Using the Whale Optimization Algorithm to Solve the Optimal Reactive Power Dispatch Problem

Abstract

The optimal reactive power dispatch (ORPD) is a complex, optimal non-meritorious control issue with continuous and discontinuous control variables. This article exhibits a whale optimization algorithm (WOA) motivated by the whale’s bubble-net hunting tactic to resolve ORPD. The intention is to comply with certain constraints to promote the voltage transmission quality by adequately altering the parameters. The WOA not only equalizes exploitation and exploration to maximize the overall performance and eliminate search stagnation but also has remarkable sustainability and robustness to accomplish superior convergence speed and computation accuracy. The WOA is contrasted with MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA by diminishing the fitness value to highlight the superiority and stability. The experimental results reveal that WOA exhibits a superior convergence level and computation precision to accomplish the minimum active power loss and superior control variables.

1. Introduction

The ORPD intends to raise voltage transfer quality, diminish the loss of transmission lines, lower operating costs and strengthen the stability of transmission lines [1,2,3,4,5]. The traditional approaches have some drawbacks when resolving the ORPD, such as significant inaccuracy, dimension curse, poor computing efficiency, high time consumption and combination explosion, so it is difficult to alter the discrete variables and acquire the exact solution. Stability illustrates the algorithm’s sensitivity to control parameters and problem data, and the algorithms utilize a set of fixed control parameters to resolve an optimization problem and obtain the best solution. Reliability illustrates the algorithm’s strong optimization ability and high calculation accuracy to resolve a design optimization problem, and a series of data comparative experiments ensure the reliability of the algorithms. Therefore, some swarm intelligence approaches have been advised to resolve the ORPD, such as the moth flame optimization (MFO) [6], bat algorithm (BA) [7], grasshopper optimization algorithm (GOA) [8], grey wolf optimization (GWO) [9], movable damped wave algorithm (MDWA) [10], slime mold algorithm (SMA) [11], student psychology based optimization (SPBO) [12] and salp swarm algorithm (SSA) [13].

PG et al. combined the properties of differential evolution and ant colonies to resolve the ORPD; this technique gained the complementary benefits to inhibit search stagnation and identify the best solution [14]. Da Silva et al. introduced a convex programming model to resolve the ORPD; this technique was practicable and workable [15]. Sharma et al. designed a backtracing-assisted elephant herding approach to resolve the ORPD; this technique utilized exploration and exploitation to generate less power loss [16]. Khan et al. utilized an altered marine predator algorithm to resolve the ORPD; this technique switched between exploitation and exploration to acquire better optimization results [17]. Abd-EI Wahab applied a modified water-based optimization method to resolve the ORPD; this technique had remarkable sustainability to accomplish superior convergence accuracy [18]. Saddique et al. established a sine–cosine approach to resolve the ORPD; this technique exhibited excellent resilience and stability to discover the finest results [19]. Zhou et al. submitted an accelerated augmented Lagrangian method to resolve the ORPD; this technique had strong stability to discover the ideal control variables and objective function value [20]. Niu et al. introduced an upgraded differential evolution to resolve the ORPD; this technique had a significant optimization ability to identify the ideal solution [21]. López et al. created a semidefinite relaxation approach to resolve the ORPD; this technique had certain validity and feasibility [22]. Khan et al. introduced a modified salp swarm technique to address the ORPD; the technique boosted population variety and broadened the search field to produce high computational accuracy [23]. Chen et al. utilized a combination of the co-evolution approach and bi-level planning to resolve the ORPD; this technique exhibited adequate reliability to decrease the active power loss [24]. Hosseini-Hemati et al. indicated a superior grey wolf optimization to resolve the ORPD; this technique conveyed excellent dependability to arrive at the finest global solution [25]. Davoodi et al. used a rapid semidefinite programming-based technique to resolve the ORPD; this technique had strong robustness to achieve better optimization results [26]. Yapici et al. deployed a pathfinder method to address the ORPD; this technique showed superior optimization results [27]. Suresh et al. utilized a self-balanced differential evolution algorithm to resolve the ORPD; this technique utilized exploration and exploitation to acquire the minimum active power loss [28]. Ebeed et al. suggested an upgraded lightning attachment procedure to resolve the ORPD; this technique balanced exploitation and exploration to discover the final solution [29]. Dai et al. created the distributed reinforcement learning algorithm to resolve the dynamic economic dispatch; this technique had certain effectiveness and robustness to obtain better parameters [30]. To summarize, these algorithms have some advantages for resolving the ORPD, such as strong search ability, high optimization efficiency, strong robustness, good parallelism and excellent scalability. The effectiveness and feasibility of these algorithms have been confirmed.

