# Optimal Power Flow Solution of Power Systems with Renewable Energy Sources Using White Sharks Algorithm

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## Abstract

**:**

## 1. Introduction

- The white shark optimizer (WSO), a novel meta-heuristic algorithm proposed in [25], is employed to efficiently solve the OPF issue in this study;
- The proposed algorithm is developed to test an IEEE-30 bus system with double wind farms and a single solar photovoltaic to determine if it can discover the best results for the OPF problem using renewable power in both unpractical and real scenarios;
- The four algorithms were prepared to find the optimal solution after 10 runs and 300 iterations. The best three solutions are selected among the ten to be presented in the results and used to study the convergence for each case in terms of reaching the optimal solution for scheduling and using power from different power plants;
- In addition, the solutions of the proposed method are compared with the method (ESMAOPF) that the authors used in [18], where the enhanced slime mold algorithm was used instead of conventional SMA, for solving the optimal power flow problem. The comparison showed the success of WSO over ESMAOPF in terms of achieving a lower total generating cost.

## 2. Optimal Power Flow Mathematical Models

_{i}, h

_{j}are the equality and inequality constraints, and m, n are the number of equality and inequality constraints.

_{G1}is the generated power of slack bus, V

_{L}is the voltage of load bus. NPQ is the number of load buses. Q

_{G}is the reactive power output of generators. NPV is the number of generation buses. S

_{TL}is the apparent power flow in transmission line. N

_{TL}is the number of transmission lines.

_{G}is the output active power of generator, NG is the number of generators. VG is the voltage of the generation bus. Q

_{C}is the injected reactive power of the shunt compensator. NC is the number of shunt compensators. T is the tap setting of transformers. NT is the number of transformers.

#### 2.1. Thermal Power Generation Cost

_{TG}. Because valve point loading influences the overall cost of fuel, accounting for it results in a more accurate and reliable cost, as shown in Equation (7).

#### 2.2. Direct Cost of Wind and Solar Photovoltaic

#### 2.3. Cost Estimation for Wind Power with Uncertain Output

#### 2.4. Cost Estimation for Solar Power with Uncertain Output

#### 2.5. Emission

_{x}and NO

_{x}are two of these hazardous gases, and their emissions rise in tandem with the amount of power generated by traditional power producers. Equation (14) illustrates the link between emissions in (t/h) and output power (in p.u. MW):

_{i}, β

_{i}, γ

_{i}, ω

_{i}, and μ

_{i}are emission coefficients. Table 2 shows the values of emission coefficients connected to the thermal power generator, which are close to the numbers in [30].

#### 2.6. Equality Constraints

_{ij}= (δ

_{i}− δ

_{j}) denotes the angle of voltage difference, NB = network number, P

_{Di}, Q

_{Di}, P

_{Gi}, and Q

_{Gi}denote active and reactive components of load demand and generation, respectively, for any of the energy resources, whether traditional or renewable. G

_{ij}and B

_{ij}, = conductance and susceptance, respectively.

#### 2.7. Inequality Constraints

#### 2.7.1. Generator’s Constraints

_{G}stands for the number of generator buses or generators.

#### 2.7.2. Safety Constraints

_{Lp}is line voltage limits subjected to PQ buses (load buses). When studying the OPF problem, power losses (P

_{loss}) in transmission lines and voltage deviation (V

_{d}) are additional and important elements to consider. It is impossible to prevent power losses from power transmission lines due to their persistent resistance. The grid’s P

_{loss}is determined by:

_{i}and V

_{j}represent the voltages applied to buses i and j, respectively. The voltage deviation (V

_{d}) in a power system specifies how far load bus voltages diverge from the nominal voltage level, which is typically 1 p.u. The voltage deviation (V

_{d}) is a measurement of how far load bus voltages deviate from a nominal voltage value, which is commonly 1 p.u. V

_{d}indicates the voltage quality of the power system.

## 3. Estimation of the Uncertain Power of Renewable Energy Sources

#### 3.1. Renewable Energy Source Probability Distribution

#### 3.2. Power Models for Wind Generator and Solar Photovoltaic

_{r}= rated speed, v

_{out}= cut-out speed, v

_{in}= cut-in speed of the wind turbine. P

_{wr}= rated output power according to the product datasheet for (Enercon E82-E4) wind turbine. The generation of solar PV energy is measured in terms of solar radiation (R) [33].

_{Sn}and P

_{Sn}

_{+}, respectively.

