p.u., respectively.

Table 11. Objective function parameters for the 85-bus test system.

Parameter	Value	Parameter	Value
$k_p$ [Rs/kW]	4000	$K_e$ [Rs/kWh]	5
LSF	0.2	T [hours]	8760
i [%]	8	n [year]	25
IDF	0.1	C [${\mathrm{Rs}/(\mathrm{mm}}^2·\mathrm{km})$]	500
$K_f$	1000	${\alpha }_c$	0.15

presents the parameters used in [27] to compute the objective function. These parameters were also adopted in the present study to calculate objective functions $f_1$ and $f_3$ for comparative purposes.

To compare the costs of capacitor bank placement in this test system, the value of ${\alpha }_c$ was set to zero. Therefore, the cost of placing capacitor banks was simply given by the product of the cost per unit of capacitance ($C_{ca}$) and the total amount of reactive power ($Q_{i,ca}$) required, denoted as $f_3=C_{ca}·Q_{i,ca}$.

Table 12. Conductor parameters for the 85-bus test system.

Conductor	Conductor	Reference	R	X	Imax	Area
Type	Name	Color	[Ω/km]	[Ω/km]	[A]	[${\mathbf{mm}}^2$]
1	Squirrel		1.3760	0.3896	115	13
2	Gopher		1.0980	0.3100	138	16
3	Weasel		0.9108	0.3797	150	20
4	Ferret		0.6795	0.2980	180	24
5	Rabbit		0.5441	0.3673	208	30
6	Mink		0.4565	0.2850	226	40
7	Beaver		0.3841	0.2795	250	45
8	Raccoon		0.3657	0.3579	470	48

lists the conductor types and their electrical specifications reported in [27], which were used for all simulations. Its worth noting that this system includes 8 types of conductors with varying characteristics. The third column provides a reference color for each conductor to better visualize the proposed solution (Figure 5). Finally, the information presented in Table 1 was used to carry out the OPCB.

The results obtained for the 85-bus test system are summarized in

Table 13. Results of the 85-bus test system.

Branch Number	Initial Conductorth	Only OCS	Branch Number	InitialOCS Conductor	Simultaneous OCS and OPCB
14;15; 21; 23; 35; 41; 42; 46; 55; 70; 74; 76; 78; 83; 84	1	1	14; 15; 21; 23; 35; 41; 42; 46; 55; 70; 74; 76; 78; 83; 84	1	1
22; 36; 37; 38; 45; 50; 53; 54; 58; 61; 65; 71; 73; 75; 77; 81; 82	1	2	22; 36; 37; 38; 44; 45; 50; 53; 54; 58; 61; 64; 65; 71; 73; 75; 77; 81; 82	1	2
13; 20; 40; 44; 64	1	4	13; 20; 40; 48; 49	1	4
12; 16; 19; 39; 43; 48; 49; 52; 60; 69; 72; 80	1	5	12; 16; 19; 39; 43; 51; 52; 60; 69; 72; 80	1	5
51; 68; 79	1	7	17; 18; 79	1	7
1–7	8	8	1–7	8	8
8–11; 17; 18; 24–34; 47; 56; 57; 59; 62; 63; 66; 67	1	8	8–11; 24–34; 47; 56; 57; 59; 62; 63; 66; 67; 68	1	8

. Note that conductor type 1 is selected in the same branches for OCS and simultaneous OCS and OPCB. This conductor is the one with the lowest capacity and is used mostly for buses at the end of the feeders. Conductor type 2 is the second with the lowest capacity and is also used at the end of some feeders.

Table 13 presents the conductor selection for both the OCS and simultaneous OCS and OPCB scenarios. The first and fourth columns indicate the branches for which the conductors were modified. Note that conductor type 1 is selected for the same branches in both OCS and simultaneous OCS and OPCB scenarios. This conductor is the one with the lowest capacity and is mostly used for buses at the end of the feeders. Conductor type 2, which has the second-lowest capacity, was also used at the end of some feeders. Different branches were selected for OCS in both scenarios. For instance, when only OCS was considered, conductor type 4 was used at branches 44 and 64. However, in the simultaneous OCS and OPCB scenario, the same branches were upgraded with conductor type 2. This pattern was also observed for other branches, such as 51, which had conductor type 7 and used conductor type 5 after simultaneous OCS and OPCB. Branches 17 and 18 had conductor type 8, and after simultaneous OCS and OPCB, they used conductor type 7. Only branch 68 had an increased conductor type, from 7 to 8.

