Simultaneous Feeder Routing and Conductor Selection in Rural Distribution Networks Using an Exact MINLP Approach

Highlights

What are the main findings?

• A novel MINLP model was developed to simultaneously solve the feeder routing and conductor sizing problem in rural distribution networks, ensuring radial topology and minimizing total system costs.
• Compared to metaheuristic methods based on the minimum spanning tree and tabu search, the proposed approach achieved a reduction in total planning costs of up to 65.28% and energy loss costs of over 61% in the 25-node test feeder.

What is the implication of the main finding?

• The integration of route and conductor selection into a unified MINLP framework enables globally optimal and long-term cost-effective solutions for rural electrification.
• This model provides a scalable and reliable methodology for utility companies and planners aiming to expand distribution networks under geographical, economic, and technical constraints.

Abstract

This article addresses the optimal network expansion problem in rural distribution systems using a mixed-integer nonlinear programming (MINLP) model that simultaneously performs route selection and conductor sizing in radial distribution systems. The proposed methodology was validated on 9- and 25-node test systems, comparing the results against approaches based on the minimum spanning tree (MST) formulation and metaheuristic approaches (the sine-cosine and tabu search algorithms). The MINLP model significantly reduced the total costs. For the nine-node system, the total cost decreased from USD 131,819.33 (MST-TSA) to USD 77,129.34 (MINLP), saving USD 54,689.99 (41.48%). Similarly, the costs of energy losses dropped from USD 111,746.73 to USD 63,764.12, a reduction of USD 47,982.61 (42.94%). In the 25-node system, the total costs fell by over 65% from USD 371,516.59 to USD 128,974.72, while the costs of energy losses decreased by USD 210,057.16 (61.06%). Despite requiring a higher initial investment in conductors, the MINLP model led to substantial long-term savings due to reduced operating costs. Unlike previous methods which separate network topology design and conductor sizing, our proposal integrates both aspects, ensuring globally optimal solutions. The results demonstrate its scalability and effectiveness for long-term distribution planning in complex power networks. The experimental implementation was carried out in Julia (v1.10.2) using JuMP (v1.21.1) and BONMIN.

1. Introduction

1.1. General Context

The design of electrical distribution networks for rural applications plays a fundamental role in fostering economic development, improving living standards, and ensuring access to essential services in remote areas [1,2]. Rural electrification is a key challenge faced by utilities and governments worldwide, as it involves extending the power infrastructure to geographically dispersed communities with low population densities [3,4]. Unlike urban networks, rural distribution systems must be optimized to minimize investment costs while ensuring reliability and efficiency, all while considering the higher per-unit cost of infrastructure deployment over long distances [5,6].

The push to expand electrical networks for remote users is driven by the need to provide equitable energy access, support agricultural and industrial activities, and promote socioeconomic growth in under-served regions [7]. However, the complexities associated with terrain constraints, environmental impact, and financial limitations require advanced planning methodologies that integrate optimization techniques to design cost-effective and technically feasible solutions [8].

1.2. Motivation

Due to the geographical dispersion of rural communities, selecting an optimal network topology is a complex decision-making process that must consider multiple technical, economic, and environmental factors [9]. Figure 1 highlights the critical role of extending electrical energy services to rural areas, demonstrating its impact on development and social integration. Access to electricity facilitates essential services such as education, healthcare, and communication, significantly improving the quality of life and fostering economic growth in these communities [10].

The design and optimization of distribution networks for rural areas require advanced mathematical models that can effectively determine the optimal feeder routes to guarantee an energy supply for all users [11]. Traditional network planning approaches often rely on predefined layouts or heuristic methods, which may not ensure the most cost-effective or technically feasible solution [1]. By formulating optimization models that explicitly incorporate the selection of distribution routes, it becomes possible to minimize infrastructure costs while satisfying electrical demand constraints. These models leverage mixed-integer nonlinear programming (MINLP) techniques to simultaneously determine the best conductor allocation and feeder routing, ensuring efficient energy delivery while reducing the impact of geographical barriers and challenging terrain conditions [12,13].

1.3. Literature Review

The optimal planning of distribution systems, encompassing both feeder routing and conductor assignment, has been extensively studied in the specialized literature. This subsection discusses some of the most relevant research contributions in the field.

The authors of [14] explored the integration of radiality constraints into optimization models for distribution systems, emphasizing their necessity in planning and operational models due to the radial nature of these networks. Their study provided a literature review, a critical analysis, and a systematic approach to incorporating these constraints into mathematical formulations. This research demonstrated that radiality constraints can be efficiently formulated and applied to network reconfiguration and expansion planning while ensuring the required topology. Furthermore, it generalizes these constraints, extending their applicability to various optimization problems in power distribution systems.

In [15], the authors analyzed the expansion of power distribution systems driven by increasing electricity demand in urban and industrial sectors. They proposed a multi-criteria decision-making approach using the CRITIC method to optimize network design by evaluating voltage profiles, power losses, current levels, and conductor costs. The methodology was validated on the IEEE 34-bus system using Matpower in Matlab, generating a decision matrix with 210 alternatives. The results demonstrated that the proposed method effectively balances quality, efficiency, and cost in distribution network planning.

Another study [1] addressed the optimal expansion of AC medium-voltage distribution grids for rural applications using a heuristic approach. Feeder routes were selected via a minimum spanning tree (MST) formulation, while conductor calibers were initially assigned based on the maximum expected load current. A tabu search algorithm (TSA) refined the solution using three-phase power flow simulations under varying load conditions. Numerical tests on 9- and 25-node feeders showed that the proposed method effectively reduced planning costs, with the heuristic conductor selection covering at least 70% of the final optimal calibers, enhancing the TSA’s performance.

In [8], a constructive heuristic algorithm was proposed to solve the complex mixed binary nonlinear programming problem of distribution system planning. This approach incorporates a local improvement phase and a branching technique to enhance solution quality. A sensitivity index, derived from a relaxed distribution system planning formulation, guides the addition of circuits or substations. Numerical tests on two benchmark systems and a real distribution network demonstrated the effectiveness of the proposed method.

The authors of [16] developed a bi-objective optimization model for distribution network expansion that integrates system reconfiguration and operational reliability. Using the non-dominated sorting genetic algorithm II and heuristic techniques, this approach efficiently solved the embedded optimization subproblem. Tested on IEEE 33- and 70-bus systems, it achieved over 200 times faster computation than existing methods.

In [17], the authors analyzed the distribution system reconfiguration problem, adjusting interconnection switches to optimize technical-economic benefits, such as loss minimization. A comparative assessment of classical optimization models and metaheuristic approaches was conducted using standardized metrics to eliminate implementation and hardware discrepancies. The results on the test systems (comprising 33, 136, and 417 buses) revealed that linear and conic programming models efficiently solved the problem in small-to-medium networks, while metaheuristics outperformed classical methods in large-scale systems due to their lower computational effort.

