A Mixed-Strategy-Based Whale Optimization Algorithm for Parameter Identification of Hydraulic Turbine Governing Systems with a Delayed Water Hammer Effect

Abstract

For solving the parameter optimization problem of a hydraulic turbine governing system (HTGS) with a delayed water hammer (DWH) effect, a Mixed-Strategy-based Whale Optimization Algorithm (MSWOA) is proposed in this paper, in which three improved strategies are designed and integrated to promote the optimization ability. Firstly, the movement strategies of WOA have been improved to balance the exploration and exploitation. In the improved movement strategies, a dynamic ratio based on improved JAYA algorithm is applied on the strategy of searching for prey and a chaotic dynamic weight is designed for improving the strategies of bubble-net attacking and encircling prey. Secondly, a guidance of the elite’s memory inspired by Particle swarm optimization (PSO) is proposed to lead the movement of the population to accelerate the convergence speed. Thirdly, the mutation strategy based on the sinusoidal chaotic map is employed to avoid prematurity and local optimum points. The proposed MSWOA are compared with six popular meta-heuristic optimization algorithms on 23 benchmark functions in numerical experiments and the results show that the MSWOA has achieved significantly better performance than others. Finally, the MSWOA is applied on parameter identification problem of HTGS with a DWH effect, and the comparative results confirm the effectiveness and identification accuracy of the proposed method.

1. Introduction

The hydraulic turbine governing system (HTGS) is the core control system of hydroelectric generating units (HGUs), which undertakes the tasks of load and frequency adjustment. The HTGS is a non-linear and non-minimum phase system [1,2,3,4,5,6,7]. Traditional identification methods, such as least squares method [8], input response method [9], and maximum likelihood estimate (MLE) [10], have been applied in parameter identification of HTGS in the past years, but there are too many restrictions on these methods. For example, the least squares method demands enough system input, and MLE easily gets trapped in local optima points. In addition, most of those methods are not global optimization methods and not suitable for parameter identification of nonlinear or complicated systems. Therefore, it is difficult to apply these methods in HTGS parameter identification if nonlinearities have been considered. To conquer these problems, identification methods based on meta-heuristic algorithms have been developed in recent years, which treat the problem of parameter identification as an optimization problem [11]. Because meta-heuristic algorithms are global optimization methods, they can directly identify the parameters by optimizing an objective function built for estimating the bias of outputs between the real system and the identified system. These methods avoid complicated mathematic modeling and deduction of the real system, which makes them easy to implement. Compared with traditional identification methods, meta-heuristic algorithms are more suitable for parameter identification of complex systems. As for these methods, the identification performance relies on the optimization ability of the meta-heuristic algorithms. To enhance the identification performance of nonlinear HTGS, new meta-heuristic algorithms are worth studying. At present, a number of meta-heuristic algorithms have been proposed, such as Genetic algorithms (GAs) [12], Particle Swarm Optimization (PSO) [13], Gravitational Search Algorithm (GSA) [14], Continuous Flock of Starlings Optimization (CFSO) and the closed forms for the CFSO [15], Sine Cosine Algorithm(SCA) [16], Ant Lion Optimization (ALO) [17], Grey Wolf Optimization (GWO) [18], Artificial Sheep Algorithm (ASA) [19,20], Whale Optimization Algorithm (WOA) [21] and so on. Among these algorithms, PSO, GSA, ALO, Ant Colony Optimization algorithm (ACO) and GWO have already been employed to deal with the problem of parameter identification [22,23,24,25,26,27,28,29,30,31]. The PSO was applied to identify parameters of nonlinear dynamic hysteretic models, in which a novel fitness function with reciprocal of the mean square error is designed [22]. In [23], an improved ALO mixed with chaotic mutation was utilized to identify the parameters of a photovoltaic cell model. In [24], GWO was used in parameter identification for polymer electrolyte membrane fuel cell models. In [25], an approach that combined fuzzy logic and ACO was proposed for system identification. In [26], a T-S fuzzy approach based on a modified inter type-2FRCM algorithm was used to solve the identification problem. In [2,4,27,28,29,30,31], different GSA strategies were used in parameter identification of HTGS. Although these algorithms have been successfully applied in different kinds of identification problems, the universal drawbacks of meta-heuristic algorithms, like local optimum points and prematurity, may still exist. Researchers have spent great efforts to promote the search ability of algorithms and thus enhance the identification performance. In [2,27,28,29,30,31], several improved GSA strategies were proposed and applied to identify HTGS parameters, and the identification accuracies were remarkably promoted compared to the original algorithm.

In these studies, it is also observed that the identification performances are affected by improvement strategies on meta-heuristic algorithms and the model complexity. In this paper, a delayed water hammer model, which is a hyperbolic tangent function, is proposed for the penstock system modeling to fully exhibit the complex characteristics of a HTGS. The model, which is superior to other penstock system models in model accuracy, greatly reduces the errors caused by the model simplification. To the best of our knowledge, the parameter identification problem of HTGS with the delayed water hammer model has never been studied before. This new problem may cause huge obstacles for existing parameter identification methods and algorithms. Hence it is interesting to develop a more effective meta-heuristic algorithm for parameter identification of this complicated HTGS.

Compared with those traditional meta-heuristic algorithms mentioned above, the Whale Optimization Algorithm (WOA) based on imitation of the predatory behavior of humpback whales, which was proposed by Mirjalili in 2016 [21], has been proved to be an excellent global optimization algorithm. Yet there are always some drawbacks in the standard WOA, which concentrates on the balance between exploration and exploitation, speed of convergence, and prematurity. Although some researchers have attempted to improve the standard WOA, few have succeeded in comprehensively solving these drawbacks.

The movement strategy of WOA, which is composed of three strategies, namely the strategy of encircling prey, the strategy of bubble-net attacking and the strategy of searching for prey, primarily dominates the search capability of WOA. In WOA, population agents are driven by the movement strategy, in which the strategy of searching for prey determines the global search ability and the convergence speed, while the bubble-net attack strategy determines the ability of fine searching around the promising area in search space. Mafarjaa et al. [32] attempted to apply a mutation strategy on the strategies of encircling prey and searching for prey, respectively. In addition, there is a real possibility that a combination of different movement strategies or introduction of a new movement strategy are advantageous to improve the search performance of WOA. Two hybridization models, namely Low-level Team work Hybrid (LTH) and High-level Relay Hybrid (HRH), were proposed in [33]. In LTH, the strategies of encircling prey and searching for prey were changed by using the local search algorithm (SA) embedded in WOA, which searched for the best solution in the neighborhood of both the randomly selected solutions (to replace the strategy of searching for prey) and the neighborhood of the best known solution (to replace the strategy of encircling prey). In HRH, a SA algorithm was used as the new movement strategy to enhance the final solution. The Lévy Flight Trajectory-Based Whale Optimization Algorithm (LWOA) is an improved WOA, in which the Lévy Flight Trajectory is used as the movement strategy to guide the agent’s movement [34]. The strategy of searching for prey was changed by applying the insert-reversed block operation on a randomly chosen search agent, and the best search agent was guided by the strategy of local search in [35]. In short, more effective movement strategies endow WOA with better search performance.

The balance between exploration and exploitation in the movement of agents significantly affects the search ability and efficiency. In WOA, a random probability value was designed to allow WOA to effectively transit between exploration and exploitation. To improve the strategy, the random probability value was replaced by a dynamic ratio of the current iteration number and the maximum iteration [36]. In the adaptive walk WOA (AWOA) [37], two types of walks were brought into WOA to make the whale agents switch between exploration and exploitation instead of a random probability value.

Prematurity is the most common potential defect of WOA. For the prevention of prematurity, some new mutation operators are employed in WOA. In [35], the swap mutation was applied on the whale agents’ population to search for best solution. It is proved that a new mutation operator is an effective approach to prevent premature and local convergence. In Chaotic Whale Optimization Algorithm (CWOA) [38], at each iteration, the chaotic sequences from a Singer map were generated by the whale agents’ population. Between them, the one which has a smaller value of objection function is retained to generate a new whale agents population for the next iteration.

Although the strategies mentioned above are proved to effectively enhance the search capability of WOA, for a complex system such as HTGS, a single improvement strategy cannot promote all aspects of WOA but a mixed strategy can comprehensively improve the performance of the WOA algorithm.

Consequently, a new mixed strategy algorithm named MSWOA, which integrates three improvements strategies in WOA, is proposed in this paper. Firstly, a hybrid movement strategy is applied on MSWOA, in which a dynamic ratio based on improved JAYA algorithm is applied on the strategy of searching for prey and a chaotic dynamic weight is applied on the strategies of bubble-net attacking and encircling prey. Secondly, a guidance of the elite’s memory inspired by PSO is applied on the movement of the whale agents of the population. Thirdly, the mutation strategy based on the sinusoidal chaotic map is employed to avoid prematurity and local optimum points. These improvements enhance the search ability of MSWOA and could make it powerful enough to assess the parameters of HTGS with a delayed water hammer effectively and with high accuracy.

The standard WOA algorithm and MSWOA algorithm are introduced in Section 2. In Section 3, the model of HTGS with a delayed water hammer and the parameter identification of HTGS using MSWOA are described. In Section 4, the simulation results show the effectiveness of MSWOA.

2. The Mixed-Strategy Based Whale Optimization Algorithm

2.1. Brief Introduction of WOA

The whale optimization algorithm was proposed by Mirjalili and Lewis in 2016 [21]. It mathematically models the predatory mechanism of humpback whales. When humpback whales prey, they firstly search for prey, then encircle the prey and finally attack the prey. Hence the whale optimization algorithm has three phases according to the predatory process of humpback whales. Symbols used in WOA and MSWOA are listed in

Table 1. The meaning of symbols proposed in this paper.

