# A Discrete-Continuous PSO for the Optimal Integration of D-STATCOMs into Electrical Distribution Systems by Considering Annual Power Loss and Investment Costs

^{1}

^{2}

^{3}

^{4}

^{5}

^{6}

^{7}

^{*}

## Abstract

**:**

## 1. Introduction

- A new Discrete-Continuous Particle Swarm Optimization method (i.e., DCPSO) to solve problems with discrete and continuous variables;
- A new master–slave methodology (i.e., DCPSO/HSA) to solve the problem of optimal integration of D-STATCOMs into EDSs that reduces operating and investment costs;
- The best results reported in the literature (in terms of solution quality, repeatability, and processing times) in the solution to the problem of the optimal location and sizing of D-STATCOMs in EDSs.

- A complete description of a mathematical model and codification that represents the problem of the optimal integration of D-STATCOMs in electrical networks for reducing the annual operating cost, which can then be used as basis for developing new mathematical solution methods;
- An efficient comparison method (DCPSO) for evaluating the performance of the new mathematical method developed in terms of quality solution, repeatability of the solution, and processing times.

- A methodology that allows the integration of electrical grid operators within their electrical networks reactive power by using a distributed static compensator that considers the reduction in energy purchasing and investment costs;
- A numerical validation of the positive effect for installing distributed static compensators in electrical grids in economical and technical terms;
- A fast and efficient methodology that allows the electrical grid’s operators to solve the problem of the optimal integration of D-STATCOMs in electrical networks, allowing the exploration of multiple load scenarios in short processing times, which is vital inside the public and private processes for obtaining contracts around the world due to the short time-period offers for carrying out this kind of process.

## 2. Mathematical Formulation

## 3. Proposed Solution Methodology

#### 3.1. Discrete-Continuous Particle Swarm Optimization Method

Algorithm 1: Pseudo-code for the DCPSO algorithm |

#### 3.2. Hourly Power Flow Method Based on Successive Approximations

Algorithm 2: Slave stage algorithm: HSA |

## 4. Test Systems, Methods Used for Comparison, and Considerations

#### 4.1. 33-Bus Test System

#### 4.2. 69-Bus Test System

#### 4.3. Methods Used for Comparison and Parameters

## 5. Simulation Results

#### 5.1. 33-Bus Test System

#### 5.2. 69-Bus Test System

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DCPSO | Discrete-continuous particle Swarm optimization algorithm. |

D-STATCOMs | Distribution static compensators. |

EDS | Electrical Distribution Systems. |

HSA | Hourly power flow Method based on successive approximations. |

GAMS | General algebraic modeling system. |

BONMIN | Solver of GAMS. |

COUNNE | Solver of GAMS. |

GA | Genetic algorithm. |

PSO | Particle swarm optimization algorithm. |

DCVSA | Discrete-continuous version of the Vortex Search Algorithm. |

$\alpha $ | Constant of quadratic term of investment cost of the integration of the D-STATCOMs. |

$\beta $ | Constant of lineal term of investment cost of the integration of the D-STATCOMs. |

${\Delta}_{h}$ | Fraction of time that has passed when the data about power demand for one day of operation are obtained. |

${\delta}_{ih}$ | Voltages angle at nodes i at hour h. |

${\delta}_{jh}$ | Voltages angle at nodes j at hour h. |

$\gamma $ | Constant of investment cost of the integration of the D-STATCOMs. |

$\mathcal{H}$ | Set that contains all the periods of time used. |

$\mathcal{N}$ | Sets that contains he buses that compose the electrical network. |

${\theta}_{ij}$ | Angle of the nodal admittance matrix associated with buses i and j. |

${A}_{cost}$ | Weighted multi-objective function to reduce the annual operating cost. |

${k}_{1}$ | Constant used to annualize the operating cost. |

${k}_{2}$ | Constant used to annualize the useful life. |

${N}_{\mathrm{avalaible}}^{\mathrm{D}-\mathrm{STATCOM}}$ | Variable that limit the maximum number of D-STATCOMs that can be installed in the distribution syste. |

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

${P}_{jh}^{g}$ | Active power generation injected at node i in period of time h. |

${Q}_{ih}^{d}$ | Reactive power generation consumed at node i in period of time h. |

${Q}_{ih}^{g}$ | Reactive power generation injected at node i in period of time h. |

${Q}_{k}^{\mathrm{D}-\mathrm{STATCOM}}$ | Reactive power assigned to each D-STATCOM located in the electrical network. |

${Q}_{\mathrm{max}}^{\mathrm{D}-\mathrm{STATCOM}}$ | Maximum limit for the injection of reactive power by D-STATCOMs. |

${Q}_{\mathrm{min}}^{\mathrm{D}-\mathrm{STATCOM}}$ | Minimum limit for the injection of reactive power by D-STATCOMs. |

