# Optimal Reactive Power Compensation via D-STATCOMs in Electrical Distribution Systems by Applying the Generalized Normal Distribution Optimizer

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## Abstract

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## 1. Introduction

- We provide a new solution method based on application of the generalized normal distribution optimizer (GNDO) for locating and sizing D-STATCOMs in distribution networks with radial and meshed topologies while using a discrete–continuous codification.
- We combine the GNDO approach with an efficient power flow multi-period approach that allows solving of the technical constraints of the optimization problem, i.e., power balance, voltage regulation, and device capabilities, among others.
- We improve the final solution obtained with the proposed master–slave optimizer by using the set of nodes where the D-STATCOMs must be located as inputs for the optimal reactive power flow (ORPF) problem, aiming to further minimize the final expected annual operating costs of the distribution grid.

- The GNDO is a metaheuristic method inspired by the classical theory of normal probability distributions. It considers an initial population that evolves throughout the iterative process, considering the means and the standard deviation as advance parameters.
- The computational implementation of the GNDO is simple and requires only a few mathematical programming skills to adapt it to any optimization problem that includes binary and continuous variables.
- Multiple reports in the specialized literature have confirmed that the GNDO approach is efficient at solving complex optimization problems such as the placement location of renewable energy resources in AC and DC networks [24,25] and parameter extraction for photovoltaic models [26], among other optimization problems.

## 2. Mathematical Formulation

#### 2.1. Objective Function Structure

**Remark 1.**

#### 2.2. Set of Constraints

## 3. Proposed Solution Method

#### 3.1. Master–Slave Optimization Algorithm

#### Slave Stage: Successive Approximations Power Flow Method

**Remark 2.**

#### 3.2. Master Stage: GNDO

#### 3.2.1. Local Exploration

**Remark 3.**

- The parameters a, b, ${\lambda}_{1}$, and ${\lambda}_{2}$ are values generated with a uniform distribution in the interval $\left[01\right]$;
- ${x}_{\mathrm{best}}^{t}$ represents the best current solution reached as of iteration t;
- M is a vector that represents the average (mean value) associated with all the individuals in the current population.

#### 3.2.2. Global Exploration

#### 3.2.3. Implementation of the GNDO Approach

Algorithm 1: General application of the GNDO approach to select nodes and define sizes for D-STATCOMs in distribution networks. |

#### 3.3. Optimal Reactive Power Flow Solution

**Remark 4.**

## 4. Test System Characterization

## 5. Results and Discussion

- The GNDO approach, combined with the successive approximations power flow method, is used to obtain the initial nodal locations and sizes for the D-STATCOMs with fixed reactive power injection. Thus, the location of these devices is fixed in the exact MINLP model while aiming for additional gains for operating these reactive power compensators with variable reactive power throughout the day.
- For comparison, the MINLP model’s exact solutions obtained with the BONMIN and COUENNE solvers and the recently developed Salp Swarm Algorithm (SSA) [22] are considered in order to confirm the effectiveness and robustness of the proposed optimization approach.

- The solution of the multi-period optimal power flow problem for the benchmark case was implemented in MATLAB with our own scripts. However, to ensure the effectiveness of this approach, a comparative analysis with the DIgSILENT software was performed.
- The convergence analysis of the power flow problem for the proposed successive approximations power flow method was proven by the authors of [30], which implies that under the studied conditions this method always converges to the power flow solution.
- The solution of the MINLP model with he GNDO approach combined with the successive approximations method was also validated via GAMS and DIgSILENT.

#### 5.1. Radial Configuration Results

- The GAMS solvers (COUENNE and BONMIN) are stuck in locally optimal solutions, with reductions of 11.22 and 16.12%, respectively, with respect to the benchmark case. This can be attributed to the fact that the MINLP structure of the model (1)–(8) causes exact solution methods based on Branch and Bound and combined with interior point methods to get stuck in local solutions without the ability to escape from them because of the non-convexity of the solution space.
- The proposed GNDO approach improved the best solution reported in the literature, which was obtained with the SSA approach, i.e., by about USD $33.42$ per year of operation. This improvement was reached since the GNDO detected that node 14 is a better location for one of the D-STATCOMs, instead of node 13 as per the SSA approach.
- The total installed reactive power obtained with the GNDO approach was 869.84 kvar, while the solution found with the SSA approach installed 891.99 kvar. This implies that the proposed GNDO approach improved the objective function value by selecting a better set of nodes to place the D-STATCOMs, with the main advantage being that less investment is required for these devices.