## 4. The Basis of the Proposed Method’s Development

#### 4.1. White Shark Optimizer (WSO)

#### 4.2. Initialization of WSO

#### 4.3. Speed of Movement towards Prey

^{i}

_{k}

_{+1}. v

^{i}is the ith index vector of sharks attaining the optimal location, as indicated by Equation (40).

_{min}and p

_{max}denote the starting and subordinate velocities for white shark motion. After a thorough examination, the values of p

_{min}and p

_{max}were discovered to be 0.5 and 1.5, respectively.

#### 4.4. Movement in the Direction of the Optimal Prey

_{0}and a

_{1}are location constants that are used for control exploration and exploitation.

#### 4.5. Movement in the Direction of the Optimal Shark

_{2}− 0.5) returns 1 or −1 to modify the search path, r

_{1}, r

_{2}, and r

_{3}= rand. No. in the domain of [0, 1], D

_{w}= length for both target and shark, is given in Equation (51). S

_{s}is a parameter that has been proposed to reflect the power of white sharks, as specified in Equation (52).

_{2}is a location factor used to regulate exploration and exploitation.

Algorithm 1: Code summarizing the iterative optimization process of WSO. | |

1: | Initialize the parameters of the problem |

2: | Initialize the parameters of WSO |

3: | Randomly generate the initial positions of WSO |

4: | Initialize the velocity of the initial population |

5: | Evaluate the position of the initial population |

6: | while (k < K) do |

7: | Update the parameters ν, p_{1}, p_{2}, µ, a, b, w_{0}, f, m_{v} and S_{s} using Equations (40)–(43), (45)–(49) and (52), respectively. |

8: | fori = 1 to n do |

9: | v^{i}_{k}_{+1}= µ [v^{i}_{k} + p_{1} (w_{gbestk} − w^{i}_{k}) × c_{1} + p_{2}(w′^{vk}_{best} − w^{i}_{k}) × c_{2}] |

10: | end for |

11: | fori = 1 to n do |

12: | if rand < mv then |

13: | w^{i}_{k+}_{1} = w^{i}_{k}·− ⊕ w_{0} + u·a + l·b |

14: | else |

15: | w^{i}_{k+}_{1} = w^{i}_{k} + v^{i}_{k}/f |

16: | end if |

17: | end for |

18: | fori = 1 to n do |

19: | if rand ≤ S_{s} then |

20: | ${\overrightarrow{D}}_{w}=\left|rand\times \left({w}_{gbest}-{w}_{k}^{i}\right)\right|$ |

21: | if i == 1 then |

22: | ${w}_{k+1}^{i}={w}_{gbestk}+{r}_{1}{\overrightarrow{D}}_{w}\mathrm{sgn}({r}_{2}-0.5)$ |

23: | else |

24: | ${{w}^{\prime}}_{k=1}^{i}={w}_{gbestk}+{r}_{1}{\overrightarrow{D}}_{w}\mathrm{sgn}({r}_{2}-0.5)$ |

25: | ${w}_{k=1}^{i}=\frac{{w}_{k}^{i}+{{w}^{\prime}}_{k+1}^{i}}{2\times rand}$ |

26: | end if |

27: | end if |

28: | end for |

29: | Adjust the position of the white sharks that proceed beyond the boundary |

30: | Evaluate and update the new positions |

31: | k = k + 1 |

32: | end while |

33: | Return the optimal solution obtained so far |

#### 4.6. Fish School Behavior

#### 4.7. Implementation and Analysis of WSO

## 5. Simulation Results and Comparison

^{®}Core

^{(TM)}i5-3210M processor, 2.5 GHz, and 8 GB of RAM.

#### 5.1. White Shark Optimizer (WSO)

#### 5.2. Northern Goshawk Optimization (NGO)

#### 5.3. Pelican Optimization Algorithm (POA)

#### 5.4. Smell Agent Optimization (SAO)

#### 5.5. Comparison of the Results of Different Optimization Techniques and Discussion

## 6. Conclusions

## 7. Future Work

- Apply the proposed methods to solve optimal power flow in a larger power system;
- Incorporate more renewable energy sources (such as hydropower plants) into the larger power system;
- Apply of more than one case study that takes into account factors other than the minimization of generation cost, such as emissions, carbon tax, reserve cost, penalty cost, and the dynamic nature of generating facilities’ control;
- Studying multi-objective optimal power flow in a power system with renewable energy sources.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**The boxplots for all algorithms for the statistical results: (

