# Application of Ordinal Optimization to Reactive Volt-Ampere Sources Planning Problems

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## Abstract

**:**

## 1. Introduction

## 2. Mathematical Formulation

_{w}is the cost per MVAR capacitance; q

_{ow}is the installation cost at bus w; and M is the overall investment budget. The considered problem is difficult to solve because ${C}_{J,i}$ and ${\overline{C}}_{J}$ belong to discrete variables while ${\delta}_{J}=({\delta}_{w},w\in J)$ belong to integer variables.

## 3. Simulation Optimization

#### 3.1. Five Stages in the OS Algorithm

#### 3.2. Flow Diagram of the Ordinal Search (OS) Algorithm

## 4. Experiment Results

#### 4.1. Test Examples and Results

#### 4.2. Comparison with the Competing Methods

#### 4.3. Multiplicity of Configurations

#### 4.4. Discussion About System Losses and Actual Investment

_{e}denotes energy cost per unit ($/Kwh) and actual investment is of the installation costs.

_{e}is large, the optimal design of the classical objective function is located in the turning point of Figure 7, which is the objective value of system losses with the budget M = $80,000. When energy is shortage and the price of electricity is very high, we can disregard the installation costs. Details of the analyses and test results (~M = $90,000) are shown in Table 13, in which “∗” represents the optimal design of classical goal function under constant K

