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A Mixed-Strategy-Based Whale Optimization Algorithm for Parameter Identification of Hydraulic Turbine Governing Systems with a Delayed Water Hammer Effect Tan Ding, Li Chang, Chaoshun Li *, Chen Feng and Nan Zhang School of Hydropower and Information Engineering, Huazhong University of Science and Technology, Wuhan 430074, China; [email protected] (T.D.); [email protected] (L.C.); [email protected] (C.F.); [email protected] (N.Z.) * Correspondence: [email protected]; Tel.: +86-27-8754-3992 Received: 25 July 2018; Accepted: 4 September 2018; Published: 7 September 2018







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. Keywords: whale optimization algorithm; meta-heuristic; improved JAYA algorithm; chaotic mutation; HTGS; parameter identification

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 [1–7]. 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, Energies 2018, 11, 2367; doi:10.3390/en11092367

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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 [22–31]. 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 [2,4,27–31], 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 [2,27–31], 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

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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.

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

Unit

Meaning

t tmax X Xrand X* Xworst Xgbest A C a b l a1 p p1 r r2 –r9 c2 , c3 , c6 –c12 c1 c4 , c5 q1 σ c x bp y1 y Td Kp Ki Kd Ty1 Ty mt q h hw Tr Ta eg

time time / / / / / / / / / / / / / / / / / / / / / / / / / second / / / second second / / / / second second /

Current iteration The maximum iteration Each agent position of population The agent position selected randomly The best position in the current iteration The worst position in the current iteration The best position obtained so far A variable in [−2,2] A random number in [0,2] A variable in [0,2] A positive constant A random number in [−1,1] Chaotic constant a1 = 2.3 A random probability value in [0,1] A random probability value in [0,1] A random in [0,1] Randoms in [0,1] Coefficients in [0,2] A coefficients in [0,1] A variable in [0,1] A dynamic ratio in [0,0.5] Output of PID Controller Relative value of given speed Relative value of speed Permanent transition coefficient Relative value of position of the auxiliary servomotor Relative value of position of the main servomotor The differential time The proportional gain, The integral gain The differential gain The response time constant of auxiliary servomotor The response time constant of main servomotor Relative value of moment Relative value of water flow Relative value of water head The pipeline characteristic coefficient The reflection time of the water hammer pressure wave The inertial time constant of the generator The adjusting coefficient of the generator →

Attention: to any variable w, 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 |A| is greater → than 1. r is a vector which is composed of a random number in [0,1], and a is evaluated according to Equation (5) and it varies within [0,2]: → −−−→ → D = | C · Xrand − X (t)|

(1)

−−−→ → → X (t + 1) = Xrand − A· D

(2)

A = 2·a·r − a

(3)

→

→

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C = 2·r   tmax − t a = 2· tmax

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2.1.2. Modeling the Course of Encircling Preya = 2 ∙

(4) 5 (5) of 29

(5)

In this phase, each whale of the population updates its position with Equations (6) and (7). A and 2.1.2. Modelingwith the Course of Encircling C are calculated Equations (3) and (4):Prey

In this phase, each whale of the → population → → updates → its position with Equations (6) and (7). A D and = | C(4): ·X∗ (t) − X (t)| (6) and C are calculated with Equations (3) →∗⃗(t) − X → → ⃗(t) D⃗ = C⃗ ∙ X X (t + 1 ) = X∗ (t ) − A · D X⃗(t + 1) = X ∗⃗(t) − A⃗ ∙ D⃗ 2.1.3. Modeling the Course of Bubble-Net Attacking (Getting) Prey →

(6) (7) (7)

In Modeling this phase, each whale of the population updates its position using Equations (8) and (9). 2.1.3. the Course of Bubble-Net Attacking (Getting) Prey The flowchart of WOA is shown in Figure 1. The term b is a constant to which a positive number is In this phase, each whale of the population updates its position using Equations (8) and (9). The assigned in this paper, l is a random number in [−1.1]. 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]. X (t + 1) = D0 ·ebl · cos 2πl + X∗ (t) (8) (8) X⃗(t + 1) = D⃗ ∙ e ∙ cos 2πl + X ∗⃗(t) →

→

→

D0 D=⃗ = |X∗X(t∗⃗)(t) − −X X (⃗t(t) )|

(9) (9)

Figure 1. 1.The ofof WOA. Figure Theflowchart flowchart WOA.

2.2. 2.2.The TheMixed-Strategy Mixed-StrategyBased BasedWOA WOA The which is is composed ofofthree not only Themixed mixedstrategy, strategy, which composed threeimprovements, improvements, not onlyenhances enhancesthe thealgorithm algorithm comprehensively but also balances exploration and exploitation. Compared with the standard comprehensively but also balances exploration and exploitation. Compared with the standardWOA WOA ininFigure MSWOA 2.2.The Figure1,1,how howthe theimprovements improvementsstrategies strategiesenhance enhance MSWOAis isillustrated illustratedininFigure Figure Thedetails details are asas follows: aredescribed described follows:

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Figure Improvement strategies in MSWOA. Figure 2.2.Improvement strategies in MSWOA.

2.2.1. Improvement 1: Hybrid Movement Strategy 2.2.1. Improvement 1: Hybrid Movement Strategy The basic movement strategy of WOA is composed of three strategies, namely the strategy of

The encircling basic movement strategy of WOA is composed of three strategies, namely prey, the strategy of searching for prey and the strategy of bubble-net attacking the prey.the strategy of encircling prey, MSWOA the strategy of searching fordifferent prey and the strategy of bubble-net attacking the To enhance comprehensively, three improvements are applied on the three movement strategies of WOA, respectively: prey. To enhance MSWOA comprehensively, three different improvements are applied on the three (1) strategies The dynamic for strategy of searching for prey. movement of ratio WOA, respectively: (1)

In standard WOA, a random probability value named p was employed to allow WOA to The dynamic ratio for strategy of searching for prey.

effectively transit between exploration and exploitation. In this paper, besides p, another random

probability value anamed p1 and a dynamicvalue ratio named q1 pwhich is defined asto Equation (10), are In standard WOA, random probability named was employed allow WOA to effectively applied on strategy of searching for prey: transit between exploration and exploitation. In this paper, besides p, another random probability q = cq1∙ which 1− value named p1 and a dynamic ratio named is defined as Equation (10), (10) are applied on strategy of searching for prey: where c1 is a coefficient in [0,1]; in this paper c1 = 0.5  and q1 varies  in [0,0.5]. t q1 = c1 · 1 − (10) tmax

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

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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. →    −−−→ −−−→ −−−→ −−−→ −−−→ 0 Xrand = Xrand + r2 ·c2 · X∗ (t) − Xrand − r3 ·c3 · Xworst (t) − Xrand

(11)

−−−→ −−−→ −−−→ 0 Xrand = (1 − c4 )· Xrand + Xgbest (t)

(12)

where c2 , c3 are coefficients in [0,2], p, c4 =

t tmax , r2 , r3

are random numbers in [0,1]. −−−→ 0 In each iteration, q1 is compared with p1 . If q1 < p1 , Xrand is upgraded according to Equation (11), −−−→ 0 otherwise Xrand 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]: ω(t + 1) = 4ω(t)·(1 − ω(t)), t = 1, · · · , tmax

