A Mixed-Integer Convex Optimization Framework for Cost-Effective Conductor Selection in Radial Distribution Networks While Considering Load and Renewable Variations

Abstract

The optimal selection of conductors (OCS) in radial distribution networks is a critical aspect of system planning, directly impacting both investment costs and energy losses. This paper proposed a mixed-integer convex (MI-Convex) optimization framework to solve the OCS problem under balanced operating conditions, integrating the costs of conductor investment and energy losses into a single convex objective. This formulation leveraged second-order conic constraints and was solved using a combination of branch-and-bound and interior-point methods. Numerical validations on standard 27-, 33-, and 85-bus test systems confirmed the effectiveness of the proposal. In the 27-bus grid, the MI-Convex approach achieved a total cost of $550,680.25, outperforming or matching the best results reported by state-of-the-art metaheuristic algorithms, including the vortex search algorithm (VSA), Newton’s metaheuristic algorithm (NMA), the generalized normal distribution optimizer (GNDO), and the tabu search algorithm (TSA). The MI-Convex method demonstrated consistent and repeatable results, in contrast to the variability observed in heuristic techniques. Further analyses considering three-period and daily load profiles led to cost reductions of up to 27.6%, and incorporating distributed renewable generation into the 85-bus system achieved a total cost of $705,197.06—approximately 22.97% lower than under peak-load planning. Moreover, the methodology proved computationally efficient, requiring only 1.84 s for the 27-bus and 12.27 s for the peak scenario of the 85-bus. These results demonstrate the superiority of the MI-Convex approach in achieving globally optimal, reproducible, and computationally tractable solutions for cost-effective conductor selection.

1. Introduction

The optimal design of electrical distribution systems is a key challenge in ensuring an adequate operation that features high energy efficiency and reduced power losses [1]. In this context, the proper selection of conductors for the distribution lines plays a fundamental role, as it has a direct impact on initial investment, power losses, voltage regulation, and load capacity. Hence, in these systems, the selected conductor size must balance costs, voltage regulation, and operational efficiency. To address this problem, optimization models are required which can find efficient and technically and economically feasible solutions [2].

On the other hand, the growing demand for electricity and the integration (optimal placement and sizing) of various systems and devices into distribution networks have intensified the need for optimizing the electrical infrastructure [3]. Among the most commonly integrated systems are distributed generation sources, including wind and photovoltaic generators [4,5,6]. Additionally, energy storage systems have been employed to enhance the operational performance of distribution networks [7]. Devices for reactive power compensation, e.g., distribution static VAR compensators [8], thyristor-controlled series capacitors [9], and other similar technologies [10], are also frequently integrated. The incorporation of these systems underscores the fact that improper conductor sizing can result in excessive power losses, line overloads, and increased operating costs.

Although there are traditional methodologies for conductor selection, many of them rely on heuristic or non-convex optimization approaches, which can lead to suboptimal or computationally inefficient solutions [11]. Various approaches have been proposed in the literature to address this problem. For instance, methods based on mixed-integer linear programming (MILP) have been explored to minimize costs and power losses, although their application is limited by the non-convexity of the problem and its computational complexity. In [11], a MILP model was proposed to solve the conductor size selection and reconductoring problems in radial distribution systems. This model represents the steady-state operation of the system using linear expressions, ensuring optimal convergence through existing optimization solvers. Additionally, a heuristic approach was employed alongside the MILP model to generate a Pareto front while considering two objective functions. The effectiveness of the proposed methodology was demonstrated through simulations on one test system and two real distribution networks, highlighting its accuracy and computational efficiency.

Other works have addressed this problem using heuristic techniques, which, although they can provide acceptable solutions, do not guarantee global optimality. For example, in [12], a sine-cosine optimization algorithm (SCA) was developed for the optimal selection of conductors in radial distribution networks. In addition, the authors of [12] utilized a database of 20 conductor types taken from actual manufacturer data to ensure practical applicability. The proposed approach was tested on a real Egyptian distribution system, demonstrating its effectiveness in minimizing power losses while observing system constraints. The results also validated the method’s ability to handle high annual load growth over a ten-year period.

A hybrid optimization approach was proposed in [13] to solve the optimal conductor size selection (CSS) problem in distribution networks with a high penetration of distributed generation (DG). This study employed an adaptive genetic algorithm (AGA) as the primary optimization strategy to determine the optimal conductor sizes while aiming to minimize the sum of life-cycle costs (LCC) and total energy procurement costs over the expected operation period. Additionally, an alternating current optimal power flow (AC-OPF) analysis was incorporated as a secondary optimization step to refine economic dispatch decisions. The proposed methodology was validated through simulations on modified IEEE 33- and 69-bus distribution systems, demonstrating its effectiveness in addressing CSS under increasing DG capacity.

In [14], a methodology was introduced to minimize power losses in unbalanced distribution systems through phase balancing. The problem was formulated as a mixed-integer nonlinear programming model and addressed using a discretely encoded vortex search algorithm (DVSA). Simulations conducted on IEEE 8-, 25-, and 37-node test systems validated the effectiveness of this approach, exhibiting a superior performance in comparison with the classical Chu & Beasley genetic algorithm (CBGA). Additionally, the proposed DVSA achieved optimal solutions within seconds, demonstrating its efficiency, reliability, and robustness. The computational implementation was carried out in MATLAB^®, and all results were verified using the PowerFactory 2021 DIgSILENT© software GmbH Heinrich-Hertz-StraBe 9 72810 Gomaringen (Germany) to ensure the accuracy of the three-phase unbalanced power-flow method.

The authors of [15] tackled the phase-balancing problem in three-phase radial power grids using a master-slave optimization approach. The master stage employed an enhanced version of the CBGA, incorporating a multi-point mutation operator as well as solution generation based on a Gaussian normal distribution, as inspired by the exploration and exploitation mechanisms of the vortex search algorithm (VSA). This stage determined the phase configuration through integer encoding, while the slave stage utilized a three-phase power flow approach based on the successive approximations method to evaluate the daily costs of energy losses.

In [16], another master-slave optimization strategy was proposed to address conductor selection and phase balancing in electrical networks, which integrated the CBGA and the successive approximations methods for power flow analysis. This approach was tested on two distribution systems under different demand scenarios and compared against three metaheuristic algorithms. According to the results, the CBGA outperformed the other methods, achieving lower costs, faster processing times (11.766 s and 94.494 s), and minimal standard deviations (0.161% and 0.199%) for the 8- and 25-node test systems considered. These findings confirmed the efficiency and practicality of the proposed strategy for optimizing electrical distribution networks.