The WOA performs shrinkage encircling, bubble-net attacking and random searching to discover a suitable solution [31]. The WOA has some advantages for resolving the ORPD, such as simple implementation, small storage space, a fast convergence rate and great computation precision, which equalizes exploitation and exploration to disrupt premature convergence and identify superior convergence accuracy. The WOA is superior to MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA; the WOA has remarkable sustainability and robustness.

Section 2 summarizes the ORPD formulation. Section 3 displays the WOA. Section 4 proposes a WOA-based ORPD. Section 5 provides the simulation results and analysis. Lastly, the conclusions are analyzed and future research directions are discussed in Section 6.

2. ORPD Formulation

The ORPD is a complex, optimal non-meritorious control problem with continuous and discontinuous control variables, which has the remarkable characteristics of multiple constraints, multiple control variables and discreteness. The intention is to adjust the reactive equipment to acquire the best control variables and the lowest fitness value according to certain constraints [5,22,28]. The mathematical formula is expressed as:

$$\min {\displaystyle \sum _{k\in N_E}{P}_{kloss}={\displaystyle \sum _{k\in N_E}{g}_k\times (v_i^2+v_j^2-2\times v_i\times v_j\times \cos {\theta }_{ij}}})$$

(1)

where $P_{kloss}$ denotes the active power loss of $k$ branches, $g_k$ denotes the conductively of $k$ branches, $N_E$ denotes all branches, ${\theta }_{ij}$ denotes the voltage phase difference, $v_i$ denotes the voltage amplitude of the $i$ node and $v_j$ denotes the voltage amplitude of $j$-node.

The power constraint equations (equality constraints) and the variable constraints (inequality constraints) make up the majority of the constraints in the ORPD. The equality constraints are known as the load flow limitations. The mathematical formulas are expressed as:

$$P_{gi}-P_{di}-v_i{\displaystyle \sum _{j=N_i}{v}_j(g_{ij}\cos {\theta }_{ij}+B_{ij}\sin {\theta }_{ij})=0}$$

(2)

$$Q_{gi}-Q_{di}-v_i{\displaystyle \sum _{j=N_i}{v}_j(g_{ij}\sin {\theta }_{ij}-B_{ij}\cos {\theta }_{ij})=0}$$

(3)

where $P_{gi}$ and $Q_{gi}$ denote the $i$ node’s active and reactive power, respectively, $P_{di}$ and $Q_{di}$ denote the $i$ load node’s active power and the reactive power, respectively, $g_{ij}$, $B_{ij}$ and ${\theta }_{ij}$ denote the conductively, the susceptance and the voltage phase difference between $i$ node and $j$ node, respectively, and $N_i$ denotes a group of nodes that are linked to the $i$ node.

The bound constraints are expressed as follows.

