# Application of the Gradient-Based Metaheuristic Optimizerto Solve the Optimal Conductor Selection Problemin Three-Phase Asymmetric Distribution Networks

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review and Contributions

## 3. Mathematical Formulation

#### 3.1. Objective Function

#### 3.2. Set of Constraints

#### 3.3. Solution Space Analysis

## 4. Solution Methodology

#### 4.1. Slave Stage: Three-Phase Power Flow

**Note**

**1.**

#### 4.2. Master Stage: MGbMO Metaheuristic Algorithm

#### 4.3. Implementation of the MGbMO

Algorithm 1: Implementation of the MGbMO algorithm |

Data: Select the AC electrical network under study;Obtain the equivalent in values per unit of the distribution system; Define the maximum number of iterations (${t}_{max}$); Define the size of the initial population (${N}_{i}$); Generate the initial population ${X}^{t}$ and make $t=0$; Define the solution space’s initial center (${\mu}_{0}$); $\mathbf{for}$
$t\le {t}_{max}$
$\mathbf{do}$ Obtain the value of the objective function for each of the individuals of the initial population ${X}_{i}^{t}$ with the help of the slave stage; Find the best current solution ${X}_{best}^{t}$; Generate a random number between 0 and 1 for $\mathit{e}$; $\mathbf{if}$ $e<\frac{1}{2}$ $\mathbf{then}$ $\mathbf{for}$ $i=1:{N}_{i}$ $\mathbf{do}$ Apply the evolution rule of Equation (19); Verify and adjust the values of ${X}_{i}^{t}$ so that it is within its limits; Evaluate ${X}_{i}^{t}$ in the slave stage to find the value of the objective function; $\mathbf{end}$ Update ${X}_{best}^{t+1}$ to the current best; $\mathbf{else}$ Calculate the radius ${r}_{t}$ as expressed in (24); Generate the descendant population ${X}_{i}^{t}$ using Equation (21); Verify and adjust the values of ${X}_{i}^{t}$ so that it is within its limits; $\mathbf{for}$ $i=1:{N}_{i}$ $\mathbf{do}$ Evaluate ${X}_{i}^{t}$ in the slave stage to find the value of the objective function; $\mathbf{end}$ Update ${X}_{best}^{t+1}$ to the current best; $\mathbf{end}$ $\mathbf{end}$ Show the best solution ${X}_{best}^{{t}_{max}}$; |

## 5. Characteristics of the Test Systems

#### 5.1. General Parameters

#### 5.2. 8-Bus Test System

#### 5.3. 27-Bus Test System

## 6. Results and Discussion

^{®}R2021a software. The computer has an Intel (R) Core (TM) i7-8565U CPU @1.80 GHz 1.99 GHz, 8 Gb RAM (Intel, Santa Clara, CA, USA), and Windows 11 Home operating system of ×64 bits. The results are analyzed with the help of Microsoft Power BI Desktop

^{®}Version 2.109.642.0 ×64 bits (Redmond, WA, USA).

#### 6.1. Results in the 8-Bus Test System

#### 6.1.1. Balanced Case for the 8-Bus Test System

- The MGbMO algorithm obtained the best numerical result for the balanced 8-bus system, yielding annual costs of USD 455,969,791, with USD 227,826 in the investment of the conductors and USD 228,143,791 in the costs associated with the losses.
- The result obtained by the MGbMO algorithm improved by 10.3% compared to the VSA and NMA algorithms, which obtained the second-best solutions with USD 508,357.959 for annual costs, which represents a reduction of USD 52,388.2 for the study case.
- The cost of the investment of the conductors, according to the set of gauges selected by the MGbMO algorithm, presents the highest cost compared to the rest of the algorithms; however, the cost of annual losses is lower, which thus ends up compensating for the high costs in the investment of the conductors.

#### 6.1.2. Unbalanced Case for the 8-Bus Test System

- The MGbMO algorithm equaled the results of the VSA and NMA algorithms for the unbalanced 8-bus system, yielding annual costs of USD 558,758,394, with USD 289,713 in the investment of the conductors and USD 269,045,394 in the costs associated with the losses.

#### 6.2. Results in the 27-Bus Test System

#### 6.2.1. Balanced Case for the 27-Bus Test System

- The MGbMO algorithm obtained the best numerical result for the balanced 27-bus system, yielding annual costs of USD 549,883.572, with USD 319,768.08 in the investment of the conductors and USD 230,115.492 in the costs associated with losses.
- The result obtained by the MGbMO algorithm improved by 1.4% compared to the NMA algorithm, which obtained the second-best solution with USD 557,695.255 for annual costs, representing a reduction of USD 7811.68 for the case study.
- The cost of the investment of the conductors, according to the set of gauges selected by the MGbMO algorithm, presents the lowest cost compared to the rest of the algorithms, thus increasing the costs of annual losses slightly. In the end, the costs are compensated, as shown in the annual costs, allowing a lower value in the objective function.

#### 6.2.2. Unbalanced Case for the 27-Bus Test System

- The MGbMO algorithm obtained the best numerical result for the unbalanced 27-bus system, yielding annual costs of USD 589,018.8, with USD 331,828.08 in the investment of the conductors and USD 257,190.72 in the costs associated with the losses.
- The result obtained by the MGbMO algorithm improved by 1.432% compared to the NMA algorithm, which obtained the second-best solution with USD 597,579.008 for annual costs, thus representing a reduction of USD 8560.21 for the case study.
- The cost of the investment of the conductors, according to the set of gauges selected by the MGbMO algorithm, presents the lowest cost compared to the rest of the algorithms, thus increasing the costs of annual losses slightly but without exceeding the costs presented by the VSA version. These costs are compensated in the end, as shown in the annual costs, allowing a lower value in the objective function.

