# Simultaneous Integration of D-STATCOMs and PV Sources in Distribution Networks to Reduce Annual Investment and Operating Costs

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. General Context

#### 1.2. Motivation

#### 1.3. Literature Review

#### 1.4. Contribution and Scope

- A general MINLP formulation of the problem regarding the simultaneous integration of PVs and D-STATCOMs in medium-voltage distribution networks in order to minimize the expected annual grid investment and operating costs while considering variable active and reactive power demand curves. This formulation is based on the combinations existing in the literature for the efficient, independent integration of each device.
- The application of a leader-follower optimization methodology based on hybridizing the VSA and the successive approximations power flow method. Based on the VSA approach, the leader stage uses a discrete continuous-codification vector to define the nodes where the PVs and D-STATCOMs must be placed (discrete part) and their optimal sizes (continuous part). On the other hand, the follower component of the optimization approach is associated with the technical evaluation of each solution provided by the leader, i.e., the calculation of voltages and powers, among other variables.

#### 1.5. Document Structure

## 2. Mathematical Model

#### 2.1. Mathematical Model for Locating and Sizing PV Generation Units

#### 2.1.1. Objective Function Formulation

#### 2.1.2. General Set of Constraints

#### 2.2. Mathematical Model for Locating and Sizing D-STATCOMS

#### 2.2.1. Objective Function Formulation

#### 2.2.2. General Set of Constraints

#### 2.3. Mathematical Model for Simultaneously Locating and Sizing PV Generation Units and D-STATCOMs

#### 2.3.1. Objective Function Formulation

#### 2.3.2. General Set of Constraints

## 3. Proposed Leader-Follower Optimization Approach

- In the leader stage, a combinatorial optimization method is implemented to evolve an initial set of individuals, i.e., a set of a potential solutions, where the discrete codification implemented allows evaluating the components ${f}_{2}$ and ${f}_{3}$ regarding the installation costs of the PV and D-STATCOM systems.
- The follower stage determines the expected energy purchasing costs at the terminals of the substation (i.e., ${f}_{1}$) and the operating and maintenance costs of the PV systems.

#### 3.1. Follower Stage: Successive Approximations Power Flow Method

**Remark**

**1.**

#### 3.2. Master Stage: The Vortex Search Algorithm (VSA)

#### 3.2.1. Generating the Initial Solution

**Remark**

**2.**

#### 3.2.2. Generating the Candidate Solutions

**Remark**

**3.**

#### 3.2.3. Correcting the Candidate Solutions

#### 3.2.4. Selecting the New Hyper-Ellipse Center

#### 3.2.5. Radius Reduction Criterion

#### 3.2.6. Stopping Criteria

- ✔
- if the maximum number of iterations ${t}_{max}$ is reached, or

- ✔
- if, after ${\tau}_{max}$ consecutive iterations, the center of the hyper-ellipse has not been modified.

#### 3.3. General Implementation of the Proposed Leader-Follower Optimization Algorithm

Algorithm 1: Application of the VSA approach for locating and sizing PVs and D-STATCOMs in distribution networks. | ||||

Data: Read data of the distribution network under analysis | ||||

Obtain the per-unit equivalent of the distribution network; | ||||

Define the initial center and radius of the hyper-ellipse ${\mu}_{t}$ and ${r}_{t}$; | ||||

Generate the initial set of candidate solutions ${s}_{i}^{t}$ using (41); | ||||

Check and correct each potential solution ${s}_{i}^{t}$ using (43); | ||||

Evaluate each potential solution ${s}_{i}^{t}$ in the follower stage, i.e., the power flow | ||||

formula defined in (36); | ||||

Find the best current solution ${s}_{i,\mathrm{best}}^{t}$; | ||||

for $t=1:{t}_{max}$ do | ||||

Update the hyper-ellipse center making ${\mu}_{t+1}={s}_{i,best}^{t}$; | ||||

Calculate the new radius of the hyper-ellipse ${r}_{t+1}$ as in (44); | ||||

Generate the new set of candidate solutions ${s}_{i}^{t}$ using (41); | ||||

Check and correct each potential solution ${s}_{i}^{t}$ using (43); | ||||

Evaluate each potential solution ${s}_{i}^{t}$ in the follower stage, i.e., the power flow | ||||