Reactive power constraint of each generator node:

$$Q_{gi,\min }\le Q_{gi}\le Q_{gi,\max },i\in N_g$$

(4)

Node voltage safety constraint:

$$v_{i,\min }\le v_i\le v_{i,\max },i\in N_B$$

(5)

Transformer tap position constraint:

$$T_{k,\min }\le T_k\le T_{k,\max },k\in N_T$$

(6)

Reactive power compensation capacity constraint:

$$S_{l,\min }\le S_l\le S_{l,\max },l\in N_L$$

(7)

where $Q_{gi}$ denotes the reactive power of the $i$ node, $Q_{gi,\min }=0.90$ and $Q_{gi,\max }=1.10$; $v_i$ denotes the voltage amplitude of the $i$ node, $v_{i,\min }=0.90$ and $v_{i,\max }=1.10$; $T_k$ denotes the transformer ratio, $T_{k,\min }=0.95$ and $T_{k,\max }=1.05$; $S_l$ denotes the reactive power compensation capacity, $S_{l,\min }=1$ and $S_{l,\max }=20$; and $N_g=6$, $N_B=22$, $N_T=4$ and $N_L=3$ denote the number of generators, load nodes, transformer adjustable taps and reactive power compensation nodes, respectively.

The purpose of optimization, while taking into account the system load parameters and structural factors, is to satisfy multiple equality and inequality requirements and identify the system’s ideal control variables to achieve the minimum active power loss [5]. Therefore, a penalty function is established to resolve the ORPD, and the augmented objective function is expressed as:

$$F_P={\displaystyle \sum _{k\in N_E}{P}_{kloss}+k_1}\times {\displaystyle \sum _{i=1}^{N_G}f(Q_{gi}})+k_2\times {\displaystyle \sum _{i=1}^{N_B}f(v_i)+k_3\times {\displaystyle \sum _{m=1}^{N_L}f(S_l}})$$

(8)

$$f(x)=\left\{\begin{array}{ll}0& if{x}_{\min }\le x\le x_{\max }\\ {(x-x_{\max })}^2& ifx>x_{\max }\\ {(x_{\min }-x)}^2& ifx<x_{\min }\end{array}\right.$$

(9)

where $k_1$, $k_2$ and $k_3$ denote the penalty function coefficient, which is set to 10,000. $x_{\min }$ and $x_{\max }$ denote the value ranges of the generator.

The WOA is utilized to resolve the ORPD, which has significant robustness and stability to boost the convergence accuracy and realize the global finest value. For ORPD, the WOA can reasonably modify the switching of the reactive power compensation device and the terminal voltage of the generator to accomplish the optimization of the power system operating status, which minimizes active power loss, strengthens voltage quality and maintains the system reliability. The WOA has certain validity and reliability to resolve the ORPD.

3. WOA

The WOA imitates shrinkage encircling, bubble-net attacking and random searching to conduct exploitation and exploration. The WOA has the benefits of a straightforward structure, strong stability, good optimization ability and easy implementation [31]. In the WOA, each whale symbolizes a search agent, and the humpback whale refreshes its position to hunt prey.

3.1. Shrinkage Encircling

The WOA utilizes each whale’s foraging behavior to conduct a targeted search, which is beneficial for altering the position and discovering the ideal solution. In the WOA, the humpback whale utilizes a ‘9’ shaped path to observe and capture its intended prey. This route makes use of echolocation technology to efficiently convey a position and conduct a global search. Assuming that the target prey’s position and the ideal whale are the finest solutions to the optimization problem, the remaining humpback whales immediately condense and encircle the prey, and the position is expressed as:

$$D=\left|\mathit{C}\cdot X^{\ast }(t)-X(t)\right|$$

(10)

$$X(t+1)=X^{\ast }(t)-A\cdot D$$

(11)

where $t$ denotes the iteration, $X^{\ast }$ denotes the optimal prey’s position and $X$ denotes the prey’s position. $A$ and $C$ are two adjustable parameters, and they are expressed as:

$$A=2a\cdot r-a$$

(12)

$$C=2\cdot r$$

(13)

$$a=2-2t/T$$

(14)

where $r$ denotes a stochastic value in [0,1], $a$ is decremented from 2 to 0 and $T$ denotes the maximum iteration.