The authors of [18] studied the impact of reliability considerations on the configuration and planning costs of radial networks. A direct search technique was implemented to minimize planning costs while leveraging the principle of optimality to enhance computational efficiency by reducing the total number of radial paths. Reliability indices were calculated to assess different feeder configurations, and the methodology was tested for optimal feeder routing with varying numbers of substations. This study provides insights into the trade-off between optimality and reliability in complex feeder configurations, demonstrating the effectiveness of the proposed approach for multi-substation distribution networks.

In [19], the authors presented a novel dynamic programming approach for the multi-objective planning of electrical distribution systems. The proposed method simultaneously optimizes feeder routes and branch conductor sizes by minimizing two key objectives: (i) installation and operating costs and (ii) interruption costs. The first objective accounts for new feeder and substation installation costs, maintenance expenses, and energy losses, while the second evaluates network reliability through non-delivered energy, repair, and customer damage costs. A dynamic programming-based algorithm was developed to generate a set of Pareto-optimal solutions via weighted aggregation. The approach was validated on 21-, 54-, and 100-node systems, and a comparative analysis against multi-objective evolutionary algorithms was conducted, demonstrating the proposal’s effectiveness.

The authors of [20] developed a simple algorithm for distribution system planning in large areas, focusing on optimal substation placement and feeder routing. The main objective was to determine the optimal substation location by minimizing costs while ensuring a radial system structure. The study also incorporated feeder routing with four types of conductors in order to enhance cost efficiency. Unlike traditional approaches that use a single conductor type, this method selects different conductors for various segments of the network to further reduce costs. The proposed approach ensures an optimal balance between investment and operational efficiency in distribution system expansion.

The authors of [21] presented a comprehensive methodology for rural electrification based on open data, georeferencing, and a mixed-integer linear programming model that minimizes the net present cost of the electrical system while considering investment, operation, maintenance, and residual value. The proposed approach is structured into four stages: demand estimation, hybrid microgrid sizing, internal network design, and integrated area-level optimization. It was validated in Butha-Buthe (Lesotho), where 72 communities were electrified, achieving 100% renewable energy solutions in more than 50% of the cases. However, the linearization of the formulation limits the ability to accurately represent nonlinear phenomena, and the sequential network design approach may restrict the identification of global optima.

The research in [22] proposed a strategy for rural electrification planning using a geographic information system-based approach, graph theory, and topographic analysis, with the goal of generating realistic electrical network topologies. The objective function aimed to minimize the total investment cost of the grid while considering both line routing and conductor selection under capacity constraints. The methodology was implemented in the GISEle tool, which combines population density analysis via clustering, Steiner tree generation, and balanced power flow analysis. The model was validated through a case study in Cavalcante (Brazil), achieving a 47% reduction in deployment costs compared with solutions based solely on MSTs. However, by structuring the optimization process in sequential stages (first topology design and then electrical component allocation), the model may limit the search for global optima, particularly in scenarios where routing and electrical sizing decisions are strongly interdependent.

On the other hand, the authors of [23] proposed a mixed-integer conic programming model for routing in balanced distribution systems, which was solved using the AMLP solver in CPLEX. This study focused on minimizing the system’s daily energy losses. Simulations on test systems with up to 203 nodes demonstrated the model’s effectiveness in reducing losses. Moreover, comparisons with other techniques highlighted the robustness and efficiency of the proposed approach in terms of solution quality and processing times.

For their part, the authors of [24] proposed a methodology based on the Harris hawks optimizer for distribution system routing, which focused on minimizing power losses under specific load conditions while maintaining a radial configuration. Validated on balanced systems with 33, 85, and 295 nodes, the strategy demonstrated efficiency compared with other metaheuristic techniques. The authors also analyzed computational times, although they did not perform a statistical analysis to assess the repeatability of the method.

In [25], the routing problem for distribution systems was addressed using a genetic algorithm, ensuring the feasibility of individuals through a heuristic stage that guaranteed a radial configuration. The objective was to minimize both energy losses and the line loading index. This methodology was validated on systems with 33, 1760, and 4400 nodes, demonstrating efficiency and applicability. Although the results were compared against those of other methods and computational times were analyzed, the authors did not include a statistical analysis to ensure repeatability of the approach.

In this literature review, the following key aspects were observed:

1.

Multiple studies underscore the importance of preserving a radial network structure in optimization models for distribution planning, as it provides both computational tractability and operational feasibility.

2.

Techniques such as multi-criteria decision making, dynamic programming, and metaheuristic methods have been proven to be effective in balancing cost, reliability, and operational efficiency for network expansion and reconfiguration problems.

3.

Heuristic algorithms, spanning tree formulations, and sensitivity-driven procedures are frequently employed for feeder routing and conductor selection, enabling cost reductions while enhancing network performance across a range of applications.

1.4. Novelty and Contributions

While the methodologies presented in the previous subsection have advanced the state of knowledge in distribution network planning, several methodological gaps remain. In particular, many approaches treat network topology design and conductor sizing as decoupled subproblems, which can lead to suboptimal configurations when the interdependence between routing decisions and electrical parameters is not fully captured. Moreover, although some studies incorporated operational factors such as reliability or load variation, they often relied on linear or piecewise-linear models, which cannot accurately represent the nonlinear nature of power flow equations, especially in medium- and low-voltage networks.

Another critical limitation is the reliance on heuristics and metaheuristics which, while useful for large-scale problems, do not guarantee global optimality and are often sensitive to parameter tuning and the initial conditions. In practice, these techniques may overlook high-quality solutions in highly combinatorial decision spaces, particularly when dealing with unbalanced, multi-phase rural distribution systems and diverse conductor options.

As a result, there is a need for a more comprehensive approach that integrates expansion planning, conductor assignment, and operational constraints while ensuring optimality and scalability for rural distribution networks.

In light of these observations, this study makes the following key contributions:

1.

A novel and comprehensive MINLP formulation is proposed to simultaneously solve the distribution feeder routing and conductor sizing problems in rural distribution networks. Unlike existing methods, which decouple these stages or rely on approximations, the proposed model accurately captures the nonlinearities of power flows and preserves the radial topology, providing a mathematically rigorous and integrated optimization framework.

2.

The proposed approach demonstrates superior performance in comparison with state-of-the-art heuristics, particularly the tabu search algorithm combined with MST topology generation [1]. According to numerical experiments conducted on 9- and 25-node test systems with 14 and 42 candidate lines as well as seven available conductor types, the MINLP model achieved substantial reductions regarding total planning costs and energy losses over a 20-year horizon.

