Optimal Allocation and Sizing of Electrical Substations Using an Improved Black Widow Algorithm

Abstract

The allocation and sizing of electrical substations are critical for the efficient planning and expansion of distribution networks. This study presents the application of an enhanced Black Widow Algorithm (BWA) to solve this complex problem, considering multiple variables and constraints. The BWA, inspired by the reproductive behavior of black widows, was employed to optimize the placement and sizing of new substations and connection of load centers. Two mutation methods were evaluated: the original BWA mutation and a genetic algorithm-inspired mutation incorporated into the BWA algorithm (GA mutation). Four scenarios with varying load center distributions were tested to assess the algorithm’s adaptability and performance. The results showed that the GA mutation consistently outperformed the original mutation in more complex scenarios, reducing total costs by up to 14.75%. The proposed GA mutation enabled greater flexibility in escaping local optima, leading to improved solutions in scenarios involving numerous new load centers. Additionally, increasing the number of generations and black widows enhanced convergence and solution stability, particularly in challenging cases. This study demonstrates the feasibility of using the enhanced BWA for real-world applications, offering a valuable tool for electrical distribution planning.

1. Introduction

The economic growth of a country is directly associated with increased consumption of electricity in both industries and households [1]. This rise in electricity consumption is a clear trend that must be considered by power distribution companies when planning the expansion of the distribution network. Such planning should take into consideration the forecasted load growth in areas already served by the grid, the projected installation of new medium and large loads, such as industrial facilities, and the municipal development plans for each city within the distribution company’s service area [2].

In the context of distribution network expansion planning, the allocation and sizing of electrical substations have a significant impact on the grid, affecting both reliability and financial aspects [3].

Given that this is a multivariable, complex problem with engineering, financial, geographical, ecological, and aesthetic constraints, it is not feasible to describe the problem using a single function and apply classical mathematical methods for finding maxima and minima [4]. The sheer magnitude of the number of possible solutions in such problems also makes it impractical to compute all potential outcomes to identify the optimal one, necessitating the use of alternative optimization approaches.

In recent years, new optimization techniques have been developed, with heuristic methods being the most commonly used to find good solutions without testing all possible combinations. These optimization techniques have numerous applications, including the allocation and sizing of electrical substations using a biogeography-based optimization with population competition algorithm (BBOPC) [4], k-means algorithm [5,6], Voronoi diagram method [7], binary modified imperialist competitive algorithm (BMICA) [8], deep learning algorithm [9], artificial immune systems [10], k-mean++ algorithm [11], dynamic programming [12], genetic algorithm [13] and 0–1 optimization model [14]. In these studies, the authors address variables such as the installation and operational costs of electrical substations, costs associated with feeder reinforcement, the estimated power of loads and distributed generation units, as well as data on the network topology.

Among the various techniques proposed in the literature, one noteworthy method is an evolutionary approach based on the reproductive behavior of a species of arachnid known as black widows. In the proposal by [15], the concept of cannibalism during the reproduction phase was introduced, which helps accelerate population improvement and demonstrates strong performance in benchmark functions.

The Black Widow Algorithm (BWA) has been applied to various optimization problems, including distributed generation planning and operation [16,17], load dispatch [18], frequency regulation [19], antenna design [20], and noise reduction in medical imaging [21]. In the literature review conducted, no studies were found that addressed the use of the BWA in solving the optimal allocation and sizing problem of electrical substations.

One of the mechanisms for diversifying the solutions generated by the BWA is mutation. In the approach proposed in [15], the values of two genes are randomly swapped with each other. However, it has been observed that the performance of the BWA can be enhanced through the adoption of an alternative mutation technique.

Therefore, this article proposes the application of the BWA algorithm to the optimal allocation and sizing problem of electrical substations. Additionally, it introduces a modification to the original BWA by incorporating the classical genetic algorithm mutation into its mutation phase. This enhancement aims to improve the algorithm’s operational performance, particularly by minimizing the total cost. Additionally, this study contributes by evaluating and comparing the performance of both the original BWA and the proposed modification within the context of electrical substation allocation and sizing. The analysis highlights the scenarios in which the proposed algorithm achieves superior performance.

The remainder of this paper is organized as follows: Section 2 presents the methodology, detailing the problem modeling, the implementation, and the proposed modifications to the BWA, as well as the tests and comparisons performed. Section 3 presents the results of the seven evaluated scenarios; finally, Section 4 presents the conclusions of the paper.

2. Methodology

This section presents the methods used for problem modeling, the modifications made to the original BWA, and the approach employed to test and compare the various proposed scenarios.