_{e}and different K

_{e}corresponds to different installation costs in the optimal designs of classical objective function. From the above discussions, this work not only obtains the optimal budget efficiently, but also provides the decision maker integrated and circumspect suggestion.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Shaheen, A.M.; El-Sehiemy, R.A.; Farrag, S.M. A reactive power planning procedure considering iterative identification of VAR candidate buses. Neural Comput. Appl.
**2019**, 31, 653–674. [Google Scholar] [CrossRef] - Zhang, H.; Cheng, H.Z.; Liu, L.; Zhang, S.X.; Zhou, Q.; Jiang, L. Coordination of generation, transmission and reactive power sources expansion planning with high penetration of wind power. Int. J. Electr. Power Energy Syst.
**2019**, 108, 191–203. [Google Scholar] [CrossRef] - Raj, S.; Bhattacharyya, B. Optimal placement of TCSC and SVC for reactive power planning using Whale optimization algorithm. Swarm Evol. Comput.
**2018**, 40, 131–143. [Google Scholar] [CrossRef] - Shen, Y.W.; Shen, F.F.; Chen, Y.L.; Liang, L.Q.; Zhang, B.; Ke, D.P. Reactive power planning for regional power grids based on active and reactive power adjustments of DGS. Energies
**2018**, 11, 1606. [Google Scholar] [CrossRef] - Bhattacharyya, B.; Babu, R. Teaching learning based optimization algorithm for reactive power planning. Int. J. Electr. Power Energy Syst.
**2016**, 81, 48–253. [Google Scholar] [CrossRef] - Birchfield, A.B.; Xu, T.; Overbye, T.J. Power flow convergence and reactive power planning in the creation of large synthetic grids. IEEE Trans. Power Syst.
**2018**, 33, 6667–6674. [Google Scholar] [CrossRef] - Karbalaei, F.; Abbasi, S. L-index based contingency filtering for voltage stability constrained reactive power planning. Turk. J. Electr. Eng. Comput. Sci.
**2018**, 26, 3156–3167. [Google Scholar] [CrossRef] - Liu, Z.H.; Yang, J.H.; Zhang, Y.J.; Ji, T.Y.; Zhou, J.H.; Cai, Z.X. Multi-objective coordinated planning of active-reactive power resources for decentralized droop-controlled islanded Microgrids based on probabilistic load flow. IEEE Access
**2018**, 6, 40267–40280. [Google Scholar] [CrossRef] - Shaheen, A.M.; Spea, S.R.; Farrag, S.M.; Abido, M.A. A review of meta-heuristic algorithms for reactive power planning problem. AIN Shams Eng. J.
**2018**, 9, 215–231. [Google Scholar] [CrossRef] [Green Version] - Bhattacharyya, B.; Raj, S. Swarm intelligence based algorithms for reactive power planning with Flexible AC transmission system devices. Int. J. Electr. Power Energy Syst.
**2016**, 78, 158–164. [Google Scholar] [CrossRef] - Huang, W.H.; Sun, K.; Qi, J.J.; Ning, J.X. Optimisation of dynamic reactive power sources using mesh adaptive direct search. IET Gener. Transm. Distrib.
**2017**, 11, 3675–3682. [Google Scholar] [CrossRef] [Green Version] - Amaran, S.; Sahinidis, N.V.; Sharda, B.; Bury, S.J. Simulation optimization: A review of algorithms and applications. Ann. Oper. Res.
**2016**, 240, 351–380. [Google Scholar] [CrossRef] - De Sousa Junior, W.T.; Montevechi, J.A.B.; Miranda, R.C.; Campos, A.T. Discrete simulation-based optimization methods for industrial engineering problems: A systematic literature review. Comput. Ind. Eng.
**2019**, 128, 526–540. [Google Scholar] [CrossRef] - Zhu, Y.L.; Liu, C.X.; Sun, K.; Shi, D.; Wang, Z.W. Optimization of battery energy storage to improve power system oscillation damping. IEEE Trans. Sustain. Energy
**2018**, in press. [Google Scholar] [CrossRef] - Abdelaziz, M.; Moradzadeh, M. Monte-Carlo simulation based multi-objective optimum allocation of renewable distributed generation using OpenCL. Electr. Power Syst. Res.
**2019**, 170, 81–91. [Google Scholar] [CrossRef] - Roberts, J.J.; Cassula, A.M.; Silveira, J.L.; Bortoni, E.D.; Mendiburu, A.Z. Robust multi-objective optimization of a renewable based hybrid power system. Appl. Energy
**2018**, 223, 52–68. [Google Scholar] [CrossRef] [Green Version] - Ebrahimzadeh, E.; Blaabjerg, F.; Wang, X.F.; Bak, C.L. Optimum design of power converter current controllers in large-scale power electronics based power systems. IEEE Trans. Ind. Appl.
**2019**, 55, 2792–2799. [Google Scholar] [CrossRef] - Lin, S.S.; Horng, S.C. A more general parallel dual-type method and application to state estimation. Int. J. Electr. Power Energy Syst.
**2011**, 33, 799–804. [Google Scholar] [CrossRef] - Lin, S.S.; Lin, C.H.; Horng, S.C. A parallel dual-type algorithm for a class of quadratic programming problems and applications. Expert Syst. Appl.
**2009**, 36, 5190–5199. [Google Scholar] [CrossRef] - Lin, S.S. An efficient graph technique based dual-type algorithm for NMNF problems with large capacity constraints. Appl. Math. Comput.
**2007**, 190, 309–320. [Google Scholar] [CrossRef] - Yang, D.; Cheng, H.Z.; Ma, Z.L.; Yao, L.Z.; Zhu, Z.L. Dynamic VAR planning methodology to enhance transient voltage stability for failure recovery. J. Mod. Power Syst. Clean Energy
**2018**, 6, 712–721. [Google Scholar] [CrossRef] - Yang, D.; Hong, S.Y.; Cheng, H.Z.; Yao, L.Z. A novel dynamic reactive power planning methodology to enhance transient voltage stability. Int. Trans. Electr. Energy Syst.
**2017**, 27, 2390. [Google Scholar] [CrossRef] - Ho, Y.C.; Zhao, Q.C.; Jia, Q.S. Ordinal Optimization: Soft Optimization for Hard Problems; Springer: New York, NY, USA, 2007. [Google Scholar]
- Shin, D.W.; Broadie, M.; Zeevi, A. Tractable sampling strategies for ordinal optimization. Oper. Res.
**2018**, 66, 1693–1712. [Google Scholar] [CrossRef] - Horng, S.C.; Lin, S.S. Merging crow search into ordinal optimization for solving equality constrained simulation optimization problems. J. Comput. Sci.
**2017**, 23, 44–57. [Google Scholar] [CrossRef] - Horng, S.C.; Lin, S.S. Embedding advanced harmony search in ordinal optimization to maximize throughput rate of flow line. Arab. J. Sci. Eng.
**2018**, 43, 1015–1031. [Google Scholar] [CrossRef] - Horng, S.C.; Lin, S.S. Ordinal optimization based metaheuristic algorithm for optimal inventory policy of assemble-to-order systems. Appl. Math. Model.
**2017**, 42, 43–57. [Google Scholar] [CrossRef] - Horng, S.C.; Lin, S.S. Embedding ordinal optimization into tree–seed algorithm for solving the probabilistic constrained simulation optimization problems. Appl. Sci.
**2018**, 8, 2153. [Google Scholar] [CrossRef] - Wu, J.; Han, W.Q.; Chen, T.; Zhao, J.Q.; Li, B.B.; Xu, D.G. Resonance characteristics analysis of grid-connected inverter systems based on sensitivity theory. J. Power Electron.
**2018**, 18, 746–756. [Google Scholar] - Pena, I.; Martinez-Anido, C.B.; Hodge, B.M. An extended IEEE 118-Bus test system with high renewable penetration. IEEE Trans. Power Syst.
**2018**, 33, 281–289. [Google Scholar] [CrossRef] - Lin, S.S.; Horng, S.C.; Lin, C.H. Distributed quadratic programming problems of power systems with continuous and discrete variables. IEEE Trans. Power Syst.
**2013**, 28, 472–481. [Google Scholar] [CrossRef] - Alcayde, A.; Banos, R.; Arrabal-Campos, F.M.; Montoya, F.G. Optimization of the contracted electric power by means of genetic algorithms. Energies
**2019**, 12, 1270. [Google Scholar] [CrossRef] - IMSL C Math Library. IMSL C Numerical Libraries 2016.1 for Windows; Rogue Wave Software, Inc.: Louisville, CO, USA, 2016. [Google Scholar]