formula defined in (36); | ||||

Find the best current solution ${s}_{i,\mathrm{best}}^{t}$; | ||||

if $\tau \ge {\tau}_{max}$ then | ||||

Report the best current solution in ${\mu}_{t+1}$; | ||||

break; | ||||

end | ||||

end | ||||

Result: Return the best solution found |

## 4. Test Feeder Characterization

## 5. Numerical Results

- S1.
- The evaluation of a benchmark case, i.e., the original conditions of the test feeder without including PVs and D-STATCOMs.
- S2.
- S3.
- S4.

#### 5.1. Numerical Results for the IEEE 33-Bus Grid

- In S2, three D-STATCOMs with sizes of about 186.4, 121.1, and 528.6 kvar were placed at nodes 14, 25, and 30. With these devices, a reduction of about USD $\mathrm{22,602.85}$ in the expected grid operating costs corresponds to $0.64\%$. However, this is an expected result, as the D-STATCOMs provide reactive power and the energy purchasing costs at the terminals of the substation are associated with active power. This implies that the utility company must purchase less energy to support all the end users since the D-STATCOMs improve the grid efficiency.
- As expected, in S3, a reduction of about USD $\mathrm{1,252,832.78}$ (i.e., $35.26\%$ with respect to the benchmark case) in the total grid costs was achieved by installing three PV generators at nodes 10, 16, and 32, with sizes of about 833.7, 918.5, and 1668.4 kW. This significant reduction in the objective function calue can be attributed to the fact that the PV generators provide active power to all the energy sources during the part of the day that has adequate solar radiation, which means that the substation bus reduces its active power injection, i.e., less energy must be purchased in the spot market to supply the end users, which is directly related to the final value of the objective function.
- The combination of D-STATCOMs and PV sources in S4 shows different locations and sizes when compared to the individual solutions in S2 and S3. Nevertheless, the joint use of these devices allowed for a reduction of about USD $\mathrm{1,261,534.76}$ (i.e., $35.50$) in the objective function with respect to the benchmark case. This was achieved by installing $3375.8$ kW in PV sources and $798.4\phantom{\rule{3.33333pt}{0ex}}$ kvar in D-STATCOMs, which confirms that, with the simultaneous integration of these devices, the best objective function value is reached, with reduced sizes in the D-STATCOMs and PV sources, when S4 is compared against S2 and S3, i.e., 836.1 kvar and 3420.6 kW.

#### 5.2. Numerical Results for the IEEE 69-Bus Grid

- The installation of D-STATCOMs in S2 allowed for an effective reduction of about USD $\mathrm{25,629.68}$ in the objective function value, i.e., $0.69\%$ with respect to the benchmark case. This result confirms (as happened with the IEEE 33-bus grid) that the D-STATCOMs contribute to minimizing the expected grid power losses, which in turn reduces the energy purchased at the terminals of the substation and therefore the total operating costs.
- In S3, the use of PV systems showed an effective reduction of about $35.36\%$ in the total grid operating and investment costs with respect to the benchmark case, which confirms that PV plants indeed allow distribution companies to reduce the expected energy purchasing costs (by about USD $\mathrm{1,316,577.62}$).
- The combination of PV units and D-STATCOMs in S4 showed the most efficient reduction in the total grid operational cost, with a reduction of about USD $\mathrm{1,323,038.87}$, i.e., a $35.53\%$ improvement with respect to the benchmark case. This demonstrates that, for utility companies, this approach constitutes an efficient option to reduce their operating costs and improve their grid efficiency in the form of reduced total grid energy losses, which indirectly improves the grid voltage profile.