3.

In addition to delivering globally optimal solutions, the methodology allows for a detailed exploration of the trade-offs between investment and operating costs, providing planners with a long-term perspective that is critical for infrastructure development in rural and under-resourced areas. The use of exact optimization ensures transparency, repeatability, and robustness, which are essential in academic and practical applications alike.

To efficiently solve the formulated MINLP problem, this research employs an exact optimization approach that integrates the branch-and-bound method with an interior-point solver. This implementation leverages state-of-the-art optimization tools available in Julia software through the JuMP optimization environment, ensuring computational efficiency and robustness in solving problems that involve large-scale distribution networks.

Within the scope of this research, it is essential to highlight three key considerations. First, the distribution planning problem for rural networks is formulated as an optimization problem with a single distribution substation that is solely responsible for meeting the energy demands of all end users in the network. Second, the optimization process is conducted under peak load operating conditions, representing the most critical scenario, where all of the selected conductor sizes must safely accommodate the maximum current demands while maintaining their conduction properties and ensuring reliable operation. Finally, the set of possible distribution feeder routes is predefined by the distribution company based on a comprehensive feasibility study that considers the geographical constraints of the rural areas benefiting from the network expansion.

1.5. Document Organization

The remainder of this document is organized as follows. Initially, Section 2 introduces the proposed MINLP formulation, which models the simultaneous optimization of feeder route selection and conductor assignment for rural electricity distribution networks. Subsequently, the solution strategy in Section 2.3 integrates the branch-and-bound approach with an interior-point optimizer while leveraging the Julia optimization framework. Section 3 describes the characteristics of the test feeders used for numerical validation, which consist of 9 and 25 nodes, with 14 and 42 distributing branches, respectively. Section 4 presents the numerical results obtained, including comparative analyses with existing metaheuristic approaches, demonstrating the effectiveness of the proposed MINLP formulation in obtaining high-quality solutions for simultaneous feeder routing and conductor assignment. Finally, Section 5 describes the most important conclusions drawn from this work, as well as some possible future developments.

2. Proposed Methodology

This section presents the solution methodology adopted for the optimal planning of rural distribution grids. First, a comprehensive MINLP formulation that simultaneously addresses the selection of feeder routes and conductor sizes is introduced, which captures both the discrete and nonlinear nature of the problem. The model incorporates power flow equations, technical constraints, and cost-related objectives over a long-term planning horizon. Subsequently, the solution strategy used to tackle the proposed MINLP problem is described, which relies on a branch-and-bound framework coupled with an interior-point method for handling nonlinear subproblems. The implementation of this methodology was carried out in the Julia programming language using the JuMP modeling environment and the BONMIN solver.

2.1. Mathematical Formulation

The primary objective of optimal distribution network expansion is to determine the most suitable routes for ensuring the efficient supply of system loads, as well as to select the optimal conductor sizes for each of the chosen routes [26]. Furthermore, the problem regarding optimal route selection in electrical distribution systems aims to define a new radial configuration that minimizes energy losses and improves operational efficiency [26,27]. This problem belongs to the family of MINLP models, as it employs a binary vector to indicate whether or not a distribution line should be constructed. On the other hand, conductor selection in distribution networks is an optimization problem which focuses on minimizing investment and operating costs [28]. This problem is also commonly represented through an MINLP formulation, as the decision variable consists of a binary vector indicating whether a conductor of a specific caliber should be selected.

The objective function and the set of constraints of the optimization model representing the distribution network expansion problem are described below.

2.1.1. Objective Function

The objective function must balance economic and technical aspects to minimize the costs associated with conductor investment and power losses over one year of operation. The objective function used in this work is as follows:

$$min{Z}_{\mathrm{cost}}=\alpha \left(\beta Z_{\mathrm{loss}}+Z_{\mathrm{inv}}\right),$$

(1)

with

$$\begin{array}{c}\hfill \alpha =\left({\displaystyle \frac{t_a}{1-{(1+t_a)}^{-N_t}}}\right),\end{array}$$

(2)

$$\begin{array}{c}\hfill \beta =\left(\sum _{t\in \mathcal{T}}{\left({\displaystyle \frac{1+t_e}{1+t_a}}\right)}^t\right),\end{array}$$

(3)

where $Z_{\mathrm{cost}}$ represents the system’s annual operating costs, calculated by summing the energy losses ($Z_{\mathrm{loss}}$) and conductor investment costs ($Z_{\mathrm{inv}}$). The components of the objective function are defined in Equations (4) and (5). The parameter $\alpha$ is the factor used to annualize the network operator’s energy losses and conductor investment costs, while $\beta$ is the factor that accounts for energy price growth over time. Meanwhile, $i_a$ represents the internal rate of return, $N_t$ is the planning horizon in years, and $i_e$ corresponds to the annual increase in energy costs. $Z_{\mathrm{loss}}$ and $Z_{\mathrm{inv}}$ are calculated as follows:

$$Z_{\mathrm{loss}}=C_{kWh}T\mathrm{Re}\left\{\sum _{l\in \mathbf{L}}{\mathbb{V}}_l^{\top }{\mathbb{J}}_l^{★}\right\},$$

(4)

$$Z_{\mathrm{inv}}=\sum _{l\in \mathbf{L}}\left(\sum _{c\in \mathbf{C}}{C}_c{\mathbf{Y}}_{lc}\right)L_l,$$

(5)

where $C_{kwh}$ is the average cost of energy; T is the number of hours in a year (i.e., 8760); ${\mathbb{V}}_l$ is the voltage drop across the distribution line l; ${\mathbb{J}}_l$ is the current flowing through the distribution line l; $C_c$ is the cost of installing one kilometer of type c conductors; $L_l$ is the length of the distribution line l; ${\mathbf{Y}}_{lc}$ is the binary variable entrusted with selecting the type c conductor to be installed in the distribution line l; and the sets $\mathbf{L}$ and $\mathbf{C}$ are associated with the lines available for construction and the conductor sizes to be installed, respectively.

2.1.2. Set of Constraints

The set of constraints modeling the optimal expansion problem for distribution systems must comply with Kirchhoff’s laws and the voltage regulation and thermal limits of the distribution lines. The set of constraints representing the problem is presented below.

2.1.3. Current Balance at the System Nodes

The equality constraint in Equation (6) represents the current balance at node k in the distribution system:

$${\mathbb{I}}_k^g-{\mathbb{I}}_k^d=\sum _{l\in \mathbf{L}}{A}_{kl}{\mathbb{J}}_l,\left\{k\in \mathbf{N}\right\},$$

(6)

where ${\mathbb{I}}_k^g$ represents the current injected by a substation connected at bus k; ${\mathbb{I}}_k^d$ corresponds to the current consumption at bus k; $A_{kl}$ is the position of the node-to-branch incidence matrix relating node k to line l; and N denotes the set containing the nodes in the distribution system.

