# Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation

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## Abstract

**:**

## 1. Introduction

## 2. Nodal Selection Strategy

- i.
- All the voltage magnitudes are assumed to be known, i.e., these can be assigned as plane voltages $(1\angle {0}^{\circ})$ or set as the power flow solution without capacitor banks (i.e., the benchmark case).
- ii.
- The magnitude of the currents through the distribution lines is mainly governed by the active and reactive power consumption, which implies that the effect of the second Kirchhoff law at each line is negligible in comparison with the first Kirchhoff law at each node.

**Remark**

**1.**

## 3. Assigning the Optimal Sizes

**Remark**

**2.**

## 4. Summary of the Solution Methodology

Algorithm 1: Proposed two-stage solution methodology to select and locate fixed-step capacitor banks in distribution networks. |

## 5. Test Feeder Information

#### 5.1. IEEE 33-Bus Grid

#### 5.2. IEEE 69-Bus Grid

#### 5.3. Parameters for the Economic Assessment

## 6. Computational Implementation

#### 6.1. IEEE 33-Bus Grid

#### 6.2. IEEE 69-Bus Grid

#### 6.3. Numerical Results Considering Daily Load Variations

#### 6.4. Applicability in Meshed Distribution Networks

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

${\mathbb{S}}_{cap,h}$ | Complex vector that contains all the power outputs |

in the fixed-step capacitor banks for each period of time h (var). | |

${\mathbb{S}}_{d,h}$ | Complex vector with the active and reactive power |

consumption in the demand nodes for each period of time (VA). | |

${\mathbb{V}}_{d,h}$ | Complex vector that contains all the voltages in the |

demanded nodes for each period of time (V). | |

${\mathbb{V}}_{s,h}$ | Complex variable associated with the voltage output at the slack source (V). |

${\mathbb{Y}}_{\mathrm{bus}}$ | Nodal admittance matrix (S). |

${\mathbb{Y}}_{dd}$ | Component of the nodal admittance matrix that associates |

demand nodes with each other (S). | |

${\mathcal{A}}_{jl}$ | Component of the node-to-branch incidence matrix that associates node j with line l. |

$\mathcal{C}$ | Set that contains all fixed-step capacitor bank types available |

for installation in the distribution grid. | |

$\mathcal{H}$ | Set that contains all hours of the operation period (typically 24 h). |

$\mathcal{L}$ | Set that contains all distribution lines of the network. |

$\mathcal{N}$ | Set that contains all the nodes of the network. |

$\epsilon $ | Parameter associated with the maximum convergence error |

admissible for the power flow solution (V). | |

${C}_{\mathrm{kWh}}$ | Expected costs of the energy losses (US$/kWh-year). |

${C}_{c}^{\mathrm{cap}}$ | Installation cost of the fixed-step capacitor bank type c (US$/kvar). |

h | Subscript associated with the set $\mathcal{H}$. |

${i}_{l,h}^{i}$ | Imaginary component of the current flowing through line l in the period of time h (A). |

${i}_{l,h}^{r}$ | Real component of the current flowing through line l in the period of time h (A). |

j | Subscript associated with the set $\mathcal{N}$. |

l | Subscript associated with the set $\mathcal{L}$. |

m | Superscript associated with the number of iterations. |

${N}_{\mathrm{ava}}^{\mathrm{cap}}$ | Number of fixed-step capacitor banks available for installation. |

${P}_{\mathrm{loss},h}$ | Active power losses in the distribution network for each period of time (W). |

${P}_{j,h}^{d}$ | Active power generation consumed at node j in period of time h (W). |

${p}_{j,h}^{g}$ | Active power generation injected at node j in period of time h (W). |

${Q}_{c}$ | Reactive power capacity of a type c fixed-step capacitor bank (kvar). |

${Q}_{j,h}^{d}$ | Reactive power generation consumed at node j in period of time h (var). |

${q}_{j,h}^{g}$ | Reactive power generation injected at node j in period of time h (var). |