3.2. Bubble-Net Attacking

The bubble-net attack contains the rocking surrounding the predator–prey mechanism and the logarithmic spiral, bubble-net predator–prey mechanism. The rocking surrounding the predator–prey mechanism is implemented using Formula (14). The $A$ is affected by $a$. The $A$ is a random number between $-a$ and $a$, where $a$ is decremented from 2 to 0. If $A$ is in [−1,1], the whale will dynamically identify the location. The WOA utilizes a logarithmic spiral, bubble-net predator–prey mechanism to estimate the distance, and then the whale starts swimming rapidly toward the surface to capture the target prey by continuously spitting out a bubble-net of different sizes in the spiral trajectory. The position is expressed as:

$$D^{\prime }=\left|X^{\ast }(t)-X(t)\right|$$

(15)

$$X(t+1)=D^{\prime }\cdot e^{bl}\cdot \cos (2\pi l)+X^{\ast }(t)$$

(16)

where $D^{\prime }$ denotes the distance, $l$ denotes a stochastic value in $[-1,1]$ and $b$ denotes a logarithmic spiral number.

In the WOA, the rocking surrounding the predator–prey mechanism makes the whale shrink, envelop and engulf the prey, and the logarithmic spiral, bubble-net predator–prey mechanism makes the individual whale spit bubbles, spiral motion and calculate the distance to hunt the prey. The probability that each mechanism will be chosen is 50%. The position is expressed as:

$$X(t+1)=\left\{\begin{array}{ll}{X}^{\ast }(t)-A\cdot D& ifp<0.5\\ D^{\prime }\cdot e^{bl}\cdot \cos (2\pi l)+X^{\ast }(t)& ifp\ge 0.5\end{array}\right.$$

(17)

where $p$ denotes a stochastic value in $[0,1]$.

3.3. Random Searching

If $\left|A\right|>1$, the whale swims beyond the shrinking, encircling circle of the bubble-net, and the humpback whale does not select the target prey to refresh its position. The humpback whale will haphazardly seek prey depending on the sharing of positional information among humpback whales. The WOA extends the search region, enhances the optimization ability, and avoids search stagnation. The position update is expressed as:

$$D=\left|C\cdot X_{rand}(t)-X(t)\right|$$

(18)

$$X(t+1)=X_{rand}(t)-A\cdot D$$

(19)

where $X_{rand}$ denotes the random position vector of a humpback whale.

The pseudo-code for the WOA is described in Algorithm 1.

Algorithm 1: WOA

1. Initialize the whale population X; ⋯ 2. Compute the fitness value of each whale; 3. Attain $X^{\ast }$; 4. While $(t<T$), do; 5. For each whale, do; 6. Modify $p$$,l$$,a$$,A$ and $C$; ⋯ 7. If $(p<0.5)$, then; 8. If $(\left|A\right|<1)$, then; 9. Modify the position using Equation (11) based on shrinkage encircling; 10. Else $(\left|A\right|\ge 1)$; 11. Choose a stochastic whale $X_{rand}$; 12. Modify the position using Equation (19) based on random searching; 13. End; 14. Else $(p\ge 0.5)$; 15. Modify the position by Equation (16) based on bubble-net attacking; 16. End; 17. End; 18. Validate if any whale exceeds the search area; 19. Compute the fitness value of each whale; 20. Modify $X^{\ast }$ if a superior option is available; 21. Then, $t=t+1$; 22. End; 23. Attain $X^{\ast }$.

4. WOA-Based ORPD

The WOA utilizes a shrinkage encircling mechanism, bubble-net attacking mechanism and random searching mechanism to resolve the ORPD. The intention is to acquire the ideal control variables and minimize active power loss.