## 6. Conclusions and Future Work

- The use of D-STATCOMs in distribution networks has a significant effect on minimizing the expected grid power losses, which influences the operating efficiency of electrical networks. In both test feeders, the use of D-STATCOMs allowed for reductions between $\mathrm{22,000}$ and $\mathrm{25,000}$ dollars with regard to the total energy purchasing costs.
- As expected, the use of PV systems also influences the total grid operating costs, as more than one and a quarter million dollars were saved by the distribution company when three PV plants were installed in the IEEE 33- and IEEE 69-bus grids. These results were expected since renewable generation from PV sources provides active power, which reduces the energy required from the terminals of the substation.
- The combination of PV units and D-STATCOMs showed the best results regarding the objective function, since, in the IEEE 33-bus grids, the expected reduction was about $35.50\%$. In the IEEE 69-bus grids, this reduction was about $35.53\%$, confirming that the simultaneous integration of these distributed energy resources in distribution networks is the best option to improve their technical and economic performance.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Ridzuan, M.I.M.; Fauzi, N.F.M.; Roslan, N.N.R.; Saad, N.M. Urban and rural medium voltage networks reliability assessment. SN Appl. Sci.
**2020**, 2, 1–9. [Google Scholar] [CrossRef] [Green Version] - Lavorato, M.; Franco, J.F.; Rider, M.J.; Romero, R. Imposing Radiality Constraints in Distribution System Optimization Problems. IEEE Trans. Power Syst.
**2012**, 27, 172–180. [Google Scholar] [CrossRef] - Celli, G.; Pilo, F.; Pisano, G.; Cicoria, R.; Iaria, A. Meshed vs. radial MV distribution network in presence of large amount of DG. In Proceedings of the IEEE PES Power Systems Conference and Exposition, New York, NY, USA, 10–13 October 2004. [Google Scholar] [CrossRef]
- Paz-Rodríguez, A.; Castro-Ordoñez, J.F.; Montoya, O.D.; Giral-Ramírez, D.A. Optimal Integration of Photovoltaic Sources in Distribution Networks for Daily Energy Losses Minimization Using the Vortex Search Algorithm. Appl. Sci.
**2021**, 11, 4418. [Google Scholar] [CrossRef] - Gnanasekaran, N.; Chandramohan, S.; Kumar, P.S.; Imran, A.M. Optimal placement of capacitors in radial distribution system using shark smell optimization algorithm. Ain Shams Eng. J.
**2016**, 7, 907–916. [Google Scholar] [CrossRef] [Green Version] - Montoya, O.D.; Gil-González, W.; Hernández, J.C. Efficient Operative Cost Reduction in Distribution Grids Considering the Optimal Placement and Sizing of D-STATCOMs Using a Discrete-Continuous VSA. Appl. Sci.
**2021**, 11, 2175. [Google Scholar] [CrossRef] - Sulaiman, M.H.; Mustaffa, Z. Optimal placement and sizing of FACTS devices for optimal power flow using metaheuristic optimizers. Results Control. Optim.
**2022**, 8, 100145. [Google Scholar] [CrossRef] - Vai, V.; Suk, S.; Lorm, R.; Chhlonh, C.; Eng, S.; Bun, L. Optimal Reconfiguration in Distribution Systems with Distributed Generations Based on Modified Sequential Switch Opening and Exchange. Appl. Sci.
**2021**, 11, 2146. [Google Scholar] [CrossRef] - Varma, R.K.; Siavashi, E.M. PV-STATCOM: A New Smart Inverter for Voltage Control in Distribution Systems. IEEE Trans. Sustain. Energy
**2018**, 9, 1681–1691. [Google Scholar] [CrossRef] - Mahmoud, K.; Abdel-Nasser, M.; Lehtonen, M.; Hussein, M.M. Optimal Voltage Regulation Scheme for PV-Rich Distribution Systems Interconnected with D-STATCOM. Electr. Power Components Syst.