Symbol	Unit	Meaning
t	time	Current iteration
tmax	time	The maximum iteration
X	/	Each agent position of population
Xrand	/	The agent position selected randomly
X*	/	The best position in the current iteration
Xworst	/	The worst position in the current iteration
Xgbest	/	The best position obtained so far
A	/	A variable in [−2,2]
C	/	A random number in [0,2]
a	/	A variable in [0,2]
b	/	A positive constant
l	/	A random number in [−1,1]
a1	/	Chaotic constant a1 = 2.3
p	/	A random probability value in [0,1]
p1	/	A random probability value in [0,1]
r	/	A random in [0,1]
r2–r9	/	Randoms in [0,1]
c2, c3, c6–c12	/	Coefficients in [0,2]
c1	/	A coefficients in [0,1]
c4, c5	/	A variable in [0,1]
q1	/	A dynamic ratio in [0,0.5]
σ	/	Output of PID Controller
c	/	Relative value of given speed
x	/	Relative value of speed
bp	/	Permanent transition coefficient
y1	/	Relative value of position of the auxiliary servomotor
y	/	Relative value of position of the main servomotor
Td	second	The differential time
Kp	/	The proportional gain,
Ki	/	The integral gain
Kd	/	The differential gain
Ty1	second	The response time constant of auxiliary servomotor
Ty	second	The response time constant of main servomotor
mt	/	Relative value of moment
q	/	Relative value of water flow
h	/	Relative value of water head
hw	/	The pipeline characteristic coefficient
Tr	second	The reflection time of the water hammer pressure wave
Ta	second	The inertial time constant of the generator
eg	/	The adjusting coefficient of the generator

Attention: to any variable w, $\overrightarrow{\mathrm{w}}$ means w is a vector or w is calculated with vectors.

.

2.1.1. Modeling the Course of Searching for Prey

In this phase, whales in the population update their positions using Equations (1) and (2). In Equations (1) and (2), A and C are calculated with Equations (3) and (4) when $\left|\mathrm{A}\right|$ is greater than 1. $\overrightarrow{\mathrm{r}}$ is a vector which is composed of a random number in [0,1], and $\mathrm{a}$ is evaluated according to Equation (5) and it varies within [0,2]:

$$\overrightarrow{\mathrm{D}}=|\overrightarrow{\mathrm{C}}·\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}-\overrightarrow{\mathrm{X}}\left(\mathrm{t}\right)|$$

(1)

$$\overrightarrow{\mathrm{X}}\left(\mathrm{t}+1\right)=\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}-\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{D}}$$

(2)

$$\mathrm{A}=2·\mathrm{a}·\mathrm{r}-\mathrm{a}$$

(3)

$$\mathrm{C}=2·\mathrm{r}$$

(4)

$$\mathrm{a}=2·\left(\frac{{\mathrm{t}}_{\max }-\mathrm{t}}{{\mathrm{t}}_{\max }}\right)$$

(5)

2.1.2. Modeling the Course of Encircling Prey

In this phase, each whale of the population updates its position with Equations (6) and (7). A and C are calculated with Equations (3) and (4):

$$\overrightarrow{\mathrm{D}}=|\overrightarrow{\mathrm{C}}·\overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{t}\right)-\overrightarrow{\mathrm{X}}\left(\mathrm{t}\right)|$$

(6)

$$\overrightarrow{\mathrm{X}}\left(\mathrm{t}+1\right)=\overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{t}\right)-\overrightarrow{\mathrm{A}}·\overrightarrow{\mathrm{D}}$$

(7)

2.1.3. Modeling the Course of Bubble-Net Attacking (Getting) Prey

In this phase, each whale of the population updates its position using Equations (8) and (9). The flowchart of WOA is shown in Figure 1. The term b is a constant to which a positive number is assigned in this paper, l is a random number in [−1.1].

$$\overrightarrow{\mathrm{X}}\left(\mathrm{t}+1\right)=\overrightarrow{{\mathrm{D}}^{\prime }}·{\mathrm{e}}^{\mathrm{bl}}·\cos 2\mathsf{\pi }\mathrm{l}+\overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{t}\right)$$

(8)

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

(9)

2.2. The Mixed-Strategy Based WOA

The mixed strategy, which is composed of three improvements, not only enhances the algorithm comprehensively but also balances exploration and exploitation. Compared with the standard WOA in Figure 1, how the improvements strategies enhance MSWOA is illustrated in Figure 2. The details are described as follows:

2.2.1. Improvement 1: Hybrid Movement Strategy

The basic movement strategy of WOA is composed of three strategies, namely the strategy of encircling prey, the strategy of searching for prey and the strategy of bubble-net attacking the prey. To enhance MSWOA comprehensively, three different improvements are applied on the three movement strategies of WOA, respectively:

(1) The dynamic ratio for strategy of searching for prey.

In standard WOA, a random probability value named p was employed to allow WOA to effectively transit between exploration and exploitation. In this paper, besides p, another random probability value named p₁ and a dynamic ratio named q₁ which is defined as Equation (10), are applied on strategy of searching for prey:

$${\mathrm{q}}_1={\mathrm{c}}_1·\left(1-\frac{\mathrm{t}}{{\mathrm{t}}_{\max }}\right)$$

(10)

where c₁ is a coefficient in [0,1]; in this paper c₁ = 0.5 and q₁ varies in [0,0.5].

In Figure 2, q₁ is compared with p₁ in each iteration. If q₁ < p₁, the improved JAYA algorithm is employed to update the agent randomly selected, otherwise we use a strategy of randomly selecting an agent. The usage of q₁ and p₁ is helpful to search the space thoroughly.

(2) Strategy of searching for prey based on an improved JAYA algorithm

Because the dynamic ratio q₁ and p₁ are applied on strategy of searching for prey, it makes MSWOA transit between the improved JAYA algorithm and the strategy of randomly selecting an agent, which are described as Equations (11) and (12), respectively, in the phase of searching for prey. Compared with the JAYA algorithm [39], Equation (11) is proposed, which is named the improved JAYA algorithm. In Equation (11), c₂ and c₃ are employed to enlarge the search space. This is useful to not only enhance global search capability but also to avoid prematurity.

$$\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}^{\prime }}=\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}+{\mathrm{r}}_2·{\mathrm{c}}_2·\left(\overrightarrow{{\mathrm{X}}^{*}}\left(\mathrm{t}\right)-\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}\right)-{\mathrm{r}}_3·{\mathrm{c}}_3·\left(\overrightarrow{{\mathrm{X}}_{\mathrm{worst}}}\left(\mathrm{t}\right)-\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}\right)$$

(11)

$$\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}^{\prime }}=\left(1-{\mathrm{c}}_4\right)·\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}+\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}\left(\mathrm{t}\right)$$

(12)

where c₂, c₃ are coefficients in [0,2], p, ${\mathrm{c}}_4=\frac{\mathrm{t}}{{\mathrm{t}}_{\max }}$, r₂, r₃ are random numbers in [0,1].

In each iteration, q₁ is compared with p₁. If q₁ < p₁, $\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}^{\prime }}$ is upgraded according to Equation (11), otherwise $\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}^{\prime }}$ is upgraded according to Equation (12). The step of q₁ < p₁ originates from the acceptance/rejection step in the Markov Chain Monte Carlo Techniques which is helpful to remove some samples by some mechanism [39,40].

(3) Strategies of encircling prey and bubble-net attacking based on chaotic dynamic weight

In this section, the logistic chaotic map is applied on the strategies of encircling prey and bubble-net attacking. The logistic chaotic map, which is ergodic, sensitive, non-repetitive and helps MSWOA to avoid local optimum points or prematurity. ω(t), which denotes value of the logistic chaotic dynamic weight in the t-th iteration, is evaluated using Equation (13) [41]. The initial value, ω(1), is a random number in [0,1]:

$$\mathsf{\omega }\left(\mathrm{t}+1\right)=4\mathsf{\omega }\left(\mathrm{t}\right)·\left(1-\mathsf{\omega }\left(\mathrm{t}\right)\right),\mathrm{t}=1,\cdots ,{\mathrm{t}}_{\max }$$

(13)

Each position of the whale agents is updated by Equation (14) in the strategy of encircling prey and is updated with Equation (15) in the strategy of bubble-net attacking while each whale agent position is updated with Equations (7) and (8) in the standard WOA. In Equations (14) and (15), $\mathsf{\omega }(\mathrm{t})$ is the chaotic dynamic weight. The global best position so far $\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}$ and the best position in current iteration $\overrightarrow{{\mathrm{X}}^{*}}$ are both taken into account, respectively. The memory of previous iterations is helpful for MSWOA to enhance the global optimum finding capability and avoid local optimum points and prematurity:

$$\overrightarrow{\mathrm{X}}(\mathrm{t}+1)=\mathsf{\omega }(\mathrm{t})·{\mathrm{c}}_5·\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}-\overrightarrow{\mathrm{A}}·({\mathrm{c}}_6·{\mathrm{r}}_4·\overrightarrow{{\mathrm{D}}_1}+{\mathrm{c}}_7·{\mathrm{r}}_5·\overrightarrow{{\mathrm{D}}_2})$$

(14)

$$\overrightarrow{\mathrm{X}}(\mathrm{t}+1)={\mathrm{c}}_8·\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}+({\mathrm{c}}_9·{\mathrm{r}}_6·\overrightarrow{{\mathrm{D}}_3}+{\mathrm{c}}_{10}·{\mathrm{r}}_7·\overrightarrow{{\mathrm{D}}_4})·{\mathrm{e}}^{\mathrm{bl}}·\cos 2\mathsf{\pi }\mathrm{l}$$

(15)

where $\overrightarrow{{\mathrm{D}}_1}=|\overrightarrow{\mathrm{C}}·\overrightarrow{{\mathrm{X}}^{*}}(\mathrm{t})-\overrightarrow{\mathrm{X}}(\mathrm{t})|$, $\overrightarrow{{\mathrm{D}}_2}=|\overrightarrow{\mathrm{C}}·\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}-\overrightarrow{\mathrm{X}}(\mathrm{t})|$, $\overrightarrow{{\mathrm{D}}_3}=|\overrightarrow{{\mathrm{X}}^{*}}-\overrightarrow{\mathrm{X}}(\mathrm{t})|$, $\overrightarrow{{\mathrm{D}}_4}=|\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}-\overrightarrow{\mathrm{X}}(\mathrm{t})|$, ${\mathrm{c}}_5=\frac{{\mathrm{t}}_{\max }-\mathrm{t}}{{\mathrm{t}}_{\max }}$, ${\mathrm{c}}_6$ to ${\mathrm{c}}_{10}$ are coefficients in [0,2], r₄ to r₇ are random values in [0,1].