The constraint in Equation (6) is a direct application of Kirchhoff’s current law for each node k. Specifically, the current entering the node (i.e., the sum of generation and imports through the lines) minus the current leaving the node (i.e., the sum of demand and exports through the lines) must be equal to zero. Alternatively, this equation can be interpreted as a nodal flow balance, where all generation and consumption are offset by the incoming or outgoing current flow within the network.

2.1.4. Voltage Drop Across the Distribution Lines

The equality constraint in Equation (7) represents the voltage drop in a distribution line l that interconnects nodes k and j:

$${\mathbb{V}}_l={\mathbf{X}}_l\sum _{k\in \mathbf{N}}{A}_{kl}{\mathbb{V}}_k=L_l\sum _{c\in \mathbf{C}}{\mathbf{Y}}_{lc}{\mathbb{Z}}_c{\mathbb{J}}_l,\left\{l\in \mathbf{J}\right\},$$

(7)

where ${\mathbb{V}}_k$ is the voltage at node k; ${\mathbf{X}}_l$ is the binary variable that determines whether a distribution line l is constructed; and ${\mathbb{Z}}_c$ is the impedance of the conductor size c selected for line l.

The constraint in Equation (7) ensures electrical consistency in the voltage drop of line l, establishing that it can be determined in two equivalent ways:

• Nodal voltage difference: The voltage drop in line l is equal to the voltage difference between nodes k and j, which the line interconnects (left-hand side of the equation).
• Application of Ohm’s law: The voltage drop can also be expressed as the product of the impedance of line l and the current flowing through it (right-hand side of the equation).

From a modeling perspective, the constraint in Equation (7) incorporates the binary variable ${\mathbf{X}}_l$, which indicates whether a line l is constructed or not. When ${\mathbf{X}}_l=1$, line l is installed, and the voltage drop equation holds, which allows expressing ${\mathbb{V}}_l$ either as the nodal voltage difference or as the product of the effective impedance and current. When ${\mathbf{X}}_l=0$, line l is not constructed, which is modeled by setting the voltage drop ${\mathbb{V}}_l$ to zero, effectively nullifying any current flow since the connection does not physically exist. Additionally, the binary variable ${\mathbf{Y}}_{lc}$ determines the conductor size c to be installed on line l, which is selected by ${\mathbf{X}}_l$ while ensuring compliance with the voltage drop and thermal limit requirements.

2.1.5. Load Current at the Demand Nodes

The equality constraint in Equation (8) allows calculating the current demanded by the system loads. In this case, constant power loads are assumed, making it possible to use Tellegen’s second theorem to relate the complex power at node k to its voltage and current:

$${\mathbb{S}}_k^{d,\ast }={\mathbb{V}}_k^{★}{\mathbb{I}}_k^d,\left\{k\in \mathbf{N}\right\},$$

(8)

where ${\mathbb{S}}_k^d$ is the complex power demanded at node k by a load, which is a known parameter, and ${\left(·\right)}^{★}$ denotes the complex conjugate of the argument.

2.1.6. Voltage Regulation

The constraint in Equation (9) represents the voltage regulation in the distribution system. The voltage magnitude at node k must remain within predefined limits set by the network operator:

$$V^{min}\le \left|{\mathbb{V}}_k\right|\le V^{max},\left\{k\in \mathbf{N}\right\},$$

(9)

where $V^{min}$ and $V^{max}$ represent the minimum and maximum voltage regulation limits established by the network operator’s regulatory policies.

2.1.7. Thermal Limit of the Conductors

The constraint in Equation (10) states that the magnitude of the current flowing through line l must not exceed the thermal limit of the type c conductor selected by the binary variable ${\mathbf{Y}}_{lc}$:

$$\left|{\mathbb{J}}_l\right|\le \sum _{c\in \mathbf{C}}{I}_c^{max}{\mathbf{Y}}_{lc},\left\{l\in \mathbf{L}\right\},$$

(10)

where $I_c^{max}$ is the maximum current that can flow through a type c conductor. Note that if line l is not selected by the binary variable ${\mathbf{X}}_l$, then the current flowing through it will be zero, and therefore, the constraint in Equation (10) will be satisfied.

2.1.8. Substation Node Conditions

The equality constraint in Equation (11) states that the only system node allowed to inject current is the substation node (i.e., the slack node), while the constraint in Equation (12) sets the substation node voltage to the system’s nominal voltage, which is generally set as 1 p.u.:

$${\mathbb{I}}_k^g=0,k\ne \mathrm{slack}\mathrm{node}$$

(11)

$${\mathbb{V}}_k={\mathbb{V}}_{\mathrm{nom}},k=\mathrm{slack}\mathrm{node}.$$

(12)

2.1.9. Radial System Topology

The constraint in Equation (13) ensures that all nodes in the distribution system remain connected, preventing the presence of isolated nodes or disconnected areas. To this effect, each node must have at least one incoming or outgoing connection. Meanwhile, the constraint in Equation (14) states that the total number of constructed system lines must always be $n_n-1$, ensuring that the resulting topology remains radial:

$$\sum _{l\in \mathbf{L}}\left|A_{kl}\right|{\mathbf{X}}_l\ge 1,\left\{k\in \mathbf{N}\right\},$$

(13)

$$\sum _{l\in \mathbf{L}}{\mathbf{X}}_l=n_n-1,$$

(14)

where $n_n$ is the number of nodes in the system. It should be mentioned that the radial network constraint was incorporated because the vast majority of rural distribution systems operate under this topology, in accordance with technical regulations and operational criteria established by network operators [29]. Radial configurations facilitate system protection, reduce infrastructure investment costs, and simplify both operation and maintenance [30,31].

2.1.10. Conductor Selection

The constraint in Equation (15) ensures that only one conductor size c is selected for distribution line l by the binary variable $X_l$, preventing the installation of multiple conductor sizes per line and ensuring that conductors are not assigned to lines that will not be constructed:

$$\sum _{c\in \mathbf{C}}{\mathbf{Y}}_{lc}={\mathbf{X}}_l,\left\{l\in \mathbf{L}\right\},$$

(15)

It is worth highlighting that the mathematical model presented from Equation (1) to Equation (15) can be adapted to consider the integration of distributed generation (DG) units within distribution networks. To this effect, the formulation must incorporate new constraints that reflect the impact of DG on the system’s power balance. In particular, the nodal current balance equation, as shown in Equation (6), should be modified to include the current injected by these sources. This adjustment implies the definition of an additional expression to calculate the injected current based on the complex power output of the DG units, following a similar formulation to that used in Equation (8). Additionally, it is necessary to impose constraints that establish the upper and lower bounds of active and reactive power generation, ensuring that the optimization respects the operational capabilities of the generation devices.