${R}_{l}$ | Resistive parameter of the distribution line l ($\mathsf{\Omega}$). |

T | Length of the planning period (days). |

${V}_{j,h}^{i}$ | Imaginary component of the voltage magnitude at node j in the period of time h (V). |

${V}_{j,h}^{r}$ | Real component of the voltage magnitude at node j in the period of time h (V). |

${x}_{\mathrm{sol}}$ | Solution vector that contains the nodes where the fixed-step capacitor |

banks will be located along with their possible sizes. | |

${x}_{jc}$ | Binary variable associated with the installation (${x}_{jc}=1$) or not (${x}_{jc}=0$) |

of a fixed-step capacitor bank type c at node j. | |

${z}_{\mathrm{approx}}$ | Approximate objective function value associated with the |

expected annual grid operating costs (US$). | |

${z}_{\mathrm{cos}\mathrm{ts}}$ | Expected annual operating costs of the network (US$). |

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**Table 1.**Characterization of the MIQC model (1)–().

Variables | Type | Number |

Capacitor locations and sizes | Binary | $nc$ |

Currents (real and imaginary parts) | Real | $2lh$ |

Active and reactive power generation | Real | $2nh$ |

Objective function | Real | 1 |

Total number of variables | Real + binary | $2\left(l+n\right)h+nc+1$ |

Constraints | Type | Number |

Active power balance | Equality | $np$ |

Reactive power balance | Equality | $np$ |

Capacitors per node | Inequality | n |

Number of capacitors available | Inequality | 1 |

Objective function | Equality | 1 |

Total number of constraints | Equalities + inequalities | $\left(2p+1\right)n+1$ |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{X}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{X}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) |
---|---|---|---|---|---|---|---|---|---|---|---|

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 | 1.2860 | 1.7210 | 60 | 20 | 32 | 33 | 0.3410 | 0.5302 | 60 | 40 |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{X}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{X}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) |
---|---|---|---|---|---|---|---|---|---|---|---|

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 |

3 | 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 |

Option | ${\mathit{Q}}_{\mathit{c}}$ (kvar) | Cost ($/kvar-Year) | Option | ${\mathit{Q}}_{\mathit{c}}$ (kvar) | Cost ($/kvar-Year) |
---|---|---|---|---|---|

1 | 150 | 0.500 | 8 | 1200 | 0.170 |

2 | 300 | 0.350 | 9 | 1350 | 0.207 |

3 | 450 | 0.253 | 10 | 1500 | 0.201 |

4 | 600 | 0.220 | 11 | 1650 | 0.193 |

5 | 750 | 0.276 | 12 | 1800 | 0.870 |

6 | 900 | 0.183 | 13 | 1950 | 0.211 |

7 | 1050 | 0.228 | 14 | 2100 | 0.176 |

**Table 5.**Optimal location, sizes, and annual expected costs for the IEEE 33-bus system under peak load conditions.

Method | Size (Node) (Mvar) | Losses (kW) | C. Caps. US$ | C. Total US$ |
---|---|---|---|---|

GAMS | {0.30(14), 0.45(24), 1.05(30)} | 139.292 | 458.25 | 23,859.313 |

MIQC (sol. 1) | {0.45(13), 0.45(24), 1.05(30)} | 138.473 | 467.10 | 23,747.317 |

MIQC (sol. 2) | {0.45(13), 0.60(24), 0.90(30)} | 138.917 | 410.55 | 23,748.531 |

MIQC (sol. 3) | {0.45(13), 0.45(24), 0.90(30)} | 139.075 | 392.40 | 23,757.083 |

**Table 6.**Optimal location, sizes, and annual expected costs for the IEEE 69-bus system under peak load conditions.

Method | Size (Node) (Mvar) | Losses (kW) | C. Caps. US$ | C. Total US$ |
---|---|---|---|---|