**2020**, 48, 2130–2143. [Google Scholar] [CrossRef] - Chen, Q.; Kuang, Z.; Liu, X.; Zhang, T. The tradeoff between electricity cost and CO
_{2}emission in the optimization of photovoltaic-battery systems for buildings. J. Clean. Prod.**2023**, 386, 135761. [Google Scholar] [CrossRef] - Sirjani, R.; Jordehi, A.R. Optimal placement and sizing of distribution static compensator (D-STATCOM) in electric distribution networks: A review. Renew. Sustain. Energy Rev.
**2017**, 77, 688–694. [Google Scholar] [CrossRef] - Jiménez, J.; Cardona, J.E.; Carvajal, S.X. Location and optimal sizing of photovoltaic sources in an isolated mini-grid. Tecnológicas
**2019**, 22, 61–80. [Google Scholar] [CrossRef] - Cortés-Caicedo, B.; Molina-Martin, F.; Grisales-Noreña, L.F.; Montoya, O.D.; Hernández, J.C. Optimal Design of PV Systems in Electrical Distribution Networks by Minimizing the Annual Equivalent Operative Costs through the Discrete-Continuous Vortex Search Algorithm. Sensors
**2022**, 22, 851. [Google Scholar] [CrossRef] [PubMed] - Saxena, V.; Kumar, N.; Nangia, U. Recent Trends in the Optimization of Renewable Distributed Generation: A Review. Ing. Investig.
**2022**, 42, e97702. [Google Scholar] [CrossRef] - Setiawan, A.; Qashtalani, H.; Pranadi, A.D.; Ali, C.F.; Setiawan, E.A. Determination of Optimal PV Locations and Capacity in Radial Distribution System To Reduce Power Losses. Energy Procedia
**2019**, 156, 384–390. [Google Scholar] [CrossRef] - Grisales-Noreña, L.; Restrepo-Cuestas, B.; Cortés-Caicedo, B.; Montano, J.; Rosales-Muñoz, A.; Rivera, M. Optimal Location and Sizing of Distributed Generators and Energy Storage Systems in Microgrids: A Review. Energies
**2022**, 16, 106. [Google Scholar] [CrossRef] - Bhumkittipich, K.; Phuangpornpitak, W. Optimal Placement and Sizing of Distributed Generation for Power Loss Reduction Using Particle Swarm Optimization. Energy Procedia
**2013**, 34, 307–317. [Google Scholar] [CrossRef] [Green Version] - Roslan, M.F.; Al-Shetwi, A.Q.; Hannan, M.A.; Ker, P.J.; Zuhdi, A.W.M. Particle swarm optimization algorithm-based PI inverter controller for a grid-connected PV system. PLoS ONE
**2020**, 15, e0243581. [Google Scholar] [CrossRef] [PubMed] - Koutsoukis, N.C.; Georgilakis, P.S.; Hatziargyriou, N.D. A Tabu search method for distribution network planning considering distributed generation and uncertainties. In Proceedings of the 2014 International Conference on Probabilistic Methods Applied to Power Systems (PMAPS), Durham, UK, 7–10 July 2014. [Google Scholar] [CrossRef]
- Ngo, V.Q.B.; Latifi, M.; Abbassi, R.; Jerbi, H.; Ohshima, K.; khaksar, M. Improved krill herd algorithm based sliding mode MPPT controller for variable step size P&O method in PV system under simultaneous change of irradiance and temperature. J. Frankl. Inst.
