# Optimal Selection and Location of Fixed-Step Capacitor Banks in Distribution Networks Using a Discrete Version of the Vortex Search Algorithm

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## Abstract

**:**

## 1. Introduction

- ✓
- The proposal of a discrete version of the VSA (DVSA) to solve the problem of the optimal selection and location of fixed-step capacitor banks in AC distribution networks with low computational effort.
- ✓
- The use of a discrete codification implements integer numbers as decision variables that simplify the dimension of the classical binary vectors used in the literature to represent this optimization problem. This codification compacts in a unique stage, the location and sizing problems of capacitor banks, which substantially reduces the processing times.
- ✓
- An improvement of the current results reported in the specialized literature for the problem of the optimal location of fixed-step capacitor banks.

## 2. Mathematical Formulation

#### 2.1. Objective Function

#### 2.2. Set of Constraints

**Remark**

**1.**

## 3. Solution Methodology

#### 3.1. Vortex Search Algorithm

Algorithm 1: Proposed master optimization algorithm based on the VSA to define the location and sizing of capacitor banks in distribution networks. |

1. Inputs: |

2. Determine the initial center ${\mu}_{0}$ from (7); |

3. Calculate the initial radius ${r}_{0}$ (or the standard deviation ${\sigma}_{0}$) with (12); |

4. Set the initial best fitness function as $z\left({s}_{\mathrm{best}}\right)=\infty $ (minimization problem); |

5. Make $t=0$; |

6. while ($t\le {t}_{max}$) |

7. Generate the candidate solutions using a Gaussian distribution around the center ${\mu}_{t}$ with a |

standard deviation (radius) ${r}_{t}$ as defined in (9) to obtain ${C}_{t}\left(s\right)$ with d-dimension columns and m rows; |

8. If ${C}_{t}\left(s\right)$ crosses any upper or lower bound, place it within its bounds using (11); |

9. Evaluate the successive approximation power flow problem (see (16)) for each ${s}_{h}$ in ${C}_{t}\left(s\right)$ and calculate |

its corresponding fitness function as (13); |

10. Select the best solution ${s}^{\star}$ as the argument that produces the minimum ${z}_{f}$ contained in ${C}_{t}\left(s\right)$; |

11. if (${z}_{f}\left({s}^{\star}\right)<{z}_{f}\left({s}_{\mathrm{best}}\right)$) |

12. ${s}_{\mathrm{best}}={s}^{\star}$; |

13. ${z}_{f}\left({s}_{\mathrm{best}}\right)={z}_{f}\left({s}^{\star}\right)$; |

14. else |

15. Retain the best solution attained so far ${s}_{\mathrm{best}}$; |

16. end |

17. Make the center ${\mu}_{t+1}$ equal to the best solution ${s}_{\mathrm{best}}$; |

18. Update the current radius ${r}_{t+1}$ as given by (12); |

19. $t=t+1$; |

20. end |

21. Output: |

22. The best solution is found for ${s}_{\mathrm{best}}$ and its fitness function is ${z}_{f}\left({s}_{\mathrm{best}}\right)$; |

**Remark**

**2.**

**Remark**

**3.**

#### 3.2. Successive Approximation Power Flow Approach

**Remark**

**4.**

## 4. Test Systems

#### 4.1. Thirty-Three-Node Test Feeder

#### 4.2. Sixty-Nine-Node Test Feeder

#### 4.3. Capacitor Banks Information

## 5. Computational Validation

#### 5.1. Results in the 33-Node Test Feeder

- ✓
- The proposed DVSA reaches a better solution regarding final power losses in the 33-node test feeder with 138.416 kW when compared to the results in the literature. This implies a total net savings of about US$$12,191.17$, which is an additional savings of US$$502.26$ in comparison to the results reached by the FPA.
- ✓
- The FRCGA method requires at least six capacitor banks to reduces the total power losses about $33.05\%$, while the proposed approach with only three capacitor banks reaches $34.40\%$, regarding the base case; this implies an additional improvement of $1.35\%$.