#### 6.3. Performance of the Algorithm

#### 6.4. A Daily Operation Scenario

- Case 1: The evaluation of the optimization methodology during the peak load condition, i.e., under the same simulation conditions used for the 8- and 27-bus grids.
- Case 2: The selection of the conductors only considers the daily load variations without penetration of renewable generation.
- Case 3: The evaluation of the optimization methodology considering the daily demand and generation curves.

- i.
- Case 1 is an operation scenario that invests more in conducting material to minimize the effect of the energy loss costs in the objective function as soon as possible. This is accomplished by increasing the conductor sizes in some strategic distribution lines, which reduces the energy loss since these are proportional to the resistive parameter and the square value of the current. Additionally, the expected energy loss costs for this scenario can be far from the real operative scenario for a distribution grid where all the demands vary along the day.
- ii.
- Case 2 demonstrates a more realistic operative scenario for medium-voltage distribution networks, where the energy consumption varies along the day, considering a typical demand curve. In this case, the expected energy losses are about USD 312,264.9263 per year of operation, which implies a reduction of about USD 91,652.7653 concerning the peak load operation scenario. This implies that it is highly probable for a distribution company that the energy loss costs will be near the daily demand variation scenario compared to the peak load condition. However, the most important result in simulation Case 2 is the reduction in the investment costs concerning the peak load scenario of about 40.0691%. This implies that when the energy calculation is more realistic, the algorithm finds conductors with small sizes (see Cases 1 and 2 in Table 16), which can be considered a more convenient design scenario for the distribution company.
- iii.
- Case 3 presents the well-known benefits of the usage of renewable generation in distribution networks since these allow important reductions in the expected energy loss costs. This scenario showcases that the energy loss costs are reduced by about USD 62,738.9098 with respect to Case 2. Additionally, the inclusion of renewables has reduced the investment costs in conducting material by about USD 27,179.0850 through the reduction in the conductor sizes in some strategy lines (lines near to the substation bus as can be seen for Cases 2 and 3 in Table 16).

## 7. Conclusions and Future Work

- i.
- The MGbMO algorithm implemented yields the best current solutions for three of the four systems studied, where for the balanced 8-bus system, an objective function of USD 455,969.791 is obtained, achieving a reduction of 10.3% over the value obtained with NMA, which has the second-best result; regarding the unbalanced case, the results of the other algorithms are matched with a value of USD 558,758.394, and it can, thus, be inferred that the optimal global solution is found (this affirmation is based on the small dimension of the solution space for the 8-bus grid, i.e., ${c}^{l}={8}^{7}$ = 2,097,152, which can be easily explored with any combinatorial optimization method or exhaustive search). In the balanced 27-bus system, a value of USD 549,883.572 is obtained as total annual costs, achieving an improvement of 1.4% compared to the NMA, while in the unbalanced case, a value of USD 589,018.800 is obtained, which is 1.432% more than in the case of the NMA.
- ii.
- For the four case studies, the findings ascertain that since the current levels that pass through the branches are far from their limits established by each gauge (less than 35%), the energy losses are low, and the loss costs are thus reduced as well, ensuring good operation and adapting to future new load connections.
- iii.
- The voltage profiles are adequate for all test systems. In this regard, the lowest and most distant one is evident in the 27-bus system for the balanced case, where a shift of 4.61% is presented in phase C, which is very low and positive for the system.
- iv.
- The MGbMO algorithm only needs a single evaluation in each test system to obtain the best numerical results, which are previously reported in this research, due to the exploration and exploitation characteristics of the solution space, which shortened simulation times, making this method one of the most efficient methods reported in the specialized literature.
- v.
- The numerical results obtained with the MGbMO in the three-phase version of the IEEE 85-bus grid indicate that the peak demand scenario is a simulation case where the energy losses costs are overestimated, which leads the optimization algorithm to increase the conductor sizes to find an adequate equilibrium between investment and operating costs. However, when daily demand curves and renewable generation are considered, the expected annual costs of the energy losses reduce significantly, allowing the MGbMO to find conductor sizes with lower costs in comparison to the peak scenario of operation. These results imply that considering daily demand curves is the most realistic scenario for the studied problem. Ultimately, this simulation case must be considered the benchmark case for any new study in this research area.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 12.**Convergence of the objective function for each test system (note that the objective function was normalized by using a dividing factor of 1000).

**Table 1.**Summary of the literature approaches used in the optimal selection of conductors for electrical distribution networks.