**2021**, 358, 3491–3511. [Google Scholar] [CrossRef] - Kaced, K.; Larbes, C.; Ramzan, N.; Bounabi, M.; elabadine Dahmane, Z. Bat algorithm based maximum power point tracking for photovoltaic system under partial shading conditions. Sol. Energy
**2017**, 158, 490–503. [Google Scholar] [CrossRef] [Green Version] - Ali, M.H.; Kamel, S.; Hassan, M.H.; Tostado-Véliz, M.; Zawbaa, H.M. An improved wild horse optimization algorithm for reliability based optimal DG planning of radial distribution networks. Energy Rep.
**2022**, 8, 582–604. [Google Scholar] [CrossRef] - Sarwar, S.; Hafeez, M.A.; Javed, M.Y.; Asghar, A.B.; Ejsmont, K. A Horse Herd Optimization Algorithm (HOA)-Based MPPT Technique under Partial and Complex Partial Shading Conditions. Energies
**2022**, 15, 1880. [Google Scholar] [CrossRef] - Ebeed, M.; Kamel, S.; Youssef, A.R. Optimal Integration of D-STATCOM in RDS by a Novel Optimization Technique. In Proceedings of the 2018 Twentieth International Middle East Power Systems Conference (MEPCON), Cairo, Egypt, 18–20 December 2018. [Google Scholar] [CrossRef]
- Gil-González, W. Optimal Placement and Sizing of D-STATCOMs in Electrical Distribution Networks Using a Stochastic Mixed-Integer Convex Model. Electronics
**2023**, 12, 1565. [Google Scholar] [CrossRef] - Shaheen, A.M.; El-Sehiemy, R.A.; Ginidi, A.; Elsayed, A.M.; Al-Gahtani, S.F. Optimal Allocation of PV-STATCOM Devices in Distribution Systems for Energy Losses Minimization and Voltage Profile Improvement via Hunter-Prey-Based Algorithm. Energies
**2023**, 16, 2790. [Google Scholar] [CrossRef] - Santamaria-Henao, N.; Montoya, O.D.; Trujillo-Rodríguez, C.L. Optimal Siting and Sizing of FACTS in Distribution Networks Using the Black Widow Algorithm. Algorithms
**2023**, 16, 225. [Google Scholar] [CrossRef] - Cai, L.; Erlich, I.; Stamtsis, G. Optimal choice and allocation of FACTS devices in deregulated electricity market using genetic algorithms. In Proceedings of the IEEE PES Power Systems Conference and Exposition, New York, NY, USA, 10–13 October 2004; pp. 1–7. [Google Scholar] [CrossRef]
- Balamurugan, K.; Srinivasan, D. Review of power flow studies on distribution network with distributed generation. In Proceedings of the 2011 IEEE Ninth International Conference on Power Electronics and Drive Systems, Singapore, 5–8 December 2011. [Google Scholar] [CrossRef]
- Shen, T.; Li, Y.; Xiang, J. A Graph-Based Power Flow Method for Balanced Distribution Systems. Energies
**2018**, 11, 511. [Google Scholar] [CrossRef] [Green Version] - Marini, A.; Mortazavi, S.; Piegari, L.; Ghazizadeh, M.S. An efficient graph-based power flow algorithm for electrical distribution systems with a comprehensive modeling of distributed generations. Electr. Power Syst. Res.
**2019**, 170, 229–243. [Google Scholar] [CrossRef] - Montoya, O.D.; Gil-González, W. On the numerical analysis based on successive approximations for power flow problems in AC distribution systems. Electr. Power Syst. Res.
**2020**, 187, 106454. [Google Scholar] [CrossRef] - Cavraro, G.; Caldognetto, T.; Carli, R.; Tenti, P. A Master/Slave Approach to Power Flow and Overvoltage Control in Low-Voltage Microgrids. Energies
**2019**, 12, 2760. [Google Scholar] [CrossRef] [Green Version] - Doğan, B.; Ölmez, T. A new metaheuristic for numerical function optimization: Vortex Search algorithm. Inf. Sci.
**2015**, 293, 125–145. [Google Scholar] [CrossRef] - Doğan, B. A Modified Vortex Search Algorithm for Numerical Function Optimization. arXiv
**2016**, arXiv:1606.02710. [Google Scholar] [CrossRef] - Doğan, B.; Ölmez, T. Modified Off-lattice AB Model for Protein Folding Problem Using the Vortex Search Algorithm. Int. J. Mach. Learn. Comput.
**2015**, 5, 329–333. [Google Scholar] [CrossRef] [Green Version]