(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), −−−→ ω(t) is the chaotic dynamic weight. The global best position so far Xgbest and the best position in →

current iteration 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: → → → −−−→ → X (t + 1) = ω(t)·c5 · Xgbest − A·(c6 ·r4 ·D1 + c7 ·r5 ·D2 ) (14) → → −−−→ X (t + 1) = c8 · Xgbest + (c9 ·r6 ·D3 + c10 ·r7 ·D4 )·ebl · cos 2πl

→

(15)

→ → → → → → → −−−→ → → → → −−−→ → where D1 = | C ·X∗ (t) − X (t)|, D2 = | C · Xgbest − X (t)|, D3 = |X∗ − X (t)|, D4 = | Xgbest − X (t)|, −t c5 = tmax tmax , c6 to c10 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:     → → −−−→ −−−→ → → vi (k + 1) = ω1 ·vi (k) + c11 ·r8 · Xpbest (k) − Xi (k) + c12 ·r9 · Xgbest (k) − Xi (k) →

→

→

Xi (k + 1) = Xi (k) + vi (k + 1)

(16) (17)

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where ω1 is a constant in [0,1], c11 and c12 are coefficients in [0,2],r8 , r9 are random numbers in [0,1], −−−→ −−−→ Xpbest denotes the global best position so far, Xgbest 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]: xi (t + 1) = a1·x2i (t)· sin(π·xi (t)), t = 1, · · · , tmax ; i = 1, · · · , 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 Energies 2018, 11, x FOR PEER REVIEW

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The pseudo-code of the algorithm is shown as follows (Algorithm 1): Algorithm 1. Pseudo-Code. 1. Initialize the agents population X, and x (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 x to x (1) Evaluate objective function value of each agent of the population and select the worst solution and the best solution and update X ∗ , X , X in the current iteration; (2) Update A, C, a, l, p and calculate the chaotic weight ω in the current iteration; (3) if 1 p < 0.5 (p and p1are random in [0,1], q1 = c1·(1 − t/tmax)) if 2 |A| ≥ 1 if 3 q1 < p1 select X in Equation (11) and update the position of each agent of population as in Equations (1) and (2) else if 3 q1 > p1 pdate X in Equation (12) and update the position of each agent of population as in Equations (1) and (2) end if 3 else if 2 |A| < 1 update the position of each agent of population as in Equation (14) end if 2 else if 1 p > 0.5 update the position of each agent of population as in Equation (15) end if 1 end for For x to x Use Equations (16) and (17) to update the position of each agent of population end for For x to x Use chaotic mutation as in Equation (18) to achieve new positions x to x . if obj(x ) < obj(x ) x = x else x = x 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 Tables 2–4. 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. To prove that MSWOA outperforms other algorithms in Table 5, the

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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 Tables 2–4. 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. 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. Table 2. Unimodal test functions with dimensions = 30. Function

Dimension

Range

fmin

30

[−100,100]

0

30

[−10,10]

0

30

[−100,100]

0

30

[−100,100]

0

30

[−30,30]

0

30

[−100,100]

0

30

[−1.28,1.28]

0

n

F1 (x) = ∑ x2i i=1 n n F2 (x) = ∑ |xi | + ∏ |xi | i=1 i=1! 2 n i F3 (x) = ∑ ∑ xj i=1 j−1 F4 (x) = max{|xi |, 1 ≤ i ≤ n} i i n−1 h
2 F5 (x) = ∑ 100 xi+1 − x2i + (xi − 1)2 i=1

n

F6 (x) = ∑ ([xi + 0.5])2 n

F7 ( x ) = ∑

i=1

i=1 i·x4i

+ random[0, 1]

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 Tables 2–4, 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 Tables 2–4, 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. Table 3. Multimodal test functions with dimensions = 30. Function n

F8 (x) = ∑ −xi sin n

F9 (x) = ∑ si=1 1 n

p





n

| xi | i=1  2  xi − 10 cos(2πxi ) + 10 n

∑ x2i

!

F12 (x) =

π n



i=1 n−1

− exp

1 n

i−1


 10 sin(πy1 ) + ∑ (y1 − 1)2 1 + sin2 πyi+1 + (yn − 1)2 i=1

Range

fmin

30

[−500,500]

−12569

30

[−5.12,5.12]

0

30

[−32,32]

0

30

[−600,600]

0

30

[−50,50]

0

–

–

–

30

[−50,50]

0



∑ cos(2πxi ) + 20 + e i=1 i=1   n n x 1 F11 (x) = 4000 ∑ x2i − ∏ cos √ii + 1

F10 (x) = −20 exp −0.2

Dimension

n

+ ∑ u(xi , 10, 100, 4) i=1 m  xi > a  k ( xi − a ) xi + 1 yi = 1 + 4 , u(xi , a, k, m) = 0 − a < xi < a   k x − a x (− ) i i < −a  n  2 1 2 2 F13 (x) = 10 sin (3πxi ) + ∑ (xi − 1) 1 + sin (3πxi + 1) i=1 n  o + (xn − 1)2 1 + sin2 (2πxn ) + ∑ u(xi , 5, 100, 4) i=1



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Table 4. Multimodal test functions with fixed dimensions. Function 1 500

dimension

Range

fmin

2

[−65,65]

1

4

[−5,5]

0.0003

2

[−5,5]

−1.0316

2

[−5,5]

0.398

2

[−2,2]

3

3

[0,1]

−3.86

6

[0,1]

−3.32

4

[0,10]

−10.1532

4

[0,10]

−10.4028

4

[0,10]

−10.5363

! −1

25

1 + ∑ 6 2 j=1 j+∑i=1 (xi −aij )  2 2 11 x1 (b +bi x2 ) F15 (x) = ∑ ai − 2 i b +b x +x

F14 (x) =

i 3 4 i i=1 F16 (x) = 4x21 − 2.1x41 + 31 x61 + x1 ·x2 − 4x22 + 4x42  2   1 5.1 2 5 cos(x1 ) + 10 + 10 1 − 8π F17 (x) = x2 − 4π 2 x1 + π x1 − 6
i 2 2 F18 (x) = [1+ (x1 + x2 + 1) 19 − 14x1 + 3x1 − 14x2 + 6x1 ·x2 + 3x22 h × 30 + (2x1 − 3x2 )2
 × 18 − 32x1 + 12x21 + 48x2 − 36x!1 ·x2 + 27x22   4 3 2 F19 (x) = − ∑ ci · exp − ∑ aij xj − pij i=1 j=1 !

4

6

i=1 5

j=1



F20 (x) = − ∑ ci · exp − ∑ aij xj − pij h

2

i −1

F21 (x) = − ∑ (x − ai )(x − ai )T + ci i=1 i −1 7 h F22 (x) = − ∑ (x − ai )(x − ai )T + ci i=1 i −1 10 h F23 (x) = − ∑ (x − ai )(x − ai )T + ci i=1

Table 5. Parameter settings for all algorithms in test. Algorithm PSO12 GSA13 ALO16 WOA20 CWOA37 EWOA35 MSWOA

Parameter Settings c1 = 2, c2 = 2, inertia weight ω = 1, inertia weight damping ratio ω2 = 0.99 G0 = 100, α = 20 All parameter settings are recommended in [15] a decrease linearly from 2 to 0 a decrease linearly from 2 to 0, other settings are same to [19] a decrease linearly from 2 to 0, c = 0.3, other settings are same to [19] 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]

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. 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.