Sol. Methodology | Objective Function | Year | Ref. |
---|---|---|---|

A heuristic approach based on power flow solutions | Saving in cost of conducting material and cost of energy losses | 2002 | [14] |

A recursive power flow solution methodology | Minimizing investment and operating costs and improving voltage profile | 2005 | [15] |

A constructive algorithm based on power flow solutions | Saving in cost of conducting material and energy losses | 2006, 2010 | [16,24] |

Evolutionary optimization strategies | Minimizing investment and operating costs | 2006 | [22] |

Optimization with genetic algorithms | Saving in cost of conducting material and active and minimizing reactive power loss | 2011, 2013 | [18,19] |

Differential evolution algorithm | Saving in cost of conducting material and energy losses | 2016 | [23] |

Adaptive optimizer based on genetic algorithms | Minimizing the sum of the life-cycle cost in conductors and the total energy procurement cost | 2019 | [20] |

Grasshopper optimization algorithm | Minimizing the annual cost of energy loss and investment cost of the conductors | 2018 | [21] |

Mixed-integer non-linear programming optimization method | Minimizing investment and operating costs | 2012 | [5] |

Crow search algorithm | Minimizing investment and operating costs | 2017 | [10] |

Sine cosine algorithm | Minimizing investment and operating costs | 2017 | [30] |

Vortex search algorithm | Minimizing investment and operating costs | 2021 | [29] |

Tabu search algorithm | Minimizing investment and operating costs | 2021 | [31] |

Newton-based metaheuristic algorithm | Minimizing investment and operating costs | 2022 | [32] |

Variable Name | Number of Variables | Variable Name | Number of Variables |
---|---|---|---|

Voltages | $3n$ | Angles | $3n$ |

Currents | $3l$ | Dis. Gen. Powers | $3d$ |

Objective func. | 1 | Slack powers | 3 |

Gauge (c) | $\mathit{r}$ ($\mathsf{\Omega}$/km) | $\mathit{x}$ ($\mathsf{\Omega}$/km) | ${\mathit{I}}^{\mathit{c},\mathit{m}\mathit{a}\mathit{x}}$ (A) | ${\mathit{C}}^{\mathit{c}}$ (USD/km) |
---|---|---|---|---|

1 | 0.8763 | 0.4133 | 180 | 1986 |

2 | 0.6960 | 0.4133 | 200 | 2790 |

3 | 0.5518 | 0.4077 | 230 | 3815 |

4 | 0.4387 | 0.3983 | 270 | 5090 |

5 | 0.3480 | 0.3899 | 300 | 8067 |

6 | 0.2765 | 0.3610 | 340 | 12,673 |

7 | 0.0966 | 0.1201 | 600 | 23,419 |

8 | 0.0853 | 0.0950 | 720 | 30,070 |

Parameter | Value | Unit |
---|---|---|

Energy cost | 0.1390 | (USD/kWh) |

Iterations | 1000 | - |

Population size | 30 | - |

Tolerance | $1\times {10}^{-10}$ | - |

**Table 5.**Connection of lines and load levels of the system for the balanced case in the 8-bus test system.

Line | Bus $\mathit{i}$ | Bus $\mathit{j}$ | ${\mathit{L}}_{\mathbf{ij}}$ (km) | ${\mathit{P}}_{\mathit{j},\mathit{h}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{h}}^{\mathit{D}}$ (kvar) |
---|---|---|---|---|---|

1 | 1 | 2 | 1.00 | 1054.2 | 0 |

2 | 2 | 3 | 1.00 | 806.5 | 0 |

3 | 1 | 4 | 1.00 | 2632.5 | 0 |

4 | 1 | 5 | 1.00 | 609 | 0 |

5 | 5 | 6 | 1.00 | 2034.5 | 0 |

6 | 3 | 7 | 1.00 | 932.8 | 0 |

7 | 3 | 8 | 1.00 | 1731.4 | 0 |

Bus $\mathit{j}$ | ${\mathit{P}}_{\mathit{j},\mathit{a}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{a}}^{\mathit{D}}$ (kvar) | ${\mathit{P}}_{\mathit{j},\mathit{b}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{b}}^{\mathit{D}}$ (kvar) | ${\mathit{P}}_{\mathit{j},\mathit{c}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{c}}^{\mathit{D}}$ (kvar) |
---|---|---|---|---|---|---|

2 | 3162.6 | 0 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 2419.5 | 0 | 0 | 0 |

4 | 0 | 0 | 0 | 0 | 7897.5 | 0 |

5 | 913.5 | 0 | 913.5 | 0 | 0 | 0 |

6 | 0 | 0 | 3051.6 | 0 | 3051.6 | 0 |

7 | 2798.4 | 0 | 0 | 0 | 0 | 0 |

8 | 1298.55 | 0 | 2597.1 | 0 | 1298.55 | 0 |

**Table 7.**Connection of lines and load levels of the system for the balanced case in the 27-bus test system.

Line | Bus $\mathit{i}$ | Bus $\mathit{j}$ | ${\mathit{L}}_{\mathbf{ij}}$ (km) | ${\mathit{P}}_{\mathit{j},\mathit{h}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{h}}^{\mathit{D}}$ (kvar) |
---|---|---|---|---|---|