2.1. Problem Modeling

The problem was modeled using a geographic representation on the $X-Y$ plane, with loads represented by load centers defined by their positions on the plane, and electrical substations with power limitations corresponding to their installed capacities. Additionally, constraints were imposed on restricted areas for new electrical substations, which could include geographic, environmental, historical, or other restrictions that prevent the construction of a new electrical substation.

Four scenarios ($A$, $B$, $C$, and $D$) were adopted for testing, aiming to evaluate the algorithm’s performance and adaptability. These scenarios are based on the final scenario presented in [4], where the coordinates of the loads and the electrical substation were used, and the load demands were adjusted to address distribution issues.

In Scenarios $A$ and $B$, eight pre-existing load centers and one electrical substation were considered, with the addition of six new load centers located either closer to (Scenario $A$) or farther from (Scenario $B$) the pre-existing load centers. These two scenarios test the algorithm’s response to new load centers in different regions, with the expectation that the placement of the new electrical substation will correspond to the location of the new load centers.

For Scenarios $C$ and $D$, two electrical substations and 10 load centers were considered. Scenario $C$ introduces five new load centers with demands that can be met by the pre-existing electrical substations, testing the algorithm’s response without requiring a new electrical substation or in identifying a non-trivial optimal allocation. Scenario $D$ presents 10 new load centers, aiming to analyze more complex situations where one or more new electrical substations may be needed.

The data used are presented in Appendix A, and Figure 1 illustrates the scenarios, with the pre-existing electrical substations represented by green triangles, the pre-existing load centers by blue circles, the projected load centers by yellow circles, and the restricted areas marked by red rectangles. Their locations are indicated by coordinates on the $X-Y$ plane.

Based on the models described in the theoretical framework, the variables used by most of the authors reviewed [5,13,22,23,24,25,26] were selected for the cost function, with these variables being expressed in monetary terms. The cost function can be divided into three components: the installation cost of new electrical substations ($CS$), the cost of implementing new feeders ($CF$), and the cost of power losses in supplying the load centers over the considered time horizon ($CL$). Equation (1) defines the cost function.

$$Cost=\sum _{ns=1}^{N_{ns}}{CS}_{ns}+\sum _{nf=1}^{N_{nf}}{CF}_{nf}+\sum _{c=1}^{N_c}{CL}_c$$

(1)

where:

${CS}_{ns}$	Installation cost of the electrical substation $ns$.
$N_{ns}$	Number of new electrical substations.
${CF}_{nf}$	Installation cost of the new feeder $nf$.
$N_{nf}$	Number of new feeders to be installed.
${CL}_c$	Cost of electrical losses in supplying load center $c$.
$N_c$	Number of load centers, both new and pre-existing.

The installation cost of a new electrical substation is obtained by referencing the tabulated values according to its installed capacity, without accounting for regional cost variations. Equation (2) provides the calculation of the installation cost of a new feeder, where the distance between the load center and the electrical substation is calculated as 1.55 times the Euclidean distance between the points ($DF=1.55\sqrt{\mathsf{\Delta }{X}^2+\mathsf{\Delta }{Y}^2}$), as approximated in [25]. In this paper, only one type of feeder is considered, and each feeder is used to supply only one load center.

$${CF}_{nf}=C_{km}\cdot D{F}_{nf}$$

(2)

where:

$C_{km}$	Cost per kilometer of the new feeder.
$D{F}_{nf}$	Estimated distance covered by feeder $nf.$

Equation (3) defines the calculation of loss costs, considering balanced three-phase load centers. An analysis horizon of 10 years was established, and it is assumed that all new load centers are connected to the system at the beginning of this period.

$${CL}_c=3\cdot R_{km}\cdot D{F}_{nf}\cdot {\left({\displaystyle \frac{S_c}{\sqrt{3}\cdot V_l}}\right)}^2\cdot {\displaystyle \frac{C_{kWh}}{{10}^9}}\cdot \left(N_{Years}\cdot 365\cdot 24\right)\cdot P_{factor\_c}$$

(3)

where:

$R_{km}$	Electrical resistance per kilometer of the feeder.
$D{F}_{nf}$	Estimated distance covered by the feeder from load center $c$.
$S_c$	Apparent power demanded by load center $c$.
$V_l$	Line voltage of the feeder.
$C_{kWh}$	Cost of energy.
$N_{Years}$	Analysis horizon.
$P_{factor\_c}$	Loss factor for load center $c$.