#### 5.2. Meshed Configuration Results

- Once again, the COUENNE and BONMIN solvers got stuck in locally optimal solutions, which is attributed to the nonlinear non-convex nature of the exact MINLP model. The COUENNE solver only reached a reduction of $4.21\%$ with respect to the benchmark case, while the BONMIN solver yielded a better local solution, with a reduction of $6.61\%$ with respect to the benchmark.
- The proposed GNDO approach found a solution with an expected reduction in the annual operative costs of about 10.45%, i.e., 0.04% better than the solution reported by the SSA approach. However, the main characteristic of both solutions is that these located the D-STATCOMs in the same set of nodes (14, 30, and 32). However, their sizes differ, which explains the difference in the final objective function value.
- The total installed size of the D-STATCOMs with the GNDO is 731.83 kvar, whereas the SSA installed 742.90 kvar, which implies that with a better definition of the D-STATCOM sizes it is also possible to reach better final objective function values. Nevertheless, the main characteristic of this behavior is that based on its probability functions, the GNDO explored and exploited the solution space with better sensitivity than the SSA approach.

#### 5.3. Additional Results

- The difference between the minimum and maximum values reached in the radial configuration was about USD 8776.59, which implies that in the worst simulation case, the expected reductions with respect to the benchmark case would be 10.43%. When compared to the results shown in Table 5, this is better than both GAMS solutions. Even if the maximum result (worst result of the GNDO method) for the radial simulation case is a local solution, it has better characteristics than the local optimal solutions found with the MINLP BONMIN and COUENNE solvers in GAMS.
- As for the meshed configuration, the difference between the maximum and minimum objective function values was USD 2397.22, i.e., the worst solution reached by the GNDO approach reduces the expected annual operating costs by about 7.69%. This result confirms that for the meshed configuration case, the worst solution of the GNDO approach is also better than both solutions found with the GAMS software and with the BONMIN and COUENNE solvers.
- The mean values for the radial and meshed configurations are very close to the minimum values, i.e., most of the solutions are closer to each other in a closed ball with a radius equivalent to the standard deviation. These results confirm the effectiveness of the GNDO approach at solving the complex optimization problem involving the optimal location and size of D-STATCOMs in distribution grids via combinatorial optimization, with the main advantage that less than 43 s is required to reach a solution in both simulation cases.