1 | 1 | 2 | 0.55 | 0 | 0 |

2 | 2 | 3 | 1.50 | 0 | 0 |

3 | 3 | 4 | 0.45 | 297.5 | 184.4 |

4 | 4 | 5 | 0.63 | 0 | 0 |

5 | 5 | 6 | 0.70 | 255 | 158 |

6 | 6 | 7 | 0.55 | 0 | 0 |

7 | 7 | 8 | 1.00 | 212.5 | 131.7 |

8 | 8 | 9 | 1.25 | 0 | 0 |

9 | 9 | 10 | 1.00 | 266.1 | 164.9 |

10 | 2 | 11 | 1.00 | 85 | 52.7 |

11 | 11 | 12 | 1.23 | 340 | 210.7 |

12 | 12 | 13 | 0.75 | 297.5 | 184.4 |

13 | 13 | 14 | 0.56 | 191.3 | 118.5 |

14 | 14 | 15 | 1.00 | 106.3 | 65.8 |

15 | 15 | 16 | 1.00 | 255 | 158 |

16 | 3 | 17 | 1.00 | 255 | 158 |

17 | 17 | 18 | 0.60 | 127.5 | 79 |

18 | 18 | 19 | 0.90 | 297.5 | 184.4 |

19 | 19 | 20 | 0.95 | 340 | 210.7 |

20 | 20 | 21 | 1.00 | 85 | 52.7 |

21 | 4 | 22 | 1.00 | 106.3 | 65.8 |

22 | 5 | 23 | 1.00 | 55.3 | 34.2 |

23 | 6 | 24 | 0.40 | 69.7 | 43.2 |

24 | 8 | 25 | 0.60 | 255 | 158 |

25 | 8 | 26 | 0.60 | 63.8 | 39.5 |

26 | 26 | 27 | 0.80 | 170 | 105.4 |

Bus $\mathit{j}$ | ${\mathit{P}}_{\mathit{j},\mathit{a}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{a}}^{\mathit{D}}$ (kvar) | ${\mathit{P}}_{\mathit{j},\mathit{b}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{b}}^{\mathit{D}}$ (kvar) | ${\mathit{P}}_{\mathit{j},\mathit{c}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{c}}^{\mathit{D}}$ (kvar) |
---|---|---|---|---|---|---|

2 | 0 | 0 | 0 | 0 | 0 | 0 |

3 | 0 | 0 | 0 | 0 | 0 | 0 |

4 | 892.5 | 553.2 | 0 | 0 | 0 | 0 |

5 | 0 | 0 | 0 | 0 | 0 | 0 |

6 | 0 | 0 | 765 | 474 | 0 | 0 |

7 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 0 | 0 | 0 | 0 | 637.5 | 395.1 |

9 | 0 | 0 | 0 | 0 | 0 | 0 |

10 | 0 | 0 | 0 | 0 | 798.3 | 494.7 |

11 | 0 | 0 | 255 | 158.1 | 0 | 0 |

12 | 1020 | 632.1 | 0 | 0 | 0 | 0 |

13 | 446.25 | 276.6 | 446.25 | 276.6 | 0 | 0 |

14 | 0 | 0 | 286.95 | 177.75 | 286.95 | 177.75 |

15 | 159.45 | 98.7 | 0 | 0 | 159.45 | 98.7 |

16 | 0 | 0 | 382.5 | 237 | 382.5 | 237 |

17 | 1 | 0 | 765 | 474 | 0 | 0 |

18 | 382.5 | 237 | 0 | 0 | 0 | 0 |

19 | 446.25 | 276.6 | 446.25 | 276.6 | 0 | 0 |

20 | 0 | 0 | 510 | 316.05 | 510 | 316.05 |

21 | 127.5 | 79.05 | 0 | 0 | 127.5 | 79.05 |

22 | 0 | 0 | 159.75 | 98.7 | 159.75 | 98.7 |

23 | 165.9 | 102.6 | 0 | 0 | 0 | 0 |

24 | 0 | 0 | 0 | 0 | 209.1 | 129.6 |

25 | 255 | 158 | 255 | 158 | 255 | 158 |

26 | 63.8 | 39.5 | 63.8 | 39.5 | 63.8 | 39.5 |

27 | 170 | 105.4 | 170 | 105.4 | 170 | 105.4 |

Method | Gauges | Investment in Conductors (USD) | Losses (USD) | Annual Costs (USD) |
---|---|---|---|---|

TGA | {6, 5, 3, 4, 4, 1, 4} | 125,433 | 406,222.461 | 531,655.461 |

CBGA | {6, 6, 4, 4, 4, 1, 4} | 143,076 | 373,155.965 | 516,231.965 |

GAMS | {6, 4, 4, 5, 4, 1, 2} | 122,358 | 416,681.580 | 539,039.580 |

TSA | {6, 5, 4, 4, 4, 1, 3} | 125,433 | 397,754.442 | 523,187.442 |

VSA | {6, 6, 5, 5, 4, 2, 4} | 163,350 | 345,007.959 | 508,357.959 |

NMA | {6, 6, 5, 5, 4, 2, 4} | 163,350 | 345,007.959 | 508,357.959 |

MGbMO | {7, 7, 5, 5, 4, 2, 4} | 227,826 | 228,143.791 | 455,969.791 |

Method | Gauges | Investment in Conductors (USD) | Losses (USD) | Annual Costs (USD) |
---|---|---|---|---|

VSA | {7, 7, 7, 5, 5, 4, 4} | 289,713 | 269,045.394 | 558,758.394 |

NMA | {7, 7, 7, 5, 5, 4, 4} | 289,713 | 269,045.394 | 558,758.394 |

MGbMO | {7, 7, 7, 5, 5, 4, 4} | 289,713 | 269,045.394 | 558,758.394 |

Method | Gauges | Investment in Conductors (USD) | Losses (USD) | Annual Costs (USD) |
---|---|---|---|---|