Although several metaheuristic and exact approaches such as MILP, the SCA, the AGA, and others have been proposed to address the optimal conductor selection (OCS) problem in distribution networks, they exhibit notable limitations. Specifically, metaheuristic techniques may achieve high-quality solutions, but they cannot ensure global optimality or reproducibility, and their performance often depends on carefully tuned parameters. These characteristics reduce their reliability and generalizability across different test systems. On the other hand, exact methods like MILP may sacrifice model accuracy by oversimplifying nonlinearities, especially under multi-period or nonlinear load conditions.

In this context, convex formulations emerge as a promising alternative for solving planning problems in distribution systems. They guarantee convergence to a globally optimal solution, ensure result reproducibility across multiple runs, and do not require algorithm-specific parameter tuning. These features make convex approaches highly attractive for practical deployment.

This work develops a novel mixed-integer convex (MI-Convex) optimization model for OCS in balanced radial distribution networks. The proposed formulation leverages quadratic and second-order cone relaxations to preserve the nonlinear characteristics of the system while ensuring convexity. The model integrates operational constraints, such as current carrying limits, voltage regulation margins, and the nodal power balance, and aims to minimize the total cost by simultaneously considering investment and long-term operating expenses. The effectiveness of this proposal was validated through tests on two benchmark networks, where it consistently outperformed state-of-the-art metaheuristic methods in terms of solution quality and computational efficiency.

Table 1. Comparative analysis of existing optimization approaches and the proposed MI-Convex method.

Method	Limitations	Advantages of the MI-Convex Approach
MILP [11]	Simplifies the nonlinear power flow via linear approximations; may compromise accuracy; not scalable for multi-period or nonlinear load conditions.	Maintains nonlinear power losses modeling through quadratic and conic approximations; supports realistic multi-period demand profiles.
SCA [12]	Heuristic; results are not reproducible; requires parameter tuning; no guarantee of global optimality.	Deterministic solution; no parameter tuning; guarantees global optimality due to convex formulation.
AGA [13]	Dependent on random initialization; requires calibration of crossover/mutation; high computational cost in large-scale grids.	Solves convex subproblems using interior-point methods; scalable to large networks (e.g., the 85-bus feeder).
TSA [17]	Requires large number of iterations; sensitive to stopping criteria; lacks consistent convergence.	Repeatable results; robust convergence using branch-and-bound and interior-point techniques.
VSA, GNDO, and NMA	Inconsistent performance across test cases; computational results vary between runs.	Consistent across all runs; reliably minimizes total planning costs; reduced processing times for medium-sized networks.

provides a comparative summary of the reviewed optimization techniques, their main limitations, and the advantages offered by the proposed MI-Convex methodology.

The remainder of this paper is organized as follows. Section 2 presents the mathematical formulation of the MI-Convex model for OCS. Furthermore, this section details the convex approximations and the solution method. Section 3 describes the test systems and scenarios considered for validation, and it discusses and analyzes the numerical results obtained. Finally, Section 4 outlines the main conclusions of this research and suggests directions for future work.

2. Methodology

OCS in radial distribution networks is among the most classical optimization problems in the field of distribution system planning an operation [18]. Its exact formulation corresponds to a mixed-integer nonlinear programming (MINLP) model that combines power flow variables in the continuous domain with integer variables representing the gauge selected for each distribution branch [17]. This research proposed a MI-Convex formulation with a quadratic structure to determine the set of conductors that must be selected for a radial distribution network operating under balanced load conditions. With the purpose of validating the effectiveness of the proposed solution, a classical power flow methodology was used to evaluate all the technical and operational constraints related to the power flow solution.

2.1. The MI-Convex Formulation

To formulate the MI-Convex approximation used to select the subset of conductors in a radial and balanced distribution network, the following assumptions were made:

Assumption 1.

Under normal operating conditions, an electrical distribution network must comply with all system constraints, especially with the voltage regulation bounds, which implies that all the voltages must be contained between $V_{min}$ and $V_{max}$. In a per-unit representation, all voltage values should be close to 1.0 pu [19]. Considering these typical operating conditions, ideal voltages were assumed for all nodes.

Assumption 2.

Under ideal operating conditions (perfect voltage profiles), the active and reactive power flow is equivalent in magnitude to the real and imaginary parts of the current flowing through a particular distribution branch.

Considering these assumptions, the proposed MI-Convex approach for conductor selection in radial electrical distribution networks under balanced operating conditions is described below.

2.1.1. Objective Function

The main idea of OCS in distribution networks is to minimize the total equivalent investment and operating costs for a planning period, which is typically assumed to be one year [20]. The total investment expenses are associated with the cost per type of conductor, which is a function of the current-carrying capacity and the total length. Meanwhile, the total operating costs are associated with the expected costs of energy losses [21]. The objective function and its two components are defined from (1) to (3).

$$\begin{array}{c}\hfill min{z}_3=z_1+z_2,\end{array}$$

(1)

with

$$\begin{array}{cc}\hfill z_1& =3C_{kWh}\sum _{h\in \mathcal{H}}\sum _{i\in \mathcal{N}}{R}_l\left(P_{lt}^2+Q_{lt}^2\right)F_t,\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill z_2& =3\sum _{l\in \mathcal{L}}\left(\sum _{c\in \mathcal{C}}{C}_c{x}_{lc}\right)L_l.\hfill \end{array}$$

(3)

where $z_1$ represents the approximate cost of energy losses, calculated using Equation (2); $z_2$ quantifies the total investment made in conductors, as defined in Equation (3); $z_3$ defines the total economic objective function value, as shown in Equation (1); $C_{\mathrm{kWh}}$ is a constant parameter associated with the average cost of a kilowatt-hour; $R_l$ is a semi-definite positive variable related to the resistance assigned to branch l; $P_{lt}$ and $Q_{lt}$ are the active and reactive power flows through branch l in period t, respectively; $F_t$ is a parameter associated with the total number of load duration periods in a single year; $C_c$ is a constant parameter related to the expected cost per kilometer of a conductor of gauge type c; $x_{lc}$ is a binary variable that defines which gauge is assigned to branch l; and $L_l$ is the total length of branch l in kilometers. Note that $\mathcal{C}$, $\mathcal{L}$, and $\mathcal{T}$ denote the sets containing all conductor gauges, distribution lines, and periods of analysis, respectively.