**a**) boxplot for WSO, (

**b**) boxplot for NGO, (

**c**) boxplot for POA (red + = suspected outliers), (

**d**) boxplot for SAO, (

**e**) boxplots for all.

**Figure 6.**The WSO results for the best 3 runs: (

**a**) characteristics of convergence for the best three runs of WSO algorithm; (

**b**) profile voltages of load buses for the best three runs of WSO.

**Figure 7.**The NGO results for the best 3 runs: (

**a**) characteristics of convergence for the best three runs of NGO algorithm; (

**b**) profile voltages of load buses for the best three runs of NGO.

**Figure 8.**The POA results for the best 3 runs: (

**a**) characteristics of convergence for the best three runs of POA algorithm; (

**b**) profile voltages of load buses for the best three runs of POA.

**Figure 9.**The SAO results for the best 3 runs: (

**a**) characteristics of convergence for the best three runs of SAO algorithm; (

**b**) profile voltages of Load buses for the best three runs of SAO.

**Figure 13.**The optimal scheduled active and reactive power for different optimization techniques: (

**a**) optimal scheduled active power (MW) with different optimization techniques; (

**b**) optimal scheduled reactive power (MVAr) with different optimization techniques.

**Table 1.**The basic specifications of the modified IEEE network with 30 buses [1].

Items | Quantity | Details |
---|---|---|

Buses | 30 | [29] |

Branches | 41 | [29] |

Thermal Generators (TG1, TG2, TG3) | 3 | Bus 1 (Swing), Bus 2, and Bus 8. |

Wind Generators (WPG1, WPG2) | 2 | Bus 5 and bus 11. |

Solar PV (SPG) | 1 | Bus 13. |

Control variables | 11 | The planned power of five generators (TG2, TG3, WPG1, WPG2, and SPG), as well as six generating bus voltages. |

Connected load | - | 283.4 MW, 126.2 MVAR |

Allowable voltage range for load buses | 24 | [0.95–1.05] p.u. |

**Table 2.**Cost and emission coefficients of thermal power generators [18].

Gen. | Bus | a | b | c | l | m | α | β | γ | ω | μ | P^{0}_{TGi}(MW) | DR_{i}(MW) | UR_{i}(MW) |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

TG1 | 1 | 0 | 2 | 0.00375 | 18 | 0.037 | 4.091 | −5.554 | 6.49 | 0.0002 | 6.667 | 99.211 | 20 | 15 |

TG2 | 2 | 0 | 1.75 | 0.0175 | 16 | 0.038 | 2.543 | −6.047 | 5.638 | 0.0005 | 3.333 | 80 | 15 | 10 |

TG3 | 8 | 0 | 3.25 | 0.00834 | 12 | 0.045 | 5.326 | −3.55 | 3.38 | 0.002 | 2 | 20 | 8 | 4 |

Wind Power Farm | Solar Power Plant | ||||||
---|---|---|---|---|---|---|---|

Wind Power Generator | No. of Turbines | Rated Power (MW) | Weibull PDF Parameters | Weibull Mean, M_{wbl} | Rated Power (MW) | Lognormal PDF Parameters | Lognormal |

1 (at bus 5) | 25 | 75 | K = 2, c = 9 | v = 7.976 (m/s) | 50 (at bus 13) | μ = 6, δ = 0.6 | R = 483 W/m^{2} |

2 (at bus 11) | 20 | 60 | K = 2, c = 10 | v = 8.862 (m/s) |

**Table 4.**The results of total generation cost and statistical results for the proposed WSO algorithm and other recent algorithms.

The Results of Total Generation Cost for 10 Runs and 300 Iterations | ||||

WSO | NGO | POA | SAO | |

1st Run | 782.6698 | 782.313 | 782.8485 | 907.0226 |

2nd Run | 783.7659 | 782.9944 | 785.3441 | 846.683 |

3rd Run | 782.5018 | 782.5711 | 782.5089 | 843.0099 |

4th Run | 781.8318 | 782.1243 | 782.4893 | 905.917 |

5th Run | 781.7552 | 781.8438 | 782.1865 | 872.6064 |

6th Run | 782.3621 | 782.2253 | 782.3702 | 851.711 |

7th Run | 781.733 | 782.9235 | 782.2994 | 811.187 |

8th Run | 783.3357 | 782.1109 | 782.4652 | 887.9602 |

9th Run | 782.4915 | 782.1162 | 782.9073 | 949.768 |

10th Run | 782.1152 | 782.1276 | 783.033 | 823.5738 |

The Statistical Results of Total Generation Cost for 10 Runs and 300 Iterations | ||||

