# Optimal Power Flow Incorporating FACTS Devices and Stochastic Wind Power Generation Using Krill Herd Algorithm

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## Abstract

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## 1. Introduction

- Modeling and including the stochastic nature of wind power generation in the problem formulation.
- Unlike the other research studies, in this paper, the OPF problem incorporating FACTS devices and stochastic wind power generation at the same time is solved.
- The KHA is used to minimize the fuel cost, active power losses across the transmission lines, emission, and CEEC, as the objective functions.

## 2. Problem Formulation

#### 2.1. General Formulation

#### 2.2. FACTS Devices Modeling

#### 2.2.1. TCSC Modeling

#### 2.2.2. TCPS Modeling

#### 2.3. Wind Power Generation Modeling

#### 2.4. Objective Functions

#### 2.4.1. Minimization of Fuel Cost

#### 2.4.2. Minimization of Active Power Losses across the Transmission Lines

#### 2.4.3. Minimization of Emission

#### 2.4.4. Minimization of the Combined Economic and Environmental Costs

#### 2.5. Constraints

#### 2.5.1. Load Flow Constraints

#### 2.5.2. Active and Reactive Power of the Generation Units

#### 2.5.3. Voltage at Each Bus

#### 2.5.4. Transformer Tap Settings

#### 2.5.5. Transmission Lines Loading

#### 2.5.6. TCSC Reactance Constraints

#### 2.5.7. TCPS Phase Shift

#### 2.6. Solution Method

## 3. Simulation Results

#### 3.1. Case 1: IEEE 30-Bus Test System

#### 3.1.1. Minimization of Fuel Cost

#### 3.1.2. Minimization of Active Power Losses across the Transmission Lines

#### 3.1.3. Minimization of Active Power Losses across the Transmission Lines

#### 3.1.4. Minimization of Combined Economic and Environmental Costs

#### 3.2. Case 2: IEEE 57-Bus Test System

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The convergence curves for the fuel cost minimization considering and neglecting the valve-point effect for the Test System 1.

**Figure 5.**The convergence curves for the minimization of power losses across the transmission lines for the Test System 1.

**Figure 8.**The convergence curves for the fuel cost minimization considering and neglecting the valve-point effect for the Test System 2.

**Figure 9.**The convergence curves for the minimization of power losses across the transmission lines for the Test System 2.

Reference | Uncertainty Model | Solution Method | Objective Functions |
---|---|---|---|

[6] | Weibull distribution function | Sequential quadratic programming PSO | Minimizing the total operating costs and minimizing emission |

[7] | Incomplete gamma function | Imperialist Competitive Algorithm (ICA) | Minimizing the fuel cost function |

[8] | Weibull probability density function | Gbest Guided-ABC | Minimizing the total operating costs |

[9] | Weibull probability density function | PSO | Minimizing the total operating and congestion costs |

Reference | Method | Objective Functions | FACTS Devices |
---|---|---|---|

[7] | Micro-genetic algorithm and hybrid method | Minimizing the fuel cost and power losses, Optimal location of FACTS devices | TCSC, TCPAR, UPFC, SVC |

[8] | PSAT software analysis | Improving voltage profile, Minimizing power losses | SVC |

[9] | Dimensional algorithm using NR load flow | Improving voltage profile, Minimizing power losses and fuel costs | TCSC, TCPR, SVC, STATCOM |

[11] | Genetic Algorithm (GA) and Differential Evolution Algorithm (DEA) | Minimizing the fuel cost and power losses, Optimal location of FACTS devices | UPFC, SVC, TCSC |

[12] | Dimensional algorithm | Heat control, Minimizing power losses, Improving power systems stability | UPFC |

[13] | GA and DEA | Minimizing the fuel cost and power losses | UPFC |

[14] | Artificial Immune Systems (AIS) | Minimizing the fuel cost | TCPS, TCSC |

[15] | GA | Minimizing the fuel cost and power losses | TCSC, TCPAR, UPFC |

[16] | DEA | Maximizing the loadability of transmission lines, Reducing the transmission lines losses | STATCOM |

[17] | Combined Tabu Search (TS) and Simulated Annealing (SA) method | Minimizing the total fuel cost | TCSC, TCPS |

[18] | GA | Minimizing the total fuel costs under security constraints | UPFC |

[19] | PSO | Reducing the FACTS devices installation costs, Reducing overload | TCSC, UPFC, SVC, TCVR |

**Table 3.**Results for fuel cost minimization without considering the valve-point effect for the Test System 1.

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 179.755 | 185.240 | 180.260 | 192.460 | 137.526 |