- ✓
- The final operating costs in the 33-node test feeder present higher variations if the number of fixed-step capacitor banks oscillates from one to three. However, after four capacitor banks, the reduction of the operating costs presents saturations. This implies that three capacitor banks are enough to reach adequate optimal solutions with minimum physical interventions inside the grid.
- ✓
- The solution of the proposed DVSA for locating two or more capacitors shows better results than approaches in the literature, which means that our approach is efficient in reducing the total operating cost and suitable to be used for utilities in real applications.

#### 5.2. Results in the 69-Node Test Feeder

#### 5.3. Additional Comments

- ✓
- To verify that the proposed DVSA can find the optimal global solution for both test feeders, we implemented an exhaustive search algorithm to evaluate each possible combination of nodes for all possible capacitor sizes by using the successive approximation power flow method embedded in nested loops. After some hours of evaluation, this exhaustive search finds the same solutions reported in Table 3 and Table 4, implying that these solutions are indeed the global optima.
- ✓
- Regarding processing times of the proposed DVSA to select and locate fixed-step capacitor banks in AC distribution networks, it can be noted that after 100 consecutive evaluations in the case of the 33-node test feeder the average processing time is about $1.33\phantom{\rule{3.33333pt}{0ex}}$s, and in the case of the 69-node test feeder this time is about $4.01\phantom{\rule{3.33333pt}{0ex}}$s. These results demonstrate the low-computational effort required by the proposed approach to achieve the optimal solution.
- ✓
- Note that the proposed DVSA for optimal location and selection of fixed-step capacitor banks in distribution networks are easily extended to multiple periods of analysis (i.e., considering load curves) by adding the sub-index t in all the voltages, angles, and power in the mathematical model (1)–(6) (see optimization model (A1)–(A6) reported in Appendix B) since the optimal location and size of the capacitors is uncoupled in time. This implies that the complication in the implementation is only associated with the number of power flow evaluations required in the slave stage. These implementations will be presented in the next section.

#### 5.4. Optimal Location of Capacitors Considering a Variable Load Curve

- ✓
- In the 33-node test feeder, the initial energy losses considering the daily load behavior (see Figure 10) are about $2508.634\phantom{\rule{3.33333pt}{0ex}}$kWh/day, which are reduced to $1720.669\phantom{\rule{3.33333pt}{0ex}}$kWh/day after the installation of three fixed-step capacitors in nodes 12, 24, and 30 with nominal rates of $300\phantom{\rule{3.33333pt}{0ex}}$kVAr, $300\phantom{\rule{3.33333pt}{0ex}}$kVAr, and $750\phantom{\rule{3.33333pt}{0ex}}$kVAr, respectively. Note that these capacitors reduce the costs of the energy losses by $31.04\%$.
- ✓
- The selected nodes in the case of the 33-node test feeder coincide for the cases of peak hour and daily load behavior. However, in the second case, their sizes have been reduced from 450 kVAr to 300 kVAr and from 1050 kVAr to 750 kVAr. It is worth mentioning that the reduction of sizes is an expected result because considering variable load behaviors, the expected reductions regarding power losses minimization are moderated in comparison to the load peak case; i.e., these pass from US$\$12191.17$ (peak load case) to US$\$5514.78$ (load daily load variation case).
- ✓
- In the case of the 69-node test feeder the initial daily energy losses without capacitors is about $2666.286\phantom{\rule{3.33333pt}{0ex}}$kWh/day, which are reduced to $1800.283\phantom{\rule{3.33333pt}{0ex}}$kWh/day. This reduction is reached after locating the fixed-step capacitor banks in nodes 11, 21, and 61 with nominal rates of $300\phantom{\rule{3.33333pt}{0ex}}$kVAr, $150\phantom{\rule{3.33333pt}{0ex}}$kVAr, and $900\phantom{\rule{3.33333pt}{0ex}}$kVAr, respectively. Observe that the annual operational costs are reduced by $32.48\%$ regarding the base case reported in Table 5.
- ✓
- When comparing the location of the capacitor banks for the cases of load peak case and time-varying load behavior, it can be observed that the nodes 11 and 69 remain constant in both scenarios. While the node 18 is changed by the node 21 in the case of variable load behavior. This variation demonstrates that it is advisable for the optimal location of capacitor banks in distribution networks to consider a realistic load scenario instead of just the peak load case, since the solutions in the peak case could not be the same when taking charge variations into account.