VSA | {7, 7, 5, 4, 4, 3, 3, 1, 1, 4, 4, 2, 3, 2, 1, 4, 4, 2, 2, 2, 1, 1, 2, 2, 1, 1} | 344,352.150 | 217,672.327 | 562,024.477 |

NMA | {7, 7, 4, 4, 4, 4, 3, 1, 1, 4, 4, 3, 3, 1, 2, 4, 3, 2, 1, 1, 1, 1, 2, 2, 1, 1} | 337,744.800 | 219,950.455 | 557,695.255 |

MGbMO | {7, 7, 4, 4, 4, 3, 3, 1, 1, 4, 4, 2, 1, 1, 1, 3, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1} | 319,768.08 | 230,115.492 | 549,883.572 |

Method | Gauges | Investment in Conductors (USD) | Losses (USD) | Annual Costs (USD) |
---|---|---|---|---|

VSA | {7, 7, 5, 4, 4, 4, 4, 2, 2, 4, 4, 3, 2, 1, 1, 2, 3, 2, 1, 2, 2, 1, 2, 2, 4, 1} | 350,392.95 | 257,999.185 | 608,392.135 |

NMA | {7, 7, 4, 4, 4, 3, 4, 2, 1, 4, 4, 4, 2, 1, 1, 4, 3, 2, 2, 1, 1, 1, 2, 2, 2, 1} | 344,954.40 | 252,624.608 | 597,579.008 |

MGbMO | {7, 7, 4, 4, 4, 4, 4, 1, 1, 4, 4, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1} | 331,828.08 | 257,190.720 | 589,018.800 |

Line | Bus $\mathit{i}$ | Bus $\mathit{j}$ | ${\mathit{L}}_{\mathbf{ij}}$ (km) | Line | Bus $\mathit{i}$ | Bus $\mathit{j}$ | ${\mathit{L}}_{\mathbf{ij}}$ (km) |
---|---|---|---|---|---|---|---|