Another important factor in performance evaluation is the verification of constraints, specifically the installed capacity constraints of electrical substations and the restricted areas for new installations. The installed capacity constraints are evaluated through a summation, as shown in Equation (4), where the power demands of the load centers connected to electrical substation “s” are summed. This constraint is verified for all pre-existing and proposed electrical substations.

$$P{D}_s=\sum _{c=1}^{Nc}{S}_c$$

(4)

where:

$P{D}_s$	Power demanded from electrical substation $s$.
$S_c$	Power demanded by load center $c$.

The second constraint pertains to the location of new electrical substations, ensuring that no new electrical substations are installed in restricted areas. Such areas may include environmental protection zones, historically or culturally significant sites, geographic obstacles such as rivers, lakes, or mountains, or any other regions that are not available for development.

Based on Equations (1)–(3) and the constraints, the optimization problem can be formulated according to Equation (5), which is to be minimized.

$$\begin{array}{ccc}\hfill Min Cost=& {\displaystyle \sum _{ns=1}^{N_{ns}}}{CS}_{ns}+{\displaystyle \sum _{nf=1}^{N_{nf}}}{C}_{km}\cdot D{F}_{nf}\hfill & \hfill \\ \hfill & \hfill +{\displaystyle \sum _{c=1}^{N_c}}3\cdot R_{km}\cdot D{F}_{nf}\cdot {\left({\displaystyle \frac{S_c}{\sqrt{3}\cdot V_l}}\right)}^2& \cdot {\displaystyle \frac{C_{kWh}}{{10}^9}}\hfill \\ & \cdot \left(N_{Years}\cdot 365\cdot 24\right)\cdot P_{factor\_c}\hfill & \hfill \end{array}$$

(5)

Complying with the installed capacity constraint—Equation (6):

$$P{D}_s\le S_s \forall s\in {\mathbb{N}}^{\ast }\le N_{sub}$$

(6)

where:

$S_s$	Installed capacity of electrical substation $s$.
$N_{sub}$	Number of electrical substations, both new and pre-existing.

And the geographical constraint, which is satisfied if the $X$ or $Y$ coordinates of the new electrical substations do not fall within the boundaries of any restricted regions, with these regions modeled as rectangles—Equation (7):

$$X_{ns}\notin \left[X_r^{min},X_r^{max}\right]\vee Y_{ns}\notin \left[Y_r^{min},Y_r^{max}\right] \forall r\in {\mathbb{N}}^{\ast }\le N_{reg}\forall ns\in {\mathbb{N}}^{\ast }\le N_{ns}$$

(7)

where:

$X_{ns}/Y_{ns}$	Coordinates of the new electrical substation $ns$.
$X_r^{min}$, $X_r^{max}$, $Y_r^{min}$, and $Y_r^{max}$	Boundary coordinates of the restricted region $r$.
$N_{reg}$	Number of restricted regions.

This study does not apply proximity constraints between the newly allocated electrical substations and the existing load centers. If necessary, the algorithm can include additional constraints to ensure the minimum required distance between the load centers and the proposed substations.

2.2. Algorithm Implementation

2.2.1. Variable Encoding

This paper focuses on analyzing a group of pre-existing loads and electrical substations, considering a group of new loads to be included in the system, and aims to allocate new electrical substations and optimize the connection of loads to them.

Based on these considerations, the encoding of a black widow $V_n$ (from the BWA population) consists of a vector, where each position of the vector corresponds to a variable. The first four values refer to a potential new electrical substation, where $V_n^0$ indicates its existence or non-existence, assuming values of 0 or 1. $V_n^1$ and $V_n^2$ represent the coordinates on the $X$ and $Y$ axes, respectively, constrained by the maximum and minimum values of $X$ and $Y$ among all pre-existing and planned load centers.

The value in $V_n^3$ represents the installed capacity of the new electrical substation, assuming values from a predefined table, as shown in

Table A10. Costs of new electrical substations.

Installed Capacity (MVA)	Installation Cost (mi USD)
10	1.6
17.5	2.0
25	2.4
40	2.8
50	3.2
66	3.6
75	4.0

in Appendix A. This group of four values is repeated $ts$ times, with $ts$ being the maximum number of new electrical substations that can be proposed by the algorithm. In this paper, $ts$ was arbitrarily set to 5, meaning that the solutions can contain up to 5 new electrical substations, which is considered a satisfactory number to cover the expected outcomes in the proposed scenarios.

The remaining values of the vector correspond to which electrical substation each load center is connected to, taking the identification values of either pre-existing or new electrical substations. Considering $tc$ as the total number of load centers (pre-existing and new), the vector always has a size of $4ts+tc$. Figure 2 illustrates a black widow with its possible values.