**Figure 3.**The objective values vs. budget on the IEEE 118-bus system for each of these investment examples.

**Figure 4.**The objective values vs. budget on the IEEE 244-bus system for each of these investment examples.

**Figure 6.**Relationship between the system losses and installation costs of various budget examples of investment on the IEEE 118-bus system.

**Figure 7.**Relationship between the system losses and various budget examples of investment on the IEEE 118-bus system.

Budget M ($) | Actual Investment ($) | Objective Values O_{1} (MW) | CPU Time (seconds) | $\left|\widehat{\mathit{J}}\right|$ |
---|---|---|---|---|

40,000 | 37,100 | 66.1228 | 1.65 | 11 |

50,000 | 48,200 | 58.8867 | 1.68 | 14 |

60,000 | 59,300 | 52.9694 | 1.78 | 17 |

70,000 | 64,900 | 49.5524 | 1.79 | 19 |

80,000 | 76,000 | 48.1205 | 1.82 | 22 |

Budget M ($) | Actual Investment ($) | Objective Values O_{1} (MW) | CPU Time (seconds) | $\left|\widehat{\mathit{J}}\right|$ |
---|---|---|---|---|

40,000 | 39,800 | 138.3324 | 7.56 | 11 |

50,000 | 48,100 | 124.7231 | 7.74 | 13 |

60,000 | 59,200 | 114.9530 | 8.22 | 16 |

70,000 | 69,500 | 109.0737 | 8.30 | 20 |

80,000 | 79,600 | 107.1801 | 8.45 | 22 |

Budget M ($) | 40,000 | 50,000 | 60,000 | 70,000 | 80,000 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Location no. | Bus no. | Installed Banks | Bus no. | Installed Banks | Bus no. | Installed Banks | Bus no. | Installed Banks | Bus no. | Installed Banks |