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Table 6. Test results of 23 benchmark functions. Function Index F1

F2

F3

F4

F5

F6

F7

F8

F9

F10

F11

F12

F13

F14

F15

F16

F17

F18

F19

F20

F21

F22

F23

Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc Ave Std Anc

PSO

GSA

ALO

WOA

CWOA

EWOA

MSWOA

1.36 × 10−8 2.55 × 10−8 500 0.0403 0.0690 411 96.484 85.763 500 2.4409 1.0333 500 45.171 41.859 500 3.11 × 10−8 4.56 × 10−8 500 0.0261 0.0121 498 −6185.6 635.16 207 46.863 12.539 308 1.6484 0.7689 380 0.0255 0.0280 325 0.0882 0.2436 394 0.0585 0.1165 463 4.3772 4.1508 120 6.21 × 10−4 5.28 × 10−4 500 −1.0316 2.2 × 10−16 42 0.3979 0 45 3 9 × 10−16 52 −3.8241 0.1729 41 −3.2863 0.0559 60 −6.8967 3.4496 70 −7.6093 3.5471 69 −8.2082 3.6499 72

3.29 × 10−16 3.74× 10−16 500 0.0361 0.1127 351 905.77 409.34 456 6.9393 1.5297 394 58.224 38.846 445 6.1 4.5986 100 0.0770 0.0268 231 −2793.1 611.86 11 27.71 7.242 250 1.3 × 10−8 4.01 × 10−8 500 27.322 6.5334 319 2.2046 1.2486 325 7.6479 5.7737 384 5.0336 3.2134 37 0.0032 0.0015 254 −1.0316 1.2 × 10−16 133 0.3979 0 132 3 3.8 × 10−15 193 −3.8628 1.8 × 10−15 254 −3.3220 2.9 × 10−16 198 −7.1798 3.7368 198 −9.4954 2.0522 195 −10.1333 1.8030 233

7.52 × 10−9 3.3× 10−9 500 0.5907 1.0355 487 0.1921 0.4005 499 0.2225 0.9838 497 194.96 436.28 500 1.26 × 10−8 2.03 × 10−8 500 0.0306 0.0162 333 −2267.1 445.86 401 22.436 13.417 391 0.6613 0.8640 488 0.2039 0.0829 457 2.3717 2.5530 478 0.0032 0.0076 390 1.7899 1.3790 67 0.0028 0.0060 242 −1.0316 6.9 × 10−14 132 0.3979 9.4 × 10−14 127 3 1.4 × 10−12 251 −3.8628 1.9 × 10−12 125 −3.2800 0.0587 348 −7.1339 3.2138 269 −7.4842 3.4103 376 −7.6188 3.6877 352

6.13 × 10−74 2.18× 10−73 500 4.06× 10−51 6.45× 10−51 500 45,988 8114 500 40.874 26.297 428 27.964 0.4795 257 0.41 0.2589 274 0.0023 0.0032 480 −10655 1702.5 414 0 0 212 3.91 × 10−15 2.38 × 10−15 299 0.0231 0.0807 240 0.0344 0.0321 255 0.5345 0.3071 402 2.6678 3.0266 211 7.79 × 10−4 5.59 × 10−4 494 −1.0316 1.01 × 10−9 36 0.3979 2.5 × 10−5 274 3.0001 1.06 × 10−4 133 −3.8528 0.0128 256 −3.2540 0.0865 310 −8.3351 2.8253 266 −9.1804 2.4700 254 −8.2293 3.2614 251

0 0 64 0 0 39 33,120 7478.3 500 40.173 17.893 490 26.869 1.1668 447 0.0951 0.1292 456 8.66 × 10−5 7.72 × 10−5 499 −11,487 1102.5 500 0 0 95 8.8818 × 10−16 0 48 0.0062 0.0251 217 0.0119 0.0123 419 0.3736 0.3249 443 1.2956 0.7269 166 6.82 × 10−4 2.33× 10−4 483 −1.0316 3.1 × 10−10 18 0.3979 1.14 × 10−6 90 3 5.3 × 10−13 20 −3.8617 0.0022 248 −3.2428 0.1183 379 −8.6193 2.3941 463 −8.5395 2.5988 444 −8.9107 2.5404 473

2.22 × 10−32 6.01× 10−32 500 7.24× 10−26 8.73× 10−26 500 47,649 13702 500 69.994 14.454 283 18.794 12.561 500 0.0026 0.0014 500 0.0151 0.0120 498 −11,916 1070.2 500 90.101 70.216 500 1.28 × 10−14 4.79 × 10−15 494 0.0120 0.0236 491 2.6187 3.7315 500 2.3549 10.0492 500 1.8336 2.2266 274 5.8 × 10−4 3.69× 10−4 500 −1.0316 3.5 × 10−16 47 0.3979 1.2 × 10−11 127 3 1.5 × 10−11 47 −3.8628 1.32 × 10−5 310 −3.2744 0.0599 482 −9.4009 2.3154 432 −8.8258 2.8416 499 −9.1198 2.9524 341

0 0 14 0 0 11 0.0218 0.0819 500 0 0 16 24.253 0.2296 500 4.11 × 10−12 4.35 × 10−12 500 6.63 × 10−5 7.16 × 10−5 406 −9203.3 1295 499 0 0 9 8.8818 × 10−16 0 10 0.0032 0.0071 402 3.05 × 10−4 2.06 × 10−4 500 0.0502 0.0461 500 1.1964 0.6107 264 3.46 × 10−4 6.44 × 10−5 500 −1.0316 7.7 × 10−16 33 0.3979 4.3 × 10−14 88 3 1.2 × 10−14 27 −3.8617 0.0024 482 −3.2920 0.0533 427 −9.8983 1.1399 279 −10.4029 3.98 × 10−10 217 −10.5364 8.79 × 10−10 262

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

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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, “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. 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 F2 F3 F4 F5 F6 F7 F8 F9 F10 F11 F12 F13 F14 F15 F16 F17 F18 F19 F20 F21 F22 F23

MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA Tie Tie Tie MSWOA MSWOA MSWOA MSWOA MSWOA

MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA Tie Tie Tie GSA MSWOA MSWOA MSWOA MSWOA

MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA ALO MSWOA MSWOA Tie Tie Tie MSWOA MSWOA MSWOA MSWOA MSWOA

MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA Tie Tie MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA

Tie Tie MSWOA MSWOA MSWOA MSWOA MSWOA Tie Tie MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA Tie Tie Tie MSWOA MSWOA MSWOA MSWOA MSWOA

MSWOA MSWOA MSWOA MSWOA EWOA MSWOA MSWOA EWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA MSWOA Tie Tie Tie 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

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 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 Tables 7 and 8, we can conclude that MSWOA has a better performance in a