The difference in the magnitudes of the variables makes the reproduction and mutation phases more challenging. Therefore, the black widow, as shown in Figure 2, is used only to verify constraints and calculate the total cost. During the other phases of the algorithm, the black widow is populated with decimal values between 0 and 1, standardizing all variables to the same range and making them resemble the individuals in the original algorithm proposal [15]. An example of a possible black widow is presented in Figure 3.

The conversion is performed linearly, with rounding applied to the discrete values. The detailed conversion process for the different types of variables are described in Appendix B.

2.2.2. Performance Calculation

After converting the black widow values for the problem, the performance of each black widow in the population is determined by applying the constraints and computing the total cost. First, the algorithm checks the constraints related to installed capacity and restricted regions. If any of these constraints are violated, the signaling variable ($R_s$) is set to 1, indicating a constraint violation. The evaluation of performance relies on the cost function modeled by Equation (1), where a lower cost value indicates better efficiency. Equation (8) defines the verification of constraints and their impact on the cost function ($f_c$).

$$f_c=\left\{\begin{array}{cc}Cost\hfill & \hfill ,if R_s=0\\ {10}^{12}\hfill & \hfill ,if R_s=1\end{array}\right.$$

(8)

When $R_s$ = 1, the assigned cost corresponds to a penalty value significantly higher than the expected optimal cost for the scenario. As a result, the penalized black widow receives the lowest ranking in the population, since a lower cost indicates a better-performing solution.

2.2.3. Reproduction, Cannibalism, and Mutation

Once the costs of all the black widows in a population are calculated, they are ranked in ascending order of cost, and the top 60% are selected for the reproduction and mutation phases. This percentage was determined based on preliminary convergence tests of the algorithm.

At the start of the reproduction phase, pairs are formed randomly and without repetition from the selected black widows. In each pair, the black widow with the lower cost is called the female ($x_1$), and the one with the higher cost is called the male ($x_2$). Reproduction is carried out by generating offspring $y_1$ and $y_2$.

These black widows are then used in the cannibalism phase, where the male always dies during reproduction, and two possible scenarios occur for the female and the offspring. If both offspring have a lower cost than the mother, the female dies, and both offspring move on to the next generation. Otherwise, the offspring with the higher cost dies, and the female and the better-performing offspring continue.

Thus, the algorithm selects 60% of the population, which then generates an equal number of black widows through the reproduction and cannibalism phases. The remaining 40% undergo mutation, ensuring a consistent population size across all generations. Figure 4 illustrates how a new generation is created from the previous one through these three mechanisms.

The mutation in the original BWA (referred to as BW mutation), where the values of two random positions are swapped [15], can be questioned regarding its effectiveness in overcoming local optima since no new values are introduced. Therefore, this paper proposes the incorporation of the classic mutation from the genetic algorithm, described in [27], into the BWA algorithm, where it is referred to as the GA mutation. In this approach, a new random value is assigned to a randomly chosen characteristic, determining the mutation method to be applied. This process allows for a systematic comparison of different mutation strategies, enabling an assessment of their impact on the results.

2.3. Tests and Comparisons

After the modeling and implementation, the four scenarios (A, B, C, and D) and the two mutation methods were tested, evaluating the installed capacity and consistency in achieving good solutions. A population size of 100 and a total of 100 generations were arbitrarily chosen. These parameters were selected empirically, as they were observed to be reasonable for the convergence of the solutions and, given the low complexity of the proposed scenarios, allowed for multiple executions in a short time for the validation of the tests, as previously mentioned.

For each combination of scenario and mutation type, 50 runs were conducted to reduce the impact of the intrinsic randomness of the method. This approach enabled the assessment of the dispersion of the new electrical substations, as well as calculating the average and standard deviation of the costs of the optimal solutions from each run.

Considering the low complexity of the first three scenarios, it was proposed to increase the population size and the number of generations to 200 in Scenario D to improve the chances of solution convergence, with comparisons made to the previous values.

Another complexity-related test was conducted by reducing the implementation cost of the electrical substations to 20% of the initially assigned value in Scenario D. This allowed for testing the algorithm’s ability to find solutions with multiple new electrical substations of higher complexity.

Table 1. Tested scenarios and their parameters.

Scenario	Number of Generations and Black Widows	Electrical Substation Costs
A	100	100%
B	100	100%
C	100	100%
D	100	100%
D—200	200	100%
D mod.	100	20%
D mod.—200	200	20%

provides a summary of the tested scenarios, including the respective number of generations, black widows, and the cost values of the electrical substations used, with both BW and GA mutations applied in each scenario.

The proposed methodology was implemented in the Python programming language and executed on a computer equipped with an Intel Core i5 processor (3.30 GHz), 16 GB of RAM, and Windows operating system.