T | Constant used to evaluate the implemented time horizon. |

${V}_{ih}$ | Voltages at nodes i at hour h. |

${V}_{jh}$ | Voltages at nodes j at hour h. |

${V}_{\mathrm{max}}$ | Maximum bus voltage allowable in the electrical system. |

${V}_{\mathrm{min}}$ | Minimum bus voltage allowable in the electrical system. |

${Y}_{ij}$ | Magnitude of the nodal admittance matrix associated with buses i and j. |

${C}_{kwh}$ | Energy cost per kWh. |

${F}_{1}$ | Cost of the annual power loss. |

${F}_{2}$ | Annual investment cost associated with the integration of D-STATCOMs. |

${x}_{i}$ | Binary variable that takes a value of 1 when a D-STATCOM is located at a bus, and 0 when it is not. |

## References

- Kirkerud, J.; Nagel, N.O.; Bolkesjø, T. The role of demand response in the future renewable northern European energy system. Energy
**2021**, 235, 121336. [Google Scholar] [CrossRef] - Akorede, M.F.; Hizam, H.; Pouresmaeil, E. Distributed energy resources and benefits to the environment. Renew. Sustain. Energy Rev.
**2010**, 14, 724–734. [Google Scholar] [CrossRef] - López González, D.M.; Garcia Rendon, J. Opportunities and challenges of mainstreaming distributed energy resources towards the transition to more efficient and resilient energy markets. Renew. Sustain. Energy Rev.
**2022**, 157, 112018. [Google Scholar] [CrossRef] - Copp, D.A.; Nguyen, T.A.; Byrne, R.H.; Chalamala, B.R. Optimal sizing of distributed energy resources for planning 100% renewable electric power systems. Energy
**2022**, 239, 122436. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Montoya, O.D.; Hincapié-Isaza, R.A.; Echeverri, M.G.; Perea-Moreno, A.J. Optimal location and sizing of DGs in DC networks using a hybrid methodology based on the PPBIL algorithm and the VSA. Mathematics
**2021**, 9, 1913. [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] - Siddiqui, A.S.; Sarwar, M.; Althobaiti, A.; Ghoneim, S.S. Optimal Location and Sizing of Distributed Generators in Power System Network with Power Quality Enhancement Using Fuzzy Logic Controlled D-STATCOM. Sustainability
**2022**, 14, 3305. [Google Scholar] - Tolabi, H.B.; Ali, M.H.; Rizwan, M. Simultaneous reconfiguration, optimal placement of DSTATCOM, and photovoltaic array in a distribution system based on fuzzy-ACO approach. IEEE Trans. Sustain. Energy
**2014**, 6, 210–218. [Google Scholar] [CrossRef] - Gupta, A.R.; Kumar, A. Energy saving using D-STATCOM placement in radial distribution system under reconfigured network. Energy Procedia
**2016**, 90, 124–136. [Google Scholar] [CrossRef] - Salkuti, S.R. An efficient allocation of D-STATCOM and DG with network reconfiguration in distribution networks. Int. J. Adv. Technol. Eng. Explor.
**2022**, 9, 299. [Google Scholar] - Rani, K.R.; Rani, P.S.; Chaitanya, N.; Janamala, V. Improved Bald Eagle Search for Optimal Allocation of D-STATCOM in Modern Electrical Distribution Networks with Emerging Loads. J. Intell. Eng. Syst.
**2021**, 15, 554–563. [Google Scholar] - 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] - Choudhury, S.; Bhowmik, P.; Rout, P.K. Economic load sharing in a D-STATCOM integrated islanded microgrid based on fuzzy logic and seeker optimization approach. Sustain. Cities Soc.
**2018**, 37, 57–69. [Google Scholar] [CrossRef] - Mehouachi, I.; Abbes, M.; Chebbi, S. Design of a high power D-STATCOM based on the isolated dual-converter topology. Int. J. Electr. Power Energy Syst.
**2019**, 106, 401–410. [Google Scholar] [CrossRef] - Roy, N.K.; Hossain, M.J.; Pota, H.R. Voltage profile improvement for distributed wind generation using D-STATCOM. In Proceedings of the IEEE Power and Energy Society General Meeting, Detroit, MI, USA, 24–29 July 2011; pp. 1–6. [Google Scholar] [CrossRef]
- Grisales-Noreña, L.F.; Montoya, O.D.; Ramos-Paja, C.A. An energy management system for optimal operation of BSS in DC distributed generation environments based on a parallel PSO algorithm. J. Energy Storage