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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 Energies algorithms 2018, 11, 2367in the same one benchmark function. In Table 6, the results of each meta-heuristic 13 of 29 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. statistical manner than the other six algorithms, especially for WOA, CWOA and EWOA. Based on In Table 8, the test results of multiple-problem statistical analysis by the Wilcoxon signed-rank WOA, test MS-WOA is improved greatly on the can capability of stability, exploitation and exploration are listed. The R+, R− and the p-values be evaluated by the Wilcoxon signed-rank test appliedand avoidance of local optimum pointsanalysis. and prematurity. in multiple-problem statistical 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 Table 8. than The Wilcoxon signed-rank test for statistically significance levelperformance at alfa = 0.05.in better another. From Tablesmultiple-problem 7 and 8, we can conclude that MSWOA has a better a statistical manner than the other six algorithms, especially for WOA, CWOA and EWOA. Based on Comparison R+ R− p-Value Significant WOA, MS-WOA is improved greatly on the capability of stability, exploitation and exploration and − 5 MSWOA vs. PSOpoints 270 6 Yes 3.173 × 10 avoidance of local optimum and prematurity. MSWOA vs. GSA 253 23 Yes 2.484 × 10−4 vs. ALO 16 test for Yes 1.098 × 10−4significance Table 8.MSWOA The Wilcoxon signed-rank260 multiple-problem statistically level at alfa = 0.05. MSWOA vs. WOA 255 21 Yes 3.261 × 10−4 Comparison R+ R− p-Value Significant MSWOA vs. −5 258 18 6 3.173 0.0010 Yes MSWOA vs. PSO 270 × 10 Yes CWOA MSWOA vs. GSA 253 23 2.484 × 10−4 Yes MSWOA vs. 224 Yes 260 52 16 1.098 0.0047 × 10−4 Yes EWOA MSWOA vs. ALO

3.2.2. Box and Whisker

MSWOA vs. WOA MSWOA vs. CWOA MSWOA vs. EWOA

255 258 224

21 18 52

3.261 × 10−4 0.0010 0.0047

Yes Yes Yes

The stability is Whisker a concept used to evaluate an algorithm by checking the randomness of solutions. 3.2.2. Box and In addition to the standard deviation statistical index, the box and whisker plot is also effective in The stability is a concept used to evaluate an algorithm by checking the randomness of solutions. estimation of algorithm stability. The box and whisker methods used in this section to evaluate the In addition to the standard deviation statistical index, the box and whisker plot is also effective in distribution of results of the repeated 20 runs on benchmark functions F1 to F23. A box and whisker estimation of algorithm stability. The box and whisker methods used in this section to evaluate the plot can illustrate the variation in repeated a set of data andon provide morefunctions information It provides distribution of results of the 20 runs benchmark F1 toabout F23. Athe boxdata. and whisker five statistics, second the median value, the the third quartile plot can namely, illustratethe theminimum variation invalue, a set the of data and quartile, provide more information about data. It and theprovides maximum value. The second is value, the value below quartile, which the 25% of the data are five statistics, namely, thequartile minimum the second thelower median value, the third quartile andquartile the maximum second quartile is the value below whichare thecontained. lower 25% of the contained. Third is the value. value The above which the upper 25% of the data 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 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 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 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 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 obtained from MSWOA after 20 times are very small and the stability of MSWOA is superior to that of the of other algorithms. Compared algorithms, WOA, CWOA and EWOA, the other algorithms. Comparedwith withhomogenous homogenous algorithms, likelike WOA, CWOA and EWOA, the the stability of MSWOA has been greatly improved. stability of MSWOA has been greatly improved.

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Figure 3. Cont.

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3. and Box and whisker sevenalgorithms algorithms on Box and whisker test test on F1; FigureFigure 3. Box whisker testtest forfor thethe seven on F1 F1totoF23. F23.(a)(a) Box and whisker on F1; (b) Box and whisker test on F2; (c) Box and whisker test on F3; (d) Box and whisker test on F4; (e) Box (b) Box and whisker test on F2; (c) Box and whisker(w) test on F3; (d) Box and whisker test on F4; (e) Box and whisker testF5; on F5; (f) Box and whiskertest teston on F6; F6; (g) (g) Box test onF7; (h) (h) BoxBox and and and whisker test on (f) Box and whisker Boxand andwhisker whisker test onF7; Figure 3.test Box and (i) whisker testwhisker for the test seven algorithms on F1 to F23.test (a) onF10; Box and on F1; whisker onF8; Box and onF9; (j) Box and whisker (k)whisker Box andtest whisker whisker test onF8; (i) Boxtest andonwhisker test onF9; (j) Box and whisker test onF10; (k) BoxF4; and whisker (b) Box and(l)whisker F2; test (c) Box and(m) whisker test whisker on F3; (d)test Box and whisker (e) test Box test onF11; Box and whisker onF12; Box and onF13; (n) Box test andon whisker test onF11; (l) Box and whisker test onF12; (m) Box and whisker test onF13; (n) Box and whisker test and whisker test on F5; (f) Box and whisker test on F6; (g) Box and whisker test onF7; (h) Box and onF14; (o) Box and whisker test onF15; (p) Box and whisker test onF16; (q) Box and whisker test onF17; onF14;(r) (o) Box and whisker test onF15; (p) Box and whisker test onF16; (q) Box and whisker test onF17; whisker testwhisker onF8; (i)test BoxonF18; and whisker (j) Box whisker testand onF10; (k) Box whisker Box and (s) Boxtest andonF9; whisker testand onF19; (t) Box whisker testand onF20; (u) (r) BoxBox and whisker test onF18; (s) and whisker test onF19; Boxand and whisker test onF20; (u) test onF11; (l) Box and whisker test and onF12; (m) Box and whisker test onF13; (n) Box whisker test Box and whisker test onF21; (v) Box Box whisker test onF22; (w)(t)Box whisker testand onF23. onF14; (o) and whisker onF15; (p) Box whisker onF16; (q) Box and whisker and whisker testBox onF21; (v) Boxtest and whisker testand onF22; (w)test Box and whisker test onF23.test onF17; (r) Box and Comparison whisker test onF18; (s) Box and whisker test onF19; (t) Box and whisker test onF20; (u) 3.3. Performance

3.3. Performance Comparison Box and whisker test onF21; (v) Box and whisker test onF22; (w) Box and whisker test onF23.

According to the test results in Tables 6 to 8, a conclusion is drawn that the MSWOA algorithm

According thepaper test results Tables 6–8, a conclusion drawn thatcompared the MSWOA proposed intothe has beeninimproved significantly on itsisperformance to thealgorithm other 3.3. Performance Comparison proposed in the paper has been improved significantly on its performance compared to based the other algorithms in Table 5. In this subsection, the performance of MSWOA will be analyzed further the testsubsection, results in Tables 6 to 8, a conclusion is drawnwill thatbe theanalyzed MSWOAfurther algorithm on theAccording results 23In benchmark functions. algorithms in Tableof5.to this the performance of MSWOA based proposed in the paper has been improved significantly on its performance compared to the other on the results of 23 benchmark functions.