The ORPD, which demands a reasonable approach to discrete variables, is resolved using the WOA. According to the coexistence characteristics of discrete variables and continuous variables, this paper uses the mixed coding method of integers and real numbers. The tap-changing transformers and the reactive compensation devices are both represented as integers. Real numbers are used to represent the continuously variable generator terminal voltage [5]. In the coding method of the WOA, the total coding length of the search agent is the quantity of the control variables. The power system contains $n_1$ generators, $n_2$ reactive compensation points and $n_3$ transformer branches, so the coding length is $n=n_1+n_2+n_3$. The specific expression is expressed as:

$$X=[V_G|T|Q_C]=[V_{G1},V_{G2},\dots ,V_{G13}|T_{6–9},T_{6–10},\dots ,T_{28–27}|Q_{C3},Q_{C10},Q_{C24}]$$

(20)

where $V_{G1},V_{G2},V_{G5},V_{G8},V_{G11},V_{G13}$ denote the generator bus voltages, $T_{6–9},T_{6–10},T_{4–12},T_{28–27}$ denote the tap ratios of transformers and $Q_{C3},Q_{C10},Q_{C24}$ denote the reactive power output of shunt compensators.

For discrete variables, some integers are obtained according to the mixed coding method. However, in practice, what the power system needs is the actual value of the control variable. The specific decoding form is expressed as:

$$\left\{\begin{array}{c}Ta{p}_j=Ta{p}_{j,\min }+T_j\times T_{step}\\ B_k=Q_{Ck}\times C_{step}\end{array}\right.$$

(21)

where $T_{step}$ denotes the transformer tap adjustment step size, $C_{step}$ denotes the input capacitor step size, $B_k$ denotes the susceptance value of the compensation capacitor and $Ta{p}_j$ denotes the transformer’s actual ratio.

Single binary coding can easily generate errors when dealing with the discretization of continuous variables. Hybrid coding can overcome this shortcoming, and the accuracy of the solution does not depend on the length of the encoded string, which has the advantage of accomplishing a greater solution accuracy in a larger search space. Hybrid coding has strong practicability and stability to reflect the various characteristics of the ORPD control variables.

The WOA-based ORPD is described in Algorithm 2. A flowchart for the ORPD is shown in Figure 1.

Algorithm 2: WOA-based ORPD

1. Input system data, bus data, line data and unit data. Establish control variables $(V_{G1},V_{G2},V_{G5},V_{G8},V_{G11},V_{G13},T_{6–9},T_{6–10},T_{4–12},T_{28–27},Q_{C3},Q_{C10},Q_{C24})$ with bounds. Initialize the whale population X; 2. Map control variables, compute the fitness value of the whale using Equation (8) and attain the best $X^{\ast }$; 3. While $(t<T$), do; 4. For each whale do; 5. Modify $p$$,l$$,a$$,A$$\mathrm{and}C$; 6. If $(p<0.5)$, then; 7. If $(\left|A\right|<1)$ then; 8. Modify the position using Equation (11) based on shrinkage encircling; 9. Else $(\left|A\right|\ge 1)$; 10. Choose a stochastic individual whale $X_{rand}$; 11. Modify the position using Equation (19) based on random searching; 12. End; 13. Else $(p\ge 0.5)$; 14. Modify the position using Equation (16) based on bubble-net attacking; 15. End; 16. End; 17. Validate if any whale exceeds the search area; 18. Compute the fitness value of each whale using Equation (8); 19. Modify $X^{\ast }$ if a superior option is available; 20. Then, $t=t+1$; 21. End; 22. Attain $X^{\ast }$; 23. Output the optimal active power loss.

Complexity Analysis

The WOA includes five operation stages: initialization, a bubble-net attacking mechanism (exploitation stage), a random searching mechanism (exploration stage), position iteration update and search stagnation judgment. If the population is $N$, the maximum is $T$, and the dimension is $D$. Initialization involves a duplicate circulation ($N$ and $D$ times); it is $O(N\ast D)$. The bubble-net attacking mechanism (exploitation stage), random searching mechanism (exploration stage) and position iteration update involve a tripe circulation ($N$, $D$ and $T$ times); it is $O(N\ast D\ast T)$. The time complexity of search stagnation judgment is $O(1)$. The total time complexity is $O(N\ast D\ast T)$. For space complexity, the WOA has strong robustness and reliability, and the space complexity is $O(N\ast D)$.