1 | 11 | 3 | 11 | 3 | 11 | 3 | 11 | 3 | 11 | 3 |

2 | 19 | 2 | 19 | 2 | 19 | 3 | 19 | 3 | 19 | 3 |

3 | 23 | 2 | 23 | 3 | 23 | 3 | 23 | 3 | 23 | 3 |

4 | 27 | 3 | 27 | 3 | 27 | 3 | 27 | 3 | 27 | 3 |

5 | 37 | 3 | 37 | 3 | 37 | 3 | 37 | 3 | 37 | 3 |

6 | 56 | 3 | 56 | 3 | 56 | 3 | 56 | 3 | 56 | 3 |

7 | 62 | 2 | 62 | 2 | 62 | 2 | 62 | 2 | 62 | 3 |

8 | 77 | 3 | 77 | 3 | 77 | 3 | 77 | 3 | 77 | 3 |

9 | 92 | 3 | 92 | 3 | 92 | 3 | 92 | 3 | 92 | 3 |

10 | 96 | 2 | 96 | 3 | 96 | 3 | 96 | 3 | 96 | 3 |

11 | 106 | 3 | 106 | 3 | 106 | 3 | 106 | 3 | 106 | 3 |

12 | – | – | 5 | 2 | 5 | 2 | 5 | 2 | 5 | 3 |

13 | – | – | 64 | 2 | 64 | 3 | 64 | 3 | 64 | 3 |

14 | – | – | 88 | 3 | 88 | 3 | 88 | 3 | 88 | 3 |

15 | – | – | – | – | 70 | 2 | 70 | 2 | 70 | 3 |

16 | – | – | – | – | 68 | 3 | 68 | 3 | 68 | 3 |

17 | – | – | – | – | 105 | 2 | 105 | 2 | 105 | 2 |

18 | – | – | – | – | – | – | 12 | 2 | 12 | 2 |

19 | – | – | – | – | – | – | 51 | 2 | 51 | 2 |

20 | – | – | – | – | – | – | – | – | 48 | 2 |

21 | – | – | – | – | – | – | – | – | 75 | 2 |

22 | – | – | – | – | – | – | – | – | 110 | 2 |

Budget M ($) | 40,000 | 50,000 | 60,000 | 70,000 | 80,000 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Location no. | Bus no. | Installed Banks | Bus no. | Installed Banks | Bus no. | Installed Banks | Bus no. | Installed Banks | Bus no. | Installed Banks |

1 | 30 | 3 | 30 | 3 | 30 | 3 | 30 | 3 | 30 | 3 |

2 | 51 | 3 | 51 | 3 | 51 | 3 | 51 | 3 | 51 | 3 |

3 | 61 | 3 | 61 | 3 | 61 | 3 | 61 | 3 | 61 | 3 |

4 | 77 | 3 | 77 | 3 | 77 | 3 | 77 | 3 | 77 | 3 |

5 | 91 | 3 | 91 | 3 | 91 | 3 | 91 | 3 | 91 | 3 |

6 | 92 | 3 | 92 | 3 | 92 | 3 | 92 | 3 | 92 | 3 |

7 | 111 | 3 | 111 | 3 | 111 | 3 | 111 | 3 | 111 | 3 |

8 | 166 | 2 | 166 | 3 | 166 | 3 | 166 | 3 | 166 | 3 |

9 | 216 | 3 | 216 | 3 | 216 | 3 | 216 | 3 | 216 | 3 |

10 | 225 | 3 | 225 | 3 | 225 | 3 | 225 | 3 | 225 | 3 |

11 | 227 | 3 | 227 | 3 | 227 | 3 | 227 | 3 | 227 | 3 |

12 | – | – | 79 | 3 | 79 | 3 | 79 | 3 | 79 | 3 |

13 | – | – | 236 | 2 | 236 | 3 | 236 | 3 | 236 | 3 |

14 | – | – | – | – | 1 | 3 | 1 | 3 | 1 | 3 |

15 | – | – | – | – | 102 | 3 | 102 | 3 | 102 | 3 |

16 | – | – | – | – | 106 | 3 | 106 | 3 | 106 | 3 |

17 | – | – | – | – | – | – | 189 | 2 | 189 | 3 |

18 | – | – | – | – | – | – | 224 | 1 | 224 | 3 |

19 | – | – | – | – | – | – | 53 | 2 | 53 | 3 |

20 | – | – | – | – | – | – | 173 | 2 | 173 | 3 |

21 | – | – | – | – | – | – | – | – | 130 | 2 |

22 | – | – | – | – | – | – | – | – | 209 | 2 |

Methods | Objective Values O_{1} (MW) | Actual Investment ($) | Power Loss Reduced Rate |
---|---|---|---|