2.1.2. Problem Constraints

OCS in distribution networks involves a set of technical constraints which, in the proposed MI-Convex model, correspond to a simplification of the power balance equations by means of a reduced transportation model [22]. This set of constraints is presented below [23].

$$\begin{array}{c}\hfill R_l=\left(\sum _{c\in C}{R_{c}x}_{lc}\right)L_l,\left\{\forall l\in L\right\}\end{array}$$

(4)

$$\begin{array}{c}\hfill P_{kt}^g-P_{kt}^d={\beta }_k\sum _{l\in L}{A}_{kl}{P}_{lt},\left\{\forall k\in N,\forall t\in T\right\}\end{array}$$

(5)

$$\begin{array}{c}\hfill Q_{kt}^g-Q_{kt}^d={\beta }_k\sum _{l\in L}{A}_{kl}{Q}_{lt},\left\{\forall k\in N,\forall t\in T\right\}\end{array}$$

(6)

$$\begin{array}{c}\hfill \sum _{c\in C}{x}_{lc}=1,\left\{\forall l\in L\right\}\end{array}$$

(7)

$$\begin{array}{c}\hfill P_{lt}^2+Q_{lt}^2\le \sum _{c\in C}{x}_{lc}{\left(I_{lc}^{max}\right)}^2,\left\{\forall k\in N,\forall t\in T\right\}\end{array}$$

(8)

$$\begin{array}{c}\hfill {{\left(P_{kt}^g\right)}^2+\left(Q_{kt}^g\right)}^2\le {\left(S_k^g\right)}^2,\left\{\forall k\in N,\forall t\in T\right\}\end{array}$$

(9)

where $R_c$ corresponds to the resistive value associated with each available conductor gauge; $P_{kt}^g$ and $P_{kt}^d$ represent the active power generation and demand at bus k in period t, while $Q_{kt}^g$ and $Q_{kt}^d$ denote the same parameters with regard to reactive power; $A_{kl}$ is a component of the node-to-branch incidence matrix that relates node k to branch l; ${\beta }_k$ is a constant parameter that emulates a possible voltage drop at each node in the network, caused by its radial structure, assuming that the loads have lagging power factors; $I_{lc}^{max}$ represents the maximum thermal limit associated with gauge type c; and $S_k^g$ is the nominal power transfer capacity associated with the generator located at bus k. Note that $\mathcal{N}$ is the set containing all the nodes in the network, where N is the total number.

Regarding the definition and validation of the ${\beta }_k$ coefficients used in Equations (5) and (6), it should be clarified that this parameter is introduced to emulate the typical voltage drop behavior observed in radial distribution networks. The values of ${\beta }_k$ are recursively defined based on the topological structure of the feeder, i.e.,

$${\beta }_1=1,{\beta }_m={\beta }_k-{\displaystyle \frac{{\Delta }_v}{N-1}},\mathrm{for}\mathrm{each}\mathrm{branch}(k,m)$$

where ${\Delta }_v=0.20$ represents the expected voltage drop per line, expressed as a percentage. This formulation ensures a monotonic decrease in the values of ${\beta }_k$ along the radial path, thereby approximating voltage decay from the substation to the downstream nodes.

The set of constraints defined in Equations (4)–(9) ensures the physical, operational, and technical feasibility of the conductor selection problem for radial distribution networks. Here, Constraint (4) establishes the equivalent resistance $R_l$ of each branch based on the conductor type selected via the binary decision variable $x_{lc}$, scaled by the branch length $L_l$. This is crucial because the branch resistance directly affects the energy losses; omitting it would lead to incorrect losses estimation and suboptimal cost calculations. Constraints (5) and (6) enforce the active and reactive power balance at each node and time, incorporating the nodal-to-branch incidence relationship and a ${\beta }_k$ factor to capture the radial network’s behavior with approximate voltage drops. This ensures energy conservation and a realistic network operation; relaxing them would produce infeasible solutions that violate Kirchhoff’s laws. Constraint (7) states that exactly one conductor gauge must be assigned for each branch, a fundamental design rule to avoid ambiguity; omitting it would allow for unrealistic or undefined solutions. Inequality (8) ensures that the current in each branch does not exceed the thermal limit of the selected conductor, preventing overheating and potential network failure; relaxing this would compromise network safety and reliability. Lastly, Constraint (9) limits the apparent power of the generators to their rated capacities, ensuring operational feasibility and preventing equipment damage; its omission would imply unrealistic generator dispatch beyond physical ratings. Altogether, these constraints form a coherent model that captures both the economic and technical realities of radial distribution network planning under mixed-integer convex optimization.

2.1.3. Model Characterization

Within the MI-Convex optimization model described from (1) to (8), for each combination of binary variables (i.e., $x_{lc}$) the resulting nonlinear programming model belongs to the family of quadratic programming with linear constraints, since the objective function defined in (2) results in a convex squared function. In addition, the entire set of constraints defined from (4) to (7) includes linear equations for each binary variable combination. Finally, the Inequality Constraints (8) and (9) are second-order conic in nature, i.e., convex constraints.

This model has a very large solution space: for d conductor gauges available and b branches, the number of possible binary variable combinations grows exponentially, as defined in Equation (10) [14].

$$\begin{array}{c}\hfill s=d^b.\end{array}$$

(10)

To contextualize the magnitude of the solution space, consider a distribution network with 27 nodes and eight available conductor types. Given its radial structure, the network consists of 26 branches, resulting in approximately $3.0223145\times {10}^{23}$ possible binary variable combinations. This demonstrates the large scale and high complexity of the solution space. Furthermore, for each combination, it is necessary to solve a quadratic-conic convex optimization model, further complicating the computational challenge.

2.2. Solution Strategy

Considering that, for each combination of binary variables, the resulting model belongs to the family of convex optimization [24], in this research, the branch-and-bound (B&B) method was implemented in combination with interior-point optimization (IPO) [25]. This was done with the tools available in the Julia software v1.11.5 (14 April 2025) for this type of problem. To assess the feasibility of the final solution provided by the aforementioned combination of methods, the power flow problem was solved for the distribution grids under analysis [26]. This evaluation procedure is depicted in Figure 1.

It is worth noting that the power flow plays an important role in determining the feasibility of OCS. In light of this, the successive approximations power flow method reported by authors of [27] was adopted, as it can be easily implemented in Julia, with the main advantage that convergence is ensured via the Banach fixed-point theorem [28].

3. Results and Discussion

The effectiveness of the proposed MI-Convex formulation to address the problem regarding OCS in radial distribution networks was validated in two test feeders composed of 27 and 33 nodes [17]. The main characteristics of these systems, as well as their numerical analysis and a discussion of the results obtained, are presented in this section.

3.1. Test Feeder Characterization

The validation of the proposed optimization methodology considered two medium-voltage distribution networks composed of 27 and 33 nodes. The electrical configuration of both test feeders is depicted in Figure 2 and Figure 3 [17].