5. Experimental Results and Analysis

5.1. Experimental Setup

The simulation is established using a machine with an Intel Core i7-8750H 2.2 GHz CPU, a GTX1060 and 8 GB memory. All algorithms are written in MATLAB R2018b.

5.2. Parameter Selection

All control parameters are derived from some typical empirical values in the original article. The WOA is compared to various algorithms to confirm the overall optimization performance, such as MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA. The population size, maximum iteration and independent run for each algorithm are 50, 200 and 30, correspondingly. The parameter selection is displayed in

Table 1. Parameter selection.

Algorithm	Parameter	Value
MFO	The constant of logarithmic spiral $b$	1
	Stochastic value $t$	[−1,1]
	Convergence value $r$	[−2,−1]
BA	Pulse frequency $f$	[0,2]
	Echo loudness $A$	0.25
	Decreasing value $\gamma$	0.5
GOA	Stochastic value $r_1$	[0,1]
	Stochastic value $r_2$	[0,1]
	Stochastic value $r_3$	[0,1]
	Intensity attraction $f$	0.5
	Attractive length scale $l$	1.5
	Gravitational constant $g$	9.8
	The maximum $c\max$	1
	The minimum $c\min$	0.00001
GWO	Convergence factor $\alpha$	[0,2]
	Stochastic value $r_1$	[0,1]
	Stochastic value $r_2$	[0,1]
	Coefficient vector $A$	[−1,1]
	Coefficient vector $C$	[0,2]
MDWA	A constant $a_{\max }$	1
	A constant $a_{\min }$	0
	Cycle value $\gamma$	1,−2,2
	Constant value $\alpha$	1
	Constant value $\beta$	1
SMA	Stochastic value $r$	[0,1]
	Stochastic value $rand$	[0,1]
	Coefficient vector $vc$	[−1,1]
	Constant value $z$	0.03
SPBO	Stochastic value $rand$	[0,1]
	Constant value $k$	1 or 2
SSA	Stochastic value $c_1$	[0,1]
	Stochastic value $c_2$	[0,1]
	Stochastic value $c_3$	[0,1]
	Stochastic value $v_0$	0
WOA	Stochastic value $r$	[0,1]
	Convergence factor $\alpha$	[0,2]
	Constant coefficient $b$	1
	Stochastic value $l$	[−1,1]
	Stochastic value $p$	[0,1]
	Stochastic value $A$	[−1,1]

.

5.3. Simulation Results and Analysis

For the IEEE 30-bus system, the generator voltages are located at nodes 1, 2, 5, 8, 11 and 13, where node 1 denotes the equilibrium node, nodes 2, 5, 8, 11 and 13 denote the PV nodes and the rest of the nodes denote the PQ nodes. The generator voltage is in [0.90, 1.10]. The tap locations in tap changing transformers are located at nodes 6–9, 6–10, 4–12 and 27–28, the adjustable ratio is in [0.95, 1.05]. The reactive compensation devices are located at nodes 3, 10 and 24, which are in [1,20]. The control variables are displayed in

Table 2. Control variables.

Control Variables	Min	Max	Type
Generator node	0.90	1.10	Continuous
Load node	0.90	1.10	Continuous
Transformer—tap	0.95	1.05	Discrete
Shunt reactive compensator	1	20	Discrete

.

Some evaluation criteria are adopted to depict superiority and resilience. The ranking is based on the std. The results of the ORPD are displayed in

Table 3. Results of the ORPD.