1 | 1 | 2 | 0.468 | 43 | 34 | 44 | 0.985 |

2 | 2 | 3 | 0.449 | 44 | 44 | 45 | 0.441 |

3 | 3 | 4 | 0.323 | 45 | 45 | 46 | 0.225 |

4 | 4 | 5 | 0.870 | 46 | 46 | 47 | 0.184 |

5 | 5 | 6 | 0.883 | 47 | 35 | 48 | 0.698 |

6 | 6 | 7 | 0.343 | 48 | 48 | 49 | 0.875 |

7 | 7 | 8 | 0.953 | 49 | 49 | 50 | 0.768 |

8 | 8 | 9 | 0.309 | 50 | 50 | 51 | 0.261 |

9 | 9 | 10 | 0.931 | 51 | 48 | 52 | 0.985 |

10 | 10 | 11 | 0.161 | 52 | 52 | 53 | 0.448 |

11 | 11 | 12 | 0.952 | 53 | 53 | 54 | 0.344 |

12 | 12 | 13 | 0.878 | 54 | 52 | 55 | 0.673 |

13 | 13 | 14 | 0.507 | 55 | 49 | 56 | 0.564 |

14 | 14 | 15 | 0.018 | 56 | 9 | 57 | 0.545 |

15 | 2 | 16 | 0.778 | 7 | 57 | 58 | 0.564 |

16 | 3 | 17 | 0.359 | 58 | 58 | 59 | 0.650 |

17 | 5 | 18 | 0.848 | 59 | 58 | 60 | 0.237 |

18 | 18 | 19 | 0.433 | 60 | 60 | 61 | 0.626 |

19 | 19 | 20 | 0.605 | 61 | 61 | 62 | 0.530 |

20 | 20 | 21 | 0.377 | 62 | 60 | 63 | 0.514 |

21 | 21 | 22 | 0.721 | 63 | 63 | 64 | 0.701 |

22 | 19 | 23 | 0.316 | 64 | 64 | 65 | 0.373 |

23 | 7 | 4 | 0.949 | 65 | 65 | 66 | 0.614 |

24 | 8 | 5 | 0.849 | 66 | 64 | 67 | 0.307 |

25 | 25 | 26 | 0.775 | 67 | 67 | 68 | 0.322 |

26 | 26 | 27 | 0.299 | 68 | 68 | 69 | 0.662 |

27 | 27 | 28 | 0.872 | 69 | 69 | 70 | 0.150 |

28 | 28 | 29 | 0.375 | 70 | 70 | 71 | 0.384 |

29 | 29 | 30 | 0.798 | 71 | 67 | 72 | 0.320 |

30 | 30 | 31 | 0.816 | 72 | 68 | 73 | 0.291 |

31 | 31 | 32 | 0.999 | 73 | 73 | 74 | 0.213 |

32 | 32 | 33 | 0.045 | 74 | 73 | 75 | 0.721 |

33 | 33 | 34 | 0.236 | 75 | 70 | 76 | 0.705 |

34 | 34 | 35 | 0.430 | 76 | 65 | 77 | 0.715 |

35 | 35 | 36 | 0.847 | 77 | 10 | 78 | 0.361 |

36 | 26 | 37 | 0.386 | 78 | 67 | 79 | 0.546 |

37 | 27 | 38 | 0.122 | 79 | 12 | 80 | 0.357 |

38 | 29 | 39 | 0.915 | 80 | 80 | 81 | 0.940 |

39 | 32 | 40 | 0.182 | 81 | 81 | 82 | 0.444 |

40 | 40 | 41 | 0.590 | 82 | 81 | 83 | 0.110 |

41 | 41 | 42 | 0.812 | 83 | 83 | 84 | 0.865 |

42 | 41 | 43 | 0.261 | 84 | 13 | 85 | 0.300 |

Bus $\mathit{j}$ | ${\mathit{P}}_{\mathit{j},\mathit{a}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{a}}^{\mathit{D}}$ (kvar) | ${\mathit{P}}_{\mathit{j},\mathit{b}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{b}}^{\mathit{D}}$ (kvar) | ${\mathit{P}}_{\mathit{j},\mathit{c}}^{\mathit{D}}$ (kW) | ${\mathit{Q}}_{\mathit{j},\mathit{c}}^{\mathit{D}}$ (kvar) |
---|---|---|---|---|---|---|