## 6. Conclusions and Future Works

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Electrical Parameters

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 0.0922 | 0.0477 | 100 | 60 | 17 | 18 | 0.7320 | 0.5740 | 90 | 40 |

2 | 3 | 0.4930 | 0.2511 | 90 | 40 | 2 | 19 | 0.1640 | 0.1565 | 90 | 40 |

3 | 4 | 0.3660 | 0.1864 | 120 | 80 | 19 | 20 | 1.5042 | 1.3554 | 90 | 40 |

4 | 5 | 0.3811 | 0.1941 | 60 | 30 | 20 | 21 | 0.4095 | 0.4784 | 90 | 40 |

5 | 6 | 0.8190 | 0.7070 | 60 | 20 | 21 | 22 | 0.7089 | 0.9373 | 90 | 40 |

6 | 7 | 0.1872 | 0.6188 | 200 | 100 | 3 | 23 | 0.4512 | 0.3083 | 90 | 50 |

7 | 8 | 1.7114 | 1.2351 | 200 | 100 | 23 | 24 | 0.8980 | 0.7091 | 420 | 200 |

8 | 9 | 1.0300 | 0.7400 | 60 | 20 | 24 | 25 | 0.8960 | 0.7011 | 420 | 200 |

9 | 10 | 1.0400 | 0.7400 | 60 | 20 | 6 | 26 | 0.2030 | 0.1034 | 60 | 25 |

10 | 11 | 0.1966 | 0.0650 | 45 | 30 | 26 | 27 | 0.2842 | 0.1447 | 60 | 25 |

11 | 12 | 0.3744 | 0.1238 | 60 | 35 | 27 | 28 | 1.0590 | 0.9337 | 60 | 20 |

12 | 13 | 1.4680 | 1.1550 | 60 | 35 | 28 | 29 | 0.8042 | 0.7006 | 120 | 70 |

13 | 14 | 0.5416 | 0.7129 | 120 | 80 | 29 | 30 | 0.5075 | 0.2585 | 200 | 600 |

14 | 15 | 0.5910 | 0.5260 | 60 | 10 | 30 | 31 | 0.9744 | 0.9630 | 150 | 70 |

15 | 16 | 0.7463 | 0.5450 | 60 | 20 | 31 | 32 | 0.3105 | 0.3619 | 210 | 100 |

16 | 17 | 1.2890 | 1.7210 | 60 | 20 | 32 | 33 | 0.3410 | 0.5302 | 60 | 40 |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{X}}_{\mathbf{ij}}$ [$\mathbf{\Omega}$] | ${\mathit{P}}_{\mathit{j}}$ [kW] | ${\mathit{Q}}_{\mathit{j}}$ [kW] |
---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 0.0005 | 0.0012 | 0 | 0 | 3 | 36 | 0.0044 | 0.0108 | 26 | 18.55 |

2 | 3 | 0.0005 | 0.0012 | 0 | 0 | 36 | 37 | 0.0640 | 0.1565 | 26 | 18.55 |

3 | 4 | 0.0015 | 0.0036 | 0 | 0 | 37 | 38 | 0.1053 | 0.1230 | 0 | 0 |

4 | 5 | 0.0251 | 0.0294 | 0 | 0 | 38 | 39 | 0.0304 | 0.0355 | 24 | 17 |

5 | 6 | 0.3660 | 0.1864 | 2.6 | 2.2 | 39 | 40 | 0.0018 | 0.0021 | 24 | 17 |

6 | 7 | 0.3810 | 0.1941 | 40.4 | 30 | 40 | 41 | 0.7283 | 0.8509 | 1.2 | 1 |

7 | 8 | 0.0922 | 0.0470 | 75 | 54 | 41 | 42 | 0.3100 | 0.3623 | 0 | 0 |

8 | 9 | 0.0493 | 0.0251 | 30 | 22 | 42 | 43 | 0.0410 | 0.0475 | 6 | 4.3 |

9 | 10 | 0.8190 | 0.2707 | 28 | 19 | 43 | 44 | 0.0092 | 0.0116 | 0 | 0 |

10 | 11 | 0.1872 | 0.0619 | 145 | 104 | 44 | 45 | 0.1089 | 0.1373 | 39.22 | 26.3 |