The main characteristics of these test feeders are listed below:

• The 27-bus grid operates using a line-to-ground voltage of $13,800\mathrm{V}$ at the substation terminals. The total active and reactive power per phase is $4131.5\mathrm{kW}$ and $2560.0\mathrm{kvar}$, respectively.
• The 33-bus grid operates using a line-to-ground voltage of $12,660\mathrm{V}$ at the substation terminals. The total active and reactive power per phase is $3715.0\mathrm{kW}$ and $2290.0\mathrm{kvar}$, respectively.
• The regulatory policies applicable to these feeders define a maximum voltage regulation bound of about $\pm 10\%$.

The electrical parameters of these distribution networks are listed in

Table 2. Electrical parameters of the 27-bus grid.

l	k	m	${\mathit{L}}_{\mathit{l}}$ (km)	${\mathit{P}}_{\mathit{mt}}^{\mathit{d}}$ (kW)	${\mathit{Q}}_{\mathit{mt}}^{\mathit{d}}$ (kvar)
1	1	2	0.55	0.0	0.0
2	2	3	1.50	0.0	0.0
3	3	4	0.45	297.5	184.4
4	4	5	0.63	0.0	0.0
5	5	6	0.70	255.0	158.0
6	6	7	0.55	0.0	0.0
7	7	8	1.00	212.5	131.7
8	8	9	1.25	0.0	0.0
9	9	10	1.00	266.1	164.9
10	2	11	1.00	85.0	52.7
11	11	12	1.23	340.0	210.7
12	12	13	0.75	297.5	184.4
13	13	14	0.56	191.3	118.5
14	14	15	1.00	106.3	65.8
15	15	16	1.00	255.0	158.0
16	3	17	1.00	255.0	158.0
17	17	18	0.60	127.5	79.0
18	18	19	0.90	297.5	184.4
19	19	20	0.95	340.0	210.7
20	20	21	1.00	85.0	52.7
21	4	22	1.00	106.3	65.8
22	5	23	1.00	55.3	34.2
23	6	24	0.40	69.7	43.2
24	8	25	0.60	255.0	158.0
25	8	26	0.60	63.8	39.5
26	26	27	0.80	170.0	105.4

and

Table 3. Electrical parameters of the 33-bus grid.

l	k	m	${\mathit{L}}_{\mathit{l}}$ (km)	${\mathit{P}}_{\mathit{mt}}^{\mathit{d}}$ (kW)	${\mathit{Q}}_{\mathit{mt}}^{\mathit{d}}$ (kvar)
1	1	2	0.0699	100	60
2	2	3	0.3720	90	40
3	3	4	0.2762	120	80
4	4	5	0.2876	60	30
5	5	6	0.7630	60	20
6	6	7	0.4030	200	100
7	7	8	1.4733	200	100
8	8	9	0.8850	60	20
9	9	10	0.8900	60	20
10	10	11	0.1308	45	30
11	11	12	0.2491	60	35
12	12	13	1.3115	60	35
13	13	14	0.6272	120	80
14	14	15	0.5585	60	10
15	15	16	0.6457	60	20
16	16	17	1.5050	60	20
17	17	18	0.6530	90	40
18	2	19	0.1603	90	40
19	19	20	1.4298	90	40
20	20	21	0.4439	90	40
21	21	22	0.8231	90	40
22	3	23	0.3798	90	40
23	23	24	0.8035	420	200
24	24	25	0.7985	420	200
25	6	26	0.1532	60	25
26	26	27	0.2145	60	25
27	27	28	0.9963	60	20
28	28	29	0.7524	120	70
29	29	30	0.3830	200	600
30	30	31	0.9687	150	70
31	31	32	0.3362	210	100
32	32	33	0.4356	60	40

.

In addition, the set of conductors available for installation in both distribution networks are listed in

Table 4. Conductors available for installation in the 27- and 33-bus grids.

Gauge c	${\mathit{R}}_{\mathit{c}}$ ($\mathbf{\Omega }$/km)	${\mathit{X}}_{\mathit{c}}$ ($\mathbf{\Omega }$/km)	${\mathit{I}}_{\mathit{c}}^{max}$ (A)	${\mathit{C}}_{\mathit{c}}$ (US$/km)
1	0.8763	0.4133	180	1986
2	0.6960	0.4133	200	2790
3	0.5518	0.4077	230	3815
4	0.4387	0.3983	270	5090
5	0.3480	0.3899	300	8067
6	0.2765	0.3610	340	12,673
7	0.0966	0.1201	600	23,419
8	0.0853	0.0950	720	30,070

.

3.2. Numerical Validations, Analysis, and Discussion

The computational implementation was performed using Julia, version 1.9.2 [29], which was executed on a PC equipped with an AMD Ryzen 7 3700 processor (2.3 GHz) and 16.0 GB of RAM, running a 64-bit version of Microsoft Windows 10 Single Language. The proposed MI-Convex model was formulated and solved via the JuMP optimization framework [30], leveraging the Ipopt solver for nonlinear programming. Additionally, the HiGHS solver was integrated through the Juniper optimizer to efficiently handle the mixed-integer nature of the problem [31]. This computational setup ensured a robust and efficient solution of the optimization model.

In the numerical validations, the following procedures were implemented:

I

A comparative assessment against literature-reported results using metaheuristic optimization algorithms under peak load conditions.

II

An evaluation of various load curves in order to determine the expected load consumption in the selected conductors.

3.2.1. Peak Load Operating Scenarios

The proposed MI-Convex formulation was tested on the aforementioned 27- and 33-bus grids while considering recent literature reports that employed metaheuristic algorithms (the NMA, VSA, GNDO, and TSA) as their solution methodologies [17].

Table 5. Solutions reported for the 27-bus grid.

Method	Gauges	${\mathit{z}}_1$ (US$)	${\mathit{z}}_2$ (US$)	${\mathit{z}}_3$ (US$)
VSA	[7,7,5,4,4,3,3,1,1,4,4,2,3,2,1,4,4,2,2,2,1,1,2,2,1,1]	217,066.25	344,352.15	561,418.40
NMA	[7,7,4,4,4,4,3,1,1,4,4,3,3,1,2,4,3,2,1,1,1,1,2,2,1,1]	219,343.86	337,744.80	557,088.66
GNDO	[7,7,4,4,4,3,3,1,1,4,4,2,1,1,1,3,2,2,1,1,1,1,1,1,1,1]	230,953.18	319,768.08	550,721.26
TSA	[7,7,4,4,4,3,3,1,1,4,4,2,1,1,1,4,2,2,1,1,1,1,1,1,1,1]	227,087.17	323,593.08	550,680.25
MI-Convex	[7,7,4,4,4,3,3,1,1,4,4,2,1,1,1,4,2,2,1,1,1,1,1,1,1,1]	227,087.17	323,593.08	550,680.25

presents a comparative analysis for the 27-bus grid that considers four different metaheuristics optimizers.