**2020**, 29, 101488. [Google Scholar] [CrossRef] - Khaleel, M.M.; Adzman, M.R.; Zali, S.M. An Integrated of Hydrogen Fuel Cell to Distribution Network System: Challenging and Opportunity for D-STATCOM. Energies
**2021**, 14, 7073. [Google Scholar] [CrossRef] - Rohouma, W.; Metry, M.; Balog, R.S.; Peerzada, A.A.; Begovic, M.M.; Zhou, D. Analysis of the Capacitor-Less D-STATCOM for Voltage Profile Improvement in Distribution Network with High PV Penetration. IEEE Open J. Power Electron.
**2022**, 3, 255–270. [Google Scholar] [CrossRef] - Chakraborty, S.; Mukhopadhyay, S.; Biswas, S.K. Coordination of D-STATCOM & SVC for Dynamic VAR Compensation and Voltage Stabilization of an AC Grid Interconnected to a DC Microgrid. IEEE Trans. Ind. Appl.
**2021**, 58, 634–644. [Google Scholar] - Rohouma, W.; Balog, R.S.; Peerzada, A.A.; Begovic, M.M. D-STATCOM for harmonic mitigation in low voltage distribution network with high penetration of nonlinear loads. Renew. Energy
**2020**, 145, 1449–1464. [Google Scholar] [CrossRef] - Castiblanco-Pérez, C.M.; Toro-Rodríguez, D.E.; Montoya, O.D.; Giral-Ramírez, D.A. Optimal Placement and sizing of D-STATCOM in radial and meshed distribution networks using a discrete-continuous version of the genetic algorithm. Electronics
**2021**, 10, 1452. [Google Scholar] [CrossRef] - Montoya, O.D.; Alvarado-Barrios, L.; Hernández, J.C. An Approximate Mixed-Integer Convex Model to Reduce Annual Operating Costs in Radial Distribution Networks Using STATCOMs. Electronics
**2021**, 10, 3102. [Google Scholar] [CrossRef] - Grisales-Noreña, L.F.; Gonzalez Montoya, D.; Ramos-Paja, C.A. Optimal sizing and location of distributed generators based on PBIL and PSO techniques. Energies
**2018**, 11, 1018. [Google Scholar] [CrossRef] [Green Version] - Montoya, O.D.; Fuentes, J.E.; Moya, F.D.; Barrios, J.Á.; Chamorro, H.R. Reduction of Annual Operational Costs in Power Systems through the Optimal Siting and Sizing of STATCOMs. Appl. Sci.
**2021**, 11, 4634. [Google Scholar] [CrossRef] - Montoya, O.D.; Garrido, V.M.; Gil-González, W.; Grisales-Noreña, L.F. Power flow analysis in DC grids: Two alternative numerical methods. IEEE Trans. Circuits Syst. II Express Briefs
**2019**, 66, 1865–1869. [Google Scholar] [CrossRef] - Moradi, M.H.; Abedini, M. A combination of genetic algorithm and particle swarm optimization for optimal DG location and sizing in distribution systems. Int. J. Electr. Power Energy Syst.
**2012**, 34, 66–74. [Google Scholar] [CrossRef] - Gomez-Gonzalez, M.; López, A.; Jurado, F. Optimization of distributed generation systems using a new discrete PSO and OPF. Electr. Power Syst. Res.
**2012**, 84, 174–180. [Google Scholar] [CrossRef] - Soroudi, A.; Afrasiab, M. Binary PSO-based dynamic multi-objective model for distributed generation planning under uncertainty. IET Renew. Power Gener.
**2012**, 6, 67–78. [Google Scholar] [CrossRef] [Green Version] - Diaz-Acevedo, J.A.; Escobar, A.; Grisales-Norena, L.F. Optimization of corona ring for 230 kV polymeric insulator based on finite element method and PSO algorithm. Electr. Power Syst. Res.
**2021**, 201, 107521. [Google Scholar] [CrossRef] - Mozafar, M.R.; Moradi, M.H.; Amini, M.H. A simultaneous approach for optimal allocation of renewable energy sources and electric vehicle charging stations in smart grids based on improved GA-PSO algorithm. Sustain. Cities Soc.
**2017**, 32, 627–637. [Google Scholar] [CrossRef] - Jafari, A.; Ganjehlou, H.G.; Khalili, T.; Mohammadi-Ivatloo, B.; Bidram, A.; Siano, P. A two-loop hybrid method for optimal placement and scheduling of switched capacitors in distribution networks. IEEE Access
**2020**, 8, 38892–38906. [Google Scholar] [CrossRef] - Grisales, L.F.; Grajales, A.; Montoya, O.D.; Hincapie, R.A.; Granada, M.; Castro, C.A. Optimal location, sizing and operation of energy storage in distribution systems using multi-objective approach. IEEE Lat. Am. Trans.
**2017**, 15, 1084–1090. [Google Scholar] [CrossRef]