${P}_{{G}_{2}}$ (MW) | 47.8185 | 46.3300 | 49.3200 | 48.3800 | 41.9122 |

${P}_{{G}_{5}}$ (MW) | 18.5154 | 20.8800 | 20.8200 | 19.5400 | 19.3311 |

${P}_{{G}_{8}}$ (MW) | 16.0965 | 15.6400 | 17.6100 | 11.6000 | 15.5927 |

${P}_{{G}_{11}}$ (MW) | 10.0000 | 11.1200 | 11.0500 | 10.0000 | 20.9936 |

${P}_{{G}_{13}}$ (MW) | 19.3238 | 12.5800 | 12.6900 | 12.0000 | 17.7110 |

Total Generation (MW) | 291.509 | 291.790 | 291.750 | 294.000 | 253.067 |

${P}_{wind}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 34.4126 |

Fuel Cost ($/h) | 779.393 | 796.930 | 797.290 | 803.840 | 683.646 |

Emission (ton/h) | 0.42496 | 0.39020 | 0.37560 | 0.00000 | 0.28904 |

Power Losses (MW) | 8.10960 | 8.39000 | 8.35000 | 10.6000 | 4.08011 |

Computation Time (s) | 184.400 | 479.200 | 487.300 | 265.800 | 188.100 |

**Table 4.**Results for fuel cost minimization considering the valve-point effect for the Test System 1.

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 191.690 | 199.850 | 199.130 | 198.810 | 130.246 |

${P}_{{G}_{2}}$ (MW) | 34.4058 | 38.2000 | 38.3200 | 38.9600 | 39.4530 |

${P}_{{G}_{5}}$ (MW) | 15.0000 | 20.1600 | 20.1700 | 19.1600 | 32.9689 |

${P}_{{G}_{8}}$ (MW) | 10.0000 | 11.1500 | 11.4300 | 10.6400 | 29.5335 |

${P}_{{G}_{11}}$ (MW) | 19.2954 | 10.1300 | 10.4300 | 13.5600 | 11.8032 |

${P}_{{G}_{13}}$ (MW) | 21.0191 | 12.6600 | 12.6600 | 12.0300 | 12.0000 |

Total Generation (MW) | 291.410 | 292.150 | 292.140 | 293.160 | 256.005 |

${P}_{wind}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 32.6020 |

Fuel Cost ($/h) | 824.150 | 825.890 | 826.540 | 831.030 | 676.762 |

Emission (ton/h) | 0.44373 | 0.44124 | 0.43830 | 0.43660 | 0.30525 |

Power Losses (MW) | 8.01050 | 8.75000 | 8.74000 | 9.76000 | 5.20730 |

Computation Time (s) | 185.700 | 503.120 | 505.600 | 714.800 | 189.000 |

**Table 5.**Results for minimizing the active power losses across the transmission lines for the Test System 1.

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 98.0937 | 74.6900 | 77.5900 | 77.5800 | 15.7459 |

${P}_{{G}_{2}}$ (MW) | 53.5641 | 67.3000 | 67.3000 | 69.5800 | 80.0000 |

${P}_{{G}_{5}}$ (MW) | 50.0000 | 50.0000 | 50.0000 | 49.9800 | 50.0000 |

${P}_{{G}_{8}}$ (MW) | 35.0000 | 34.6600 | 34.8500 | 34.9600 | 35.0000 |

${P}_{{G}_{11}}$ (MW) | 16.5549 | 27.2600 | 27.0400 | 23.6900 | 30.0000 |

${P}_{{G}_{13}}$ (MW) | 32.8586 | 32.2200 | 32.3600 | 30.4300 | 40.0000 |

Total Generation (MW) | 286.071 | 286.130 | 285.140 | 286.220 | 250.745 |

${P}_{wind}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 34.4212 |

Fuel Cost ($/h) | 992.050 | 992.180 | 992.300 | 985.210 | 918.639 |

Emission (ton/h) | 0.21091 | 0.21090 | 0.21090 | 0.21440 | 0.21031 |

Power Losses (MW) | 2.92130 | 2.98000 | 2.99000 | 3.07000 | 1.76710 |

Computation Time (s) | 170.150 | 482.100 | 497.400 | 711.700 | 174.300 |

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 51.3924 | 64.5200 | 63.5000 | 63.9800 | 45.9204 |