3. Results and Discussion

This section presents the results obtained for each of the proposed scenarios, discussing the algorithm’s performance in each case and comparing the original BW mutation method with the proposed GA mutation.

3.1. Scenario A

Scenario A includes only one initial electrical substation, with new load centers planned near the pre-existing ones, whereas the total demanded power cannot be supplied by the pre-existing electrical substation. Both mutation methods found the same optimal solution after the test rounds, as shown in Figure 5, with one new electrical substation allocated and a redistribution of the load centers within the system. The load centers are represented by blue circles, the pre-existing electrical substations by green triangles, the new electrical substations by pink triangles, the feeders by black lines, and the restricted areas indicated by red lines.

Table 2. Costs of the optimal solution found in Scenario A.

Cost	Value (mi USD)
New electrical substations	2.80
New feeders	2.54
Power losses	8.08
Total value	13.42

details the values associated with each type of cost in the optimal solution found. Notably, 60% of the total cost is attributed to electrical losses in the feeders over the 10-year analysis period.

Figure 6 confirms the similarity in performance between the two mutations, showing the evolution of one of the tests over the generations through a graph displaying the lowest cost value of each generation for each mutation type. It can be observed that the algorithm achieved convergence quickly in this scenario, encountering no significant difficulties in reaching the final value.

Figure 7 shows the distribution of solutions after 50 runs for each mutation type, with red circles indicating the position of new electrical substations. It can be observed that the solutions are tightly clustered.

All 50 solutions from each mutation method found results with a new electrical substation having an installed capacity of 40 MVA. This behavior demonstrates consistency in identifying such an important discrete parameter for the cost function. The optimal values found by each mutation, along with the averages and standard deviations of the optimal solutions after 50 runs, are shown in

Table 3. Cost values obtained after 50 runs for Scenario A.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW	13.42	13.75	0.23
GA	13.42	13.71	0.38

. It can be observed that, on average, the cost of the GA mutation was 0.33% lower than that of the BW mutation, indicating no significant performance differences in this scenario.

3.2. Scenario B

Scenario B also features only one initial electrical substation, but the new planned load centers are farther from the pre-existing ones, and the total demanded power cannot be supplied by the pre-existing electrical substation. Both mutations found the same optimal solution, which is presented in Figure 8.

This solution includes a new electrical substation supplying only the new load centers. The detailed cost breakdown is shown in

Table 4. Costs of the optimal solution found in Scenario B.

Cost	Value (mi USD)
New electrical substations	2.80
New feeders	2.08
Power losses	8.51
Total value	13.39

, where the cost of electrical losses accounts for a large portion of the total cost, amounting to 63.5% in this case.

The rapid convergence and similarity between the mutations are also observed in this scenario. Figure 9 shows examples where the optimal solution was found in approximately 40 generations for both mutation types.

In Figure 10, one can see the dispersion of solutions, where it is noticeable that the BW mutation has a wider spread compared to the GA mutation, which is clustered closer to the optimal solution.

Table 5. Cost values obtained after 50 runs for Scenario B.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW	13.39	13.47	0.16
GA	13.39	13.39	0.004

provides a summary of the costs found by each mutation, showing that the average cost of the GA mutation was 0.51% lower than the average cost of the BW mutation, indicating minimal difference between the mutation methods.

3.3. Scenario C

In Scenario C, there are two electrical substations that have the installed capacity to meet the demand of the new load centers. Figure 11 shows the optimal solution found by each mutation.

Table 6. Costs of the optimal solution found in Scenario C.

Cost	Value (mi USD)
	BW Mutation	GA Mutation	Δ (GA–BW)
New electrical substations	0.00	1.60	1.60
New feeders	3.05	1.94	−1.11
Power losses	10.24	9.74	−0.50
Total value	13.29	13.28	−0.01

provides the detailed costs of the optimal solution found using each mutation type. The solution from the GA mutation includes the cost of installing a new electrical substation, but the costs of new feeders and electrical losses are lower. As a result, the overall balance shows that the total cost is approximately USD 10,000 less than the solution without installing a new electrical substation.

In Figure 12, one can observe convergence in fewer generations than in Scenarios A and B, which is the expected behavior since this scenario is less complex than the previously mentioned ones.

Figure 13 shows the distribution of solutions after 50 runs for each mutation type. One solution using the BW mutation found a new electrical substation, while two solutions using the GA mutation identified a new electrical substation.

These three solutions suggested a new electrical substation with an installed capacity of 10 MVA, serving only a few load centers.