The numerical results in Table 5 allow stating that:

• The total costs of all the plans for the grid vary from US$ 550,680.25 to US$ 561,418.40 for all the optimization methodologies, with the TSA and the proposed MI-Convex approach exhibiting the minimum values and the VSA reporting the maximum value. These results confirm the effectiveness of our proposal in obtaining excellent numerical results, which coincide with the minimum values reported in the specialized literature [17], i.e., they are equal to the solution reported by the TSA.
• The total expected costs of energy losses ($z_1$) exhibit an inverse relationship with the expected investment in conductors ($z_2$), as observed in the solutions obtained using the VSA and the NMA. This correlation arises because higher investments in conductors lead to the selection of larger gauges, reducing the resistive effects and consequently lowering power losses. Conversely, lower investments result in smaller gauges, increasing the resistive effects and energy losses, as seen in the solutions provided by the GNDO, the TSA, and the MI-Convex method.
• An assessment of the best solution approaches, namely the TSA and MI-Convex within the proposed solution methodology (see Figure 1), reveals that the minimum voltage profile in the 27-bus grid occurs at bus 10, with a voltage value of approximately 0.9745 p.u. This indicates a voltage regulation of about 2.5473% for the test feeder.
• Regarding the expected branch loadability, under peak operating conditions, the distribution line connecting buses 1 and 2 experiences the highest current flow, with a magnitude of approximately 358.164 A. The conductor gauge assigned to this branch is No. 7 (see the first element in the MI-Convex row, the Gauges column), which has a thermal limit of about 600 A. This results in a loadability of approximately 59.6940% for this branch.

Considering this analysis in the 27-bus grid, the proposed MI-Convex and the TSA are the best optimization methods to determine the subset of gauges for all system branches. In both solutions, the costs of energy losses correspond to 41.2376% of the total planning costs, whereas the conductor investment expenses represent 58.7624%.

Table 6. Solutions reported for the 33-bus grid.

Method	Gauges	${\mathit{z}}_1$ (US$)	${\mathit{z}}_2$ (US$)	${\mathit{z}}_3$ (US$)
TSA	[7,7,5,5,5,4,3,2,1,1,1,1,1,1,1,1	215,137.5583	209,773.4628	424,911.0211
	1,1,1,1,1,3,2,1,4,4,4,3,3,1,1,1]
MI-Convex	[7,7,7,5,5,4,3,2,1,1,1,1,1,1,1,1	199,221.9755	225,372.0600	424,594.0354
	1,1,1,1,1,3,2,1,4,4,4,3,3,1,1,1]

presents a comparative analysis between the proposed MI-Convex approach and the TSA reported by the authors of [17].

These results show that:

• The proposed MI-Convex approach improves upon the numerical results reported by the TSA in [17], achieving an additional reduction of approximately US$ 316.99 in the total objective function. Although this reduction may seem minimal, it confirms the effectiveness of the MI-Convex approach in solving the OCS problem. The key advantage of MI-Convex is its reliability, as it consistently ensures the same numerical solution in every evaluation—which is something that the TSA cannot guarantee.
• The only difference between the TSA and MI-Convex lies in the conductor gauge assigned to the line between nodes 3 and 4 (i.e., line 3). The proposed MI-Convex approach selects gauge No. 7, whereas the TSA uses gauge No. 5. This variation results in a higher conductor investment cost for MI-Convex, totaling US$ 225,372.06, compared to the US$ 209,773.46 required by the TSA. However, as observed in the analysis of the 27-bus grid, higher investment costs lead to reduced power losses, effectively compensating for the increased expenditure and favoring the MI-Convex approach in terms of overall cost-effectiveness.
• The final solution reached with MI-Convex reports its minimum voltage profile at bus 18, with a magnitude of about 0.9629 p.u., which implies that its voltage regulation under peak load conditions is about 3.7093%. This confirms the feasibility of the final solution and an adequate voltage regulation performance.
• Regarding the loadability of all distribution branches for the 33-bus grid under peak load conditions, the maximum value occurs in the branch that connects nodes 4 and 5 (i.e., line 4), with a value of 70.0597%. This confirms the effectiveness of the proposed MI-Convex approach in providing an adequate and effective solution for the OCS problem.

The solution obtained with MI-Convex allocates 46.9206% of the total costs to energy losses, while the remaining 53.0794% corresponds to conductor investment costs. This demonstrates that, under peak load conditions, the optimal solution prioritizes investment in conductor gauges to effectively minimize the expected costs of energy losses.

3.2.2. Load Profile Evaluation

This subsection evaluates the effect of defining the subset of gauges to be installed along the distribution grid when analyzing different load profiles instead of the typical peak load operation scenario. Two cases were evaluated. The first case supposed that the expected load consumption could be divided into three main ranges, i.e., 1000, 6760, and 1000 h with power consumptions of 100%, 70%, and 40% of the peak load demand. The second case assumed a typical daily load curve composed of 24 periods in order to evaluate the effect of daily demand consumptions on the distribution network plan. These cases are listed in

Table 7. Expected daily power profile consumption.

Time (t)	Power (p.u.)	Time (t)	Power (p.u.)
1	0.6845	13	0.8706
2	0.6441	14	0.8343
3	0.6131	15	0.8165
4	0.5997	16	0.8194
5	0.5889	17	0.8741
6	0.5980	18	1.0000
7	0.6268	19	0.9836
8	0.6517	20	0.9364
9	0.7060	21	0.8876
10	0.7870	22	0.8093
11	0.8390	23	0.7459
12	0.8527	24	0.7335

.

The numerical simulations in

Table 8. Variations in the investment and operating costs of the 27-bus grid under different load operating conditions.

Load Case	Gauges	${\mathit{z}}_1$ (US$)	${\mathit{z}}_2$ (US$)	${\mathit{z}}_3$ (US$)	Min. Voltage (p.u.)
Peak load scenario	[7,7,4,4,4,3,3,1,1,4,4,2,1,	227,087.17	323,593.08	550,680.25	0.9745 (Bus 10)
	1,1,4,2,2,1,1,1,1,1,1,1,1]
Three-period case	[7,5,4,3,3,2,2,1,1,3,2,1,1,	183,788.84	220,016.58	403,805.42	0.9622 (Bus 10)
	1,1,2,1,1,1,1,1,1,1,1,1,1]
Daily operation case	[7,5,4,3,3,2,2,1,1,3,3,1,1,	212,715.20	226,873.83	439,589.03	0.9622 (Bus 10)
	1,1,3,1,1,1,1,1,1,1,1,1,1]

enable a comparative analysis of the different expected load profiles in the 27-bus grid.