Table 14. Optimal solution for only OPCB and simultaneous OCS and OPCB (85-bus test system).

Bus	Capacitor Size for Only OPCB [kVAr]	Bus	Capacitor Size for OCS and OPCB [kVAr]
9	1050	8	1200
34	750	32	450
68	450	64	300
Total reactive power [kVAr]	2250		1950

shows the reactive power injected by the capacitor banks for both OPCB and simultaneous OCS and OPCB scenarios. The amount of reactive power for the simultaneous case is lower than for the OPCB alone. This reduction in reactive power (13.33%) is due to the effect of the conductor selection employed in the simultaneous scenario.

Table 15. Results of the 85-bus test system.

Description	Base Case	Proposed Solution			Proposed Solution
		OCS	OPCB	OCS and OPCB	in [27] for OCS and OPCB
Minimum voltage [p.u]	0.8628	0.8892	0.9283	0.9473	0.9452
Active power losses [kW]	315.71	223.81	151.37	112.75	118.14
Active power losses cost [Rs]	4,028,459.60	2,855,827.61	1,931,503.87	1,438,792.75	1,507,466.40
Conductor cost [Rs/year]	36,379.00	60,625.50	0.00	59,950.00
Capacitor cost [Rs/year]	0.00	0.00	710.25	572.00
Total cost [Rs/year]	4,064,838.60	2,916,453.11	1,932,214.12	1,499,315.60	1,603,875.14
Total economic benefit [Rs/year]	–	1,148,385.49	2,113,699.47	2,565,523.00

Rs: Rupees.

summarizes the results obtained for each simulation and provides a comparison with [27] which considered the simultaneous OCS and OPCB. The results show that the simultaneous OCS and OPCB improved the minimum voltage magnitude by 9.79%, reduced power losses up to 64.28%, and total investment cost up to 63.1% compared to the base case. The proposed model achieved a 4.56% reduction in active power losses compared to the results reported in [27], which presented the lowest power losses. Moreover, there was a 6.52% reduction in the total cost of the solution compared to [27]. These results demonstrate the superiority of the proposed approach over the metaheuristics implemented in [27].

The conductor selection and location of capacitor banks for the simultaneous OCS and OPCB are illustrated in Figure 5. As expected, a higher capacity conductor was used for the main feeder, and the capacity of the selected conductors decreased towards the end of the feeders. Three capacitor banks were allocated at buses 8, 32, and 64 to minimize the cost of network reinforcements, while also selecting the type of conductor to minimize power losses.

Figure 6 shows the voltage profile of the 85-bus test system for each case studied. It is worth noting that the OCS alone does not provide significant improvements in the voltage profile. However, the voltage profile was noticeably improved with the OPCB, and even more so when OCS was combined with OPCB. In general, all nodes exhibit improvement in voltage magnitudes.

4. Conclusions

The OCS is a critical step in improving the technical performance of electrical power systems. The optimal selection of new conductors to replace outdated or degraded cables can significantly reduce power losses and ameliorate voltage profiles, improving the overall system performance. Nonetheless, the OCS alone is not always enough to reduce technical losses and improve voltage profile; therefore, it is usually combined with other technical solutions. In this paper, the OCS has been combined with the OPCB which consists of selecting the best locations within the network to install reactive power compensation.

Both OCS and OPCB were merged in a single MILP model that allowed obtaining global optimal solutions. This constitutes the main contribution of the paper since the combination of both problems has not been proposed in the specialized literature through an MILP approach. Furthermore, the problems can be solved either jointly or separately, providing versatility to the decision-making process.

The effectiveness and applicability of the proposed model were tested in three distribution test systems of 27, 69, and 85 buses and compared with previous results reported in the specialized literature. In all cases under study, the proposed model was able to find better results. It was found that although the OCS was crucial in reducing power losses, the improvement of the voltage profile responds better to the OPCB.

In the three distribution test systems, when considering simultaneous OCS and OPCB, the selected conductors had a lower current capacity compared to the case with only OCS. This reduction in capacity was caused by the effect of the capacitor banks and had a high impact on the investment cost.

Future work may include the hourly, daily, or seasonally profile of load demand. This would require a multi-period optimization approach that would also consider switchable capacitor banks.