In addition to the integration of DG, the model can also be extended to address demand uncertainty or variations. One approach involves defining multiple demand scenarios based on probabilistic forecasts or historical variability. These scenarios can be incorporated into the model through Equation (8), as demand may vary over time. In this regard, it becomes important to include the time variable as a set within the optimization model. This allows planners to evaluate network configurations that remain cost-effective and technically feasible under variable load conditions. Such an extension would enhance the model’s applicability to real-world planning environments, where both generation and demand exhibit significant temporal and spatial variability.

By incorporating these adjustments, the model could effectively support the design of modern active distribution networks, combining optimal routing, conductor sizing, and generation and demand uncertainty.

2.2. Impact of Conductor and Route Set Reduction on Computational Complexity

The distribution network expansion model presented from Equation (1) to Equation (15) is mathematically formulated as an MINLP problem due to the simultaneous presence of binary decision variables (for route and conductor selection) and nonlinear relationships (load current at demand nodes). The combinatorial nature of choosing which lines to build (while ensuring a radial network) and which conductor size to install for each line leads to exponential growth in the possible configurations. All of the possible solutions are defined by a combination with the following structure:

$$C\left(n_l,n_n-1\right)·n_c^{n_n-1}={\displaystyle \frac{\left(n_l\right)!}{\left(n_l-n_n-1\right)!\left(n_n-1\right)!}}·n_c^{n_n-1},$$

(16)

where $n_l$ corresponds to the number of lines available for construction in the distribution system and $n_c$ is the number of conductors available for installation. For example, in a nine-node system with 14 possible lines for construction, where each line can select from seven available conductor sizes, there are approximately $1.7312\times {10}^{10}$ possible combinations. Similarly, for a 25-node test system with 42 possible lines for construction, where each constructed line can be assigned one of seven conductors, there are approximately $6.7762\times {10}^{31}$ possible combinations. Finally, if the same exercise is repeated for a 100-node test system with 180 possible lines for construction, there are approximately $1.7158\times {10}^{136}$ possible combinations. This clearly illustrates the combinatorial explosion inherent in this problem.

Thus, the computational time required to solve the problem scales exponentially with the parameters $n_n$, $n_l$, and $n_c$. Specifically, the solution space is defined by the number of ways to choose $n_n-1$ lines from $n_l$ candidates while ensuring a radial structure, as well as by the allocation of one of the $n_c$ conductor types to each selected line. This combinatorial explosion underpins the NP-hard classification of the problem (for which a polynomial-time algorithm is not expected) and explains why exhaustive enumeration or exact solution methods become computationally prohibitive as the system size increases.

Furthermore, the presence of nonlinear constraints such as Kirchhoff’s laws, voltage regulation bounds, and thermal capacity limits intensifies the computational burden. While the use of exact optimization techniques (e.g., the branch-and-bound method with nonlinear solvers) is necessary to ensure global optimality, these methods are computationally intensive for large-scale instances.

Importantly, the number of available lines and conductor options directly influences the size of the solution space and, consequently, the computational load. As illustrated in Equation (16), restricting the number of conductor types or candidate routes can significantly reduce the search space, allowing for faster solution times. This trade-off between modeling flexibility and tractability is particularly relevant in practical planning contexts, where utilities may adopt a limited set of standardized conductor types to streamline procurement and reduce complexity. In such cases, the proposed MINLP formulation remains applicable and can produce near-optimal solutions with lower computational effort, increasing its relevance for real-world rural network planning.

2.3. Solution Strategy

To address the MINLP problem dealing with the simultaneous selection of distribution feeder routes and conductor sizes in radial medium-voltage networks, this study proposes an efficient optimization approach that leverages free and open-source optimization tools, specifically the Julia programming language and its dedicated MINLP solvers [32]. The methodology follows a two-stage strategy, integrating the branch-and-bound (B&B) technique with an interior-point method (IPM) [33]. The interaction between both stages is shown in Figure 2.

As shown in Figure 2, the B&B algorithm is particularly suited for handling the discrete nature of route and conductor selection, ensuring that only feasible integer solutions are explored while systematically reducing the search space [34]. At each branching node, a continuous relaxation of the problem is solved using the IPM, temporarily treating integer variables as continuous ones to facilitate efficient exploration of the solution space and leverage the convex properties of the relaxed problem, which ultimately enhances computational efficiency [35].

The IPM, widely recognized for its robustness in solving large-scale nonlinear problems, is employed to efficiently handle the continuous relaxations within the B&B framework [36]. This approach transforms the constrained optimization problem into a sequence of barrier subproblems, where the barrier parameter is iteratively reduced until convergence and an optimal solution is achieved. The synergy between the B&B framework and the IPM ensures that the MINLP problem is systematically decomposed into manageable subproblems, improving both convergence rates and solution quality [37]. By integrating these optimization techniques within the Julia environment, the proposed methodology effectively addresses the inherent complexities of MINLP, ensuring optimal selection of conductors and feeder routes while maintaining the feasibility, reliability, and cost-efficiency of the electrical distribution network.

Table 1. General structure of the solution methodology for solving a generic MINLP problem by combining the B&B method and the IPM.

Step	Description
1. Problem definition	Define the MINLP problem for simultaneous distribution feeder routing and conductor selection in radial medium-voltage networks. Specify the objective function, constraints, and integer decision variables.
2. Relax integer variables	Convert the MINLP model into an NLP one by treating the integer variables (conductor selection and network topology) as continuous. This enables application of the IPM for solving relaxed subproblems.
3. Solve relaxed problem	Use the IPM to solve the relaxed problem, finding an initial feasible solution while efficiently handling nonlinear constraints.
4. Check integer constraints	Verify if the current solution satisfies all integer constraints (e.g., the conductor choices must be discrete values).
5. Store feasible solution	If the solution is integer-feasible and improves the objective function, then store it as a candidate optimal solution.
6. Bounding process	Evaluate whether the obtained solution improves the current best bound. If not, discard the branch (pruning).
7. Branching step	If the integer constraints are not met, then split the problem into new subproblems by selecting a branching variable and adding constraints to enforce integer feasibility.
8. Check for remaining subproblems	If more subproblems need to be explored, then return to Step 2 and repeat the process.
9. Optimal solution found	If no further subproblems remain, then the best stored solution is declared the optimal feeder routing and conductor selection plan.

illustrates the main steps involved in using the B&B method in combination with the IPM to solve the MINLP formulation regarding optimal feeder routing and conductor selection for rural distribution network applications.