WSO | NGO | POA | SAO | |

Best | 781.733 | 781.8438 | 782.1865 | 811.187 |

Worst | 783.7659 | 782.9944 | 785.3441 | 949.768 |

Mean | 782.4562 | 782.335 | 782.8452 | 869.9439 |

Std | 0.6722 | 0.3766 | 0.9205 | 42.9133 |

Average time of one run (s) | 524 | 1024 | 1610 | 1790 |

Control Variables | Min | Max | WSO | NGO | POA | SAO | ESMA [18] |

PTG1 (MW) | 50 | 140 | 134.9165 | 134.9041 | 134.9076 | 113.404 | 134.9143 |

PTG2 (MW) | 20 | 80 | 27.57455 | 29.36321 | 26.6666 | 26.64171 | 27.688 |

PTG3 (MW) | 10 | 35 | 10.00492 | 10.0109 | 10.02834 | 18.70488 | 10.0125 |

PwG1 (MW) | 0 | 75 | 43.05681 | 43.73514 | 43.27867 | 46.21081 | 43.5782 |

PwG2 (MW) | 0 | 60 | 36.12732 | 36.71591 | 38.29961 | 43.67977 | 37.4508 |

PsG1 (MW) | 0 | 50 | 37.52105 | 34.45914 | 35.97858 | 39.90458 | 35.5275 |

V1 (p.u) | 0.95 | 1.1 | 1.070752 | 1.071513 | 1.072656 | 1.017665 | 1.0699 |

V2 (p.u) | 0.95 | 1.1 | 1.056697 | 1.056241 | 1.056156 | 1.004626 | 1.0568 |

V5 (p.u) | 0.95 | 1.1 | 1.034903 | 1.034296 | 1.029249 | 1.095148 | 1.0334 |

V8 (p.u) | 0.95 | 1.1 | 1.0401 | 1.041903 | 1.043762 | 1.105398 | 1.088 |

V11 (p.u) | 0.95 | 1.1 | 1.099687 | 1.099531 | 1.099998 | 1.084872 | 1.097 |

V13 (p.u) | 0.95 | 1.1 | 1.056971 | 1.054164 | 1.049375 | 1.109471 | 1.052 |

Parameters | Min | Max | WSO | NGO | POA | SAO | ESMA [18] |

QTG1 (MVAr) | −20 | 150 | −4.43953 | −1.8199 | 1.326721 | −6.41329 | −6.588 |

QTG2 (MVAr) | −20 | 60 | 14.28056 | 10.49797 | 12.42993 | −3.53766 | 16.436 |

QTG3 (MVAr) | −15 | 40 | 23.31959 | 22.32038 | 17.85702 | 35 | 40 |

QwG1 (MVAr) | −30 | 35 | 35.7025 | 39.00757 | 40 | 40 | 21.181 |

QwG2 (MVAr) | −25 | 30 | 30 | 30 | 30 | 30 | 29.548 |

QwG1 (MVAr) | −20 | 25 | 18.19068 | 17.01745 | 15.63321 | 25 | 16.472 |

Total power cost (USD/h) | 781.733 | 781.8438 | 782.1865 | 811.1871 | 781.9375 | ||

Emissions (t/h) | 1.763241 | 1.761466 | 1.7625 | 0.510854 | 1.7629 | ||

P_{loss} (MW) | 5.80114 | 5.788415 | 5.759369 | 5.145789 | 5.7715 | ||

V_{d} (p.u) | 0.468569 | 0.465172 | 0.44752 | 0.68291 | 0.45868 |

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**MDPI and ACS Style**

Ali, M.A.; Kamel, S.; Hassan, M.H.; Ahmed, E.M.; Alanazi, M.
Optimal Power Flow Solution of Power Systems with Renewable Energy Sources Using White Sharks Algorithm. *Sustainability* **2022**, *14*, 6049.
https://doi.org/10.3390/su14106049

**AMA Style**

Ali MA, Kamel S, Hassan MH, Ahmed EM, Alanazi M.
Optimal Power Flow Solution of Power Systems with Renewable Energy Sources Using White Sharks Algorithm. *Sustainability*. 2022; 14(10):6049.
https://doi.org/10.3390/su14106049

**Chicago/Turabian Style**

Ali, Mahmoud A., Salah Kamel, Mohamed H. Hassan, Emad M. Ahmed, and Mohana Alanazi.
2022. "Optimal Power Flow Solution of Power Systems with Renewable Energy Sources Using White Sharks Algorithm" *Sustainability* 14, no. 10: 6049.
https://doi.org/10.3390/su14106049