**Figure 3.**Active and reactive power demand in a Colombian EDS [12].

Par. | Value | Unit | Par. | Value | Unit |
---|---|---|---|---|---|

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

${\Delta}_{h}$ | 0.50 | h | $\alpha $ | 0.30 | USD/MVAr${}^{3}$ |

$\beta $ | −305.10 | USD/MVAr${}^{2}$ | $\gamma $ | 127,380 | USD/MVAr |

${k}_{1}$ | 6/2190 | 1/Days | ${k}_{2}$ | 10 | Years |

Method | GA | PSO | DCPSO |
---|---|---|---|

Number of particles | 12 | 30 | 99 |

Selection method | Tournament | Cognitive and social component: 1.4 | Cognitive and social component: 1.93 and 1.79 |

Population update method | Cross over: simple | Speed (max–min): (0.1–0.1) Inertia (max–min): (0.7–0.001) | Speed (max–min): (0.05–2) Inertia (max–min): (0.1–0.2) |

Mutation | Binary simple | R1 = R2: Random | R1 = R2: Random |

Stopping criterion | Generational cycles: (200) | Maximum iterations: (200) | Maximum iterations: (219) |

Methodology | Bus/Power (MVAr) | ${\mathit{A}}_{\mathbf{cost}}$ (US D/year) Reduction (%) | Time (s) | STD (%) | ${\mathit{V}}_{\mathbf{worst}}$ (p.u) | ${\mathit{I}}_{max}$ [A] |
---|---|---|---|---|---|---|

Without STATCOMs | [0–2] | 112,740.90 | - - - | - - - | [0.9–1.1] | 380 |

COUENNE | 16/0.0109 17/0.0224 18/0.2065 | 107,589.50/4.56 | 3.03 | 0 | 0.92 | 336 |

BONMIN | 17/0.0339 18/0.0227 30/0.2395 | 102,447.29/9.13 | 7.59 | 0 | 0.91 | 334 |

GA/PSO | 14/0.1599 30/0.3497 31/0.11166 | 98,511.63/12.62 | 6417.91 | 0.0974 | 0.92 | 323 |

DCVSA | 14/0.1599 30/0.3591 31/0.1072 | 98,497.90/12.63 | 59.64 | 0.0414 | 0.92 | 323 |

DCPSO/HSA | 14/0.1599 30/0.3591 32/0.1072 | 98,497.90/12.63 | 23.05 | 0.2555 | 0.92 | 323 |

Methodology | Bus/Power (MVAr) | ${\mathit{A}}_{\mathbf{cost}}$
(US D/Year) Reduction (%) | Time (s) | STD (%) | ${\mathit{V}}_{\mathbf{worst}}$ (p.u) | ${\mathit{I}}_{\mathbf{max}}$ (A) |
---|---|---|---|---|---|---|

Without D-STATCOMs | [0–2] | 119,715.63 | - - - | - - - | [0.9–1.1] | 380 |

GA/PSO | 21/0.0.0839 61/0.4600 64/0.1139 | 102,990.79/13.97 | 6417.91 | 0.0974 | 0.92 | 344 |

DCVSA | 21/0.0839 61/0.4601 64/0.1139 | 102,990.79/13.97 | 202.66 | 0.1367 | 0.92 | 343 |

DCPSO/HSA | 21/0.0839 61/0.4601 64/0.1139 | 102,990.79/13.97 | 54.34 | 0.1444 | 0.92 | 343 |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 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**

Grisales-Noreña, L.F.; Montoya, O.D.; Hernández, J.C.; Ramos-Paja, C.A.; Perea-Moreno, A.-J.
A Discrete-Continuous PSO for the Optimal Integration of D-STATCOMs into Electrical Distribution Systems by Considering Annual Power Loss and Investment Costs. *Mathematics* **2022**, *10*, 2453.
https://doi.org/10.3390/math10142453

**AMA Style**

Grisales-Noreña LF, Montoya OD, Hernández JC, Ramos-Paja CA, Perea-Moreno A-J.
A Discrete-Continuous PSO for the Optimal Integration of D-STATCOMs into Electrical Distribution Systems by Considering Annual Power Loss and Investment Costs. *Mathematics*. 2022; 10(14):2453.
https://doi.org/10.3390/math10142453

**Chicago/Turabian Style**

Grisales-Noreña, Luis Fernando, Oscar Danilo Montoya, Jesús C. Hernández, Carlos Andres Ramos-Paja, and Alberto-Jesus Perea-Moreno.
2022. "A Discrete-Continuous PSO for the Optimal Integration of D-STATCOMs into Electrical Distribution Systems by Considering Annual Power Loss and Investment Costs" *Mathematics* 10, no. 14: 2453.
https://doi.org/10.3390/math10142453