For this research, the Julia programming language, combined with the BONMIN solver (basic open-source nonlinear mixed-integer programming), provides an efficient and flexible environment for solving MINLP problems in electrical distribution network optimization. Julia’s high-performance computing capabilities, coupled with its ability to interface seamlessly with state-of-the-art solvers, make it an ideal choice for large-scale optimization problems. BONMIN, which integrates the B&B method with IPMs, can effectively handle the discrete nature of conductor selection and feeder routing problems, as well as the nonlinear constraints of power distribution models (in the complex variable domain), thereby ensuring computational efficiency and solution accuracy [38]. This combination enables the precise formulation and resolution of complex optimization problems such as the one studied in this work.

Some of the advantages of this solution methodology are listed below:

1.

Julia’s just-in-time compilation allows for faster execution compared with traditional high-level languages, making it highly efficient for solving large-scale MINLP problems.

2.

BONMIN effectively integrates the B&B method, outer approximation, and IPMs, enabling robust solutions for MINLP models.

3.

The Julia ecosystem, particularly the JuMP package, allows for the seamless formulation of nonlinear constraints and integer variables, supporting complex power system optimization tasks.

4.

Julia and BONMIN are open-source, reducing computational costs and allowing researchers to customize and extend their optimization algorithms for specific problem structures.

3. Test Feeder

To validate the performance of the proposed mathematical model and assess its effectiveness in addressing the optimal expansion problem in distribution networks, two test systems were selected from the specialized literature (i.e., the aforementioned 9- and 25-node feeders), which have been previously used to solve this problem. These systems allowed for a comparison of our model against previously reported solutions, as well as for an analysis of how the model responded to different network configurations, system sizes, and levels of complexity. Recall that, according to Equation (16), as the number of nodes and lines available for installation increases, the number of possible combinations grows, thereby increasing the complexity of the problem.

3.1. Nine-Node Test Feeder

The nine-node test system is a system that operates at a line-to-line voltage of 13.2 kV and consists of nine nodes and eight loads, with a total installed three-phase power of 7625 kW and 3692.95 kVAr. For the optimal expansion problem, 14 possible routes (segmented lines) are available for construction, as shown in Figure 3.

The parametric information for the nine-node system is summarized in

Table 2. Three-phase power consumption at each node of the 9-node test system.

Node i	${\mathit{P}}_{\mathit{i}}$ (kW)	${\mathit{Q}}_{\mathit{i}}$ (kvar)	Node i	${\mathit{P}}_{\mathit{i}}$ (kW)	${\mathit{Q}}_{\mathit{i}}$ (kvar)
2	850	411.6738	6	1500	726.4832
3	750	363.2416	7	500	242.1611
4	925	447.9979	8	850	411.6738
5	1000	484.3221	9	1250	605.4026

and

Table 3. Possible lines for construction in the 9-node test system.

Line l	Node i	Node j	${\mathit{L}}_{\mathit{l}}$ (km)	Line l	Node i	Node j	${\mathit{L}}_{\mathit{l}}$ (km)
1	1	2	0.40	8	3	8	0.92
2	1	4	0.85	9	3	9	0.87
3	1	6	0.60	10	4	5	0.65
4	2	3	0.65	11	5	8	1.00
5	2	4	0.65	12	6	7	0.65
6	2	7	0.82	13	7	8	0.80
7	3	5	0.65	14	7	9	0.72

, which include the projected load consumption per node and the lines available for construction.

3.2. Twenty-Five-Node Test Feeder

The 25-node test system is a system that operates at a line-to-line voltage of 13.2 kV and consists of 25 nodes and 24 loads, with a total installed three-phase power of 6375 kW and 3087.55 kVAr. For the optimal expansion problem, 42 possible routes (segmented lines) are available for construction, as shown in Figure 4.

The parametric information for the 25-node system is summarized in

Table 4. Three-phase power consumption at each node of the 25-node test system.

Node i	${\mathit{P}}_{\mathit{i}}$ (kW)	${\mathit{Q}}_{\mathit{i}}$ (kvar)	Node i	${\mathit{P}}_{\mathit{i}}$ (kW)	${\mathit{Q}}_{\mathit{i}}$ (kvar)
2	625	302.7013	14	250	121.0805
3	400	193.7288	15	250	121.0805
4	250	121.0805	16	375	181.6208
5	250	121.0805	17	200	96.8644
6	125	60.5403	18	100	48.4322
7	250	121.0805	19	250	121.0805
8	250	121.0805	20	100	48.4322
9	625	302.7013	21	150	72.6483
10	400	193.7288	22	100	48.4322
11	250	121.0805	23	200	96.8644
12	400	193.7288	24	250	121.0805
13	250	121.0805	25	75	36.3242

and

Table 5. Possible lines for construction in the 25-node test system.

Line l	Node i	Node j	${\mathit{L}}_{\mathit{l}}$ (km)	Line l	Node i	Node j	${\mathit{L}}_{\mathit{l}}$ (km)
1	1	2	2.10	22	1	16	1.50
2	1	3	1.65	23	2	16	1.05
3	1	4	2.20	24	16	17	0.75
4	2	5	2.00	25	2	17	1.05
5	2	6	1.50	26	5	17	1.00
6	3	6	1.75	27	17	18	1.50
7	3	7	1.75	28	5	18	0.75
8	4	7	1.75	29	15	18	1.25
9	4	8	1.00	30	1	19	1.55
10	4	12	1.00	31	4	19	1.00
11	5	9	1.25	32	19	20	0.75
12	6	9	1.50	33	12	20	0.75
13	6	10	1.75	34	12	21	0.50
14	7	10	2.00	35	21	22	0.50
15	7	11	2.00	36	8	23	1.05
16	7	8	1.75	37	11	23	0.50
17	9	15	1.25	38	8	22	0.65
18	9	10	1.75	39	3	24	0.75
19	10	14	1.75	40	9	25	0.45
20	10	13	2.75	41	14	25	0.50
21	11	13	1.75	42	4	24	0.40

, indicating the projected load consumption per node and the lines available for construction.

Remark 1.

The test systems shown in Figure 3 and Figure 4 were considered to be balanced three-phase systems. Therefore, their single-phase equivalent representation was used. However, when working with per-unit (p.u.) quantities, three-phase base values had to be used for the voltage (i.e., line-to-line voltage) and power (i.e., three-phase power).

3.3. Available Conductor Sizes and Additional Parametric Information

To determine the numerical value of the objective function, seven different conductor sizes were considered for selection in both test systems. The investment costs used in the model were based on real data on network acquisition and deployment, which were obtained from utility companies and regional suppliers [1]. The information for each conductor size is presented in

Table 6. Conductor sizes available for the optimal distribution network expansion problem.