Without capacitors | 113.7812 | 0 | – |

Random selection | 72.8300 | 76,000 | 35.99% |

Proposed approach | 48.1205 | 76,000 | 57.71% |

Methods | Objective Values O_{1} (MW) | Actual Investment ($) | Power Loss Reduced Rate |
---|---|---|---|

Without capacitors | 227.0201 | 0 | – |

Random selection | 150.6278 | 79,600 | 33.65% |

Porposed approach | 107.1801 | 79,600 | 52.79% |

Budget M ($) | Actual Investment ($) | Objective Value O_{G} (MW) | Object. Value red. ${\scriptscriptstyle \frac{{\mathit{O}}_{\mathit{G}}-{\mathit{O}}_{\mathit{I}}}{{\mathit{O}}_{\mathit{I}}}}\times \mathbf{100}\%$ | CPU Time (Sec.) |
---|---|---|---|---|

40,000 | 25,200 | 80.0159 | 21.01% | 232.73 |

50,000 | 28,200 | 80.1924 | 36.18% | 209.50 |

60,000 | 31,000 | 78.3342 | 47.89% | 258.92 |

70,000 | 34,600 | 75.4880 | 52.34% | 250.65 |

80,000 | 33,800 | 77.9135 | 61.91% | 255.84 |

Budget M ($) | Actual Investment ($) | Objective Value O_{T} (MW) | Object. Value red. ${\scriptscriptstyle \frac{{\mathit{O}}_{\mathit{T}}-{\mathit{O}}_{\mathit{I}}}{{\mathit{O}}_{\mathit{I}}}}\times \mathbf{100}\%$ | CPU Time (Sec.) |
---|---|---|---|---|

40,000 | 18600 | 87.4152 | 32.20% | 209.00 |

50,000 | 11200 | 96.6673 | 64.16% | 206.06 |

60,000 | 9400 | 99.7849 | 88.38% | 237.86 |

70,000 | 7500 | 101.2972 | 104.42% | 202.18 |

80,000 | 7500 | 100.0942 | 109.69% | 239.31 |

**Table 9.**Comparisons of the proposed approach with the IMSL numerical libraries for various IEEE systems.

IEEE Systems | Budget M ($) | $\left|\widehat{\mathit{J}}\right|$ | Final Obj. Value (MW) | CPU Time (Sec.) | Speed-up Ratio (II/I) | ||
---|---|---|---|---|---|---|---|

Our App. | IMSL | Our App. (I) | IMSL (II) | ||||

6-bus | 5000 | 2 | 41.95 | 41.95 | 0.04 | 0.13 | 3.25 |

9-bus | 7000 | 2 | 33.65 | 33.65 | 0.11 | 0.39 | 3.54 |

11-bus | 7000 | 2 | 27.32 | 27.32 | 0.12 | 1.13 | 9.41 |

30-bus | 11,000 | 3 | 22.97 | 22.97 | 0.24 | 8.21 | 34.21 |

57-bus | 20,000 | 6 | 37.04 | 37.04 | 0.28 | 23.58 | 84.21 |

118-bus | 40,000 | 11 | 66.12 | - | 0.83 | - | - |

244-bus | 40,000 | 11 | 138.33 | - | 3.78 | - | - |

Budget M ($) | Objective Values O _{1} (MW) | Case 1 Line 14-15 | Case 2 Line 23-32 | Case 3 Line 62-66 | Case 4 Line 101-102 |
---|---|---|---|---|---|