algorithms in Table 5.Results In this subsection, 3.3.1. Analysis of Test of F1–F7 the performance of MSWOA will be analyzed further based on the results of 23 benchmark functions. 3.3.1. Analysis Results F1–F7 functions, unimodal test functions (F1–F7) are applied to test Basedof onTest character of of benchmark the capacity of of exploitation of of algorithms while the multimodal test function and 3.3.1. Analysis Test Results F1–F7 Based on character of benchmark functions, unimodal test functions (F1–F7) aremultimodal applied totest test the functions with fixed dimension (F8–F23) are employed to test the capacity of exploration of capacity ofBased exploitation of algorithms while the multimodal test function and multimodal test functions of benchmark functions, unimodal test (F1–F7) arethan applied test algorithms. on In character Table 6, its manifest that the MSWOA gives a functions superior test result the to other with fixed dimension (F8–F23) areofemployed towhile test the capacity of exploration ofand algorithms. In test Table 6, the capacity of exploitation algorithms the multimodal test function multimodal algorithms in all unimodal test functions except F5, while EWOA gives a better test result than the its manifest that the MSWOA gives a superior test result than the other algorithms in all unimodal functions with fixed dimension (F8–F23) are employed to test the capacity of exploration of test others on F5, MSWOA outperforms other algorithms on both the final test results and convergence algorithms. In Table 6, its manifest that the MSWOA gives a superior test result than the other functions except F5, while EWOA gives a better test result than the others on F5, MSWOA outperforms rate. In Figure 4a–w, the convergence curves of PSO, GSA, ALO, WOA, CWOA, EWOA and MSWOA algorithms in unimodal testtest functions except F5, EWOA a better test result than the other are algorithms onall both the results and rate.gives In Figure the convergence the average curves withfinal mean values tested onconvergence thewhile 23 benchmark functions 204a–w, times, respectively. others on F5, MSWOA outperforms other algorithms on both the final test results and convergence Figure is clearWOA, that the MSWOA has theand fastest convergence rateaverage to searchcurves for thewith global curvesInof PSO,4a–g, GSA,it ALO, CWOA, EWOA MSWOA are the mean rate. In Figurepoint. 4a–w,Inthe convergence curves of PSO,test, GSA, WOA, CWOA, EWOA and MSWOA the unimodal test functions theALO, convergence rates of the algorithms are the valuesoptimization tested on the 23 benchmark functions 20 times, respectively. In Figure 4a–g, it is clear that are the average curves with mean tested on the 23 functionsoutperforms 20 times, respectively. more vital than the final results. Asvalues for Figure 3a–g, wethe canbenchmark find that MSWOA the others MSWOA has the fastest convergence rate to search for global optimization point. In the unimodal In Figure 4a–g, it is clear that the MSWOA has the fastest convergence rate to search for theWOA, global in the box and whisker test results. That means MSWOA is more stable than PSO, GSA, ALO, test functions test, the convergence rates of the algorithms are more vital than the final results. As for optimization point. In the unimodal test functions test, the convergence rates of the algorithms are CWOA and EWOA for unimodal test functions (F1–F7). Figuremore 3a–g, wethan canthe find that MSWOA the in MSWOA the box and whiskerthe test results. vital final results. As for outperforms Figure 3a–g, we canothers find that outperforms others That means MSWOA is more GSA, ALO, CWOA for unimodal in the box and whisker teststable results.than ThatPSO, means MSWOA is WOA, more stable thanand PSO,EWOA GSA, ALO, WOA, test functions (F1–F7). CWOA and EWOA for unimodal test functions (F1–F7).

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Figure 4. Cont.

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4. Comparison of average iterationprocess process of of seven seven different for for 23 benchmark FigureFigure 4. Comparison of average iteration differentalgorithms algorithms 23 benchmark functions. (a) Curves of the fitness values for F1; (b) Curves of the fitness values for F2; (c) of of functions. (a) Curves of the fitness values for F1; (b) Curves of the fitness values for F2; Curves (c) Curves the fitness values for F3; (d) Curves of the fitness values for F4; (e) Curves of the fitness values for F5; the fitness values for F3; (d) Curves of the fitness values for F4; (e) Curves of the fitness values for F5; (f) Curves of the fitness values for F6; (g) Curves of the fitness values for F7; (h) Curves of the fitness (f) Curves of the fitness values for F6; (g) Curves of the fitness values for F7; (h) Curves of the fitness values for F8; (i) Curves of the fitness values for F9; (j) Curves of the fitness values for F10; (k) Curves values for F8; (i) Curves of the fitness values for F9; (j) Curves of the fitness values for F10; (k) Curves of the fitness values for F11; (l) Curves of the fitness values for F12; (m) Curves of the fitness values of the fitness values for F11; (l) Curves of the fitness values for F12; (m) Curves of the fitness values for for F13; (n) Curves of the fitness values for F14; (o) Curves of the fitness values for F15; (p) Curves of F13; (n) of the for fitness for of F14; Curves of the fitness valuesof forthe F15; (p) values Curvesfor of the theCurves fitness values F16; values (q) Curves the(o) fitness values for F17; (r) Curves fitness fitnessF18; values for F16; (q) fitness Curves of the F17;fitness (r) Curves fitness values (s) Curves of the values forfitness F19; (t)values Curvesfor of the valuesof forthe F20; (u) Curves of for the F18; (s) Curves the fitness values for F19; (t)fitness Curves of the values forofF20; (u) Curves the fitness fitnessofvalues for F21; (v) Curves of the values forfitness F22; (w) Curves the fitness valuesoffor F23. values for F21; (v) Curves of the fitness values for F22; (w) Curves of the fitness values for F23. 3.3.2. Analysis of Test Results of F8–F13

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

The ability which an algorithm escapefunctions from local optimum and locate global optimization point, makes can be tested by multimodal (F8–F13). Whenpoints an algorithm is runa on multimodal functions (F8–F13), results canfunctions reflect the (F8–F13). ability better than the optimization point, can be tested the by test multimodal When an convergence algorithm israte. run on In functions Table 6, it is clear thatthe MSWOA outperforms the other algorithms inthan F9, F10, and F12 in rate. multimodal (F8–F13), test results can reflect the ability better theF11 convergence theTable test results. MSWOA has the second best performance F13. Although better test In 6, it is clear that MSWOA outperforms the otherinalgorithms in F9,GSA F10,has F11a and F12 in the result than others in F13, MSWOA outperforms WOA, CWOA and EWOA on both the final test test results. MSWOA has the second best performance in F13. Although GSA has a better test result results and convergence rate. In Figure 3h–m, it is found that MSWOA outperforms the others in the than others in F13, MSWOA outperforms WOA, CWOA and EWOA on both the final test results and box and whisker test results, which means the stability of MSWOA is superior to those of PSO, GSA, convergence rate.CWOA In Figure 3h–m, for it ismultimodal found thatfunctions MSWOA outperforms the others in the box and ALO, WOA, and EWOA (F8–F13). whisker test results, which means the stability of MSWOA is superior to those of PSO, GSA, ALO, WOA,3.3.3. CWOA and of EWOA for multimodal Analysis Test Results of F14–F23 functions (F8–F13). 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 an F14algorithm to F23 apart from aF20 in the test results. achieves the second best The ability of to avoid small quantity of localMSWOA optima can be tested by multimodal performance in F20. Though ALO has a better test result than the others in F20, MSWOA outperforms functions with fixed dimensions (F14–F23). In Table 6, it is clear that MSWOA outperforms other WOA, CWOA andto EWOA on both the F20 final in test results and convergence rate. In Figure 3n–w, it is best the algorithms in F14 F23 apart from the test results. MSWOA achieves the second found that MSWOA outperforms others in test results in F14 to F23 apart from F20. That means the performance in F20. Though ALO has a better test result than the others in F20, MSWOA outperforms stability of MSWOA is superior to those of PSO, GSA, ALO, WOA, CWOA and EWOA for WOA, CWOA and EWOA on both the final test results and convergence rate. In Figure 3n–w, it is multimodal functions (F14–F23) apart from F20. In Figure 3t, the stability of MSWOA is only inferior