11 | 12 | 0.7114 | 0.2351 | 145 | 104 | 45 | 46 | 0.0009 | 0.0012 | 39.22 | 26.3 |

12 | 13 | 1.0300 | 0.3400 | 8 | 5 | 4 | 47 | 0.0034 | 0.0084 | 0 | 0 |

13 | 14 | 1.0440 | 0.3450 | 8 | 5.5 | 47 | 48 | 0.0851 | 0.2083 | 79 | 56.4 |

14 | 15 | 1.0580 | 0.3496 | 0 | 0 | 48 | 49 | 0.2898 | 0.7091 | 384.7 | 274.5 |

15 | 16 | 0.1966 | 0.0650 | 45.5 | 30 | 49 | 50 | 0.0822 | 0.2011 | 384.7 | 274.5 |

16 | 17 | 0.3744 | 0.1238 | 60 | 35 | 8 | 51 | 0.0928 | 0.0473 | 40.5 | 28.3 |

17 | 18 | 0.0047 | 0.0016 | 60 | 35 | 51 | 52 | 0.3319 | 0.1114 | 3.6 | 2.7 |

18 | 19 | 0.3276 | 0.1083 | 0 | 0 | 9 | 53 | 0.1740 | 0.0886 | 4.35 | 3.5 |

19 | 20 | 0.2106 | 0.0690 | 1 | 0.6 | 53 | 54 | 0.2030 | 0.1034 | 26.4 | 19 |

20 | 21 | 0.3416 | 0.1129 | 114 | 81 | 54 | 55 | 0.2842 | 0.1447 | 24 | 17.2 |

21 | 22 | 0.0140 | 0.0046 | 5 | 3.5 | 55 | 56 | 0.2813 | 0.1433 | 0 | 0 |

22 | 23 | 0.1591 | 0.0526 | 0 | 0 | 56 | 57 | 1.5900 | 0.5337 | 0 | 0 |

23 | 24 | 0.3460 | 0.1145 | 28 | 20 | 57 | 58 | 0.7837 | 0.2630 | 0 | 0 |

24 | 25 | 0.7488 | 0.2475 | 0 | 0 | 58 | 59 | 0.3042 | 0.1006 | 100 | 72 |

25 | 26 | 0.3089 | 0.1021 | 14 | 10 | 59 | 60 | 0.3861 | 0.1172 | 0 | 0 |

26 | 27 | 0.1732 | 0.0572 | 14 | 10 | 60 | 61 | 0.5075 | 0.2585 | 1244 | 888 |

23 | 28 | 0.0044 | 0.0108 | 26 | 18.6 | 61 | 62 | 0.0974 | 0.0496 | 32 | 23 |

28 | 29 | 0.0640 | 0.1565 | 26 | 18.6 | 62 | 63 | 0.1450 | 0.0738 | 0 | 0 |

29 | 30 | 0.3978 | 0.1315 | 0 | 0 | 63 | 64 | 0.7105 | 0.3619 | 227 | 162 |

30 | 31 | 0.0702 | 0.0232 | 0 | 0 | 64 | 65 | 1.0410 | 0.5302 | 59 | 42 |

31 | 32 | 0.3510 | 0.1160 | 0 | 0 | 11 | 66 | 0.2012 | 0.0611 | 18 | 13 |

32 | 33 | 0.8390 | 0.2816 | 14 | 10 | 66 | 67 | 0.0047 | 0.0014 | 18 | 13 |

33 | 34 | 1.7080 | 0.5646 | 19.5 | 14 | 12 | 68 | 0.7394 | 0.2444 | 28 | 20 |

34 | 35 | 1.4740 | 0.4873 | 6 | 4 | 68 | 69 | 0.0047 | 0.0016 | 28 | 20 |

## Appendix B. Time-Varying Optimization Model

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**Figure 1.**Flowchart of the successive approximation power flow method used in the slave stage of the proposed discrete version of the vortex search algorithm (DVSA).