Caliber (c)	${\mathit{R}}_{\mathit{c}}$ ($\mathbf{\Omega }$/km)	${\mathit{X}}_{\mathit{c}}$ ($\mathbf{\Omega }$/km)	${\mathit{I}}_{\mathit{c}}^{max}$ (A)	${\mathit{C}}_{\mathit{c}}$ (USD/km)
1	0.80	0.50	75	1500
2	0.72	0.45	100	2000
3	0.65	0.40	125	2750
4	0.60	0.40	150	3000
5	0.55	0.36	175	3500
6	0.45	0.32	200	4200
7	0.40	0.20	225	4700

, including its impedance, thermal limit, and cost per kilometer.

Similarly, to determine the values of the objective functions defined in Equations (1) and (5), the parametric information shown in

Table 7. Parametric information used for the system cost calculations.

Parameter	Value	Unit	Parameter	Value	Unit
$C_{kWh}$	0.1302	USD/kWh	T	8760	hours
$t_a$	10	%	$N_t$	20	years
$t_e$	2	%	$V_{min}$	−10	%
$V_{max}$	$+10$	%	–	–	–

was used.

4. Numerical Results and Discussions

For the computational implementation of the MINLP model representing the studied problem, a workstation with a 12th Gen Intel Core i7-12700T CPU @ 1.40 GHz, 32 GB of RAM, and 64-bit Windows 11 Pro was used. All scripts were programmed in Julia v1.10.2. Similarly, to solve the proposed MINLP model, the JuMP optimization environment was used in conjunction with the BONMIN solver.

To validate the effectiveness of the proposed MINLP model, the results were compared against those reported in [1]. In that study, to reduce the complexity of the mathematical model, the authors addressed the optimal grid expansion problem in two stages. In the first stage, they solved the route selection problem using MSTs, a graph theory procedure that connects all nodes in a system with the lowest sum of edge weights, thereby ensuring a radial topology without sub-tours at the minimum interconnection cost (i.e., the shortest distance) [39,40]. Subsequently, based on the radial topology determined via the shortest distance, they addressed the conductor selection problem using the sine-cosine (SCA) and tabu search (TSA) metaheuristic algorithms.

Comparing the proposed MINLP model against previous solutions is essential to demonstrate its validity and effectiveness, as this allows verifying its solution robustness and quality with respect to well-established approaches in the specialized literature. This comparison reinforces confidence in the practical applicability of the new method and provides a clear reference for its advantages and limitations in relation to existing methodologies.

4.1. 9-Node Test Feeder

The results of applying the MINLP model to the nine-node feeder are shown in Figure 5 and

Table 8. Numerical results for the 9-node system.

Method	Caliber	${\mathit{Z}}_{\mathbf{cost}}$ (USD)	${\mathit{Z}}_{\mathbf{loss}}$ (USD)	${\mathit{Z}}_{\mathbf{inv}}$ (USD)
MST-SCA	$\left[6,7,1,3,1,4,1,1\right]$	132,699.1306	112,471.8028	12,467.5000
MST-TSA	$\left[7,7,1,2,1,4,1,1\right]$	131,819.3324	111,746.7327	12,180.0000
MINLP	$\left[7,7,7,7,7,6,7,2\right]$	77,129.3388	63,764.1247	23,224.0000

. Figure 5 presents the optimal route selection for the system in comparison with the radial topology obtained in [1] via the MST approach.

Table 8 presents the numerical results obtained for the nine-node system, comparing them against three different optimization approaches. For each method, the results include the conductor sizes selected for each constructed line (Figure 5), the total costs ($Z_{\mathrm{cost}}$), the energy losses costs ($Z_{\mathrm{loss}}$), and the conductor investment costs ($Z_{\mathrm{inv}}$).

Based on the numerical results shown in Figure 5 and Table 8, the following can be stated:

• The solution obtained using the MST method ensured a radial topology with a shorter total line length (5.12 km) compared with the MINLP model (5.47 km), significantly reducing conductor investment costs. In this regard, the MST-SCA and MST-TSA approaches required conductor investments of approximately USD 12,467.50 and USD 12,180.00, respectively, whereas the MINLP model required nearly twice as much in conductors (USD 23,224.00). However, aside from minimizing the total line length, the MST-based solution did not consider technical aspects such as system energy losses.
  Additionally, it is important to highlight that in the MST-based methods, the conductor size selection varied significantly across different lines, evidencing a partial optimization focused on minimizing investment costs by selecting the shortest route. In contrast, the MINLP model tended to select larger conductor sizes more uniformly (with a predominance of sizes seven and six), reflecting a strategy aimed at reducing network losses at the expense of higher initial investment costs.
• The topology obtained using the MINLP model optimized network expansion by considering both economic (investment and operating costs) and technical aspects (voltage limits and line capacity). As observed in Figure 5b, the model prioritized incorporating a greater number of branch circuits, allowing for a more efficient distribution of current flow and, consequently, a significant reduction in system energy losses. In this case, the MINLP model achieved an energy losses cost of USD 63,764.12, which represents a reduction of USD 48,707.68 (43.31%) compared with MST-SCA and USD 47,982.61 (42.94%) compared with MST-TSA.
• Regarding the total costs, the MINLP model achieved the lowest value (USD 77,129.34), representing reductions of approximately USD 55,569.79 (41.88%) and USD 54,689.99 (41.48%) compared with MST-SCA and MST-TSA, respectively.
• As for the computation times, the MINLP model achieved global optimality in approximately 15.08 s, which was comparable to MST-TSA (14.35 s) and significantly faster than MST-SCA, which required 28.76 s. These results indicate that for small systems, the exact approach is competitive in terms of computation time while offering globally optimal solutions.

The results demonstrate that the MINLP model provided a more efficient solution in terms of total system cost, as it significantly reduced energy losses ($Z_{\mathrm{cost}}$), despite requiring a higher initial investment in conductors ($Z_{\mathrm{inv}}$). This highlights the advantage of simultaneously optimizing network topology and conductor selection, rather than addressing these problems in two separate stages, as was performed in the MST-SCA and MST-TSA methods. From a practical perspective, although the MINLP model entailed a higher initial investment cost, the reduction in energy losses compensated for this expense in the long term, making it a more cost-effective and efficient alternative for distribution network expansion.

4.2. Twenty-Five-Node Test Feeder

The 25-node system exhibited a similar behavior to that observed in the 9-node feeder, albeit amplified due to the increased complexity of the network. Figure 6 illustrates the radial topology provided by the MST method in comparison with that obtained via the MINLP model, while

Table 9. Numerical results for the 25-node system.