## 6. Conclusions and Future Work

- In the radial configuration scenario, the proposed GNDO approach reached a lower minimum objective function value than the SSA approach. In addition, the improvement stage allowed for an additional reduction of USD 1665.25 regarding the fixed injection case, which confirms that the variable reactive power scenario via the daily control of the D-STATCOMs allows the distribution company to obtain additional profits in the total annual expected operating costs of the network.
- In the meshed configuration, the proposed GNDO approach found the same nodal location for the D-STATCOMs as the SSA methodology. However, the sizing was better, which allowed for a better final objective function value. In the variable reactive power scenario, an additional profit of USD 186.39 with respect to the fixed injection scenario was also reached, thus confirming that controlling the reactive power injection as a function of the system’s requirements is the better option to operate D-STATCOMs based on utilities.
- The expected annual operative gains for the meshed topology are considerably lower than those of the radial grid, i.e., USD 9080.32 for the meshed cased vs. USD 22,397.96 for the radial case. This is an expected result since in meshed configurations the energy losses are lower when compared to radial configurations, which is attributed to better flow distribution and voltage profiles.
- Regarding the processing times, the proposed GNDO approach found the solution for the MINLP model in less than one minute for both topologies, which confirms the effectiveness and robustness of the master–slave optimization methodology to deal with complex electrical engineering problems involving solution spaces with infinite dimensions and complex nonlinear, non-convex constraints.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Siano, P.; Rigatos, G.; Piccolo, A. Active Distribution Networks and Smart Grids: Optimal Allocation of Wind Turbines by Using Hybrid GA and Multi-Period OPF. In Atlantis Computational Intelligence Systems; Atlantis Press: Amsterdam, The Netherlands, 2012; pp. 579–599. [Google Scholar] [CrossRef]
- Lakshmi, S.; Ganguly, S. Transition of Power Distribution System Planning from Passive to Active Networks: A State-of-the-Art Review and a New Proposal. In Sustainable Energy Technology and Policies; Springer: Singapore, 2017; pp. 87–117. [Google Scholar] [CrossRef]
- Goldemberg, J.; Prado, L.T. The “decarbonization” of the world’s energy matrix. Energy Policy
**2010**, 38, 3274–3276. [Google Scholar] [CrossRef] - León-Vargas, F.; García-Jaramillo, M.; Krejci, E. Pre-feasibility of wind and solar systems for residential self-sufficiency in four urban locations of Colombia: Implication of new incentives included in Law 1715. Renew. Energy
**2019**, 130, 1082–1091. [Google Scholar] [CrossRef] - López, A.R.; Krumm, A.; Schattenhofer, L.; Burandt, T.; Montoya, F.C.; Oberländer, N.; Oei, P.Y. Solar PV generation in Colombia—A qualitative and quantitative approach to analyze the potential of solar energy market. Renew. Energy
**2020**, 148, 1266–1279. [Google Scholar] [CrossRef] - Krishna, T.M.; Ramana, N.; Kamakshaiah, S. A novel algorithm for the loss estimation and minimization of radial distribution system with distributed generation. In Proceedings of the 2013 International Conference on Energy Efficient Technologies for Sustainability, Nagercoil, India, 10–12 April 2013; pp. 1289–1293. [Google Scholar]
- Nasir, M.; Shahrin, N.; Bohari, Z.; Sulaima, M.; Hassan, M. A Distribution Network Reconfiguration based on PSO: Considering DGs sizing and allocation evaluation for voltage profile improvement. In Proceedings of the 2014 IEEE Student Conference on Research and Development, Penang, Malaysia, 16–17 December 2014; pp. 1–6. [Google Scholar]
- Vita, V. Development of a Decision-Making Algorithm for the Optimum Size and Placement of Distributed Generation Units in Distribution Networks. Energies