Table 7. Cost values obtained after 50 runs for Scenario C.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW	13.29	13.31	0.10
GA	13.28	13.30	0.06

provides a summary of the 50 test runs for each mutation type, once again showing averages close to the optimal value found, approximately 0.1% higher, and small standard deviations.

3.4. Scenario D

Scenario D was designed to be more challenging for the algorithm, with two pre-existing electrical substations and several new load centers spread out. Figure 14 shows the optimal solution found by each mutation.

The optimal solution found by the BW mutation identified only one new electrical substation and allocated the remaining load centers to the pre-existing electrical substations. In contrast, the GA mutation found a solution with two new electrical substations to serve the more distant load centers. As detailed in

Table 8. Costs of the optimal solution found in Scenario D.

Cost	Value (mi USD)
	BW Mutation	GA Mutation	Δ (GA–BW)
New electrical substations	2.40	3.20	0.80
New feeders	6.07	5.01	−1.06
Power losses	12.48	11.75	−0.73
Total value	20.95	19.96	−0.99

, this solution has higher costs for electrical substation installation but lower costs for other factors, particularly new feeders, resulting in a lower total cost.

Another consequence of the greater complexity of Scenario D is observed in Figure 15, which shows that, in these examples, the optimal solution was found in the later generations, especially in the GA mutation.

Figure 16 shows the distribution of solutions after 50 runs for each mutation type, where no clear solution pattern is observed. This, along with the evaluation of solution evolution, indicates the need for a larger number of generations and black widows in the population to achieve greater consistency in the results.

Figure 17 contains histograms showing a greater diversity in the BW mutation, with a preference for 25 MVA electrical substations. In contrast, the GA mutation concentrated on electrical substations with lower installed capacity, primarily 17.5 and 25 MVA.

Table 9. Cost values obtained after 50 runs for Scenario D.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW	20.95	22.29	1.49
GA	19.96	21.69	0.88

presents a summary of the costs for Scenario D. The GA mutation showed better results, achieving an overall cost 4.74% lower than that of the BW mutation, which represents a reduction of USD 1 million over 10 years in this scenario. This pattern is consistent in the averages, with the GA mutation producing an average 2.67% lower than the average of the optimal results from the BW mutation.

Scenario D with 200 Generations and Black Widows

Since consistency in the solutions was not observed, additional test rounds were conducted, increasing the number of black widows and generations from 100 to 200 in an attempt to obtain better results. Figure 18 presents the optimal solutions found after the rounds with 200 black widows and 200 generations, along with the optimal solution for 100 black widows and 100 generations for comparison.

Once again, the BW mutation favored a solution with only one new electrical substation, while the GA mutation produced a solution with two new electrical substations as the optimal result. Despite the GA solution having USD 1.6 million higher electrical substation installation costs, the savings in new feeder installations and reduced losses outweigh this additional investment, as shown in

Table 10. Costs of the optimal solution found in Scenario D with 200 generations and black widows.

Cost	Value (mi USD)
	BW Mutation—200 Generations and Black Widows	GA Mutation—200 Generations and Black Widows	Δ (GA–BW)
New electrical substations	2.00	3.60	1.60
New feeders	6.06	4.75	−1.32
Power losses	12.48	11.44	−1.04
Total value	20.55	19.79	−0.75

.

Table 11. Costs of the optimal solution found in Scenario D with 100 and 200 generations and black widows for the GA mutation.

Cost	Value (mi USD)
	GA Mutation—100 Generations and Black Widows	GA Mutation—200 Generations and Black Widows	Δ (GA 200—GA 100)
New electrical substations	3.20	3.60	0.40
New feeders	5.01	4.75	−0.26
Power losses	11.75	11.44	−0.31
Total value	19.96	19.79	−0.17

details the costs of the optimal solution previously found compared to the solution obtained with 100 black widows and generations. The new optimal solution has higher electrical substation installation costs due to selecting a larger installed capacity. However, the electrical substations are better positioned, making them advantageous in terms of the overall cost.

Table 12. Cost values obtained after 50 runs for Scenario D with 100 and 200 generations and black widows for the BW and GA mutations.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW—100 generations and black widows	20.95	22.29	1.49
BW—200 generations and black widows	20.55	21.48	0.73
GA—100 generations and black widows	19.96	21.69	0.88
GA—200 generations and black widows	19.79	21.34	0.73

provides a summary of the results from the new test rounds and compares them with the previous results. As expected, the algorithm found lower costs and a smaller standard deviation with the increased number of generations and black widows.

Figure 19 shows the dispersion of the positions of the new electrical substations suggested by each mutation after 50 test rounds. It can be observed that the solutions are more organized after increasing the number of generations and black widows. This behavior indicates greater convergence in the algorithm, resulting in solutions that are closer to one another.