2.2.2. Improvement 2: Movement Strategy Inspired by Guidance of the Elite’s Memory

A guidance of the elite’s memory inspired by PSO is applied on the movement of the whale agents of the population. In each iteration, the movement strategy is introduced to update each position of the whale agents of the population after searching space by using the strategies proposed above. The strategy is described as Equations (16) and (17). This is helpful to improve search capability:

$$\overrightarrow{{\mathrm{v}}_{\mathrm{i}}}\left(\mathrm{k}+1\right)={\mathsf{\omega }}_1·\overrightarrow{{\mathrm{v}}_{\mathrm{i}}}\left(\mathrm{k}\right)+{\mathrm{c}}_{11}·{\mathrm{r}}_8·\left(\overrightarrow{{\mathrm{X}}_{\mathrm{pbest}}}\left(\mathrm{k}\right)-\overrightarrow{{\mathrm{X}}_{\mathrm{i}}}\left(\mathrm{k}\right)\right)+{\mathrm{c}}_{12}·{\mathrm{r}}_9·\left(\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}\left(\mathrm{k}\right)-\overrightarrow{{\mathrm{X}}_{\mathrm{i}}}\left(\mathrm{k}\right)\right)$$

(16)

$$\overrightarrow{{\mathrm{X}}_{\mathrm{i}}}\left(\mathrm{k}+1\right)=\overrightarrow{{\mathrm{X}}_{\mathrm{i}}}\left(\mathrm{k}\right)+\overrightarrow{{\mathrm{v}}_{\mathrm{i}}}\left(\mathrm{k}+1\right)$$

(17)

where ${\mathsf{\omega }}_1$ is a constant in [0,1], ${\mathrm{c}}_{11}$ and ${\mathrm{c}}_{12}$ are coefficients in [0,2],${\mathrm{r}}_8$, ${\mathrm{r}}_9$ are random numbers in [0,1], $\overrightarrow{{\mathrm{X}}_{\mathrm{pbest}}}$ denotes the global best position so far, $\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}$ denotes the personal best position so far.

2.2.3. Improvement 3: Mutation Operator Based on The Sinusoidal Chaotic Mutation

The chaotic mutation operator, which can overcome the shortcoming of local convergence or prematurity, is one of the best approaches to find the best solution through thoroughly evaluating the search space. In this section, the sinusoidal chaotic map is selected as the chaotic mutation operation. The mathematical function of sinusoidal chaotic map can be written as Equation (18) [41]:

$${\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}+1\right)=\mathrm{a}1·{\mathrm{x}}_{\mathrm{i}}^2\left(\mathrm{t}\right)·\sin \left(\mathsf{\pi }·{\mathrm{x}}_{\mathrm{i}}\left(\mathrm{t}\right)\right),\mathrm{t}=1,\cdots ,{\mathrm{t}}_{\max };\mathrm{i}=1,\cdots ,\mathrm{N}$$

(18)

where a1 is chaotic constant and a1 = 2.3, N denotes the total number of members of the population.

At each iteration, each position of the whale agents of population is brought into Equation (18) to achieve a new position. The values of objection function of each position and its new position achieved by chaotic mutation operation are calculated. Between any position and its new position achieved by Equation (18), the one whose value of objection function is better remains in the population while the other is abandoned.

2.3. Procedure of Mixed-Strategy WOA

The pseudo-code of the algorithm is shown as follows (Algorithm 1):

Algorithm 1. Pseudo-Code.

1. Initialize the agents population X, and ${\mathrm{x}}_{\mathrm{i}}$ (i = 1, 2 … N, N is total number of all agents)  denotes position of the i-th agent of X; 2. While (t < tmax)   For ${\mathrm{x}}_1$ to ${\mathrm{x}}_{\mathrm{N}}$   (1) Evaluate objective function value of each agent of the population and select the worst   solution and the best solution and update $\overrightarrow{{\mathrm{X}}^{*}}$, $\overrightarrow{{\mathrm{X}}_{\mathrm{worst}}}$, $\overrightarrow{{\mathrm{X}}_{\mathrm{gbest}}}$ in the current iteration;   (2) Update A, C, a, l, p and calculate the chaotic weight ω in the current iteration;   (3) if 1 $\mathrm{p}<0.5$ (p and p1 are random in [0,1], q1 = c1·(1 − t/tmax))       if 2 $\left|\mathrm{A}\right|\ge 1$         if 3 q1 < p1 select $\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}$ in Equation (11) and update the position of each agent              of population as in Equations (1) and (2)        else if 3 q1 > p1 pdate $\overrightarrow{{\mathrm{X}}_{\mathrm{rand}}}$ in Equation (12) and update the position of each              agent of population as in Equations (1) and (2)        end if 3       else if 2 $\left|\mathrm{A}\right|<1$ update the position of each agent of population as in Equation (14)       else if 2    else if 1 $\mathrm{p}>0.5$      update the position of each agent of population as in Equation (15)    else if 1   end for  For ${\mathrm{x}}_1$ to ${\mathrm{x}}_{\mathrm{N}}$    Use Equations (16) and (17) to update the position of each agent of population  end for  For ${\mathrm{x}}_1$ to ${\mathrm{x}}_{\mathrm{N}}$    Use chaotic mutation as in Equation (18) to achieve new positions ${\mathrm{x}}_{\mathrm{c}1}$ to ${\mathrm{x}}_{\mathrm{cN}}$.     if obj(${\mathrm{x}}_{\mathrm{i}}$) < obj(${\mathrm{x}}_{\mathrm{ci}}$) ${\mathrm{x}}_{\mathrm{i}}={\mathrm{x}}_{\mathrm{i}}$    else ${\mathrm{x}}_{\mathrm{i}}={\mathrm{x}}_{\mathrm{ci}}$  end for   Check whether the position of any agent of population is out of the boundaries.   t = t + 1  end while 3. end

3. Experiments and Result Discussion

In this section, PSO [10], GSA [13], ALO [15], WOA [19], the Enhanced Whale Optimization Algorithm (EWOA) [34] and CWOA [36] are selected to be compared with the MSWOA proposed above on 23 benchmark functions which are depicted in detail in

Table 2. Unimodal test functions with dimensions = 30.

Function	Dimension	Range	fmin
${\mathrm{F}}_1\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}^2$	30	[−100,100]	0
${\mathrm{F}}_2\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}\left|{\mathrm{x}}_{\mathrm{i}}\right|+{\displaystyle {\displaystyle \prod }_{\mathrm{i}=1}^{\mathrm{n}}}\left|{\mathrm{x}}_{\mathrm{i}}\right|$	30	[−10,10]	0
${\mathrm{F}}_3\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}{\left({\displaystyle {\displaystyle \sum }_{\mathrm{j}-1}^{\mathrm{i}}}{\mathrm{x}}_{\mathrm{j}}\right)}^2$	30	[−100,100]	0
${\mathrm{F}}_4\left(\mathrm{x}\right)=\underset{\mathrm{i}}{\max }\left\{\left|{\mathrm{x}}_{\mathrm{i}}\right|,1\le \mathrm{i}\le \mathrm{n}\right\}$	30	[−100,100]	0
${\mathrm{F}}_5\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}-1}}\left[100{\left({\mathrm{x}}_{\mathrm{i}+1}-{\mathrm{x}}_{\mathrm{i}}^2\right)}^2+{\left({\mathrm{x}}_{\mathrm{i}}-1\right)}^2\right]$	30	[−30,30]	0
${\mathrm{F}}_6\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}{\left(\left[{\mathrm{x}}_{\mathrm{i}}+0.5\right]\right)}^2$	30	[−100,100]	0
${\mathrm{F}}_7\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{i}·{\mathrm{x}}_{\mathrm{i}}^4+\mathrm{random}\left[0,1\right]$	30	[−1.28,1.28]	0

,

Table 3. Multimodal test functions with dimensions = 30.