**2017**, 10, 1433. [Google Scholar] [CrossRef][Green Version] - Verma, H.K.; Singh, P. Optimal reconfiguration of distribution network using modified culture algorithm. J. Inst. Eng. Ser. B
**2018**, 99, 613–622. [Google Scholar] [CrossRef] - Sultana, S.; Roy, P.K. Optimal capacitor placement in radial distribution systems using teaching learning based optimization. Int. J. Electr. Power Energy Syst.
**2014**, 54, 387–398. [Google Scholar] [CrossRef] - Kerrouche, K.D.E.; Lodhi, E.; Kerrouche, M.B.; Wang, L.; Zhu, F.; Xiong, G. Modeling and design of the improved D-STATCOM control for power distribution grid. SN Appl. Sci.
**2020**, 2, 1519. [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] - Rezaeian Marjani, S.; Talavat, V.; Galvani, S. Optimal allocation of D-STATCOM and reconfiguration in radial distribution network using MOPSO algorithm in TOPSIS framework. Int. Trans. Electr. Energy Syst.
**2019**, 29, e2723. [Google Scholar] [CrossRef] - 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 savings using D-STATCOM placement in radial distribution system. Procedia Comput. Sci.
**2015**, 70, 558–564. [Google Scholar] [CrossRef][Green Version] - Jazebi, S.; Hosseinian, S.H.; Vahidi, B. DSTATCOM allocation in distribution networks considering reconfiguration using differential evolution algorithm. Energy Convers. Manag.
**2011**, 52, 2777–2783. [Google Scholar] [CrossRef] - Devi, S.; Geethanjali, M. Optimal location and sizing of distribution static synchronous series compensator using particle swarm optimization. Int. J. Electr. Power Energy Syst.
**2014**, 62, 646–653. [Google Scholar] [CrossRef] - Gupta, A.R.; Kumar, A. Optimal placement of D-STATCOM in distribution network using new sensitivity index with probabilistic load models. In Proceedings of the 2015 2nd International Conference on Recent Advances in Engineering & Computational Sciences (RAECS), Chandigarh, India, 21–22 December 2015; pp. 1–6. [Google Scholar]
- Sharma, A.K.; Saxena, A.; Tiwari, R. Optimal Placement of SVC Incorporating Installation Cost. Int. J. Hybrid Inf. Technol.
**2016**, 9, 289–302. [Google Scholar] [CrossRef] - Sirjani, R. Optimal placement and sizing of PV-STATCOM in power systems using empirical data and adaptive particle swarm optimization. Sustainability
**2018**, 10, 727. [Google Scholar] [CrossRef][Green Version] - Tuzikova, V.; Tlusty, J.; Muller, Z. A novel power losses reduction method based on a particle swarm optimization algorithm using STATCOM. Energies
**2018**, 11, 2851. [Google Scholar] [CrossRef][Green Version] - Mora-Burbano, J.A.; Fonseca-Díaz, C.D.; Montoya, O.D. Application of the SSA for Optimal Reactive Power Compensation in Radial and Meshed Distribution Using D-STATCOMs. Algorithms
**2022**, 15, 345. [Google Scholar] [CrossRef] - Garrido, V.M.; Montoya, O.D.; Medina-Quesada, Á.; Hernández, J.C. Optimal Reactive Power Compensation in Distribution Networks with Radial and Meshed Structures Using D-STATCOMs: A Mixed-Integer Convex Approach. Sensors
**2022**, 22, 8676. [Google Scholar] [CrossRef] - Zhang, Y. An improved generalized normal distribution optimization and its applications in numerical problems and engineering design problems. Artif. Intell. Rev.
**2022**. [Google Scholar] [CrossRef] - Abdel-Basset, M.; Mohamed, R.; Abouhawwash, M.; Chang, V.; Askar, S. A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling. Appl. Sci.
**2021**, 11, 4837. [Google Scholar] [CrossRef] - Zhang, Y.; Jin, Z.; Mirjalili, S. Generalized normal distribution optimization and its applications in parameter extraction of photovoltaic models. Energy Convers. Manag.
**2020**, 224, 113301. [Google Scholar] [CrossRef] - Lazarou, S.; Vita, V.; Christodoulou, C.; Ekonomou, L. Calculating Operational Patterns for Electric Vehicle Charging on a Real Distribution Network Based on Renewables’ Production. Energies
**2018**, 11, 2400. [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] - Kaur, S.; Kumbhar, G.; Sharma, J. A MINLP technique for optimal placement of multiple DG units in distribution systems. Int. J. Electr. Power Energy Syst.
**2014**, 63, 609–617. [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] - Herrera-Briñez, M.C.; Montoya, O.D.; Alvarado-Barrios, L.; Chamorro, H.R. The Equivalence between Successive Approximations and Matricial Load Flow Formulations. Appl. Sci.
**2021**, 11, 2905. [Google Scholar] [CrossRef] - Liang, Y.C.; Juarez, J.R.C. A normalization method for solving the combined economic and emission dispatch problem with meta-heuristic algorithms. Int. J. Electr. Power Energy Syst.
**2014**, 54, 163–186. [Google Scholar] [CrossRef] - Ara, A.L.; Kazemi, A.; Gahramani, S.; Behshad, M. Optimal reactive power flow using multi-objective mathematical programming. Sci. Iran.
**2012**, 19, 1829–1836. [Google Scholar] [CrossRef] - Soroudi, A. Power System Optimization Modeling in GAMS; Springer International Publishing: Berlin/Heidelberg, Germany, 2017. [Google Scholar] [CrossRef]
- Saravan, M.; Slochanal, S.; Venkatesh, P.; Abraham, P. Application of PSO technique for optimal location of FACTS devices considering system loadability and cost of installation. In Proceedings of the 2005 International Power Engineering Conference, Singapore, 29 November–2 December 2005. [Google Scholar] [CrossRef]