**Table 1.**Model parameters associated with the optimal placement and sizing of PVs in distribution grids.

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

${C}_{kWh}$ | 0.1390 | USD/kWh | T | 365 | days |

${t}_{a}$ | 10 | % | ${N}_{t}$ | 20 | years |

$\mathsf{\Delta}h$ | 1 | h | ${t}_{e}$ | 2 | % |

${C}_{pv}$ | 1036.49 | USD/kWp | ${C}_{0andM}$ | 0.0019 | USD/kWh |

${N}_{pv}^{ava}$ | 3 | - | ${p}_{i}^{pv,max}$ | 2400 | kW |

${P}_{k}^{pv,min}$ | 0 | kW |

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

${\omega}_{1}$ | 0.30 | USD/Mvar${}^{3}$ | ${\omega}_{2}$ | −305.10 | USD/Mvar${}^{2}$ |

${\omega}_{3}$ | 127,380 | USD/Mvar | $\gamma $ | 1/20 | — |

${Q}_{i}^{comp,min}$ | 0 | Mvar | ${Q}_{i,h}^{comp,max}$ | 2000 | kvar |

${P}_{i}^{cg,min}$ | 0 | W | ${P}_{i}^{cg,max}$ | 5000 | kW |

${Q}_{i}^{cg,min}$ | 0 | var | ${Q}_{i}^{cg,max}$ | 5000 | kvar |

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

1 | 2 | 0.0005 | 000012 | 0.00 | 0.00 | 3 | 36 | 0.0044 | 0.0108 | 26.00 | 18.55 |

2 | 3 | 0.0005 | 0.0012 | 0.00 | 0.00 | 36 | 37 | 0.0640 | 0.1565 | 26.00 | 18.55 |

3 | 4 | 0.0015 | 0.0036 | 0.00 | 0.00 | 37 | 38 | 0.1053 | 0.1230 | 0.00 | 0.00 |

4 | 5 | 0.0251 | 0.0294 | 0.00 | 0.00 | 38 | 39 | 0.0304 | 0.0355 | 24.00 | 17.00 |

5 | 6 | 0.3660 | 0.1864 | 2.60 | 2.20 | 39 | 40 | 0.0018 | 0.0021 | 24.00 | 17.00 |

6 | 7 | 0.3810 | 0.1941 | 40.40 | 30.00 | 40 | 41 | 0.7283 | 0.8509 | 1.20 | 1.00 |

7 | 8 | 0.0922 | 0.0470 | 75.00 | 54.00 | 41 | 42 | 0.3100 | 0.3623 | 0.00 | 0.00 |

8 | 9 | 0.0493 | 0.0251 | 30.00 | 22.00 | 42 | 43 | 0.0410 | 0.0478 | 6.00 | 4.30 |

9 | 10 | 0.8190 | 0.2707 | 28.00 | 19.00 | 43 | 44 | 0.0092 | 0.0116 | 0.00 | 0.00 |

10 | 11 | 0.1872 | 0.0619 | 145.00 | 104.00 | 44 | 45 | 0.1089 | 0.1373 | 39.22 | 26.30 |

11 | 12 | 0.7114 | 0.2351 | 145.00 | 104.00 | 45 | 46 | 0.0009 | 0.0012 | 29.22 | 26.30 |

12 | 13 | 1.0300 | 0.3400 | 8.00 | 5.00 | 4 | 47 | 0.0034 | 0.0084 | 0.00 | 0.00 |

13 | 14 | 1.0440 | 0.3450 | 8.00 | 5.50 | 47 | 48 | 0.0851 | 0.2083 | 79.00 | 56.40 |

14 | 15 | 1.0580 | 0.3496 | 0.00 | 0.00 | 48 | 49 | 0.2898 | 0.7091 | 384.70 | 274.50 |

15 | 16 | 0.1966 | 0.0650 | 45.50 | 30.00 | 49 | 50 | 0.0822 | 0.2011 | 384.70 | 274.50 |