The results in Table 8 allow noting that:

• Considering the load behavior has a significant impact on the final distribution system plan, as it directly influences the expected costs of energy losses. In the three-period case, the expected energy losses amount to US$ 183,788.84, representing a reduction of US$ 43,298.33 compared to the peak load scenario. This reduction leads to a decrease of approximately US$ 103,576.50 in conductor investment, as the set of gauges required is adjusted based on the expected energy losses. These variations in both components of the objective function result in a total benefit of US$ 146,874.83 compared to the peak load case.
• The daily operation case shows a slight increase in investment costs, approximately US$ 6,857.25 higher than in the three-period case. This is because the costs of energy losses in the former amount to US$ 212,715.20, which is US$ 28,926.36 more than in the latter. Consequently, larger conductor gauges are selected to mitigate the impact of the squared current flow on the total energy losses calculations. Notably, the daily operating case achieves a cost reduction of approximately 20.17% compared to the peak load scenario, highlighting its effectiveness in optimizing operating expenses.

Table 9. Variations in the investment and operating costs of the 33-bus grid under different operating conditions.

Load Case	Gauges	${\mathit{z}}_1$ (US$)	${\mathit{z}}_2$ (US$)	${\mathit{z}}_3$ (US$)	Min. Voltage (pu)
Peak load scenario	[7,7,7,5,5,4,3,2,1,1,1,1,1,1,1,1,	201,987.52	222,494.12	424,481.65	0.9629 (Bus 18)
	1,1,1,1,1,3,2,1,4,4,4,3,3,1,1,1]
Three-period case	[7,7,5,4,4,2,1,1,1,1,1,1,1,1,1,1,	136,498.40	170,937.37	307,435.77	0.9554 (Bus 18)
	1,1,1,1,1,1,1,1,3,2,2,2,2,1,1,1]
Daily operation case	[7,7,5,5,5,3,2,1,1,1,1,1,1,1,1,1,	144,208.42	189,752.39	333,960.81	0.9588 (Bus 18)
	1,1,1,1,1,2,1,1,3,3,3,2,2,1,1,1]

presents a comparative analysis of the different expected load profiles in the 33-bus grid.

The numerical results in Table 9 indicate that:

• The expected costs of energy losses in the three-period and daily operating cases are reduced by approximately 32.41% and 28.63%, respectively, in comparison the peak load scenario. As previously mentioned, this reduction can be attributed to variations in current flow across all distribution branches and to the selection of conductor gauges, which are optimized to effectively balance investment and operating costs.
• The total cost of the distribution system plan is reduced by approximately US$ 117,045.88 and US$ 90,520.84 in the three-period and daily operation scenarios when compared to the peak load case. These reductions confirm that accurate modeling of the expected demand behavior plays a crucial role in determining the investment costs of conductor gauges. When combined with the projected power losses, this directly and significantly impacts the total cost of the distribution system plan.

The results presented in Table 8 and Table 9 highlight the significant impact of accurately modeling the expected daily load behavior on the annual costs of the distribution system plan, particularly with regard to conductor gauge selection. This effect stems from the strong dependence of total energy losses costs on the projected behavior of constant power loads. Consequently, the selection of conductor gauges across the distribution network is determined by the need to balance investment costs while effectively reducing the expected energy losses.

Additionally, it is important to highlight that, while the peak load operating scenario represents unrealistic conditions for a distribution network, it serves as a reference to determine the upper bound for the costs of conductor investment and energy losses. Conversely, the three-period case reports the lowest costs for these factors. However, the most realistic scenario, i.e., the daily operating period, provides an optimal balance between the expected costs of energy losses and the total conductor investment, making it the most efficient approach for determining the appropriate subset of gauges to be installed. This methodology allows distribution companies to strategically allocate budgets based on projected operating costs, ensuring both economic efficiency and reliable network performance.

As for the voltage profiles, in all cases evaluated for the 27- and 33-bus grids, the minimum voltages remain within acceptable operational limits, exceeding 0.90 p.u. even under the three-period and daily load variation scenarios. Although slight reductions in voltage levels are noted when moving from peak load to more realistic load conditions, the final voltage profiles confirm that the selected conductor sets ensure an adequate voltage regulation across the network, guaranteeing the quality and reliability of the service.

3.3. Large-Scale Distribution Network

To validate the effectiveness of the proposed optimization methodology in managing large-scale distribution networks under daily demand fluctuations and renewable energy integration, this subsection presents the numerical results obtained using the balanced version of a 85-bus system. Figure 4 illustrates the single-line diagram of this feeder [32]. It should be highlighted that the PV generator is installed at node 34, providing a nominal generation capacity of approximately 750 kW per phase. Similarly, the wind turbine is located at node 60, offering a total generation capacity of 600 kW.

The line connections, along with their corresponding lengths and the active and reactive power demands per node, are presented in

Table 10. Line configuration and lengths for the 85-bus grid.