3.3.3. Analysis of Test Results of F14–F23

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

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to that 2018, of GSA which the Energies 11, x FOR PEERgains REVIEW Energies 2018, 11, 2367

best value while the stability of MSWOA is superior to others 20 of 29 20 of 29 including PSO, ALO, WOA, CWOA and EWOA. to that of GSA which gains the best value while the stability of MSWOA is superior to others 4.including Parameter Identification ofCWOA the Hydraulic Turbine Governing System to others including PSO, ALO, WOA, and GSA whichPSO, gains the best value while theEWOA. stability of MSWOA is superior ALO, WOA, CWOA and EWOA. In this section, the MSWOA is applied to solve the parameter identification problem of a 4. Parameter Identification of the Hydraulic Turbine Governing System complicated with a delayed water hammer effect. The mathematical model of the HTGS is 4. Parameter HTGS Identification of the Hydraulic Turbine Governing System In this and section, applied to solve the parameter of a established, then the the MSWOA parameterisidentification methodology as wellidentification as simulationproblem experiments In this section, the MSWOA is applied to solve the parameter identification problem of ais complicated HTGS with a delayed water hammer effect. The mathematical model of the HTGS are conducted. complicated HTGS with a delayed water hammer effect. The mathematical model of the HTGS established, and then the parameter identification methodology as well as simulation experiments is established, and thenTurbine the parameter identification well asEffect simulation experiments 4.1. Model of Hydraulic Governing System with methodology a Delay Water as Hammer are conducted. are conducted. A HTGS is composed of speed governor, penstock, hydraulic turbine and generator [1,4,25,26], 4.1. Model of Hydraulic Turbine Governing System with a Delay Water Hammer Effect as shown 5. In FigureGoverning 5, X denotes the with speed of theWater generator, Y denotes 4.1. Model in of Figure Hydraulic Turbine System a Delay Hammer Effect the position of the A HTGS is composed of speed governor, penstock, hydraulic turbine and generator [1,4,25,26], main servomotor, H denotes the water head, Q denotes the flow and M denotes the moment. In this A HTGS is composed of speed governor, penstock, hydraulic turbine and generator [1,4,25,26], as shown Figure 5. In Figure 5, Xisdenotes the speed of the generator, Y denotes the from position the paper, the in penstock system model a hyperbolic tangent function which is different the of rigid as shown in Figure H 5. denotes In Figure 5, water X denotes the of the the flow generator, denotesthe themoment. position of the main servomotor, head, Q speed denotes MYdenotes In can this water hammer equation and the elastic water hammer equation. Theand hyperbolic tangent function main servomotor, H denotes the water Q denotes flow andwhich M denotes moment. Inrigid this paper, the system a head, hyperbolic tangent is different from the describe thepenstock dynamic processmodel of theispenstock system in the thefunction most precise way. Inthe Figure 6, each part 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 of the HTGS model is described. water hammer equation and elastic hammer equation. Theprecise hyperbolic function describe the dynamic process of the water penstock system in the most way. tangent In Figure 6, each can part describe the dynamic process of the penstock system in the most precise way. In Figure 6, each part of of the HTGS model is described. the HTGS model is described.

Figure 5. Diagram of the structure of HTGS. Figure 5. Diagram of the structure of HTGS. Figure 5. Diagram of the structure of HTGS.

Figure 6. Block diagram of HTGS.

4.1.1. PID Controller Model 4.1.1. PID Controller Model Figure 6. Block diagram of HTGS. The PID controller is one part of the hydraulic turbine governor. The PID controller model is The PID controller is one part of the hydraulic turbine governor. The PID controller model is 4.1.1. PIDas Controller described follows: Model described as follows:  part ofthe hydraulic turbine governor. The PID controller model is The PID controller is one
K s Ki c(s)−−x(s) x(s)−−bbp∙·yy1(s) + K ds ∙ ·x(s) x(s) (19) (s) == K (s) + Kp + + S ∙ · c(s) (19) described as follows:σσ(s) T T dss++11 K s ∙ c(s) − − b controller; ∙ y (s) + c(s) is the ∙ x(s) σ(s)transform = K + of output where σ(s) is the Laplace ofx(s) the PID Laplace transform(19) of a T s+1 given speed; x(s) is the Laplace transform of the speed; bp is a permanent transition coefficient; y1 (s)

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is the position of the auxiliary servomotor; Td is the differential time; Kp is the proportional gain, Ki is the integral gain and Kd 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:  y1 ( s )  = Ty11s+1  σ(s)    y(s) (20) = Ty s1+1 y1 ( s )      s.t. − y0 ≤ y ≤ 1 − y0 where y(s) is the position of the main servomotor; Ty1 is the response time constant of the auxiliary servomotor; Ty is the response time constant of the main servomotor, y0 is the initial value of y(s). 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: (

mt (s) = ex · x (s) + ey ·y(s) + eh ·h(s) q(s) = eqx · x (s) + eqy ·y(s) + eqh ·h(s)

(21)

where mt (s) is the Laplace transform of the moment of the hydraulic turbine, q(s) is the Laplace transform of the water flow of the hydraulic turbine, h(s) is the water head of the hydraulic turbine, ex , ey , eh , eqx , eqy , eqh 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: h(s) 1 − e − Tr s = −2hw q(s) 1 + e − Tr s

(22)

where hw is the pipeline characteristic coefficient, Tr is the reflection time of the water hammer pressure wave.

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4.1.5. Generator Generator System System Model Model 4.1.5. The dynamic dynamicequation equationofof synchronous generator is described as Equation in which The thethe synchronous generator is described as Equation (23) in(23) which mg ( s ) (s) means the load disturbance. When the HTGS is working under load conditions, m ≠ 0. 0 m means the load disturbance. When the HTGS is working under load conditions, mg (s) 6= 0 and (s) eg 6= and e HTGS ≠ 0. When HTGSunder is working under no-loadmconditions, m (s) = 0 and e = 0. When is working no-load conditions, g (s) = 0 and eg = 0. x(s) 1 = (23) x(s) 1 m (s) − m (s)= T s + e (23) mt ( s ) − mg ( s ) Ta s + en where T is the inertial time constant of the generator, e is the adjustment coefficient of the where Ta isand the einertial = e time − e . constant of the generator, eg is the adjustment coefficient of the generator generator and en = eg − ex . 4.2. Identification Strategy Based on MSWOA 4.2. Identification Strategy Based on MSWOA The proposed MSWOA is employed to identify the HTGS parameters. In the MSWOA The proposed MSWOA is employed to employed identify thetoHTGS parameters. In the MSWOA identification strategy, firstly, an input is activate the real system and the identification system to be strategy, firstly, an input is employed to activate the real system and the system to be identified. identified. The outputs of the real system and the system to be identified are input into Equation (24) The outputs the of the real of system and to evaluate value C θ . the system to be identified are input into Equation (24) to evaluate the value of COF (θˆ ). N M
2 CCOF (θˆ ) = ∑ ∑ zzj ((k) k) −−zˆzj ((k) k) (24) (24) i=1 j=1

i h i where θθisissix-dimensional six-dimensionalvector vectorand andθθ== T Ty1 T Tyh where = is x the y output mt hTw TTr e ,Taz =egx y, z m h the outputiof the identified system. N is the total number of of the real system, z = x y m is is the output of the real system, zˆ = xˆ yˆ mˆ t is the output of the identified system. N is the total samples, M is the dimension of the outputs. After that, in the MSWOA-based optimizer, the number of samples, M is the dimension of the outputs. After that, in the MSWOA-based optimizer, parameters to be identified are identified by minimizing C θ . As the optimization loop in the the parameters to be identified are identified by minimizing COF (θˆ ). As the optimization loop in the MSWOA-based optimizer goes on, the parameters to be identified accurately match the real values. 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 θ is a parameter vector of which each The process of identification is illustrated in Figure 7, where θ is a parameter vector of which each element is a real value while θ is a parameter vector of which each element is to be identified. element is a real value while θˆ is a parameter vector of which each element is to be identified. h

Figure 7. Diagram of parameter identification of HTGS based MSWOA. Figure 7. Diagram of parameter identification of HTGS based MSWOA.