${P}_{{G}_{2}}$ (MW) | 80.0000 | 66.9000 | 67.9200 | 67.7500 | 51.6969 |

${P}_{{G}_{5}}$ (MW) | 50.0000 | 50.0000 | 50.0000 | 50.0000 | 50.0000 |

${P}_{{G}_{8}}$ (MW) | 35.0000 | 35.0000 | 35.0000 | 35.0000 | 35.0000 |

${P}_{{G}_{11}}$ (MW) | 30.0000 | 30.0000 | 30.0000 | 29.9600 | 30.0000 |

${P}_{{G}_{13}}$ (MW) | 40.0000 | 40.0000 | 40.0000 | 40.0000 | 40.0000 |

Total Generation (MW) | 286.392 | 286.420 | 286.420 | 286.690 | 252.617 |

${P}_{wind}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 33.3285 |

Fuel Cost ($/h) | 1012.75 | 1014.24 | 1015.10 | 1015.80 | 906.068 |

Emission (ton/h) | 0.20469 | 0.20475 | 0.20480 | 0.20490 | 0.19587 |

Power Losses (MW) | 2.99240 | 3.02000 | 3.02000 | 3.29000 | 2.54580 |

Computation Time (s) | 169.140 | 506.100 | 511.300 | 706.000 | 173.180 |

Control Variable | Without Wind Farm | With Wind Farm | ||
---|---|---|---|---|

KHA | ALC-PSO | DEA | KHA | |

${P}_{{G}_{1}}$ (MW) | 126.476 | 115.230 | 107.980 | 110.376 |

${P}_{{G}_{2}}$ (MW) | 66.4293 | 56.5700 | 58.5700 | 63.8014 |

${P}_{{G}_{5}}$ (MW) | 29.8519 | 31.8800 | 32.3800 | 23.7588 |

${P}_{{G}_{8}}$ (MW) | 27.9298 | 27.5400 | 27.6100 | 17.1252 |

${P}_{{G}_{11}}$ (MW) | 18.0473 | 23.8900 | 29.5100 | 16.7590 |

${P}_{{G}_{13}}$ (MW) | 19.8514 | 34.2300 | 33.2700 | 23.6716 |

Total Generation (MW) | 288.585 | 289.330 | 289.320 | 255.492 |

${P}_{wind}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 32.3655 |

Fuel Cost ($/h) | 897.430 | 907.170 | 922.360 | 784.653 |

Emission (ton/h) | 0.23990 | 0.24302 | 0.23640 | 0.22423 |

Power Losses (MW) | 5.18590 | 5.92000 | 5.93000 | 4.45820 |

CEEC | 1232.80 | 1234.44 | 1238.09 | 1095.72 |

Computation Time (s) | 189.140 | 515.100 | 521.300 | 189.180 |

**Table 8.**Results for fuel cost minimization without considering the valve-point effect for the Test System 2.

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 584.6750 | 514.2600 | 520.0900 | 517.4500 | 422.6630 |

${P}_{{G}_{2}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 105.4910 |

${P}_{{G}_{5}}$ (MW) | 75.16610 | 123.5300 | 103.7400 | 94.81000 | 161.5274 |

${P}_{{G}_{6}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 182.3932 |

${P}_{{G}_{8}}$ (MW) | 166.0264 | 159.6700 | 175.6300 | 181.7500 | 0.00000 |

${P}_{{G}_{9}}$ (MW) | 253.7019 | 0.000000 | 0.000000 | 0.000000 | 125.9095 |

${P}_{{G}_{12}}$ (MW) | 211.8802 | 486.8900 | 485.2300 | 489.7700 | 256.8480 |

Total Generation (MW) | 1291.449 | 1284.350 | 1284.690 | 1283.780 | 1254.832 |

${P}_{wind}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 36.37860 |

Fuel Cost ($/h) | 7768.000 | 8103.180 | 8309.270 | 8413.430 | 6748.000 |

Emission (ton/h) | 2.379500 | 2.397820 | 2.433300 | 2.433100 | 2.018000 |

Power Losses (MW) | 40.64980 | 33.55000 | 33.89000 | 32.98000 | 40.41060 |

Computation Time (s) | 678.9000 | 680.1200 | 689.9000 | 847.9000 | 714.7000 |

**Table 9.**Results for minimizing the active power losses across the transmission lines for the Test System 2.

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 192.3159 | 303.2400 | 318.5800 | 311.3400 | 176.1450 |

${P}_{{G}_{2}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 16.85120 |

${P}_{{G}_{5}}$ (MW) | 34.34410 | 63.19000 | 45.90000 | 60.61300 | 156.9747 |

${P}_{{G}_{6}}$ (MW) | 134.0298 | 0.000000 | 0.000000 | 0.000000 | 58.62480 |

${P}_{{G}_{8}}$ (MW) | 469.7929 | 400.7500 | 407.6500 | 400.0600 | 158.5790 |

${P}_{{G}_{9}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 290.1441 |

${P}_{{G}_{12}}$ (MW) | 436.1565 | 500.0000 | 495.0300 | 495.1400 | 371.8631 |

Total Generation (MW) | 1266.639 | 1267.180 | 1267.160 | 1267.153 | 1229.181 |

${P}_{wind}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 0.000000 | 35.46800 |