GAMS | {0.45(11), 0.15(27), 1.20(61)} | 145.738 | 392.85 | 24,876.910 |

MIQC (sol. 1) | {0.45(11), 0.15(21), 1.20(61)} | 145.550 | 392.85 | 24,845.246 |

MIQC (sol. 2) | {0.30(11), 0.30(21), 1.20(61)} | 145.492 | 414.00 | 24,856.573 |

MIQC (sol. 3) | {0.60(11), 0.15(21), 1.20(61)} | 145.614 | 411.00 | 24,874.173 |

MIQC (sol. 4) | {0.45(11), 0.30(21), 1.20(61)} | 145.556 | 422.85 | 24,876.229 |

Time (h) | Active (pu) | Reactive (pu) | Time (h) | Active (pu) | Reactive (pu) |
---|---|---|---|---|---|

1 | 0.34 | 0.2954 | 25 | 0.94 | 0.6764 |

2 | 0.28 | 0.2238 | 26 | 0.94 | 0.7228 |

3 | 0.22 | 0.1964 | 27 | 0.90 | 0.7754 |

4 | 0.22 | 0.1666 | 28 | 0.84 | 0.6868 |

5 | 0.22 | 0.1478 | 29 | 0.86 | 0.7542 |

6 | 0.20 | 0.1654 | 30 | 0.90 | 0.8538 |

7 | 0.18 | 0.1662 | 31 | 0.90 | 0.8448 |

8 | 0.18 | 0.1274 | 32 | 0.90 | 0.7294 |

9 | 0.18 | 0.1404 | 33 | 0.90 | 0.8452 |

10 | 0.20 | 0.1750 | 34 | 0.90 | 0.6162 |

11 | 0.22 | 0.1456 | 35 | 0.90 | 0.5988 |

12 | 0.26 | 0.2428 | 36 | 0.90 | 0.6672 |

13 | 0.28 | 0.2462 | 37 | 0.86 | 0.7086 |

14 | 0.34 | 0.2780 | 38 | 0.84 | 0.6798 |

15 | 0.40 | 0.2820 | 39 | 0.92 | 0.8468 |

16 | 0.50 | 0.3996 | 40 | 1 | 0.8122 |

17 | 0.62 | 0.4994 | 41 | 0.98 | 0.7640 |

18 | 0.68 | 0.6448 | 42 | 0.94 | 0.7640 |

19 | 0.72 | 0.6526 | 43 | 0.90 | 0.7774 |

20 | 0.78 | 0.7322 | 44 | 0.84 | 0.5502 |

21 | 0.84 | 0.7170 | 45 | 0.76 | 0.6766 |

22 | 0.86 | 0.6632 | 46 | 0.68 | 0.4710 |

23 | 0.90 | 0.8374 | 47 | 0.58 | 0.4602 |

24 | 0.92 | 0.7304 | 48 | 0.50 | 0.3636 |

Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{X}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | Node i | Node j | ${\mathit{R}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) | ${\mathit{X}}_{\mathbf{ij}}$ ($\mathsf{\Omega}$) |
---|---|---|---|---|---|---|---|

8 | 21 | 2.0 | 2.0 | 12 | 22 | 2.0 | 2.0 |

9 | 15 | 2.0 | 2.0 | 18 | 33 | 0.5 | 0.5 |

25 | 29 | 0.5 | 0.5 | — | — | — | — |

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**MDPI and ACS Style**

Montoya, O.D.; Moya, F.D.; Rajagopalan, A.
Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation. *Mathematics* **2022**, *10*, 1600.
https://doi.org/10.3390/math10091600

**AMA Style**

Montoya OD, Moya FD, Rajagopalan A.
Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation. *Mathematics*. 2022; 10(9):1600.
https://doi.org/10.3390/math10091600

**Chicago/Turabian Style**

Montoya, Oscar Danilo, Francisco David Moya, and Arul Rajagopalan.
2022. "Annual Operating Costs Minimization in Electrical Distribution Networks via the Optimal Selection and Location of Fixed-Step Capacitor Banks Using a Hybrid Mathematical Formulation" *Mathematics* 10, no. 9: 1600.
https://doi.org/10.3390/math10091600