**Figure 4.**Percentage of energy-saving costs after locating capacitor banks in the 33-node test feeder.

**Figure 7.**Percentage of energy saving costs after locating capacitor banks in the 69-node test feeder.

**Figure 10.**Typical demand load curve in residential networks in Colombia [3].

k | 1 | 2 | 3 |

${Q}_{k}$ [kVAr] | 150 | 300 | 450 |

${C}_{k}^{\mathrm{year}}$ [$/kVAr-year] | 0.500 | 0.350 | 0.253 |

k | 4 | 5 | 6 |

${Q}_{k}$ [kVAr] | 600 | 750 | 900 |

${C}_{k}^{\mathrm{year}}$ [$/kVAr-year] | 0.220 | 0.276 | 0.183 |

k | 7 | 8 | 9 |

${Q}_{k}$ [kVAr] | 1050 | 1200 | 1350 |

${C}_{k}^{\mathrm{year}}$ [$/kVAr-year] | 0.228 | 0.170 | 0.207 |

k | 10 | 11 | 12 |

${Q}_{k}$ [kVAr] | 1500 | 1650 | 1800 |

${C}_{k}^{\mathrm{year}}$ [$/kVAr-year] | 0.201 | 0.193 | 0.870 |

k | 13 | 14 | — |

${Q}_{k}$ [kVAr] | 1950 | 2100 | — |

${C}_{k}^{\mathrm{year}}$ [$/kVAr-year] | 0.211 | 0.176 | — |

Discrete Vortex search algorithm | |
---|---|

Population size | 10 |

Number of iterations | 1000 |

Population generation | Gaussian Distribution |

Successive approximation power flow | |

Number of iterations | 1000 |

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

Experimental tests per system | |

Number of evaluations | 100 |

Method | Nodes | Size [kVAr] | Total Losses [kW] | Costs [US$] |
---|---|---|---|---|

Base case | — | — | 210.987 | 35445.91 |

GSA [17] | $\left\{9,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}30\right\}$ | $\left\{450,\phantom{\rule{0.277778em}{0ex}}800,\phantom{\rule{0.277778em}{0ex}}900\right\}$ | 171.780 | 29,358.39 |

TSM [20] | $\left\{7,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}30\right\}$ | $\left\{850,\phantom{\rule{0.277778em}{0ex}}25,\phantom{\rule{0.277778em}{0ex}}900\right\}$ | 144.040 | 24,705.87 |

FRCGA [21] | $\left\{28,\phantom{\rule{0.277778em}{0ex}}6,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}8,\phantom{\rule{0.277778em}{0ex}}30,\phantom{\rule{0.277778em}{0ex}}9\right\}$ | $\left\{25,\phantom{\rule{0.277778em}{0ex}}475,\phantom{\rule{0.277778em}{0ex}}300,\phantom{\rule{0.277778em}{0ex}}175,\phantom{\rule{0.277778em}{0ex}}400,\phantom{\rule{0.277778em}{0ex}}350\right\}$ | 141.240 | 24,221.18 |

FPA [22] | $\left\{30,\phantom{\rule{0.277778em}{0ex}}13,\phantom{\rule{0.277778em}{0ex}}24\right\}$ | $\left\{900,\phantom{\rule{0.277778em}{0ex}}450,\phantom{\rule{0.277778em}{0ex}}450\right\}$ | 139.075 | 23,757.00 |

DVSA | $\left\{24,\phantom{\rule{0.277778em}{0ex}}12,\phantom{\rule{0.277778em}{0ex}}30\right\}$ | $\left\{450,\phantom{\rule{0.277778em}{0ex}}450,\phantom{\rule{0.277778em}{0ex}}1050\right\}$ | 138.416 | 23,254.74 |