PE (parameter error) and APE (the average parameter error) which are described by Equations PE (parameter error) and APE (the parameter error) which are described by Equations (25) and (26) are employed to evaluate theaverage accuracy of the parameters identified by MSWOA: (25) and (26) are employed to evaluate the accuracy of the parameters identified by MSWOA: θk − θˆk PE = 𝜃 − 𝜃 (25) (25) 𝑃𝐸 = θk 𝜃 θ𝜃ˆk 11 m 𝜃θk − − APE (26) 𝐴𝑃𝐸== m ∑ (26) θk 𝑚 k =1 𝜃 ˆ where theparameters parametersofofthe thereal realsystem, system, θ𝜃k isisthe theparameters parametersofofthe theidentified identifiedsystem, system,m m is is the the where θ𝜃k isisthe total number of parameters of θ while k is the k-th parameter of θ. total number of parameters of 𝜃 while k is the k-th parameter of 𝜃.

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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. Ty1 , Ty ,Tr , hw , Ta , eg 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. Kp = 5.59, Ki = 1.06, Kd = 3.3, Td = 0.28, bp = 0.04, the initial h i value of the parameter vector θ is θ = 0.1 0.3 1.5 0.5 12 0.5 , the transfer coefficients of HTGS are selected referring to [2]. The transfer coefficients are listed in Table 9. Table 9. Transfer coefficients of HTGS on two working conditions. Working Condition No-load Load

Transfer Coefficients of HTGS ex

ey

eh

eqx

eqy

eqh

−1.0567 −1.4673

0.9080 0.7713

1.4191 1.7179

−0.0574 −0.4901

0.7887 0.8184

0.4571 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, eg = 0. Therefore θ has five elements, namely Ty1 , Ty , hw , Tr and Ta , 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 Tables 10 and 11, 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 hw 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. Table 10. Mean identified parameters under no-load conditions. Mean Value of Identified Parameters after 20 Repetitions θk

Ty1 Ty hw Tr Ta

Real Value

0.1 0.3 1.5 0.5 12

PSO

ALO

PE

θˆ k

0.1638 0.0508 0.0061 0.1371 0.0040

0.1164 0.2848 1.4908 0.5686 12.048

WOA

PE

θˆ k

0.182 0.0655 0.0035 0.0256 0.0018

0.1182 0.2803 1.4948 0.5128 12.022

PE

θˆ k

0.188 0.021 0.1332 0.1331 0.0022

0.1188 0.3064 1.3002 0.5666 12.026

EWOA PE

θˆ k

0.153 0.082 0.0745 0.099 0.0073

0.1153 0.2754 1.3882 0.5495 12.088

CWOA

MSWOA

PE

θˆ k

PE

θˆ k

0.1608 0.058 0.0379 0.0078 0.0018

0.1161 0.2826 1.5568 0.4961 12.022

0.1094 0.012 0.008 0.0068 0.0011

0.1109 0.2964 1.512 0.4966 12.014

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Table 11. Mean best cost and APE under no-load condition. Table 11. Mean best cost and APE under no-load condition. Index Table 11. Mean best cost and APE under condition. ALO WOA EWOAno-load CWOA MSWOA Index PSO PSO ALO WOA EWOA CWOA MSWOA Best cost 0.2831 0.0858 0.6770 0.2860 0.0958 0.0126 0.0858 0.6770 0.2860 Index Best cost PSO 0.2831ALO WOA EWOA 0.0958 CWOA0.0126 MSWOA Mean APE 0.0724 0.0557 0.0955 0.0832 0.0533 0.0275 Mean APE 0.0724 0.0557 0.0955 0.0832 0.0533 0.0275 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

In Figure 8, the ability of exploration and exploitation of WOA and EWOA are worse than those 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 of the others. PSO, which is easily trapped in local optimum points, achieves a better ability of In Figure 8, exploitation the ability ofthan exploration exploitation of WOA andCWOA EWOAand are ALO worsehave thanathose of exploration and those ofand WOA and EWOA. Though similar exploration and exploitation than those of WOA and EWOA. Though CWOA and ALO have a similar the others. PSO, which easily trapped local optimum achieves a better ability of exploration iteration process, ALOisachieves a betterinexploration andpoints, exploitation ability than CWOA. MSWOA iteration process, ALO achieves a better exploration and exploitation ability than CWOA. MSWOA and exploitation those of WOAfrom and EWOA. have a similarMSWOA iteration achieves a better than ability of escaping of local Though optimumCWOA points and thanALO others. Therefore, achieves a better ability of escaping from of local optimum points than others. Therefore, MSWOA process, ALOability achieves a better exploration and exploitation MSWOA achieves has a better of exploration, exploitation and escapingability from than localCWOA. optimum points than othera has a better ability of exploration, exploitation and escaping from local optimum points than other better ability of escaping from of local optimum points than others. Therefore, MSWOA has a better algorithms. algorithms. ability of exploration, exploitation and escaping from local optimum points than other algorithms.

Figure 8. Comparison of average iteration process under no-load condition. Figure8.8.Comparison Comparisonofofaverage averageiteration iterationprocess processunder under no-load condition. Figure no-load condition.

The outputs of HTGS with real values are compared with those of HTGS with the identified The Theoutputs outputsofofHTGS HTGSwith withreal realvalues valuesare arecompared comparedwith withthose thoseofofHTGS HTGSwith withthe theidentified identified parameters in Figure 9. The compared outputs are the turbine speed, guide vane opening and turbine parameters in 9.9. The parameters inFigure Figure Thecompared comparedoutputs outputsare arethe theturbine turbinespeed, speed,guide guidevane vaneopening openingand andturbine turbine torque, respectively. The curves of the identified system are approximately the same as those of the torque, respectively. The curves of the identified system are approximately the same as those torque, respectively. The curves of the identified system are approximately the same as thoseofofthe the real system. This verifies that MSWOA is effective for HTGS parameter identification with a delayed real realsystem. system.This Thisverifies verifiesthat thatMSWOA MSWOAisiseffective effectivefor forHTGS HTGSparameter parameteridentification identificationwith witha adelayed delayed water hammer effect effect under under no-load no-load working working conditions. conditions. water waterhammer hammer effect under no-load working conditions.