Function	Dimension	Range	fmin
${\mathrm{F}}_8\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}-{\mathrm{x}}_{\mathrm{i}}\sin \left(\sqrt{\left|{\mathrm{x}}_{\mathrm{i}}\right|}\right)$	30	[−500,500]	−12569
${\mathrm{F}}_9\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}\left[{\mathrm{x}}_{\mathrm{i}}^2-10\cos \left(2{\mathsf{\pi }\mathrm{x}}_{\mathrm{i}}\right)+10\right]$	30	[−5.12,5.12]	0
${\mathrm{F}}_{10}\left(\mathrm{x}\right)=-20\exp \left(-0.2\sqrt{\frac{1}{\mathrm{n}}{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}^2}\right)-\exp \left(\frac{1}{\mathrm{n}}{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}\cos \left(2{\mathsf{\pi }\mathrm{x}}_{\mathrm{i}}\right)\right)+20+\mathrm{e}$	30	[−32,32]	0
${\mathrm{F}}_{11}\left(\mathrm{x}\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}{\mathrm{x}}_{\mathrm{i}}^2-{\displaystyle {\displaystyle \prod }_{\mathrm{i}-1}^{\mathrm{n}}}\cos \left(\frac{{\mathrm{x}}_{\mathrm{i}}}{\sqrt{\mathrm{i}}}\right)+1$	30	[−600,600]	0
$\begin{array}{c}{\mathrm{F}}_{12}\left(\mathrm{x}\right)=\frac{\mathsf{\pi }}{\mathrm{n}}\left\{10\sin \left({\mathsf{\pi }\mathrm{y}}_1\right)+{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}-1}}{\left({\mathrm{y}}_1-1\right)}^2\left[1+{\sin }^2\left({\mathsf{\pi }\mathrm{y}}_{\mathrm{i}+1}\right)\right]+{\left({\mathrm{y}}_{\mathrm{n}}-1\right)}^2\right\}\\ +{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{u}\left({\mathrm{x}}_{\mathrm{i}},10,100,4\right)\hfill \end{array}$	30	[−50,50]	0
${\mathrm{y}}_{\mathrm{i}}=1+\frac{{\mathrm{x}}_{\mathrm{i}}+1}{4}$, $\mathrm{u}\left({\mathrm{x}}_{\mathrm{i}},\mathrm{a},\mathrm{k},\mathrm{m}\right)=\{\begin{array}{cc}\mathrm{k}{\left({\mathrm{x}}_{\mathrm{i}}-\mathrm{a}\right)}^{\mathrm{m}}& {\mathrm{x}}_{\mathrm{i}}>\mathrm{a}\\ 0& -\mathrm{a}<{\mathrm{x}}_{\mathrm{i}}<\mathrm{a}\\ \mathrm{k}\left(-{\mathrm{x}}_{\mathrm{i}}-\mathrm{a}\right)& {\mathrm{x}}_{\mathrm{i}}<-\mathrm{a}\end{array}$	–	–	–
$\begin{array}{c}{\mathrm{F}}_{13}\left(\mathrm{x}\right)=\frac{1}{10}\{{\sin }^2\left(3{\mathsf{\pi }\mathrm{x}}_{\mathrm{i}}\right)+{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}{\left({\mathrm{x}}_{\mathrm{i}}-1\right)}^2\left[1+{\sin }^2\left(3{\mathsf{\pi }\mathrm{x}}_{\mathrm{i}}+1\right)\right]\\ \hfill +{\left({\mathrm{x}}_{\mathrm{n}}-1\right)}^2\left[1+{\sin }^2\left(2{\mathsf{\pi }\mathrm{x}}_{\mathrm{n}}\right)\right]\}+{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{n}}}\mathrm{u}\left({\mathrm{x}}_{\mathrm{i}},5,100,4\right)\end{array}$	30	[−50,50]	0

and

Table 4. Multimodal test functions with fixed dimensions.

Function	dimension	Range	fmin
${\mathrm{F}}_{14}\left(\mathrm{x}\right)={\left(\frac{1}{500}+{\displaystyle {\displaystyle \sum }_{\mathrm{j}=1}^{25}}\frac{1}{\mathrm{j}+{{\displaystyle \sum }}_{\mathrm{i}=1}^2{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{a}}_{\mathrm{ij}}\right)}^6}\right)}^{-1}$	2	[−65,65]	1
${\mathrm{F}}_{15}\left(\mathrm{x}\right)={\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{11}}{\left[{\mathrm{a}}_{\mathrm{i}}-\frac{{\mathrm{x}}_1\left({\mathrm{b}}_{\mathrm{i}}^2+{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_2\right)}{{\mathrm{b}}_{\mathrm{i}}^2+{\mathrm{b}}_{\mathrm{i}}{\mathrm{x}}_3+{\mathrm{x}}_4}\right]}^2$	4	[−5,5]	0.0003
${\mathrm{F}}_{16}\left(\mathrm{x}\right)=4{\mathrm{x}}_1^2-2.1{\mathrm{x}}_1^4+\frac{1}{3}{\mathrm{x}}_1^6+{\mathrm{x}}_1·{\mathrm{x}}_2-4{\mathrm{x}}_2^2+4{\mathrm{x}}_2^4$	2	[−5,5]	−1.0316
${\mathrm{F}}_{17}\left(\mathrm{x}\right)={\left({\mathrm{x}}_2-\frac{5.1}{4{\mathsf{\pi }}^2}{\mathrm{x}}_1^2+\frac{5}{\mathsf{\pi }}{\mathrm{x}}_1-6\right)}^2+10\left(1-\frac{1}{8\mathsf{\pi }}\right)\cos \left({\mathrm{x}}_1\right)+10$	2	[−5,5]	0.398
$\begin{array}{cc}\hfill {\mathrm{F}}_{18}\left(\mathrm{x}\right)=[1+& {\left({\mathrm{x}}_1+{\mathrm{x}}_2+1\right)}^2\left(19-14{\mathrm{x}}_1+3{\mathrm{x}}_1^2-14{\mathrm{x}}_2+6{\mathrm{x}}_1·{\mathrm{x}}_2+3{\mathrm{x}}_2^2\right)]\hfill \\ & \times [30+{\left(2{\mathrm{x}}_1-3{\mathrm{x}}_2\right)}^2\hfill \\ & \times \left(18-32{\mathrm{x}}_1+12{\mathrm{x}}_1^2+48{\mathrm{x}}_2-36{\mathrm{x}}_1·{\mathrm{x}}_2+27{\mathrm{x}}_2^2\right)]\hfill \end{array}$	2	[−2,2]	3
${\mathrm{F}}_{19}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^4}{\mathrm{c}}_{\mathrm{i}}·\exp \left(-{\displaystyle {\displaystyle \sum }_{\mathrm{j}=1}^3}{\mathrm{a}}_{\mathrm{ij}}{\left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{p}}_{\mathrm{ij}}\right)}^2\right)$	3	[0,1]	−3.86
${\mathrm{F}}_{20}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^4}{\mathrm{c}}_{\mathrm{i}}·\exp \left(-{\displaystyle {\displaystyle \sum }_{\mathrm{j}=1}^6}{\mathrm{a}}_{\mathrm{ij}}{\left({\mathrm{x}}_{\mathrm{j}}-{\mathrm{p}}_{\mathrm{ij}}\right)}^2\right)$	6	[0,1]	−3.32
${\mathrm{F}}_{21}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^5}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{i}}\right]}^{-1}$	4	[0,10]	−10.1532
${\mathrm{F}}_{22}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^7}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{i}}\right]}^{-1}$	4	[0,10]	−10.4028
${\mathrm{F}}_{23}\left(\mathrm{x}\right)=-{\displaystyle {\displaystyle \sum }_{\mathrm{i}=1}^{10}}{\left[\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right){\left(\mathrm{x}-{\mathrm{a}}_{\mathrm{i}}\right)}^{\mathrm{T}}+{\mathrm{c}}_{\mathrm{i}}\right]}^{-1}$	4	[0,10]	−10.5363

. In all tests on 23 benchmark functions, the population size is 30, and the total number of iterations is 500 for all algorithms proposed in

Table 5. Parameter settings for all algorithms in test.

Algorithm	Parameter Settings
PSO12	c1 = 2, c2 = 2, inertia weight ω = 1, inertia weight damping ratio ω2 = 0.99
GSA13	G0 = 100, α = 20
ALO16	All parameter settings are recommended in [15]
WOA20	a decrease linearly from 2 to 0
CWOA37	a decrease linearly from 2 to 0, other settings are same to [19]
EWOA35	a decrease linearly from 2 to 0, c = 0.3, other settings are same to [19]
MSWOA	a decrease linearly from 2 to 0, q1 = 0.5(1 − t/tmax), inertia weight ω = 1,inertia weight damping ratio ω2 = 0.99, other settings are same to WOA [19]

. To prove that MSWOA outperforms other algorithms in Table 5, the experimental results of MSWOA are compared with those of other algorithms by two methods, namely the Wilcoxon’ test and the box and whisker method.

3.1. Experiments Setting and Benchmark Function

Each algorithm of the seven algorithms proposed in Table 5 is tested on 23 benchmark functions which can be divided into three groups in Table 2, Table 3 and Table 4, namely unimodal test functions (F1–F7), multimodal test functions (F8–F13) and multimodal test functions with fixed dimensions (F14–F23). Generally, F1–F7 are employed to calculate the exploitation ability of algorithms benchmarked by benchmark functions and F8–F23 are used to calculate exploration ability. In Table 2, Table 3 and Table 4, the first column is the expression of function, the second column is dimension of function, the third column is the domain of variable and the last column is the standard minimum value of the function.

Because the meta-heuristic algorithms proposed in Table 5 are stochastic, the experiment of any kind of benchmark function is repeated 20 times. Meanwhile the mean value and the standard deviation value of 20 repeated experiments are saved as test results. The population size of each algorithm is set at 30 and total number of iterations is 500, other parameter settings are listed in Table 5. In addition, all benchmark functions experiments are run on MATLAB R2016a (R2016a, Math Works, Natick, MA, USA).