16 | 17 | 0.3744 | 0.1238 | 60.00 | 35.00 | 8 | 51 | 0.0928 | 0.0473 | 40.50 | 28.30 |

17 | 18 | 0.0047 | 0.0016 | 60.00 | 35.00 | 51 | 52 | 0.3319 | 0.1114 | 3.60 | 2.70 |

18 | 19 | 0.3276 | 0.1083 | 0.00 | 0.00 | 9 | 53 | 0.1740 | 0.0886 | 4.35 | 3.50 |

19 | 20 | 0.2106 | 0.0690 | 1.00 | 0.60 | 53 | 54 | 0.2030 | 0.1034 | 26.40 | 19.00 |

20 | 21 | 0.3416 | 0.1129 | 114.00 | 81.00 | 54 | 55 | 0.2842 | 0.1447 | 24.00 | 17.20 |

21 | 22 | 0.0140 | 0.0046 | 5.00 | 3.50 | 55 | 56 | 0.2813 | 0.1433 | 0.00 | 0.00 |

22 | 23 | 0.1591 | 0.0526 | 0.00 | 0.00 | 56 | 57 | 1.5900 | 0.5337 | 0.00 | 0.00 |

23 | 24 | 0.3463 | 0.1145 | 28.00 | 20.00 | 57 | 58 | 0.7837 | 0.2630 | 0.00 | 0.00 |

24 | 25 | 0.7488 | 0.2475 | 0.00 | 0.00 | 58 | 59 | 0.3042 | 0.1006 | 100.00 | 72.00 |

25 | 26 | 0.3089 | 0.1021 | 14.00 | 10.00 | 59 | 60 | 0.3861 | 0.1172 | 0.00 | 0.00 |

26 | 27 | 0.1732 | 0.0572 | 14.00 | 10.00 | 60 | 61 | 0.5075 | 0.2585 | 1244.00 | 888.00 |

3 | 28 | 0.0044 | 0.0108 | 26.00 | 18.60 | 61 | 62 | 0.0974 | 0.0496 | 32.00 | 23.00 |

28 | 29 | 0.0640 | 0.1565 | 26.00 | 18.60 | 62 | 63 | 0.1450 | 0.0738 | 0.00 | 0.00 |

29 | 30 | 0.3978 | 0.1315 | 0.00 | 0.00 | 63 | 64 | 0.7105 | 0.3619 | 227.00 | 162.00 |

30 | 31 | 0.0702 | 0.0232 | 0.00 | 0.00 | 64 | 65 | 1.0410 | 0.5302 | 59.00 | 42.00 |

31 | 32 | 0.3510 | 0.1160 | 0.00 | 0.00 | 11 | 66 | 0.2012 | 0.0611 | 18.00 | 13.00 |

32 | 33 | 0.8390 | 0.2816 | 14.00 | 10.00 | 66 | 67 | 0.0470 | 0.0140 | 18.00 | 13.00 |

33 | 34 | 1.7080 | 0.5646 | 19.50 | 14.00 | 12 | 68 | 0.7394 | 0.2444 | 28.00 | 20.00 |

34 | 35 | 1.4740 | 0.4873 | 6.00 | 4.00 | 68 | 69 | 0.0047 | 0.0016 | 28.00 | 20.00 |

Scen. | ${\mathit{x}}_{\mathit{i}}^{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}$ (Node) | ${\mathit{q}}_{\mathit{i}}^{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}$ (Mvar) | ${\mathit{x}}_{\mathit{i}}^{\mathit{p}\mathit{v}}$ (Node) | ${\mathit{p}}_{\mathit{i}}^{\mathit{p}\mathit{v}}$ (MW) | ${\mathit{A}}_{{\mathbf{cost}}_{3}}$ (USD) |
---|---|---|---|---|---|