**Table 1.**Main literature reports regarding the optimal placement and sizing of D-STATCOMs in distribution grids.

Solution Method | Objective Function | Ref. | Year |
---|---|---|---|

Evolutionary programming | Power loss minimization and voltage profile improvement | [16] | 2011 |

Particle swarm optimization | Power loss minimization and voltage profile improvement | [17] | 2014 |

Sensitivity index | Power loss minimization and voltage profile improvement | [18] | 2015 |

Gravitational search algorithm | Power loss minimization, voltage profile improvement, and investment cost minimization | [19] | 2016 |

Adaptive particle swarm optimization | Energy loss and investment cost minimization | [20] | 2018 |

Particle swarm optimization | Energy loss minimization | [21] | 2018 |

Salp swarm algorithm | Energy loss and investment cost minimization | [22] | 2022 |

Two-stage convex optimization model | Energy loss and investment cost minimization | [23] | 2022 |

Node i | Node j | ${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (kW) | ${\mathit{Q}}_{\mathit{j}}$ (kvar) | Node i | Node j | ${\mathit{R}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{X}}_{\mathit{ij}}$ ($\mathbf{\Omega}$) | ${\mathit{P}}_{\mathit{j}}$ (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 | 1.2890 | 1.7210 | 60 | 20 | 32 | 33 | 0.3410 | 0.5302 | 60 | 40 |

Hour | Ind. (pu) | Res. (pu) | Com. (pu) |
---|---|---|---|

1 | 0.56 | 0.69 | 0.20 |

2 | 0.54 | 0.65 | 0.19 |

3 | 0.52 | 0.62 | 0.18 |

4 | 0.50 | 0.56 | 0.18 |

5 | 0.55 | 0.58 | 0.20 |

6 | 0.58 | 0.61 | 0.22 |

7 | 0.68 | 0.64 | 0.25 |

8 | 0.80 | 0.76 | 0.40 |

9 | 0.90 | 0.90 | 0.65 |

10 | 0.98 | 0.95 | 0.86 |

11 | 1 | 0.98 | 0.90 |

12 | 0.94 | 1 | 0.92 |

13 | 0.95 | 0.99 | 0.89 |

14 | 0.96 | 0.99 | 0.92 |

15 | 0.9 | 1 | 0.94 |

16 | 0.83 | 0.96 | 0.96 |

17 | 0.78 | 0.96 | 1 |

18 | 0.72 | 0.94 | 0.88 |

19 | 0.71 | 0.93 | 0.76 |

20 | 0.70 | 0.92 | 0.73 |

21 | 0.69 | 0.91 | 0.65 |

22 | 0.67 | 0.88 | 0.5 |

23 | 0.65 | 0.84 | 0.28 |

24 | 0.60 | 0.72 | 0.22 |

**Table 4.**Parametric information to evaluate the objective function regarding the investment costs of the D-STATCOMs.

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

${C}_{\mathrm{kWh}}$ | 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 |

**Table 5.**Optimal solutions obtained by the studied optimization methods for a radial grid configuration.

Method | Nodes | Sizes (Mvar) | Annual Costs (USD/year) | Red. (%) |
---|---|---|---|---|

Ben. Case | — | — | 130,613.90 | 0.00 |

COUENNE | $\left[\begin{array}{ccc}5& 6& 11\end{array}\right]$ | $\left[\begin{array}{ccc}0.0000& 0.53600& 0.27466\end{array}\right]$ | 115,960,99 | 11.22 |

BONMIN | $\left[\begin{array}{ccc}8& 25& 30\end{array}\right]$ | $\left[\begin{array}{ccc}0.29796& 0.09204& 0.51265\end{array}\right]$ | 109,560.85 | 16.12 |

SSA | $\left[\begin{array}{ccc}13& 25& 30\end{array}\right]$ | $\left[\begin{array}{ccc}0.25851& 0.10547& 0.52801\end{array}\right]$ | 108,249.36 | 17.12 |

GNDO | $\left[\begin{array}{ccc}14& 25& 30\end{array}\right]$ | $\left[\begin{array}{ccc}0.23083& 0.09996& 0.53905\end{array}\right]$ | 108,215.94 | 17.15 |

**Table 6.**Additional improvements reached when variable reactive power is used for daily compensation with D-STATCOMs for the radial grid configuration.