3.2. Results Analysis Based on Statistical Test Methods

All meta-heuristic algorithms presented in Table 5 are applied on the 23 benchmark functions in comparative experiments. For a fair competition, the experiments are repeated 20 times, while the average values and standard deviation are calculated and presented in

Table 6. Test results of 23 benchmark functions.

Function	Index	PSO	GSA	ALO	WOA	CWOA	EWOA	MSWOA
F1	Ave	1.36 × 10−8	3.29 × 10−16	7.52 × 10−9	6.13 × 10−74	0	2.22 × 10−32	0
	Std	2.55 × 10−8	3.74× 10−16	3.3× 10−9	2.18× 10−73	0	6.01× 10−32	0
	Anc	500	500	500	500	64	500	14
F2	Ave	0.0403	0.0361	0.5907	4.06× 10−51	0	7.24× 10−26	0
	Std	0.0690	0.1127	1.0355	6.45× 10−51	0	8.73× 10−26	0
	Anc	411	351	487	500	39	500	11
F3	Ave	96.484	905.77	0.1921	45,988	33,120	47,649	0.0218
	Std	85.763	409.34	0.4005	8114	7478.3	13702	0.0819
	Anc	500	456	499	500	500	500	500
F4	Ave	2.4409	6.9393	0.2225	40.874	40.173	69.994	0
	Std	1.0333	1.5297	0.9838	26.297	17.893	14.454	0
	Anc	500	394	497	428	490	283	16
F5	Ave	45.171	58.224	194.96	27.964	26.869	18.794	24.253
	Std	41.859	38.846	436.28	0.4795	1.1668	12.561	0.2296
	Anc	500	445	500	257	447	500	500
F6	Ave	3.11 × 10−8	6.1	1.26 × 10−8	0.41	0.0951	0.0026	4.11 × 10−12
	Std	4.56 × 10−8	4.5986	2.03 × 10−8	0.2589	0.1292	0.0014	4.35 × 10−12
	Anc	500	100	500	274	456	500	500
F7	Ave	0.0261	0.0770	0.0306	0.0023	8.66 × 10−5	0.0151	6.63 × 10−5
	Std	0.0121	0.0268	0.0162	0.0032	7.72 × 10−5	0.0120	7.16 × 10−5
	Anc	498	231	333	480	499	498	406
F8	Ave	−6185.6	−2793.1	−2267.1	−10655	−11,487	−11,916	−9203.3
	Std	635.16	611.86	445.86	1702.5	1102.5	1070.2	1295
	Anc	207	11	401	414	500	500	499
F9	Ave	46.863	27.71	22.436	0	0	90.101	0
	Std	12.539	7.242	13.417	0	0	70.216	0
	Anc	308	250	391	212	95	500	9
F10	Ave	1.6484	1.3 × 10−8	0.6613	3.91 × 10−15	8.8818 × 10−16	1.28 × 10−14	8.8818 × 10−16
	Std	0.7689	4.01 × 10−8	0.8640	2.38 × 10−15	0	4.79 × 10−15	0
	Anc	380	500	488	299	48	494	10
F11	Ave	0.0255	27.322	0.2039	0.0231	0.0062	0.0120	0.0032
	Std	0.0280	6.5334	0.0829	0.0807	0.0251	0.0236	0.0071
	Anc	325	319	457	240	217	491	402
F12	Ave	0.0882	2.2046	2.3717	0.0344	0.0119	2.6187	3.05 × 10−4
	Std	0.2436	1.2486	2.5530	0.0321	0.0123	3.7315	2.06 × 10−4
	Anc	394	325	478	255	419	500	500
F13	Ave	0.0585	7.6479	0.0032	0.5345	0.3736	2.3549	0.0502
	Std	0.1165	5.7737	0.0076	0.3071	0.3249	10.0492	0.0461
	Anc	463	384	390	402	443	500	500
F14	Ave	4.3772	5.0336	1.7899	2.6678	1.2956	1.8336	1.1964
	Std	4.1508	3.2134	1.3790	3.0266	0.7269	2.2266	0.6107
	Anc	120	37	67	211	166	274	264
F15	Ave	6.21 × 10−4	0.0032	0.0028	7.79 × 10−4	6.82 × 10−4	5.8 × 10−4	3.46 × 10−4
	Std	5.28 × 10−4	0.0015	0.0060	5.59 × 10−4	2.33× 10−4	3.69× 10−4	6.44 × 10−5
	Anc	500	254	242	494	483	500	500
F16	Ave	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316	−1.0316
	Std	2.2 × 10−16	1.2 × 10−16	6.9 × 10−14	1.01 × 10−9	3.1 × 10−10	3.5 × 10−16	7.7 × 10−16
	Anc	42	133	132	36	18	47	33
F17	Ave	0.3979	0.3979	0.3979	0.3979	0.3979	0.3979	0.3979
	Std	0	0	9.4 × 10−14	2.5 × 10−5	1.14 × 10−6	1.2 × 10−11	4.3 × 10−14
	Anc	45	132	127	274	90	127	88
F18	Ave	3	3	3	3.0001	3	3	3
	Std	9 × 10−16	3.8 × 10−15	1.4 × 10−12	1.06 × 10−4	5.3 × 10−13	1.5 × 10−11	1.2 × 10−14
	Anc	52	193	251	133	20	47	27
F19	Ave	−3.8241	−3.8628	−3.8628	−3.8528	−3.8617	−3.8628	−3.8617
	Std	0.1729	1.8 × 10−15	1.9 × 10−12	0.0128	0.0022	1.32 × 10−5	0.0024
	Anc	41	254	125	256	248	310	482
F20	Ave	−3.2863	−3.3220	−3.2800	−3.2540	−3.2428	−3.2744	−3.2920
	Std	0.0559	2.9 × 10−16	0.0587	0.0865	0.1183	0.0599	0.0533
	Anc	60	198	348	310	379	482	427
F21	Ave	−6.8967	−7.1798	−7.1339	−8.3351	−8.6193	−9.4009	−9.8983
	Std	3.4496	3.7368	3.2138	2.8253	2.3941	2.3154	1.1399
	Anc	70	198	269	266	463	432	279
F22	Ave	−7.6093	−9.4954	−7.4842	−9.1804	−8.5395	−8.8258	−10.4029
	Std	3.5471	2.0522	3.4103	2.4700	2.5988	2.8416	3.98 × 10−10
	Anc	69	195	376	254	444	499	217
F23	Ave	−8.2082	−10.1333	−7.6188	−8.2293	−8.9107	−9.1198	−10.5364
	Std	3.6499	1.8030	3.6877	3.2614	2.5404	2.9524	8.79 × 10−10
	Anc	72	233	352	251	473	341	262

Note: Ave: average value; Std: Standard deviation; Anc: average iteration number of convergence.

. To analyze these results reasonably, two statistical test methods, namely the Wilcoxon’s and box and whisker tests, are adopted. The specific results are discussed as follows.

3.2.1. The Wilcoxon’s Test

The Wilcoxon’s test, a nonparametric statistical test, is applied to test which is the more significant between two algorithms. The Wilcoxon’s test is divided into the single-problem statistical analysis and multiple-problem statistical analysis.

In the single-problem statistical analysis, any two algorithms mentioned in Table 5 are selected to be compared with each other on the same benchmark function, F1 to F23. In Table 6, “Ave” represents a mean value of a group of experiments for 20 times on one benchmark function. Hence the “Ave” is used as the test sample for the Wilcoxon’s test. If algorithm A obtains a smaller “Ave” than algorithm B, algorithm A is considered to have a better significance level than algorithm B. In other words, algorithm A is superior to algorithm B by the Wilcoxon signed-rank test at alfa = 0.05. In

Table 7. The Wilcoxon signed-rank single-problem test for statistically significance level at alfa = 0.05.

Function	MSWOA vs. PSO	MSWOA vs. GSA	MSWOA vs. ALO	MSWOA vs. WOA	MSWOA vs. CWOA	MSWOA vs. EWOA
F1	MSWOA	MSWOA	MSWOA	MSWOA	Tie	MSWOA
F2	MSWOA	MSWOA	MSWOA	MSWOA	Tie	MSWOA
F3	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F4	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F5	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	EWOA
F6	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F7	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F8	MSWOA	MSWOA	MSWOA	MSWOA	Tie	EWOA
F9	MSWOA	MSWOA	MSWOA	MSWOA	Tie	MSWOA
F10	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F11	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F12	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F13	MSWOA	MSWOA	ALO	MSWOA	MSWOA	MSWOA
F14	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F15	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F16	Tie	Tie	Tie	Tie	Tie	Tie
F17	Tie	Tie	Tie	Tie	Tie	Tie
F18	Tie	Tie	Tie	MSWOA	Tie	Tie
F19	MSWOA	GSA	MSWOA	MSWOA	MSWOA	MSWOA
F20	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F21	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F22	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
F23	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA	MSWOA
W/T/L	20/3/0	20/3/0	19/3/1	21/2/0	16/7/0	18/3/2

, “W/T/L” represents three relations, namely Win, Tie and Lose. “W” means MSWOA obtains a smaller “Ave” than another algorithm, “L” means MSWOA obtains a bigger “Ave” than another algorithm, “T” means MSWOA obtains an equal “Ave” to another algorithm. In the test, the total number of “W/T/L” is counted in the last row of Table 7.