Variables	MFO	BA	GOA	GWO	MDWA	SMA	SPBO	SSA	WOA
$V_{G1}$	1.0450	0.9235	1.0654	1.0315	1.0794	1.0208	0.9297	1.0924	1.0162
$V_{G2}$	0.9366	1.0096	1.0994	0.9781	1.0519	1.0682	1.0706	1.0448	0.9880
$V_{G5}$	0.9257	1.0643	1.0118	0.9615	0.9489	1.0754	0.9731	0.9971	0.9643
$V_{G8}$	0.9493	0.9002	1.0441	1.0953	1.0691	1.0724	0.9850	1.0297	1.0249
$V_{G11}$	1.0391	1.0314	0.9965	0.9477	1.0589	1.0495	1.0617	0.9429	0.9804
$V_{G13}$	1.0717	1.0375	1.0711	0.9748	1.0969	1.0981	0.9278	1.0790	1.0063
$T_{6–9}$	0.9881	1.0087	1.0234	0.9809	0.9832	0.9600	1.0102	1.0047	0.9933
$T_{6–10}$	1.0407	0.9864	0.9799	0.9794	0.9762	1.0178	1.0108	1.0337	0.9692
$T_{4–12}$	1.0004	0.9906	0.9694	0.9916	0.9640	0.9590	1.0476	0.9983	0.9576
$T_{27–28}$	1.0408	0.9954	1.0017	1.0324	1.0192	0.9678	1.0100	0.9510	0.9769
$Q_{C3}$	14.095	9.9213	14.211	11.294	1.4472	7.5077	7.3418	9.2494	18.942
$Q_{C10}$	8.1877	13.839	19.019	17.105	5.3207	2.4151	4.6044	9.5945	19.591
$Q_{C24}$	5.5078	1.1554	5.0776	12.114	1.8969	8.0402	14.9292	15.316	7.6008
Best	17.1311	17.0135	16.7329	16.5451	16.1752	16.3050	18.9813	17.1327	16.1729
Worst	44.0783	51.6990	19.7624	18.6759	18.7576	19.9818	58.2238	20.7303	18.1128
Mean	25.3397	29.7044	18.5261	17.6203	17.2425	17.8409	29.7783	18.5758	17.1024
Std	6.9630	7.4302	0.7834	0.5451	0.6857	0.9241	7.8469	0.7806	0.5234
Rank	7	8	5	2	3	6	9	4	1

.

The WOA is used to resolve the ORPD, and the intention is to comply with certain constraints to promote the voltage transmission quality by adequately altering the parameters. The shrinkage encircling refreshes the whale’s position according to the prey’s position. The bubble-net attack enhances the regional ability to establish the accurate finest objective value. Random searching promotes the global optimization ability to eliminate premature convergence. Three mechanisms balance exploitation and exploration to attain the ideal solution. Compared with MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA, the WOA obtains better control variable values and a smaller total active power loss, which indicates that the WOA exhibits tremendous durability and resilience when resolving the complex optimization problem. The algorithm’s total optimization performance can be directly influenced by the ideal objective value. The ideal value of the WOA is superior to those of MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA, which indicates that WOA has a productive exploration when resolving the ORPD. The optimization values of the WOA are superior to those of other algorithms, and the WOA has exceptional adaptability and optimization ability. The WOA has minimum standard deviation across all algorithms and is ranked top. The WOA integrates exploitation and exploration to recognize the appropriate solution, which is a reliable and stable method for resolving the ORPD.

The convergence curves of the algorithms are described in Figure 2. The WOA has significant abilities in exploitation, exploration and optimization. Compared with MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA, the WOA has a superior convergence accuracy. The WOA has significant dependability and durability to efficiently discover the ideal solution in the space. The ANOVA test of the algorithms is described in Figure 3. The algorithm with a lower standard deviation is more stable. The standard deviation of the WOA is superior to MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA, and the WOA has tremendous stability. To summarize, the WOA has a certain superiority and robustness for determining the best solution.

The simulation results for computational time are displayed in

Table 4. Simulation results for computational time (seconds).