l	k	m	${\mathit{L}}_{\mathit{l}}$ (km)	${\mathit{P}}_{\mathit{mt}}^{\mathit{d}}$ (kW)	${\mathit{Q}}_{\mathit{mt}}^{\mathit{d}}$ (kvar)	l	k	m	${\mathit{L}}_{\mathit{l}}$ (km)	${\mathit{P}}_{\mathit{mt}}^{\mathit{d}}$ (kW)	${\mathit{Q}}_{\mathit{mt}}^{\mathit{d}}$ (kvar)
1	1	2	0.468	0.000	0.000	43	34	44	0.985	35.28	35.99
2	2	3	0.449	0.000	0.000	44	44	45	0.441	35.28	35.99
3	3	4	0.323	56	57.13	45	45	46	0.225	35.28	35.99
4	4	5	0.870	0.000	0.000	46	46	47	0.184	14	14.28
5	5	6	0.883	35.28	35.99	47	35	48	0.698	0.000	0.000
6	6	7	0.343	0.000	0.000	48	48	49	0.875	0.000	0.000
7	7	8	0.953	35.28	35.99	49	49	50	0.768	36.28	37.01
8	8	9	0.309	0.000	0.000	50	50	51	0.261	56	57.13
9	9	10	0.931	0.000	0.000	51	48	52	0.985	0.000	0.000
10	10	11	0.161	56	57.13	52	52	53	0.448	35.28	35.99
11	11	12	0.952	0.000	0.000	53	53	54	0.344	56	57.13
12	12	13	0.878	0.000	0.000	54	52	55	0.673	56	57.13
13	13	14	0.507	35.28	35.99	55	49	56	0.564	14	14.28
14	14	15	0.018	35.28	35.99	56	9	57	0.545	56	57.13
15	2	16	0.778	35.28	35.99	7	57	58	0.564	0.000	0.000
16	3	17	0.359	112	114.26	58	58	59	0.650	56	57.13
17	5	18	0.848	56	57.13	59	58	60	0.237	56	57.13
18	18	19	0.433	56	57.13	60	60	61	0.626	56	57.13
19	19	20	0.605	35.28	35.99	61	61	62	0.530	56	57.13
20	20	21	0.377	35.28	35.99	62	60	63	0.514	14	14.28
21	21	22	0.721	35.28	35.99	63	63	64	0.701	0.000	0.000
22	19	23	0.316	56	57.13	64	64	65	0.373	0.000	0.000
23	7	4	0.949	35.28	35.99	65	65	66	0.614	56	57.13
24	8	5	0.849	35.28	35.99	66	64	67	0.307	0.000	0.000
25	25	26	0.775	56	57.13	67	67	68	0.322	0.000	0.000
26	26	27	0.299	0.000	0.000	68	68	69	0.662	56	57.13
27	27	28	0.872	56	57.13	69	69	70	0.150	0.000	0.000
28	28	29	0.375	0.000	0.000	70	70	71	0.384	35.28	35.99
29	29	30	0.798	35.28	35.99	71	67	72	0.320	56	57.13
30	30	31	0.816	35.28	35.99	72	68	73	0.291	0.000	0.000
31	31	32	0.999	0.000	0.000	73	73	74	0.213	56	57.13
32	32	33	0.045	14	14.28	74	73	75	0.721	35.28	35.99
33	33	34	0.236	0.000	0.000	75	70	76	0.705	56	57.13
34	34	35	0.430	0.000	0.000	76	65	77	0.715	14	14.28
35	35	36	0.847	35.28	35.99	77	10	78	0.361	56	57.13
36	26	37	0.386	56	57.13	78	67	79	0.546	35.28	35.99
37	27	38	0.122	56	57.13	79	12	80	0.357	56	57.13
38	29	39	0.915	56	57.13	80	80	81	0.940	0.000	0.000
39	32	40	0.182	35.28	35.99	81	81	82	0.444	56	57.13
40	40	41	0.590	0.000	0.000	82	81	83	0.110	35.28	35.99
41	41	42	0.812	35.28	35.99	83	83	84	0.865	14	14.28
42	41	43	0.261	35.28	35.99	84	13	85	0.300	35.28	35.99

, while the expected daily renewable generation and demand behaviors are listed in

Table 11. Daily demand and generation variation for the 85-bus grid.

Time (h)	Demand (pu)	Photovoltaic (pu)	Wind (pu)
1	0.684511335492475	0	0.633118295
2	0.644122690036197	0	0.607259323
3	0.613069156029720	0	0.605557422
4	0.599733282530006	0	0.684246423
5	0.588874071251667	0	0.783719339
6	0.598018670222900	0	0.790557706
7	0.626786054486569	0	0.744958950
8	0.651743189178891	0.0391233650	0.769603567
9	0.706039245570585	0.0655871790	0.826492212
10	0.787007048961707	0.2368707960	0.876523598
11	0.839016955610593	0.4550178180	0.931213527
12	0.852733854067441	0.7264402650	0.965504834
13	0.870642027052772	0.9244863260	0.972218577
14	0.834254143646409	0.9820411530	0.981135531
15	0.816536483139646	0.8294070790	0.991393173
16	0.819394170318156	0.7330632950	1
17	0.874071251666984	0.5011338490	0.987258076
18	1	0.1771175180	0.929542167
19	0.983615926843208	0	0.791155379
20	0.936368832158506	0	0.708839248
21	0.887597637645266	0	0.712881960
22	0.809297008954087	0	0.719897641
23	0.745856353591160	0	0.703007456
24	0.733473042484283	0	0.687238555

.

To assess the performance of the proposed MI-Convex optimization approach for OCS in the 85-bus grid, four distinct numerical test cases were considered:

• Case 1: An analysis of the optimization framework under peak load conditions, maintaining the same simulation parameters as those used for the 27- and 33-bus grids.
• Case 2: Conductor selection based exclusively on the three-period demand profile.
• Case 3: Conductor selection considering daily load variations, without accounting for the integration of renewable generation sources.
• Case 4: Comprehensive evaluation incorporating both daily demand variations and renewable generation profiles.

The numerical evaluation of each of these simulation cases is presented in

Table 12. Variations in the investment and operating costs of the 85-bus grid under different operating conditions.

Load Case	Gauges	${\mathit{z}}_1$ (US$)	${\mathit{z}}_2$ (US$)	${\mathit{z}}_3$ (US$)	Min. Voltage (pu)
Case 1	[7,7,7,7,7,7,7,4,1,1,1,1,1,1,1,1,1,1,1,1,1,	325,339.96	590,252.78	915,592.73	0.9414 (Bus 54)
	1,1,4,4,4,3,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,3,3,1,2,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
Case 2	[7,7,7,6,5,5,5,4,1,1,1,1,1,1,1,1,1,1,1,1,1,	305,047.05	434,737.18	739,784.23	0.9036 (Bus 54)
	1,1,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
Case 3	[7,7,7,7,5,5,5,4,1,1,1,1,1,1,1,1,1,1,1,1,1,	319,327.95	467,893.31	787,221.25	0.9136 (Bus 54)
	1,1,3,3,3,2,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
Case 4	[7,6,5,5,5,5,5,3,1,1,1,1,1,1,1,1,1,1,1,1,1,	314,328.96	390,868.10	705,197.06	0.9008 (Bus 54)
	1,1,3,3,2,2,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,
	1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]

.