The multiple-problem statistical analysis is used to compare two algorithms in the several similar benchmark functions while the single-problem statistical analysis is used to compare two algorithms in the same one benchmark function. In Table 6, the results of each meta-heuristic algorithm tested on 23 benchmark functions are listed. All mean values of the results of each algorithm tested on the 23 benchmark functions are treated as the input vectors of the Wilcoxon test.

In

Table 8. The Wilcoxon signed-rank multiple-problem test for statistically significance level at alfa = 0.05.

Comparison	R+	R−	p-Value	Significant
MSWOA vs. PSO	270	6	3.173 × 10−5	Yes
MSWOA vs. GSA	253	23	2.484 × 10−4	Yes
MSWOA vs. ALO	260	16	1.098 × 10−4	Yes
MSWOA vs. WOA	255	21	3.261 × 10−4	Yes
MSWOA vs. CWOA	258	18	0.0010	Yes
MSWOA vs. EWOA	224	52	0.0047	Yes

, the test results of multiple-problem statistical analysis by the Wilcoxon signed-rank test are listed. The R+, R− and the p-values can be evaluated by the Wilcoxon signed-rank test applied in multiple-problem statistical analysis. If MSWOA obtained the higher R+ than R− values in any one pair of comparison and the p-value is less than 0.05, it means that MSWOA is statistically significantly better than another. From Table 7 and Table 8, we can conclude that MSWOA has a better performance in a statistical manner than the other six algorithms, especially for WOA, CWOA and EWOA. Based on WOA, MS-WOA is improved greatly on the capability of stability, exploitation and exploration and avoidance of local optimum points and prematurity.

3.2.2. Box and Whisker

The stability is a concept used to evaluate an algorithm by checking the randomness of solutions. In addition to the standard deviation statistical index, the box and whisker plot is also effective in estimation of algorithm stability. The box and whisker methods used in this section to evaluate the distribution of results of the repeated 20 runs on benchmark functions F1 to F23. A box and whisker plot can illustrate the variation in a set of data and provide more information about the data. It provides five statistics, namely, the minimum value, the second quartile, the median value, the third quartile and the maximum value. The second quartile is the value below which the lower 25% of the data are contained. Third quartile is the value above which the upper 25% of the data are contained.

In Figure 3, the distributions of the objective function values of the optimal solutions of multiple runs are revealed by box and whisker plots, while the proposed algorithm is compared with the other six algorithms. According to Figure 3, it is obvious that box and whisker of MSWOA are all nearly horizontal lines except for (h) and (t). It is shown that the variation of the values of the best solution obtained from MSWOA after 20 times are very small and the stability of MSWOA is superior to that of the other algorithms. Compared with homogenous algorithms, like WOA, CWOA and EWOA, the stability of MSWOA has been greatly improved.

3.3. Performance Comparison

According to the test results in Table 6, Table 7 and Table 8, a conclusion is drawn that the MSWOA algorithm proposed in the paper has been improved significantly on its performance compared to the other algorithms in Table 5. In this subsection, the performance of MSWOA will be analyzed further based on the results of 23 benchmark functions.

3.3.1. Analysis of Test Results of F1–F7

Based on character of benchmark functions, unimodal test functions (F1–F7) are applied to test the capacity of exploitation of algorithms while the multimodal test function and multimodal test functions with fixed dimension (F8–F23) are employed to test the capacity of exploration of algorithms. In Table 6, its manifest that the MSWOA gives a superior test result than the other algorithms in all unimodal test functions except F5, while EWOA gives a better test result than the others on F5, MSWOA outperforms other algorithms on both the final test results and convergence rate. In Figure 4a–w, the convergence curves of PSO, GSA, ALO, WOA, CWOA, EWOA and MSWOA are the average curves with mean values tested on the 23 benchmark functions 20 times, respectively. In Figure 4a–g, it is clear that the MSWOA has the fastest convergence rate to search for the global optimization point. In the unimodal test functions test, the convergence rates of the algorithms are more vital than the final results. As for Figure 3a–g, we can find that MSWOA outperforms the others in the box and whisker test results. That means MSWOA is more stable than PSO, GSA, ALO, WOA, CWOA and EWOA for unimodal test functions (F1–F7).

3.3.2. Analysis of Test Results of F8–F13

The ability which makes an algorithm escape from local optimum points and locate a global optimization point, can be tested by multimodal functions (F8–F13). When an algorithm is run on multimodal functions (F8–F13), the test results can reflect the ability better than the convergence rate.

In Table 6, it is clear that MSWOA outperforms the other algorithms in F9, F10, F11 and F12 in the test results. MSWOA has the second best performance in F13. Although GSA has a better test result than others in F13, MSWOA outperforms WOA, CWOA and EWOA on both the final test results and convergence rate. In Figure 3h–m, it is found that MSWOA outperforms the others in the box and whisker test results, which means the stability of MSWOA is superior to those of PSO, GSA, ALO, WOA, CWOA and EWOA for multimodal functions (F8–F13).

3.3.3. Analysis of Test Results of F14–F23

The ability of an algorithm to avoid a small quantity of local optima can be tested by multimodal functions with fixed dimensions (F14–F23). In Table 6, it is clear that MSWOA outperforms other the algorithms in F14 to F23 apart from F20 in the test results. MSWOA achieves the second best performance in F20. Though ALO has a better test result than the others in F20, MSWOA outperforms WOA, CWOA and EWOA on both the final test results and convergence rate. In Figure 3n–w, it is found that MSWOA outperforms others in test results in F14 to F23 apart from F20. That means the stability of MSWOA is superior to those of PSO, GSA, ALO, WOA, CWOA and EWOA for multimodal functions (F14–F23) apart from F20. In Figure 3t, the stability of MSWOA is only inferior to that of GSA which gains the best value while the stability of MSWOA is superior to others including PSO, ALO, WOA, CWOA and EWOA.

4. Parameter Identification of the Hydraulic Turbine Governing System

In this section, the MSWOA is applied to solve the parameter identification problem of a complicated HTGS with a delayed water hammer effect. The mathematical model of the HTGS is established, and then the parameter identification methodology as well as simulation experiments are conducted.

4.1. Model of Hydraulic Turbine Governing System with a Delay Water Hammer Effect

A HTGS is composed of speed governor, penstock, hydraulic turbine and generator [1,4,25,26], as shown in Figure 5. In Figure 5, X denotes the speed of the generator, Y denotes the position of the main servomotor, H denotes the water head, Q denotes the flow and M denotes the moment. In this paper, the penstock system model is a hyperbolic tangent function which is different from the rigid water hammer equation and elastic water hammer equation. The hyperbolic tangent function can describe the dynamic process of the penstock system in the most precise way. In Figure 6, each part of the HTGS model is described.

4.1.1. PID Controller Model

The PID controller is one part of the hydraulic turbine governor. The PID controller model is described as follows:

$$\mathsf{\sigma }\left(\mathrm{s}\right)=\left({\mathrm{K}}_{\mathrm{p}}+\frac{{\mathrm{K}}_{\mathrm{i}}}{\mathrm{S}}\right)·\left(\mathrm{c}\left(\mathrm{s}\right)-\mathrm{x}\left(\mathrm{s}\right)-{\mathrm{b}}_{\mathrm{p}}·{\mathrm{y}}_1\left(\mathrm{s}\right)\right)+\frac{{\mathrm{K}}_{\mathrm{d}}\mathrm{s}}{{\mathrm{T}}_{\mathrm{d}}\mathrm{s}+1}·\mathrm{x}\left(\mathrm{s}\right)$$

(19)

where $\mathsf{\sigma }\left(\mathrm{s}\right)$ is the Laplace transform of output of the PID controller; $\mathrm{c}\left(\mathrm{s}\right)$ is the Laplace transform of a given speed; $\mathrm{x}\left(\mathrm{s}\right)$ is the Laplace transform of the speed; ${\mathrm{b}}_{\mathrm{p}}$ is a permanent transition coefficient; ${\mathrm{y}}_1\left(\mathrm{s}\right)$ is the position of the auxiliary servomotor; ${\mathrm{T}}_{\mathrm{d}}$ is the differential time; ${\mathrm{K}}_{\mathrm{p}}$ is the proportional gain, ${\mathrm{K}}_{\mathrm{i}}$ is the integral gain and ${\mathrm{K}}_{\mathrm{d}}$ is the differential gain.

4.1.2. Servomechanism Model

The servomechanism is another part of the hydraulic turbine governor. The model is described as follows:

$$\{\begin{array}{c}\frac{{\mathrm{y}}_1\left(\mathrm{s}\right)}{\mathsf{\sigma }\left(\mathrm{s}\right)}=\frac{1}{{\mathrm{T}}_{\mathrm{y}1}\mathrm{s}+1}\\ \frac{\mathrm{y}\left(\mathrm{s}\right)}{{\mathrm{y}}_1\left(\mathrm{s}\right)}=\frac{1}{{\mathrm{T}}_{\mathrm{y}}\mathrm{s}+1}\\ \mathrm{s}.\mathrm{t}.-{\mathrm{y}}_0\le y\le 1-{\mathrm{y}}_0\end{array}$$

(20)

where $\mathrm{y}\left(\mathrm{s}\right)$ is the position of the main servomotor; ${\mathrm{T}}_{\mathrm{y}1}$ is the response time constant of the auxiliary servomotor; ${\mathrm{T}}_{\mathrm{y}}$ is the response time constant of the main servomotor, ${\mathrm{y}}_0$ is the initial value of $\mathrm{y}\left(\mathrm{s}\right)$.