Method | Nodes | Size [kVAr] | Losses [kW] | Costs [US$] |
---|---|---|---|---|

Base case | — | — | 225.072 | 37800.00 |

GSA [17] | $\left\{11,\phantom{\rule{0.277778em}{0ex}}29,\phantom{\rule{0.277778em}{0ex}}60\right\}$ | $\left\{900,\phantom{\rule{0.277778em}{0ex}}1050,\phantom{\rule{0.277778em}{0ex}}450\right\}$ | 163.280 | 27,431.04 |

TSM [20] | $\left\{19,\phantom{\rule{0.277778em}{0ex}}62,\phantom{\rule{0.277778em}{0ex}}63\right\}$ | $\left\{225,\phantom{\rule{0.277778em}{0ex}}900,\phantom{\rule{0.277778em}{0ex}}225\right\}$ | 148.910 | 25,016.88 |

TBLO [23] | $\left\{12,\phantom{\rule{0.277778em}{0ex}}61,\phantom{\rule{0.277778em}{0ex}}64\right\}$ | $\left\{600\phantom{\rule{0.277778em}{0ex}}1050,\phantom{\rule{0.277778em}{0ex}}150\right\}$ | 146.350 | 25,033.20 |

FPA [22] | $\left\{11,\phantom{\rule{0.277778em}{0ex}}61,\phantom{\rule{0.277778em}{0ex}}22\right\}$ | $\left\{450,\phantom{\rule{0.277778em}{0ex}}1350,\phantom{\rule{0.277778em}{0ex}}150,\right\}$ | 145.860 | 24,972.78 |

DVSA | $\left\{11,\phantom{\rule{0.277778em}{0ex}}18,\phantom{\rule{0.277778em}{0ex}}61\right\}$ | $\left\{300,\phantom{\rule{0.277778em}{0ex}}300,\phantom{\rule{0.277778em}{0ex}}1200\right\}$ | 145.397 | 24,427.65 |

**Table 5.**Optimal location of capacitor banks in the 33- and 69-node test feeders considering daily load variations.

Method | Nodes | Size [kVAr] | Energy Losses [kWh/day] | Costs [US$] |
---|---|---|---|---|

33-node test feeder | ||||

Base case | — | — | 2508.634 | 17,560.44 |

DVSA | $\left\{12,\phantom{\rule{0.277778em}{0ex}}24,\phantom{\rule{0.277778em}{0ex}}30\right\}$ | $\left\{300,\phantom{\rule{0.277778em}{0ex}}300,\phantom{\rule{0.277778em}{0ex}}750\right\}$ | 1720.669 | 12,045.66 |

69-node test feeder | ||||

Base case | — | — | 2666.286 | 18,664.00 |

DVSA | $\left\{11,\phantom{\rule{0.277778em}{0ex}}21,\phantom{\rule{0.277778em}{0ex}}61\right\}$ | $\left\{300,\phantom{\rule{0.277778em}{0ex}}150,\phantom{\rule{0.277778em}{0ex}}900\right\}$ | 1800.283 | 12,652.24 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gil-González, W.; Montoya, O.D.; Rajagopalan, A.; Grisales-Noreña, L.F.; Hernández, J.C. Optimal Selection and Location of Fixed-Step Capacitor Banks in Distribution Networks Using a Discrete Version of the Vortex Search Algorithm. *Energies* **2020**, *13*, 4914.
https://doi.org/10.3390/en13184914

**AMA Style**

Gil-González W, Montoya OD, Rajagopalan A, Grisales-Noreña LF, Hernández JC. Optimal Selection and Location of Fixed-Step Capacitor Banks in Distribution Networks Using a Discrete Version of the Vortex Search Algorithm. *Energies*. 2020; 13(18):4914.
https://doi.org/10.3390/en13184914

**Chicago/Turabian Style**

Gil-González, Walter, Oscar Danilo Montoya, Arul Rajagopalan, Luis Fernando Grisales-Noreña, and Jesus C. Hernández. 2020. "Optimal Selection and Location of Fixed-Step Capacitor Banks in Distribution Networks Using a Discrete Version of the Vortex Search Algorithm" *Energies* 13, no. 18: 4914.
https://doi.org/10.3390/en13184914