The numerical results for the 85-bus grid under different operating conditions reveal key insights into the impact of considering realistic load profiles and renewable generation integration on OCS and the total planning cost (see the results in Table 12):

• Under peak load conditions, the network is large enough to simultaneously withstand the maximum possible demand across all nodes. This results in the highest total cost (US$ 915,592.73), with US$ 590,252.78 allocated to conductor investment and US$ 325,339.96 to energy losses. The network is heavily oversized because it must account for an extreme – though rare – operational situation, leading to overinvestment in conductor sizes and underutilization during most operating hours.
• When the load profile is segmented into three typical periods (high, medium, and low demand), the total cost is substantially reduced to US$ 739,784.23. Both conductor investment (US$ 434,737.18) and energy losses costs (US$ 305,047.05) decrease compared to the peak load case. This demonstrates that planning based on more realistic demand scenarios prevents unnecessary oversizing, leading to significant cost savings of approximately 19.21%.
• Considering a detailed daily load variation without renewable integration yields a total cost of US$ 787,221.25. Although slightly higher than case 2, this case captures daily consumption fluctuations more precisely, leading to a balanced investment in conductors (US$ 467,893.31) and operational losses reduction (US$ 319,327.95). This approach better reflects the network’s true usage in spite of demand peaks throughout the day.
• The most significant impact is observed when renewable generation is incorporated. The total system cost drops to US$ 705,197.06, even lower than case 3, despite an increase in the costs of energy losses (US$ 314,328.96). This decrease is mainly due to the substantial reduction in conductor investment (US$ 390,868.10)—the lowest among all cases. The presence of DG reduces the net power demand during the day, alleviating current flows across the network and enabling the selection of smaller conductors without compromising system reliability. Thus, renewables minimize infrastructure investment, achieving a cost reduction of about 22.97% compared to the traditional peak-load design.

The numerical results obtained for the 85-bus grid show that dimensioning electrical distribution networks based solely on peak load conditions leads to oversized infrastructures and economically inefficient designs. By contrast, incorporating time-dependent load profiles, such as three-period or daily variations, enables a more accurate matching between conductor sizes and the expected current flows, significantly reducing planning costs. Moreover, the integration of renewable energy sources provides dual benefits: it alleviates demand stress on the system and further lowers infrastructure investments, despite a slight increase in energy losses. Overall, case 4 underscores the economic and operational advantages of distributed energy resources, demonstrating that modern distribution networks should integrate renewables, not only to achieve sustainability goals but also to achieve greater cost efficiency.

Regarding the voltage profiles of the 85-bus system, the minimum voltages across all simulated cases remain above 0.90 p.u., even under varying operating conditions and in the face of renewables integration. Although slight voltage drops are noted when moving from peak load (0.9414 p.u.) to more realistic load and generation scenarios (lowest at 0.9008 p.u. for bus 54), the final profiles remain within acceptable operational standards for distribution networks. This confirms that the proposed conductor selection methodology not only provides cost-effective solutions but also maintains satisfactory voltage regulation across large-scale systems.

3.4. Computational Performance Characteristics

To demonstrate the effectiveness and robustness of the proposed MI-Convex optimization approach to solve the OCS problem in balanced distribution networks,

Table 13. Numerical performance of the MI-Convex model implemented in Julia.

Scenario	Proc. Time (s)	Binary Variables	Total Variables
27-bus grid
Peak scenario	1.8389	208	366
Three-period case	2.4659	208	578
Daily operation case	9.6989	208	2804
33-bus grid
Peak scenario	2.1530	256	450
Three-period case	4.9890	256	710
Daily operation case	7.1860	256	3440
85-bus grid
Peak scenario	12.2730	672	1178
Three-period case	19.0600	672	1854
Daily operation case	765.2999	672	8952
Daily operation case with DGs	724.6180	672	8952

presents the average processing times, the number of integer variables (i.e., binary variables), and the total number of variables employed.

The numerical results in Table 13 show that:

• The MI-Convex model shows an expected increase in processing times and the number of total variables as the network size grows (from 27 to 85 buses). Despite the significant rise in variables, especially for the daily operation cases, the solver remains capable of handling large problem instances within reasonable times, particularly for small- and medium-sized grids.
• As the complexity of the scenario increases (from peak load to three-period to daily operation), both the total number of variables and the processing times grow substantially. For example, in the 85-bus grid, the daily operation case requires over 765 s, compared to just 12 s for the peak load scenario, highlighting the computational impact of multi-period modeling.
• The inclusion of DGs in the daily operation case of the 85-bus grid results in a slight reduction in processing times (724.6 s vs. 765.3 s), despite using the same number of variables. This suggests that DG integration can potentially introduce structural changes in the optimization model that favor faster convergence under certain conditions.

4. Conclusions

This paper proposed a MI-Convex optimization approach for solving the OCS problem in radial distribution networks operating under balanced conditions. Comparative analyses with state-of-the-art metaheuristic techniques, such as the VSA, the NMA, the GNDO, and the TSA, demonstrated the superiority of our proposal in terms of solution quality, reproducibility, and computational efficiency. Numerical validations showed that MI-Convex consistently achieved cost-effective solutions, matching or outperforming the best solutions reported in the literature while ensuring global optimality and systematic convergence.

A sensitivity analysis considering different load profiles revealed that OCS is highly dependent on the modeling of the expected demand. In particular, considering realistic load variations (e.g., three-period or daily profiles) led to significant cost reductions compared to traditional peak-load planning. For the 27-bus grid, the daily operation profile achieved a total cost reduction of approximately 20.17%, while the 33-bus grid reported a decrease of 27.61%. These results highlight the critical importance of realistic load modeling to avoid network oversizing, reduce unnecessary investments, and improve operational efficiency.

Furthermore, a case study on a 85-bus grid underscored the advantages of integrating renewable energy sources (PV and wind generation). Incorporating distributed generation not only reduced the infrastructure investment required but also mitigated demand stress on the network, resulting in a total cost reduction of nearly 22.97% compared to peak-load designs. Across all cases, the MI-Convex method ensured that voltage profiles remained within acceptable operational limits, guaranteeing technical feasibility alongside economic benefits.

In summary, the proposed MI-Convex methodology offers a robust, globally optimal, and computationally efficient tool for planning electrical grids, providing distribution companies with a strategic means to optimize conductor sizing, minimize total costs, and enhance network reliability under dynamic operating conditions.

Future research will aim to further enhance the proposed MI-Convex optimization framework by addressing several key aspects. First, the incorporation of uncertainty modeling for renewable energy generation, demand-side variations, and component failures will be explored to improve the robustness and resilience of the network planning process. Second, the integration of dynamic reconfiguration strategies will be investigated, allowing the system to adapt to real-time operating conditions and load fluctuations. Third, the methodology will be extended to more complex network configurations, including three-phase unbalanced and meshed distribution systems, in order to broaden its applicability to practical urban and semi-urban environments. Additionally, the inclusion of multi-source supply schemes and the analysis of dynamic operational constraints such as switching operations are envisioned to further increase the practical relevance and scalability of the approach. Collectively, these advancements aim to develop more flexible, sustainable, and cost-effective distribution networks aligned with the evolving demands of modern power systems.