2 | 14.50 | 20.00 | 10.00 | 28.50 | 20.50 | 31.00 |

3 | 12.50 | 29.50 | 0.000 | 0.000 | 17.50 | 48.50 |

4 | 0.000 | 0.000 | 0.000 | 0.000 | 35.50 | 43.00 |

5 | 39.00 | 43.50 | 32.50 | 48.50 | 0.000 | 0.000 |

6 | 28.00 | 22.00 | 22.50 | 24.50 | 0.000 | 0.000 |

7 | 14.00 | 10.50 | 0.000 | 0.000 | 24.50 | 42.50 |

8 | 41.50 | 29.50 | 23.50 | 12.00 | 10.50 | 27.00 |

9 | 0.000 | 0.000 | 30.00 | 40.50 | 17.00 | 50.00 |

10 | 0.000 | 0.000 | 14.00 | 45.00 | 0.000 | 0.000 |

11 | 0.000 | 0.000 | 26.50 | 14.00 | 27.50 | 35.50 |

12 | 35.50 | 49.00 | 45.50 | 47.50 | 0.000 | 0.000 |

13 | 17.00 | 43.00 | 15.00 | 47.50 | 48.00 | 37.50 |

14 | 25.00 | 45.00 | 22.00 | 44.00 | 0.000 | 0.000 |

15 | 0.000 | 0.000 | 0.000 | 0.000 | 10.50 | 20.00 |

16 | 19.00 | 39.00 | 47.50 | 32.00 | 35.50 | 18.50 |

17 | 18.50 | 44.50 | 49.00 | 26.00 | 32.50 | 26.50 |

18 | 0.000 | 0.000 | 11.00 | 44.00 | 48.50 | 34.50 |

19 | 38.00 | 31.00 | 35.00 | 39.00 | 42.50 | 25.50 |

20 | 46.00 | 46.00 | 0.000 | 0.000 | 26.50 | 32.50 |

21 | 39.00 | 25.00 | 37.00 | 11.00 | 41.50 | 23.50 |

22 | 22.50 | 50.00 | 21.00 | 30.00 | 18.50 | 25.50 |

23 | 0.000 | 0.000 | 0.000 | 0.000 | 24.00 | 23.50 |

24 | 44.50 | 12.50 | 33.00 | 27.00 | 37.00 | 18.00 |

25 | 40.50 | 14.00 | 40.50 | 36.00 | 33.50 | 39.50 |

26 | 15.50 | 22.50 | 0.000 | 0.000 | 25.50 | 47.00 |

27 | 16.00 | 38.00 | 0.000 | 0.000 | 0.000 | 0.000 |

28 | 13.50 | 15.00 | 35.00 | 38.00 | 15.50 | 22.50 |

29 | 18.50 | 39.50 | 10.50 | 36.00 | 39.00 | 42.00 |

30 | 21.00 | 36.50 | 0.000 | 0.000 | 43.00 | 13.00 |

31 | 0.000 | 0.000 | 0.000 | 0.000 | 20.50 | 20.50 |

32 | 41.50 | 35.50 | 46.50 | 50.00 | 0.000 | 0.000 |

33 | 29.00 | 12.50 | 47.00 | 37.50 | 27.50 | 24.50 |

34 | 16.00 | 28.00 | 0.000 | 0.000 | 25.00 | 12.00 |

35 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

36 | 48.50 | 34.00 | 0.000 | 0.000 | 10.50 | 18.00 |

37 | 41.00 | 45.00 | 44.00 | 47.50 | 12.00 | 45.50 |

38 | 42.00 | 22.00 | 45.00 | 15.50 | 0.000 | 0.000 |

39 | 0.000 | 0.000 | 37.00 | 47.50 | 27.50 | 18.50 |

40 | 0.000 | 0.000 | 0.000 | 0.000 | 15.00 | 13.50 |

41 | 17.50 | 27.50 | 36.00 | 40.00 | 43.00 | 20.00 |

42 | 44.50 | 17.00 | 38.50 | 19.00 | 0.000 | 0.000 |

43 | 49.50 | 45.50 | 0.000 | 0.000 | 0.000 | 0.000 |

44 | 39.00 | 28.50 | 32.50 | 26.50 | 30.00 | 30.50 |

45 | 30.50 | 27.00 | 39.00 | 36.50 | 13.50 | 42.50 |

46 | 25.00 | 22.50 | 30.50 | 21.00 | 11.00 | 15.50 |

47 | 30.50 | 48.00 | 48.00 | 22.00 | 14.00 | 34.00 |

48 | 34.00 | 42.00 | 50.00 | 13.50 | 0.000 | 0.000 |

49 | 42.50 | 34.50 | 36.00 | 45.50 | 45.00 | 46.50 |

50 | 0.000 | 0.000 | 0.000 | 0.000 | 10.50 | 13.50 |

51 | 22.00 | 18.50 | 24.50 | 23.00 | 10.00 | 13.50 |

52 | 49.00 | 40.50 | 47.00 | 48.00 | 20.00 | 28.00 |

53 | 13.50 | 41.50 | 15.00 | 48.00 | 0.000 | 0.000 |

54 | 39.50 | 19.50 | 19.00 | 43.00 | 45.50 | 40.50 |

55 | 0.000 | 0.000 | 19.50 | 26.50 | 19.50 | 45.00 |

56 | 36.50 | 20.00 | 34.00 | 49.00 | 14.50 | 17.00 |

57 | 13.50 | 40.50 | 16.50 | 33.50 | 16.00 | 34.00 |

58 | 29.50 | 18.00 | 22.00 | 18.00 | 0.000 | 0.000 |

59 | 24.00 | 36.50 | 32.00 | 15.50 | 29.00 | 27.50 |

60 | 0.000 | 0.000 | 38.00 | 12.00 | 25.00 | 24.50 |

61 | 38.00 | 42.00 | 28.00 | 17.50 | 45.50 | 27.00 |

62 | 30.00 | 32.00 | 42.00 | 26.50 | 36.00 | 47.50 |

63 | 42.00 | 25.50 | 46.50 | 26.00 | 26.00 | 17.00 |

64 | 0.000 | 0.000 | 0.000 | 0.000 | 19.00 | 47.50 |

65 | 37.00 | 29.00 | 0.000 | 0.000 | 28.00 | 26.50 |

66 | 47.50 | 49.00 | 25.00 | 36.50 | 47.50 | 17.50 |

67 | 14.00 | 22.50 | 0.000 | 0.000 | 22.50 | 41.00 |

68 | 35.50 | 39.00 | 36.50 | 23.50 | 41.50 | 37.00 |

69 | 22.00 | 22.50 | 0.000 | 0.000 | 0.000 | 0.000 |

70 | 43.50 | 24.50 | 0.000 | 0.000 | 0.000 | 0.000 |

71 | 11.00 | 50.00 | 0.000 | 0.000 | 11.50 | 28.50 |

72 | 31.00 | 20.00 | 0.000 | 0.000 | 0.000 | 0.000 |

73 | 43.00 | 23.00 | 49.50 | 48.00 | 50.00 | 35.00 |

74 | 28.00 | 22.00 | 21.00 | 19.50 | 0.000 | 0.000 |

75 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

76 | 0.000 | 0.000 | 46.50 | 45.00 | 19.50 | 47.50 |

77 | 28.00 | 12.50 | 25.00 | 24.00 | 41.00 | 49.00 |

78 | 27.50 | 45.00 | 48.50 | 31.00 | 29.50 | 38.50 |

79 | 32.50 | 35.00 | 48.50 | 18.50 | 0.000 | 0.000 |

80 | 28.50 | 46.50 | 11.50 | 21.00 | 18.50 | 37.00 |

81 | 18.00 | 49.50 | 36.00 | 13.50 | 23.00 | 13.50 |

82 | 46.00 | 37.00 | 28.50 | 32.00 | 17.50 | 14.50 |

83 | 24.50 | 13.00 | 34.00 | 30.00 | 0.000 | 0.000 |

84 | 0.000 | 0.000 | 0.000 | 0.000 | 11.50 | 48.50 |

85 | 0.000 | 0.000 | 36.50 | 12.00 | 0.000 | 0.000 |

Time (h) | Demand (pu) | Photovoltaic (pu) | Wind (pu) |
---|---|---|---|

1 | 0.684511335492475 | 0 | 0.633118295 |

2 | 0.644122690036197 | 0 | 0.607259323 |

3 | 0.613069156029720 | 0 | 0.605557422 |

4 | 0.599733282530006 | 0 | 0.684246423 |

5 | 0.588874071251667 | 0 | 0.783719339 |

6 | 0.598018670222900 | 0 | 0.790557706 |

7 | 0.626786054486569 | 0 | 0.744958950 |

8 | 0.651743189178891 | 0.0391233650 | 0.769603567 |

9 | 0.706039245570585 | 0.0655871790 | 0.826492212 |

10 | 0.787007048961707 | 0.2368707960 | 0.876523598 |

11 | 0.839016955610593 | 0.4550178180 | 0.931213527 |

12 | 0.852733854067441 | 0.7264402650 | 0.965504834 |

13 | 0.870642027052772 | 0.9244863260 | 0.972218577 |

14 | 0.834254143646409 | 0.9820411530 | 0.981135531 |

15 | 0.816536483139646 | 0.8294070790 | 0.991393173 |

16 | 0.819394170318156 | 0.7330632950 | 1 |

17 | 0.874071251666984 | 0.5011338490 | 0.987258076 |

18 | 1 | 0.1771175180 | 0.929542167 |

19 | 0.983615926843208 | 0 | 0.791155379 |

20 | 0.936368832158506 | 0 | 0.708839248 |

21 | 0.887597637645266 | 0 | 0.712881960 |

22 | 0.809297008954087 | 0 | 0.719897641 |

23 | 0.745856353591160 | 0 | 0.703007456 |

24 | 0.733473042484283 | 0 | 0.687238555 |

Line | Case 1 | Case 2 | Case 3 | Line | Case 1 | Case 2 | Case 3 |
---|---|---|---|---|---|---|---|