40,000 | 66.1228 | 67.3026 | 66.4918 | 66.2439 | 67.8421 |

50,000 | 58.8867 | 60.7235 | 58.9831 | 60.2090 | 60.4789 |

60,000 | 52.9694 | 54.5842 | 53.0359 | 55.5249 | 54.7753 |

Budget M ($) | Objective Values O _{1} (MW) | Case 1 Line 14-15 Line 23-32 | Case 2 Line 23-32 Line 62-66 | Case 3 Line 62-66 Line 101-102 | Case 4 Line 101-102 Line 14-15 |
---|---|---|---|---|---|

40,000 | 66.1228 | 66.4842 | 66.1123 | 67.8220 | 68.3532 |

50,000 | 58.8867 | 59.7301 | 58.9490 | 61.6985 | 62.8257 |

60,000 | 52.9694 | 54.5448 | 53.3249 | 54.7271 | 56.2231 |

Budget M ($) | Objective Values O _{1} (MW) | Case 1 Line 14-15 Line 23-32 Line 62-66 | Case 2 Line 23-32 Line 62-66 Line 101-102 | Case 3 Line 14-15 Line 62-66 Line 101-102 | Case 4 Line 14-15 Line 23-32 Line 101-102 |
---|---|---|---|---|---|

40,000 | 66.1228 | 66.6751 | 68.0312 | 68.0976 | 68.5908 |

50,000 | 58.8867 | 60.3298 | 60.8323 | 62.6996 | 63.0502 |

60,000 | 52.9694 | 54.3795 | 54.3238 | 56.1794 | 55.2237 |

**Table 13.**Transform the test results into the classical objective values on the IEEE 118-bus system.

Budget M | Actual Investment | Objective Values (System Losses) | The Classical Objective Values (System Losses + (1/K _{e}) ×Actual Investment) | |||||
---|---|---|---|---|---|---|---|---|

K_{e} = 10,000 | K_{e} = 2500 | K_{e} = 1667 | K_{e} = 1250 | K_{e} = 833 | K_{e} = 625 | |||

0 | 0 | 113.7812 | 113.7812 | 113.7812 | 113.7812 | 113.7812 | 113.7812 | ^{*} 113.7812 |

30,000 | 29,600 | 73.75 | 76.71 | 85.59 | 91.51 | 97.43 | ^{*} 109.27 | 121.11 |

40,000 | 37,100 | 66.1228 | 69.8328 | 80.9628 | 88.3828 | ^{*} 95.8028 | 110.6428 | 125.4828 |

50,000 | 48,200 | 58.8867 | 63.7067 | 78.1667 | ^{*} 87.8067 | 97.4467 | 116.7267 | 136.0067 |

60,000 | 59,300 | 52.9694 | 58.8994 | 76.6894 | 88.5494 | 100.4094 | 124.1294 | 147.8494 |

70,000 | 64,900 | 49.5524 | 56.0424 | ^{*} 75.5124 | 88.4924 | 101.4724 | 127.4324 | 153.3924 |

80,000 | 76,000 | 48.1205 | ^{*} 55.7205 | 78.5205 | 93.7205 | 108.9205 | 139.3205 | 169.7205 |

90,000 | 76,000 | 48.1205 | ^{*} 55.7205 | 78.5205 | 93.7205 | 108.9205 | 139.3205 | 169.7205 |

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## Share and Cite

**MDPI and ACS Style**

Lee, W.-T.; Horng, S.-C.; Lin, C.-F.
Application of Ordinal Optimization to Reactive Volt-Ampere Sources Planning Problems. *Energies* **2019**, *12*, 2746.
https://doi.org/10.3390/en12142746

**AMA Style**

Lee W-T, Horng S-C, Lin C-F.
Application of Ordinal Optimization to Reactive Volt-Ampere Sources Planning Problems. *Energies*. 2019; 12(14):2746.
https://doi.org/10.3390/en12142746

**Chicago/Turabian Style**

Lee, Wen-Tung, Shih-Cheng Horng, and Chi-Fang Lin.
2019. "Application of Ordinal Optimization to Reactive Volt-Ampere Sources Planning Problems" *Energies* 12, no. 14: 2746.
https://doi.org/10.3390/en12142746