3.5. Scenario D with Lower Installation Cost of Electrical Substations

To test more complex solutions from the algorithm and compare the performance of the mutations, Scenario D was used with the installation cost of new electrical substations set to 20% of the initially assigned cost. This adjustment was made with the expectation that the algorithm would propose several new electrical substations. Figure 20 shows the optimal solution found by each mutation, where three new electrical substations were suggested in both cases, demonstrating the algorithm’s performance in identifying a more complex solution.

Table 13. Costs of the optimal solution found in Scenario D with reduced electrical substation costs.

Cost	Value (mi USD)
	BW Mutation	GA Mutation	Δ (GA–BW)
New electrical substations	2.08	1.36	−0.72
New feeders	5.18	4.64	−0.54
Power losses	9.50	9.35	−0.15
Total value	16.76	15.35	−1.41

details the costs of the optimal solution found by each mutation, where the GA mutation’s solution showed lower costs across all criteria. It demonstrated better performance in selecting the installed capacity of the new electrical substations, resulting in USD 0.72 million less in costs for this specific category.

Figure 21 shows the evolution graph of a population for each mutation, where once again, complete convergence of the GA mutation’s solution is not observed.

Figure 22 shows the dispersion of the solutions after 50 rounds, where no clear solution pattern is identified. This supports the discussion on the number of generations and black widows used.

Figure 23 presents the histograms of the installed capacity of the new electrical substations for each mutation, where the BW mutation favored higher installed capacities, and the GA mutation favored lower capacities. This behavior can be explained by the more efficient resource allocation achieved by the GA mutation, resulting in the better solutions found by it.

Table 14. Cost values obtained after 50 runs for Scenario D with reduced electrical substation costs.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW	16.76	19.93	1.18
GA	15.35	18.30	1.21

provides a summary of the 50 optimal solutions for each mutation, with the GA mutation achieving a solution that was 8.4% lower in cost than the optimal solution found by the BW mutation. When looking at the averages, it is again evident that the GA mutation performed better, with an average 8.2% lower than the average of the optimal solutions found by the BW mutation.

Scenario D with Reduced Installation Cost of Electrical Substations and with 200 Generations and Black Widows

As with Scenario D, additional test rounds were conducted, increasing the number of generations and black widows to 200, along with the reduced electrical substation installation costs. The optimal solutions found after 50 rounds are shown in Figure 24.

The GA mutation once again provided a better solution than the BW mutation. Although both resulted in three new electrical substations, the GA mutation achieved better positioning of the electrical substations, leading to a significant cost difference in new feeders.

Table 15. Costs of the optimal solution found in Scenario D with reduced electrical substation costs and with 200 generations and black widows.

Cost	Value (mi USD)
	BW Mutation—200 Generations and Black Widows	GA Mutation—200 Generations and Black Widows	Δ (GA–BW)
New electrical substations	1.60	1.20	−0.40
New feeders	6.06	3.44	−2.62
Power losses	9.00	9.57	0.57
Total value	16.66	14.21	−2.46

details the costs associated with the optimal solution from each mutation, with the GA mutation finding a solution approximately USD 2.46 million less expensive than that found with the BW mutation.

Table 16. Costs of the optimal solution found in Scenario D with reduced electrical substation costs and with 100 and 200 generations and black widows for the GA mutation.

Cost	Value (mi USD)
	GA Mutation—100 Generations and Black Widows	GA Mutation—200 Generations and Black Widows	Δ (GA 200—GA 100)
New electrical substations	1.36	1.20	−0.16
New feeders	4.64	3.44	−1.20
Power losses	9.35	9.57	0.23
Total value	15.35	14.21	−1.14

compares the optimal solutions with 100 and 200 generations and black widows, where once again the difference in the installation cost of new feeders is observed. Despite the higher cost in losses, this solution achieved a total cost nearly USD 1 million lower.

Figure 25 confirms the improved convergence when increasing the number of generations and black widows, particularly for the GA mutation, with solutions that are closer to one another, highlighting areas of interest identified by the algorithm for the installation of new electrical substations.

The summary of the test rounds with different mutations and numbers of generations and black widows is presented in

Table 17. Cost values obtained after 50 runs for Scenario D with reduced electrical substation costs and with 100 and 200 generations and black widows for the BW and GA mutation.