1 | 7 | 5 | 4 | 43 | 3 | 1 | 1 |

2 | 7 | 5 | 4 | 44 | 3 | 1 | 1 |

3 | 5 | 5 | 4 | 45 | 2 | 1 | 1 |

4 | 4 | 5 | 4 | 46 | 2 | 1 | 1 |

5 | 4 | 4 | 3 | 47 | 3 | 1 | 1 |

6 | 4 | 4 | 3 | 48 | 3 | 1 | 1 |

7 | 4 | 4 | 3 | 49 | 3 | 1 | 1 |

8 | 4 | 1 | 1 | 50 | 3 | 1 | 1 |

9 | 4 | 1 | 1 | 51 | 3 | 1 | 1 |

10 | 4 | 1 | 1 | 52 | 3 | 1 | 1 |

11 | 4 | 1 | 1 | 53 | 3 | 1 | 1 |

12 | 4 | 1 | 1 | 54 | 2 | 1 | 1 |

13 | 3 | 1 | 1 | 55 | 3 | 1 | 1 |

14 | 1 | 1 | 1 | 56 | 3 | 1 | 1 |

15 | 1 | 1 | 1 | 7 | 3 | 1 | 1 |

16 | 2 | 1 | 1 | 58 | 1 | 1 | 1 |

17 | 3 | 1 | 1 | 59 | 3 | 1 | 1 |

18 | 3 | 1 | 1 | 60 | 3 | 1 | 1 |

19 | 3 | 1 | 1 | 61 | 1 | 1 | 1 |

20 | 3 | 1 | 1 | 62 | 3 | 1 | 1 |

21 | 2 | 1 | 1 | 63 | 3 | 1 | 1 |

22 | 2 | 1 | 1 | 64 | 3 | 1 | 1 |

23 | 2 | 1 | 1 | 65 | 3 | 1 | 1 |

24 | 3 | 1 | 1 | 66 | 3 | 1 | 1 |

25 | 3 | 1 | 1 | 67 | 3 | 1 | 1 |

26 | 3 | 1 | 1 | 68 | 3 | 1 | 1 |

27 | 3 | 1 | 1 | 69 | 3 | 1 | 1 |

28 | 3 | 1 | 1 | 70 | 3 | 1 | 1 |

29 | 3 | 1 | 1 | 71 | 3 | 1 | 1 |

30 | 3 | 1 | 1 | 72 | 2 | 1 | 1 |

31 | 3 | 1 | 1 | 73 | 2 | 1 | 1 |

32 | 3 | 1 | 1 | 74 | 2 | 1 | 1 |

33 | 3 | 1 | 1 | 75 | 1 | 1 | 1 |

34 | 3 | 1 | 1 | 76 | 3 | 1 | 1 |

35 | 1 | 1 | 1 | 77 | 2 | 1 | 1 |

36 | 3 | 1 | 1 | 78 | 2 | 1 | 1 |

37 | 3 | 1 | 1 | 79 | 3 | 1 | 1 |

38 | 2 | 1 | 1 | 80 | 3 | 1 | 1 |

39 | 3 | 1 | 1 | 81 | 1 | 1 | 1 |

40 | 2 | 1 | 1 | 82 | 3 | 1 | 1 |

41 | 2 | 1 | 1 | 83 | 1 | 1 | 1 |

42 | 2 | 1 | 1 | 84 | 2 | 1 | 1 |

**Table 17.**Investment and operating costs found with the application of the MGbMO to the three-phase version of the IEEE 85-bus grid for each operative scenario.

Case | ${\mathit{C}}_{\mathbf{inv}}$ (USD) | ${\mathit{C}}_{\mathbf{loss}}$ (USD) | Z (USD) |
---|---|---|---|

1 | 550,998.7080 | 403,917.6916 | 954,916.3996 |

2 | 330,218.1420 | 312,264.9263 | 642,483.0683 |

3 | 303,039.0570 | 249,526.0165 | 552,565.0735 |

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## Share and Cite

**MDPI and ACS Style**

Pradilla-Rozo, J.D.; Vega-Forero, J.A.; Montoya, O.D.
Application of the Gradient-Based Metaheuristic Optimizerto Solve the Optimal Conductor Selection Problemin Three-Phase Asymmetric Distribution Networks. *Energies* **2023**, *16*, 888.
https://doi.org/10.3390/en16020888

**AMA Style**

Pradilla-Rozo JD, Vega-Forero JA, Montoya OD.
Application of the Gradient-Based Metaheuristic Optimizerto Solve the Optimal Conductor Selection Problemin Three-Phase Asymmetric Distribution Networks. *Energies*. 2023; 16(2):888.
https://doi.org/10.3390/en16020888

**Chicago/Turabian Style**

Pradilla-Rozo, Julián David, Julián Alejandro Vega-Forero, and Oscar Danilo Montoya.
2023. "Application of the Gradient-Based Metaheuristic Optimizerto Solve the Optimal Conductor Selection Problemin Three-Phase Asymmetric Distribution Networks" *Energies* 16, no. 2: 888.
https://doi.org/10.3390/en16020888