Method	Caliber	${\mathit{Z}}_{\mathbf{cost}}$ (USD)	${\mathit{Z}}_{\mathbf{loss}}$ (USD)	${\mathit{Z}}_{\mathbf{inv}}$ (USD)
MST-TSA	$\left[\begin{array}{c}1,1,1,1,1,7,7,6,2,1,1,5\\ 1,2,2,1,1,1,1,1,1,1,1,1\end{array}\right]$	371,516.5933	313,238.9341	51,270.0000
MINLP	$\left[\begin{array}{c}1,7,7,1,1,7,6,1,1,7,7,7\\ 1,1,2,1,1,1,1,1,1,1,1,1\end{array}\right]$	128,974.7251	103,181.7780	73,045.0000

presents the conductor sizes selected for each constructed line, the investment costs, the energy loss costs, and the total cost for each method.

Based on Figure 6 and Table 9, the following can be stated:

• Just like in the nine-node system, the MST-TSA solution, which minimized the total system distance to 23.95 km, resulted in a lower investment cost (USD 51,270) but exhibited extremely high energy losses (USD 313,238.9341), leading to a total cost of USD 371,516.5933.
• In contrast, the MINLP model invested more in conductors (USD 73,045) since the total system distance increased to 28.65 km. However, the network topology selected by this model (Figure 6) significantly reduced the costs of energy losses (USD 103,181.7780) by efficiently redistributing power flows and preventing branch circuit overloads. This resulted in a total cost of approximately USD 128,974.7251.
• In the MST-TSA method, the selected conductors were predominantly of sizes one and two, with some larger exceptions (e.g., sizes six, seven, and five) in specific lines. This suggests a strategy aimed at restricting investment in most branches and using larger conductors only when strictly necessary to meet the minimum operational criteria. In contrast, the MINLP model invested more and more uniformly in larger conductors (e.g., there was a predominance of sizes seven, six, and two), reflecting an approach that prioritizes the overall reduction in long-term losses, which has a greater influence on the final objective function value, as demonstrated in Table 8 and Table 9.
  In other words, while MST-TSA maximized the use of smaller conductors to minimize short-term investment costs, the MINLP model balanced investment costs and loss reduction, leading to a network design with a greater electrical capacity but lower operating costs over the planning horizon.
• This substantial reduction in total costs (over USD 188,282.1561, or 51.65% compared with the MST-TSA method) demonstrates the advantage of simultaneously optimizing route selection and conductor size via exact solution techniques, rather than solving each step separately. Simultaneously solving these problems enables the direct interaction between investment and operational decisions, ensuring that installation costs and energy losses are optimized together and thus increasing the likelihood of finding the global optimum. Conversely, addressing these aspects separately (e.g., establishing the topology first and then selecting conductor sizes) limits the ability to search for global optima. This often leads to configurations that minimize the initial costs without adequately considering the long-term impact of energy losses.
• It should be added that the standard deviation of the TSA for the 25-node test system was USD 640.21, indicating a degree of variability in the solutions obtained across multiple runs. In contrast, the standard deviation of the proposed MINLP model was zero, which confirms its ability to consistently converge to the same numerical solution regardless of the number of executions.
• In this case, the computation time of the MINLP model increased substantially to 55,546.18 s due to the exponential growth of the solution space (see Section 2.2). In contrast, the TSA method provided a feasible solution in less than 70 s. This highlights a well-known trade-off between solution optimality and computational effort, emphasizing the need for future research on hybrid or decomposition-based strategies to extend the applicability of exact methods to larger distribution systems.

The results for both test systems demonstrate that the MINLP formulation achieved a substantial reduction in the cost of energy losses, being approximately USD 47,982.61 (42.93%) and USD 210,057.16 (61.06%) compared with the MST-TSA approach for the 9- and 25-node test systems, respectively. Consequently, the total costs also exhibited significant reductions of approximately USD 54,689.99 (41.48%) and USD 242,541.87 (65.28%) compared with the MST-TSA approach for the 9- and 25-node test systems, even though the MINLP model requires a higher initial investment in conductors. In terms of computational performance, for both test systems, the MINLP model achieved a standard deviation of zero. This behavior reflects the deterministic nature of exact optimization methods and provides strong evidence for the global optimality of the results produced by the proposed formulation. Regarding processing times, although the exact model required more computation time for larger systems, it delivered solutions that were significantly lower in terms of total costs as well as superior technical performance, justifying its use in long-term planning scenarios where solution quality is a priority.

These findings highlight the importance of simultaneously optimizing both network topology and conductor selection to achieve globally efficient solutions. Furthermore, the robustness of the proposed methodology in the 9- and 25-node systems demonstrates its scalability, indicating that it could be successfully extended to even larger distribution networks with more demanding operational conditions. This makes it a valuable tool for the long-term planning of real electrical systems.

5. Conclusions and Future Works

This article formulated the optimal distribution network expansion problem as an MINLP model, simultaneously addressing route selection and conductor sizing in radial distribution systems. Case test systems with 9 and 25 nodes were used to validate the proposed approach. Furthermore, the proposed model was compared against existing methods based on metaheuristic algorithms and the construction of an MST approach.

The numerical results confirmed significant reductions in both total costs and energy losses for the MINLP formulation in comparison with methodologies based on MSTs and metaheuristic algorithms (i.e., SCA and TSA). In the nine-node system, the total cost decreased from approximately USD 131,819.33 (MST-TSA) to USD 77,129.34 (MINLP), resulting in savings of USD 54,689.99, equivalent to 41.48%. Additionally, the energy loss costs were reduced from USD 111,746.73 to USD 63,764.12, representing a decrease of USD 47,982.61 (42.94%). For the 25-node system, the total cost reduction exceeded 65%, dropping from approximately USD 371,516.59 (MST-TSA) to USD 128,974.72 (MINLP). Similarly, the energy losses costs decreased by USD 210,057.16 (61.06%) from USD 313,238.93 to USD 103,181.78. Although the MINLP model requires a higher initial investment in conductors, these savings in energy losses more than compensate for the additional investment in the medium and long term, reinforcing the importance of simultaneously optimizing both network topology and conductor selection.

As potential future work, our first proposal would be to extend the mathematical model to unbalanced distribution systems while considering loads connected in star or delta configurations as well as the asymmetry caused by the geometric arrangement of conductors. Secondly, the mathematical model could be formulated for a multi-objective context, evaluating economic, technical, and environmental criteria. Similarly, it would be interesting to incorporate distributed resources (e.g., photovoltaic generation, energy storage, or electric vehicles) to analyze the impact of decentralization on network expansion. The integration of heuristic initialization strategies or hybrid optimization approaches could improve computational efficiency in large-scale systems, allowing the model to be scaled to networks with more than 100 nodes while preserving solution quality. Finally, introducing uncertainty in demand and renewable generation is suggested. This could be accomplished by employing stochastic or robust methods to capture more realistic hourly operating scenarios.