4.1.3. Hydraulic Turbine Model

When the HTGS is working, the dynamic process of the hydraulic turbine system can’t be obtained by experiments or model tests, but if the speed of the hydraulic turbine fluctuates in a small range, a linear model of the hydraulic turbine can be given by Equation (21), which includes the express moment and flow characteristics, and is applied to depict the characteristics of a hydraulic turbine:

$$\{\begin{array}{c}{\mathrm{m}}_{\mathrm{t}}\left(\mathrm{s}\right)={\mathrm{e}}_{\mathrm{x}}·x\left(\mathrm{s}\right)+{\mathrm{e}}_{\mathrm{y}}·y\left(\mathrm{s}\right)+{\mathrm{e}}_{\mathrm{h}}·h\left(\mathrm{s}\right)\\ q\left(\mathrm{s}\right)={\mathrm{e}}_{\mathrm{qx}}·x\left(\mathrm{s}\right)+{\mathrm{e}}_{\mathrm{qy}}·y\left(\mathrm{s}\right)+{\mathrm{e}}_{\mathrm{qh}}·h\left(\mathrm{s}\right)\end{array}$$

(21)

where ${\mathrm{m}}_{\mathrm{t}}\left(\mathrm{s}\right)$ is the Laplace transform of the moment of the hydraulic turbine, $\mathrm{q}\left(\mathrm{s}\right)$ is the Laplace transform of the water flow of the hydraulic turbine, $\mathrm{h}\left(\mathrm{s}\right)$ is the water head of the hydraulic turbine, ${\mathrm{e}}_{\mathrm{x}}$, ${\mathrm{e}}_{\mathrm{y}}$, ${\mathrm{e}}_{\mathrm{h}}$, ${\mathrm{e}}_{\mathrm{qx}}$, ${\mathrm{e}}_{\mathrm{qy}}$, ${\mathrm{e}}_{\mathrm{qh}}$ are the transfer coefficients of the hydraulic turbine which are obtained from the comprehensive characteristic curve of the hydraulic turbine. Usually we can obtain the transfer coefficients of some working point to build the hydraulic turbine model. The detailed instructions of Equation (21) as well as the calculation of those transfer coefficients may be found in [42].

4.1.4. Penstock System Model

The elastic water hammer and rigid water hammer models are main expressions of the penstock system models and are frequently used to describe the characteristics of a penstock system. In recent studies, many novel expressions of elastic water hammer [27,28] and rigid water hammer [2] models are proposed. Compared with a hyperbolic tangent function water hammer model as given by Equation (22), those models only keep parts of a hyperbolic tangent function water hammer model. This inevitably reduces the accuracy of the penstock system model. In fact, the hyperbolic tangent function water hammer model is the most proximate to the real penstock system and can describe the dynamic process of the penstock system in the most precise way. Therefore, the hyperbolic tangent function water hammer model given by Equation (22) is employed as the penstock system model in this paper. The transfer function of the penstock system can be expressed as:

$$\frac{\mathrm{h}\left(\mathrm{s}\right)}{\mathrm{q}\left(\mathrm{s}\right)}=-2{\mathrm{h}}_{\mathrm{w}}\frac{1-{\mathrm{e}}^{-{\mathrm{T}}_{\mathrm{r}}\mathrm{s}}}{1+{\mathrm{e}}^{-{\mathrm{T}}_{\mathrm{r}}\mathrm{s}}}$$

(22)

where ${\mathrm{h}}_{\mathrm{w}}$ is the pipeline characteristic coefficient, ${\mathrm{T}}_{\mathrm{r}}$ is the reflection time of the water hammer pressure wave.

4.1.5. Generator System Model

The dynamic equation of the synchronous generator is described as Equation (23) in which ${\mathrm{m}}_{\mathrm{g}}\left(\mathrm{s}\right)$ means the load disturbance. When the HTGS is working under load conditions, ${\mathrm{m}}_{\mathrm{g}}\left(\mathrm{s}\right)\ne 0$ and ${\mathrm{e}}_{\mathrm{g}}\ne 0$. When HTGS is working under no-load conditions, ${\mathrm{m}}_{\mathrm{g}}\left(\mathrm{s}\right)=0$ and ${\mathrm{e}}_{\mathrm{g}}=0$.

$$\frac{\mathrm{x}\left(\mathrm{s}\right)}{{\mathrm{m}}_{\mathrm{t}}\left(\mathrm{s}\right)-{\mathrm{m}}_{\mathrm{g}}\left(\mathrm{s}\right)}=\frac{1}{{\mathrm{T}}_{\mathrm{a}}\mathrm{s}+{\mathrm{e}}_{\mathrm{n}}}$$

(23)

where ${\mathrm{T}}_{\mathrm{a}}$ is the inertial time constant of the generator, ${\mathrm{e}}_{\mathrm{g}}$ is the adjustment coefficient of the generator and ${\mathrm{e}}_{\mathrm{n}}={\mathrm{e}}_{\mathrm{g}}-{\mathrm{e}}_{\mathrm{x}}$.

4.2. Identification Strategy Based on MSWOA

The proposed MSWOA is employed to identify the HTGS parameters. In the MSWOA identification strategy, firstly, an input is employed to activate the real system and the system to be identified. The outputs of the real system and the system to be identified are input into Equation (24) to evaluate the value of ${\mathrm{C}}_{\mathrm{OF}}(\hat{\mathsf{\theta }})$.

$${\mathrm{C}}_{\mathrm{OF}}(\hat{\mathsf{\theta }})={\displaystyle \sum }_{\mathrm{i}=1}^{\mathrm{N}}{\displaystyle \sum }_{\mathrm{j}=1}^{\mathrm{M}}{\left({\mathrm{z}}_{\mathrm{j}}\left(\mathrm{k}\right)-{\hat{\mathrm{z}}}_{\mathrm{j}}\left(\mathrm{k}\right)\right)}^2$$

(24)

where $\mathsf{\theta }$ is six-dimensional vector and $\mathsf{\theta }=\left[\begin{array}{ccc}\begin{array}{cc}{\mathrm{T}}_{\mathrm{y}1}& {\mathrm{T}}_{\mathrm{y}}\end{array}& \begin{array}{cc}{\mathrm{h}}_{\mathrm{w}}& {\mathrm{T}}_{\mathrm{r}}\end{array}& \begin{array}{cc}{\mathrm{T}}_{\mathrm{a}}& {\mathrm{e}}_{\mathrm{g}}\end{array}\end{array}\right]$, $\mathrm{z}=\left[\begin{array}{ccc}\mathrm{x}& \mathrm{y}& {\mathrm{m}}_{\mathrm{t}}\end{array}\right]$ is the output of the real system, $\hat{\mathrm{z}}=\left[\begin{array}{ccc}\widehat{\mathrm{x}}& \widehat{\mathrm{y}}& \widehat{{\mathrm{m}}_{\mathrm{t}}}\end{array}\right]$ is the output of the identified system. N is the total number of samples, M is the dimension of the outputs. After that, in the MSWOA-based optimizer, the parameters to be identified are identified by minimizing ${\mathrm{C}}_{\mathrm{OF}}(\widehat{\mathsf{\theta }})$. As the optimization loop in the MSWOA-based optimizer goes on, the parameters to be identified accurately match the real values. The process of identification is illustrated in Figure 7, where $\mathsf{\theta }$ is a parameter vector of which each element is a real value while $\hat{\mathsf{\theta }}$ is a parameter vector of which each element is to be identified.

PE (parameter error) and APE (the average parameter error) which are described by Equations (25) and (26) are employed to evaluate the accuracy of the parameters identified by MSWOA:

$$PE=\frac{\left|{\theta }_k-{\widehat{\theta }}_k\right|}{{\theta }_k}$$

(25)

$$APE=\frac{1}{m}{\displaystyle \sum }_{k=1}^m\frac{\left|{\theta }_k-{\widehat{\theta }}_k\right|}{{\theta }_k}$$

(26)

where ${\theta }_k$ is the parameters of the real system, ${\widehat{\theta }}_k$ is the parameters of the identified system, m is the total number of parameters of $\theta$ while k is the k-th parameter of $\theta$.

4.3. Experiments and Analysis of Parameters Identification

In this section, MSWOA is employed to identify the parameters of a mathematical model of HTGS which is simulated in MATLAB R2016a. ${\mathrm{T}}_{\mathrm{y}1}$, ${\mathrm{T}}_{\mathrm{y}}$,${\mathrm{T}}_{\mathrm{r}}$, ${\mathrm{h}}_{\mathrm{w}}$, ${\mathrm{T}}_{\mathrm{a}}$, ${\mathrm{e}}_{\mathrm{g}}$ are the parameters to be identified. In identification experiments, two working conditions of HTGS, namely the no-load condition and the load condition, are both taken into account. The step disturbance of frequency and load are utilized to excite the system, respectively. Under no-load conditions, the amplitude of the step disturbance of the frequency is 0.1 p.u and under no-load conditions the amplitude of the step disturbance of the load is 0.1 p.u, respectively. Other parameters are set as follows: the total time of the simulation experiments is 30 s and the sampling time is 0.01 s. ${\mathrm{K}}_{\mathrm{p}}=5.59$, ${\mathrm{K}}_{\mathrm{i}}=1.06$, ${\mathrm{K}}_{\mathrm{d}}=3.3$, ${\mathrm{T}}_{\mathrm{d}}=0.28$, ${\mathrm{b}}_{\mathrm{p}}=0.04$, the initial value of the parameter vector $\mathsf{\theta }$ is $\mathsf{\theta }=\left[\begin{array}{ccc}\begin{array}{cc}0.1& 0.3\end{array}& \begin{array}{cc}1.5& 0.5\end{array}& \begin{array}{cc}12& 0.5\end{array}\end{array}\right]$, the transfer coefficients of HTGS are selected referring to [2]. The transfer coefficients are listed in

Table 9. Transfer coefficients of HTGS on two working conditions.