Fuel Cost ($/h) | 15,354.40 | 15,423.88 | 15,691.30 | 15,348.11 | 13,078.79 |

Emission (ton/h) | 1.916836 | 1.906545 | 1.966905 | 1.917299 | 1.507600 |

Power Losses (MW) | 21.93910 | 22.48000 | 22.46000 | 22.46300 | 21.11000 |

Computation Time (s) | 670.2000 | 881.3000 | 701.7000 | 691.0450 | 715.0000 |

**Table 10.**Results for minimizing the active power losses across the transmission lines for the Test System 2.

Control Variable | Without Wind Farm | With Wind Farm | |||
---|---|---|---|---|---|

KHA | ALC-PSO | DEA | RCGA | KHA | |

${P}_{{G}_{1}}$ (MW) | 333.585 | 341.910 | 298.12 | 300.23 | 143.311 |

${P}_{{G}_{2}}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 149.200 |

${P}_{{G}_{5}}$ (MW) | 170.617 | 91.9000 | 83.24 | 91.43 | 158.020 |

${P}_{{G}_{6}}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 161.346 |

${P}_{{G}_{8}}$ (MW) | 311.707 | 419.250 | 413.63 | 406.26 | 220.977 |

${P}_{{G}_{9}}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 173.139 |

${P}_{{G}_{12}}$ (MW) | 453.292 | 418.450 | 474.14 | 472.08 | 228.994 |

Total Generation (MW) | 1269.20 | 1271.51 | 1269.13 | 1270 | 1234.99 |

${P}_{wind}$ (MW) | 0.00000 | 0.00000 | 0.00000 | 0.00000 | 34.2539 |

Fuel Cost ($/h) | 15,667.9 | 15,856.1 | 15,914.3 | 15,577.3 | 15,202.6 |

Emission (ton/h) | 1.82129 | 1.88918 | 1.85870 | 1.83871 | 1.72090 |

Power Losses (MW) | 18.4031 | 20.7100 | 18.3300 | 19.2000 | 18.4442 |

Computation Time (s) | 690.100 | 878.700 | 694.200 | 690.140 | 705.510 |

Control Variable | Without Wind Farm | With Wind Farm | ||
---|---|---|---|---|

KHA | ALC-PSO | DEA | KHA | |

${P}_{{G}_{1}}$ (MW) | 346.8868 | 480.9300 | 475.6800 | 92.82350 |

${P}_{{G}_{2}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 286.8995 |

${P}_{{G}_{5}}$ (MW) | 173.0854 | 80.14000 | 80.64000 | 89.87540 |

${P}_{{G}_{6}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 193.2463 |

${P}_{{G}_{8}}$ (MW) | 157.0796 | 270.4200 | 276.0300 | 13.55130 |

${P}_{{G}_{9}}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 89.01440 |

${P}_{{G}_{12}}$ (MW) | 583.2398 | 446.0400 | 447.2000 | 459.5085 |

Total Generation (MW) | 1260.291 | 1279.530 | 1279.550 | 1224.918 |

${P}_{wind}$ (MW) | 0.000000 | 0.000000 | 0.000000 | 36.18200 |

Fuel Cost ($/h) | 9917.870 | 10,237.79 | 10,408.49 | 8481.851 |

Emission (ton/h) | 2.200089 | 2.227447 | 2.211635 | 1.620300 |

Power Losses (MW) | 9.491500 | 28.73000 | 28.75000 | 10.30090 |

CEEC | 11,410.00 | 13,032.56 | 13,183.42 | 10,060.00 |

Computation Time (s) | 690.1000 | 700.1400 | 702.2000 | 717.5100 |

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## Share and Cite

**MDPI and ACS Style**

Abdollahi, A.; Ghadimi, A.A.; Miveh, M.R.; Mohammadi, F.; Jurado, F. Optimal Power Flow Incorporating FACTS Devices and Stochastic Wind Power Generation Using Krill Herd Algorithm. *Electronics* **2020**, *9*, 1043.
https://doi.org/10.3390/electronics9061043

**AMA Style**

Abdollahi A, Ghadimi AA, Miveh MR, Mohammadi F, Jurado F. Optimal Power Flow Incorporating FACTS Devices and Stochastic Wind Power Generation Using Krill Herd Algorithm. *Electronics*. 2020; 9(6):1043.
https://doi.org/10.3390/electronics9061043

**Chicago/Turabian Style**

Abdollahi, Arsalan, Ali Asghar Ghadimi, Mohammad Reza Miveh, Fazel Mohammadi, and Francisco Jurado. 2020. "Optimal Power Flow Incorporating FACTS Devices and Stochastic Wind Power Generation Using Krill Herd Algorithm" *Electronics* 9, no. 6: 1043.
https://doi.org/10.3390/electronics9061043