S1 | — | — | — | — | 3,553,557.38 |

S2 | $[14,\phantom{\rule{4pt}{0ex}}25,\phantom{\rule{4pt}{0ex}}30]$ | $[0.1864,\phantom{\rule{4pt}{0ex}}0.1211,\phantom{\rule{4pt}{0ex}}0.5286]$ | — | — | 3,530,954.52 |

S3 | — | — | $[10,\phantom{\rule{4pt}{0ex}}16,\phantom{\rule{4pt}{0ex}}32]$ | $[0.8337,\phantom{\rule{4pt}{0ex}}0.9185,\phantom{\rule{4pt}{0ex}}1.6684]$ | 2,300,724.60 |

S4 | $[6,\phantom{\rule{4pt}{0ex}}15,\phantom{\rule{4pt}{0ex}}31]$ | $[0.3801,\phantom{\rule{4pt}{0ex}}0.0640,\phantom{\rule{4pt}{0ex}}0.3543]$ | $[9,\phantom{\rule{4pt}{0ex}}14,\phantom{\rule{4pt}{0ex}}31]$ | $[0.9844,\phantom{\rule{4pt}{0ex}}0.6312,\phantom{\rule{4pt}{0ex}}1.7602]$ | 2,292,022.62 |

Scen. | ${\mathit{x}}_{\mathit{i}}^{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}$ (Node) | ${\mathit{q}}_{\mathit{i}}^{\mathit{c}\mathit{o}\mathit{m}\mathit{p}}$ (Mvar) | ${\mathit{x}}_{\mathit{i}}^{\mathit{p}\mathit{v}}$ (Node) | ${\mathit{p}}_{\mathit{i}}^{\mathit{p}\mathit{v}}$ (MW) | ${\mathit{A}}_{{\mathbf{cost}}_{3}}$ (USD) |
---|---|---|---|---|---|

S1 | — | — | — | — | 3,723,529.52 |

S2 | $[18,\phantom{\rule{4pt}{0ex}}61,\phantom{\rule{4pt}{0ex}}64]$ | $[0.1470,\phantom{\rule{4pt}{0ex}}0.5287,\phantom{\rule{4pt}{0ex}}0.1145]$ | — | — | 3,697,899.84 |

S3 | — | — | $[24,\phantom{\rule{4pt}{0ex}}61,\phantom{\rule{4pt}{0ex}}63]$ | $[0.3688,\phantom{\rule{4pt}{0ex}}1.8879,\phantom{\rule{4pt}{0ex}}1.3167]$ | 2,406,951.90 |

S4 | $[19,\phantom{\rule{4pt}{0ex}}53,\phantom{\rule{4pt}{0ex}}63]$ | $[0.0871,\phantom{\rule{4pt}{0ex}}0.0075,\phantom{\rule{4pt}{0ex}}0.4555]$ | $[15,\phantom{\rule{4pt}{0ex}}33,\phantom{\rule{4pt}{0ex}}62]$ | $[0.8753,\phantom{\rule{4pt}{0ex}}0.5941,\phantom{\rule{4pt}{0ex}}2.0184]$ | 2,400,490.65 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Rincón-Miranda, A.; Gantiva-Mora, G.V.; Montoya, O.D.
Simultaneous Integration of D-STATCOMs and PV Sources in Distribution Networks to Reduce Annual Investment and Operating Costs. *Computation* **2023**, *11*, 145.
https://doi.org/10.3390/computation11070145

**AMA Style**

Rincón-Miranda A, Gantiva-Mora GV, Montoya OD.
Simultaneous Integration of D-STATCOMs and PV Sources in Distribution Networks to Reduce Annual Investment and Operating Costs. *Computation*. 2023; 11(7):145.
https://doi.org/10.3390/computation11070145

**Chicago/Turabian Style**

Rincón-Miranda, Adriana, Giselle Viviana Gantiva-Mora, and Oscar Danilo Montoya.
2023. "Simultaneous Integration of D-STATCOMs and PV Sources in Distribution Networks to Reduce Annual Investment and Operating Costs" *Computation* 11, no. 7: 145.
https://doi.org/10.3390/computation11070145