Working Condition	Transfer Coefficients of HTGS
	ex	ey	eh	eqx	eqy	eqh
No-load	−1.0567	0.9080	1.4191	−0.0574	0.7887	0.4571
Load	−1.4673	0.7713	1.7179	−0.4901	0.8184	0.7257

.

To prove that the MSWOA is superior to other algorithms in parameter identification, PSO, ALO, WOA, EWOA and CWOA were employed to identify the parameter of HTGS as a comparison. The identification experiments of any algorithm are independently repeated 20 times and the average values of the identified parameters of each algorithm have been obtained. The size of the population and the total number of iterations are set as 30 and 100 for each algorithm. The parameter setting of the algorithms proposed in this paper is described in details in Table 5.

4.3.1. Comparison of Different Identification Methods under No-Load Condition

In Figure 6, c and mg are the frequency disturbance and the load disturbance signals, respectively. Under no-load conditions, c is added to HTGS without loads, namely c, and mg = 0. Moreover, because the value of mg is zero, ${\mathrm{e}}_{\mathrm{g}}=0$. Therefore $\mathsf{\theta }$ has five elements, namely ${\mathrm{T}}_{\mathrm{y}1}$, ${\mathrm{T}}_{\mathrm{y}}$, ${\mathrm{h}}_{\mathrm{w}}$, ${\mathrm{T}}_{\mathrm{r}}$ and ${\mathrm{T}}_{\mathrm{a}}$, that need to be identified. In addition, because a linear model of the hydraulic turbine is used in the identification experiments, the transfer coefficients are given values.

Mean values of the identified parameters and mean best cost for 20 repetitions of PSO, ALO, WOA, EWOA, CWOA and MSWOA are listed in

Table 10. Mean identified parameters under no-load conditions.

θk	Real Value	Mean Value of Identified Parameters after 20 Repetitions
		PSO		ALO		WOA		EWOA		CWOA		MSWOA
		PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$
Ty1	0.1	0.1638	0.1164	0.182	0.1182	0.188	0.1188	0.153	0.1153	0.1608	0.1161	0.1094	0.1109
Ty	0.3	0.0508	0.2848	0.0655	0.2803	0.021	0.3064	0.082	0.2754	0.058	0.2826	0.012	0.2964
hw	1.5	0.0061	1.4908	0.0035	1.4948	0.1332	1.3002	0.0745	1.3882	0.0379	1.5568	0.008	1.512
Tr	0.5	0.1371	0.5686	0.0256	0.5128	0.1331	0.5666	0.099	0.5495	0.0078	0.4961	0.0068	0.4966
Ta	12	0.0040	12.048	0.0018	12.022	0.0022	12.026	0.0073	12.088	0.0018	12.022	0.0011	12.014

and

Table 11. Mean best cost and APE under no-load condition.

Index	PSO	ALO	WOA	EWOA	CWOA	MSWOA
Best cost	0.2831	0.0858	0.6770	0.2860	0.0958	0.0126
Mean APE	0.0724	0.0557	0.0955	0.0832	0.0533	0.0275

, respectively. It is obvious that the mean best cost and mean APE of MSWOA are smaller than those of the others. Furthermore, the mean values of the identified parameters except ${\mathrm{h}}_{\mathrm{w}}$ of MSWOA are more approximate to the real values than the others. In short MSWOA can get a comprehensively better accuracy in parameter identification than other algorithms.

In Figure 8, the ability of exploration and exploitation of WOA and EWOA are worse than those of the others. PSO, which is easily trapped in local optimum points, achieves a better ability of exploration and exploitation than those of WOA and EWOA. Though CWOA and ALO have a similar iteration process, ALO achieves a better exploration and exploitation ability than CWOA. MSWOA achieves a better ability of escaping from of local optimum points than others. Therefore, MSWOA has a better ability of exploration, exploitation and escaping from local optimum points than other algorithms.

The outputs of HTGS with real values are compared with those of HTGS with the identified parameters in Figure 9. The compared outputs are the turbine speed, guide vane opening and turbine torque, respectively. The curves of the identified system are approximately the same as those of the real system. This verifies that MSWOA is effective for HTGS parameter identification with a delayed water hammer effect under no-load working conditions.

4.3.2. Comparison of Different Identification Methods under Load Conditions

Under load condition, only a load disturbance is added to HTGS. Therefore ${\mathrm{e}}_{\mathrm{g}}\ne 0$ and the parameter vector $\mathsf{\theta }$ have six elements, namely ${\mathrm{T}}_{\mathrm{y}1}$, ${\mathrm{T}}_{\mathrm{y}}$, ${\mathrm{h}}_{\mathrm{w}}$, ${\mathrm{T}}_{\mathrm{r}}$, ${\mathrm{T}}_{\mathrm{a}}$ and ${\mathrm{e}}_{\mathrm{g}}$, that need to be identified.

In

Table 12. Mean identified parameters under load conditions.

θk	Real Value	Mean Value of Identified Parameters after 20 Repetitions
		PSO		ALO		WOA		EWOA		CWOA		MSWOA
		PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$	PE	${\widehat{\mathbf{\theta }}}_{\mathit{k}}$
Ty1	0.1	0.146	0.0854	0.1334	0.1133	0.0434	0.1043	0.1474	0.1147	0.0320	0.0968	0.1432	0.1143
Ty	0.3	0.0398	0.3120	0.0552	0.2835	0.1335	0.2600	0.1710	0.2487	0.0300	0.2910	0.0675	0.2798
hw	1.5	0.1489	1.2766	0.0249	1.4626	0.0805	1.3792	0.0565	1.4152	0.1323	1.3016	0.0237	1.4644
Tr	0.5	0.2123	0.6061	0.1147	0.5574	0.1992	0.5996	0.1566	0.5783	0.2222	0.6111	0.0451	0.5225
Ta	12	0.0077	11.908	5 × 10−4	12.006	0.0145	12.174	0.0137	12.164	0.0027	12.032	1.6 × 10−4	12.002
eg	0.5	0.0096	0.4952	0.0072	0.4964	0.0888	0.5444	0.0136	0.4932	0.0090	0.5045	0.0028	0.4986

, it is obvious that the four parameters identified by MSWOA, namely ${\mathrm{h}}_{\mathrm{w}}$, ${\mathrm{T}}_{\mathrm{r}}$, ${\mathrm{T}}_{\mathrm{a}}$ and ${\mathrm{e}}_{\mathrm{g}}$, more accurately match the real values than the other algorithms while ${\mathrm{T}}_{\mathrm{y}}$ and ${\mathrm{T}}_{\mathrm{y}1}$ are not the most accurate match with the real values, but ${\mathrm{T}}_{\mathrm{y}}$ and ${\mathrm{T}}_{\mathrm{y}1}$ are more accurately matched to the real values than those of WOA and EWOA. Furthermore, the mean best cost and mean APE achieved by MSWOA are both better than those of the other algorithms in

Table 13. Mean best cost and APE under load condition.

Index	PSO	ALO	WOA	EWOA	CWOA	MSWOA
Best cost	0.0030	8.07 × 10−4	0.0108	0.0067	0.0019	2.77 × 10−4
Mean APE	0.0941	0.0560	0.0933	0.0931	0.0714	0.0471

.

In Figure 10, PSO has a better exploration and exploitation ability in the early stages of the iteration process but gets trapped into prematurity later. MSWOA has a better ability of exploration, exploitation and escaping from local optimum points than the other algorithms.

The outputs including the turbine speed, guide vane opening and turbine torque of the real system of HTGS and the identified system acquired by MSWOA are compared in Figure 11. The curves of the identified system almost overlap with those of the real system. Water hammer effects would cause a change of flow parameters which have to be managed by the guide vane system, so the fact that the outputs of the real system match those of identified system shows that MSWOA is effective for HTGS identification parameter with a delayed water hammer effect under load working conditions. Therefore, we can draw the conclusion that MSWOA has better global optimization ability and can get better accuracy in parameter identification than the other algorithms proposed in this paper whether under no-load conditions or load conditions.

5. Conclusions

MSWOA, a novel algorithm-based WOA with a mixed strategy, is proposed in this paper. Compared with the standard WOA, three improvements have been made to enhance the searching ability. Firstly, because the strategies of bubble-net attacking and encircling prey can identify the optimization points in a local region, a hybrid movement strategy is applied on MSWOA, in which a dynamic ratio based on improved JAYA algorithm is applied on the strategy of searching for prey and a chaotic dynamic weight is applied on the strategies of bubble-net attacking and encircling prey. Secondly, a guidance of the elite’s memory inspired by PSO is applied on the movement of whale agents of the population. Thirdly, the mutation strategy based on the sinusoidal chaotic map is employed to avoid prematurity and local optimum points. Subsequently the MSWOA is compared with six meta-heuristic algorithms on 23 benchmark functions and the results show that MSWOA achieves remarkably better performance than the others, and the significance has been confirmed by box and whisker’s tests. Finally, the proposed MSWOA, together with ALO, PSO, WOA and EWOA and CWOA, are applied in parameter identification of a HTGS with a delayed water hammer effect.

The results reveal that the MSWOA demonstrates satisfactory global searching ability and dramatically promotes the identification accuracy of the complicated system studied, compared with other existing algorithms.