Mutation	Optimal Value (mi USD)	Average Value (mi USD)	Standard Deviation (mi USD)
BW—100 generations and black widows	16.76	19.93	1.18
BW—200 generations and black widows	16.66	19.37	0.88
GA—100 generations and black widows	15.35	18.30	1.21
GA—200 generations and black widows	14.21	17.63	1.65

, where the results observed for the optimal solution are also reflected in the averages, with the GA mutation showing better performance.

3.6. Discussion of Results

Based on the presented results, it is possible to verify the usability and feasibility of the algorithm for the allocation and positioning of distribution electrical substations. The algorithm found good results with solid consistency in scenarios similar to the real-world vegetative growth of cities, such as Scenarios A, B, and C.

Table 18. Comparison of the optimal solutions found in each scenario.

Scenario	Value (mi USD)
	BW Mutation	GA Mutation	Δ (GA–BW)
A	13.42	13.42	-
B	13.39	13.39	-
C	13.29	13.28	−0.08 %
D	20.95	19.96	−4.73 %
D—200	20.55	19.79	−3.67%
D mod.	16.76	15.35	−8.41%
D mod.—200	16.66	14.21	−14.75%

summarizes the optimal results found in each scenario and the percentage difference between the results of each mutation (GA—BW). The use of the GA mutation proved superior to the BW mutation, particularly as the complexity of the scenarios increased, achieving a solution 14.75% lower in the most complex case and with a higher likelihood of convergence. The total execution time for each combination of scenario and mutation ranged from 7 min and 42 s to 34 min and 18 s.

Regarding the types of mutations analyzed, there were no significant differences in results when changing the mutation method in Scenarios A, B, and C. With averages close to the optimal solutions found and showing consistency in the dispersion of the electrical substation locations, both mutations arrived at the optimal solution in Scenarios A and B. In Scenario C, only one solution from the GA mutation achieved a lower cost with a non-obvious solution.

In Scenario D, however, the GA mutation showed a considerable advantage in the results, being approximately 4% better regardless of the number of generations and black widows. This advantage was further reinforced in Scenario D with reduced electrical substation installation costs, where the optimal solution was 8.41% lower for 100 generations and black widows, and up to 14.75% lower for 200. This demonstrates that for more complex optimal solutions, the GA mutation is capable of finding better results than the original black widow mutation.

This behavior can be explained by the fact that the GA mutation introduces completely new values into the black widows, whereas the BW mutation only swaps values, giving the GA mutation a greater ability to escape local optima.

This superiority is further confirmed when observing the average results, as the average reduces the effects of randomness in the method. The GA mutation achieved averages between 8% and 9% lower than the averages from the BW mutation in the modified Scenario D, as shown in

Table 19. Comparison of the average solutions found in each scenario.

Scenario	Value (mi USD)
	BW Mutation	GA Mutation	Δ (GA–BW)
A	13.75	13.71	−0.23%
B	13.47	13.39	−0.58%
C	13.31	13.30	−0.09%
D	22.29	21.69	−2.68%
D—200	21.48	21.34	−0.66%
D mod.	19.93	18.30	−8.17%
D mod.—200	19.37	17.63	−9.00%

.

It is still possible to observe the influence of increasing the number of generations and black widows in more complex scenarios, where both mutation types showed improvement in the solutions and greater consistency in the dispersion maps, indicating a higher level of convergence as a result of the increase in generations and black widows.

Given the importance of distribution expansion planning and the magnitude of the investments involved, it is essential to run the algorithm multiple times to ensure a broader search of the solution space, as well as to carefully choose the number of generations and population size. This approach increases the likelihood that the optimal solution found is truly an optimal one.

4. Conclusions

This paper applied an enhanced Black Widow Algorithm (BWA) to solve the complex problem of allocating and sizing electrical substations within power distribution networks. The proposed modification to the BWA, a mutation approach inspired by genetic algorithms (GA), enabled the BWA to consistently deliver optimal or near-optimal results across a variety of test scenarios, demonstrating its adaptability and robustness in handling real-world constraints.

The performance comparison between the original BWA mutation and the GA-inspired mutation revealed that the latter performed significantly better in more complex scenarios, achieving cost reductions of up to 14.75%. The GA mutation showed greater flexibility in escaping local minima, which was particularly advantageous in cases involving numerous new load centers and greater complexity in the distribution of loads. Additionally, increasing the number of generations and the size of the population proved effective in enhancing convergence and solution stability.

The findings underscore the practicality of the enhanced BWA in real-world applications of electrical substation planning. The improved flexibility of the GA-inspired mutation allowed the algorithm to handle the intricate demands of substation allocation, providing substantial cost savings and better solutions in complex scenarios. The paper emphasizes the importance of carefully choosing algorithm parameters such as population size and the number of generations to achieve more consistent and reliable results.