Algorithm	MFO	BA	GOA	GWO	MDWA	SMA	SPBO	SSA	WOA
Computational time	159	62	62	104	111	107	71	109	58

. The histogram of computational time is described in Figure 4. For the computational time, MFO is 159 s, BA is 62 s, GOA is 62 s, GWO is 104 s, MDWA is 111 s, SMA is 107 s, SPBO is 71 s, SSA is 109 s and the WOA is 58 s. The computational time for the WOA is superior to MFO, BA, GOA, GWO, MDWA, SMA, SPBO, and SSA. Thus, the WOA uses less time to resolve the ORPD, which indicates that the WOA has a quick optimization speed and potent overall search ability. The Wilcoxon rank test is used to determine if there is a noticeable distinction [32], where a p-value under 0.05 indicates a noticeable distinction and a p-value above 0.05 indicates no noticeable distinction. The p-value results are displayed in

Table 5. p-value results.

Algorithm	MFO	BA	GOA	GWO	MDWA	SMA	SPBO	SSA
p-value	$1.96\times {10}^{-10}$	$2.87\times {10}^{-10}$	$1.85\times {10}^{-8}$	$6.20\times {10}^{-4}$	$6.20\times {10}^{-1}$	$9.52\times {10}^{-4}$	$3.02\times {10}^{-11}$	$3.20\times {10}^{-9}$

. The data between the WOA and other algorithms are less than 0.05, which indicates that the data are authentic and trustworthy, and the data are not generated randomly.

Statistically, the WOA is utilized to address the ORPD for the following factors. First, the optimization process of the WOA is simple, the control parameters are quite good, the time and space complexity are small and the algorithm operation and implementation are easy. The WOA has an excellent optimization ability to identify the best solution, which is advantageous for averting premature convergence and search stagnation and accomplishing the optimization problem. Second, the WOA has a unique bubble-net attacking mechanism including the rocking surrounding the predator–prey mechanism and the logarithmic spiral, bubble-net predator–prey mechanism. The WOA estimates the separation between the whale and the prey, and then the whale starts swimming quickly towards the surface to capture the target prey by repeatedly spitting out a bubble net of various sizes in a spiral trajectory. The distinctive position update mechanism promotes overall effectiveness and accelerates the convergence rate. Third, $\left|A\right|$ is a fundamental tuning parameter that allows for switching between exploration and exploitation. If $\left|A\right|\le 1$, the WOA utilizes the bubble-net attacking mechanism to shrink and surround the prey, which promotes the local search. The rocking surrounding the predator–prey mechanism makes the individual whale shrink, envelop and swallow the prey, and the logarithmic spiral, bubble-net predator–prey mechanism makes the individual whale spit bubbles, spiral motion and calculate the distance to hunt the prey. If $\left|A\right|>1$, the whale updates its current position according to peer-to-peer exchanges of position information, which promotes the global search.

6. Conclusions and Future Research

The ORPD is a complex, optimal non-meritorious control problem with continuous and discontinuous control variables. In this paper, the WOA is used to resolve the ORPD, the intention is to establish the ideal control factors and diminish the entire active power loss. The WOA utilizes three search mechanisms of shrinkage encircling, bubble-net attacking and random searching to promote the overall search performance and expand the solution’s optimization efficiency. The WOA has some advantages including simple implementation, small storage space, a fast convergence rate, great calculation accuracy and strong robustness. The WOA not only has remarkable reliability and dependability for accomplishing a superior convergence rate and computation precision but also utilizes exploitation and exploration to attain the finest solution. Compared to MFO, BA, GOA, GWO, MDWA, SMA, SPBO and SSA, the WOA has a substantial exploration to avert search stagnation. The WOA has a quicker convergence level, greater computation accuracy, lower active power loss and superior stability. The WOA is a feasible and effective approach for resolving the ORPD.

In future research, the WOA will be confirmed from two aspects. First, the WOA will be used to resolve the 57 bus and 118 bus test power systems, which will be conducive to further verify the reliability and sustainability of the WOA. Second, implementing effective search strategies, adopting unique coding techniques or combining them with other intelligence algorithms will furnish complementary benefits and promote the general search ability.