Method | Nodes | Sizes (Mvar) | Annual Costs (USD/year) | Red. (%) |
---|---|---|---|---|

Ben. Case | — | — | 130,613.90 | 0.00 |

GNDO (Fixed) | $\left[\begin{array}{ccc}14& 25& 30\end{array}\right]$ | $\left[\begin{array}{ccc}0.23083& 0.09996& 0.53905\end{array}\right]$ | 108,215.94 | 17.15 |

GNDO (Variable) | $\left[\begin{array}{ccc}14& 25& 30\end{array}\right]$ | $\left[\begin{array}{ccc}0.24092& 0.10118& 0.69257\end{array}\right]$ | 106,550.69 | 18.42 |

**Table 7.**Optimal solutions reached by the comparison and proposed optimization methods with fixed reactive power injection in a meshed grid configuration.

Method | Nodes | Sizes (Mvar) | Annual Costs (USD/year) | Red. (%) |
---|---|---|---|---|

Ben. Case | — | — | 86,914.74 | 0.00 |

COUENNE | $\left[\begin{array}{ccc}5& 6& 11\end{array}\right]$ | $\left[\begin{array}{ccc}0.0000& 0.32577& 0.23872\end{array}\right]$ | 83,254.95 | 4.21 |

BONMIN | $\left[\begin{array}{ccc}15& 16& 17\end{array}\right]$ | $\left[\begin{array}{ccc}0.11818& 0.00391& 0.34700\end{array}\right]$ | 81,171.76 | 6.61 |

SSA | $\left[\begin{array}{ccc}14& 30& 32\end{array}\right]$ | $\left[\begin{array}{ccc}0.14620& 0.39440& 0.20230\end{array}\right]$ | 77,870.17 | 10.41 |

GNDO | $\left[\begin{array}{ccc}14& 30& 32\end{array}\right]$ | $\left[\begin{array}{ccc}0.11554& 0.46482& 0.15147\end{array}\right]$ | 77,834.42 | 10.45 |

**Table 8.**Additional improvements reached when variable reactive power is used for daily compensation with D-STATCOMs for the meshed grid configuration.

Method | Nodes | Sizes (Mvar) | Annual Costs (USD/year) | Red. (%) |
---|---|---|---|---|

Ben. Case | — | — | 86,914.74 | 0.00 |

GNDO (Fixed) | $\left[\begin{array}{ccc}14& 30& 32\end{array}\right]$ | $\left[\begin{array}{ccc}0.11554& 0.46482& 0.15147\end{array}\right]$ | 77,834.42 | 10.45 |

GNDO (Variable) | $\left[\begin{array}{ccc}14& 30& 32\end{array}\right]$ | $\left[\begin{array}{ccc}0.11578& 0.49915& 0.17417\end{array}\right]$ | 77.697.53 | 10.60 |

Case | Min. (USD/year) | Max. (USD/year) | Mean (USD/year) | Std. Dev. (USD/year) | Time (s) |
---|---|---|---|---|---|

Radial | 108,215.94 | 116,992.53 | 109,373.20 | 2053.04 | 42.70 |

Meshed | 77,834.42 | 80,231.64 | 78,256.17 | 597.01 | 42.94 |

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**MDPI and ACS Style**

García-Pineda, L.P.; Montoya, O.D. Optimal Reactive Power Compensation via D-STATCOMs in Electrical Distribution Systems by Applying the Generalized Normal Distribution Optimizer. *Algorithms* **2023**, *16*, 29.
https://doi.org/10.3390/a16010029

**AMA Style**

García-Pineda LP, Montoya OD. Optimal Reactive Power Compensation via D-STATCOMs in Electrical Distribution Systems by Applying the Generalized Normal Distribution Optimizer. *Algorithms*. 2023; 16(1):29.
https://doi.org/10.3390/a16010029

**Chicago/Turabian Style**

García-Pineda, Laura Patricia, and Oscar Danilo Montoya. 2023. "Optimal Reactive Power Compensation via D-STATCOMs in Electrical Distribution Systems by Applying the Generalized Normal Distribution Optimizer" *Algorithms* 16, no. 1: 29.
https://doi.org/10.3390/a16010029