(a) (a) Figure 9. Cont.

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(b)

(c) Figure Figure9.9. Comparison Comparisonof ofsystem systemoutputs outputsusing usingMSWOA MSWOAunder underno-load no-loadconditions; conditions;(a) (a)Comparison Comparison curves turbine speed; speed;(b) (b)Comparison Comparison curves of guide vane; (c) Comparison of turbine curves of of turbine curves of guide vane; (c) Comparison curvescurves of turbine torque. torque.

4.3.2. Comparison of Different Identification Methods under Load Conditions

4.3.2. Under Comparison of Differentonly Identification Methods under Load load condition, a load disturbance is added to Conditions HTGS. Therefore eg 6= 0 and the parameter vector θ have six elements, namely T , T , h , T , T and eg , that need to be identified. y is w r ato HTGS. y1 Under load condition, only a load disturbance added Therefore e ≠ 0 and the In Table 12, it is obvious that the four parameters identified by MSWOA, namely h , Tr , Ta and w parameter vector θ have six elements, namely T , T , h , T , T and e , that need to be identified. eg , more accurately match the real values than the other algorithms while T and T are not y namely y1 h , T , the T most and In Table 12, it is obvious that the four parameters identified by MSWOA, accurate match with the real values, but T and T are more accurately matched to the real values y y1 e , more accurately match the real values than the other algorithms while T and T are not the most than those of WOA Furthermore, theTmean cost and meanmatched APE achieved accurate match withand the EWOA. real values, but T and arebest more accurately to the by realMSWOA values are both better than those of the other algorithms in Table 13. 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.

Table 12. Mean identified parameters under load conditions.

Table 12. Mean identified under load Mean Value of parameters Identified Parameters after conditions. 20 Repetitions Real PSO ALO WOA EWOA CWOA MSWOA Value Mean Value of Identified Parameters after 20 Repetitions Real ˆ ˆ ˆ ˆ ˆ PE PE PE PE PE PE θ θˆ k θ θ θ θ k k k k PSO ALO WOA k EWOA CWOA MSWOA Value PE PE 0.1147 𝜽𝒌 𝜽𝒌 0.0320PE0.0968 𝜽𝒌 0.1432PE 0.1143 𝜽𝒌 Ty1 0.1 PE0.146 𝜽0.0854 0.1334 𝜽𝒌 0.1133 PE 0.0434 0.1043 0.1474 𝒌 Ty 0.1 0.3 0.146 0.03980.0854 0.3120 0.1334 0.0552 0.1133 0.2835 0.1335 0.1043 0.2600 0.1474 0.1710 0.2487 0.2910 0.0434 0.11470.0300 0.0320 0.09680.0675 0.1432 0.2798 0.1143 hw 0.3 1.5 0.0398 0.14890.3120 1.2766 0.0552 0.0249 0.2835 1.4626 0.0805 0.2600 1.3792 0.1710 0.0565 1.4152 1.3016 0.1335 0.24870.1323 0.0300 0.29100.0237 0.0675 1.4644 0.2798 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 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 − 4 4 Ta 12 0.0077 11.908 5 × 10 12.006 0.0145 12.174 0.0137 12.164 0.0027 12.032 1.6 × 10 12.002 0.1992 0.57830.0090 0.2222 0.61110.0028 0.0451 0.4986 0.5225 eg 0.5 0.5 0.2123 0.00960.6061 0.4952 0.1147 0.0072 0.5574 0.4964 0.0888 0.5996 0.5444 0.1566 0.0136 0.4932 0.5045 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 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 θk

θk Ty1 Ty hw Tr Ta eg

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

PSO 0.0030 0.0941

ALO 8.07 × 10−4 0.0560

WOA 0.0108 0.0933

EWOA 0.0067 0.0931

CWOA 0.0019 0.0714

MSWOA 2.77 × 10−4 0.0471

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

PSO

ALO 10−4

Best cost 0.0030 8.07 × Mean APE 0.0941 0.0560 Energies 2018, 11, x FOR PEER REVIEW Energies 2018, 11, x FOR PEER REVIEW

WOA

EWOA

CWOA

MSWOA

0.0108 0.0933

0.0067 0.0931

0.0019 0.0714

2.77 × 10−4 0.0471

26 26 of of 29 29

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

Figure Comparison Figure 10. 10. Comparison Comparison of of average average iteration iteration process process under under load load conditions. conditions.

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

(a) (a) Figure 11. Cont.

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(b)

(c) Figure 11. Comparison Comparison of of system system outputs outputs using using MSWOA MSWOA under under load conditions; (a) Comparison curves of of turbine turbinespeed; speed;(b)(b) Comparison curves of guide (c) Comparison of turbine Comparison curves of guide vane;vane; (c) Comparison curvescurves of turbine torque. torque.

5. Conclusions

5. Conclusions MSWOA, a novel algorithm-based WOA with a mixed strategy, is proposed in this paper. Compared witha the standard WOA, threeWOA improvements have strategy, been made enhancein thethis searching MSWOA, novel algorithm-based with a mixed is to proposed paper. ability. Firstly, because the strategies of bubble-net attacking and encircling prey can identify the Compared with the standard WOA, three improvements have been made to enhance the searching optimization in athe local region, aofhybrid movement strategy applied on MSWOA, in which ability. Firstly,points because strategies bubble-net attacking and is encircling prey can identify the a dynamic ratio based on improved JAYA algorithm is applied on the strategy of searching for prey optimization points in a local region, a hybrid movement strategy is applied on MSWOA, in which a and a chaotic is applied on the strategies ofon bubble-net attacking and encircling dynamic ratio dynamic based on weight improved JAYA algorithm is applied the strategy of searching for prey prey.a chaotic Secondly, a guidance elite’sonmemory inspired by PSO is attacking applied on the movement of and dynamic weightofisthe applied the strategies of bubble-net and encircling prey. whale agents of the population. Thirdly, the mutation strategy based on the sinusoidal chaotic map is Secondly, a guidance of the elite’s memory inspired by PSO is applied on the movement of whale employed to avoid prematurity andthe local optimum points.based Subsequently the MSWOA is compared agents of the population. Thirdly, mutation strategy on the sinusoidal chaotic map is with six meta-heuristic algorithms on 23 benchmark functions and the results show that MSWOA employed to avoid prematurity and local optimum points. Subsequently the MSWOA is compared achieves remarkably better performance the others, and theand significance hasshow beenthat confirmed by with six meta-heuristic algorithms on 23than benchmark functions the results MSWOA box and whisker’s tests. Finally, the proposed MSWOA, with ALO, PSO, WOA and EWOA achieves remarkably better performance than the others, together and the significance has been confirmed by and CWOA, are applied in parameter identification of a HTGS with a delayed water hammer effect. box and whisker’s tests. Finally, the proposed MSWOA, together with ALO, PSO, WOA and EWOA The results reveal that the MSWOA demonstrates satisfactory global searching ability and and CWOA, are applied in parameter identification of a HTGS with a delayed water hammer effect. dramatically promotes of thesatisfactory complicatedglobal system studied,ability compared The results reveal the thatidentification the MSWOAaccuracy demonstrates searching and with other existing algorithms. dramatically promotes the identification accuracy of the complicated system studied, compared with other existing algorithms.

Author Contributions: T.D. and C.L. conceived and designed the experiments; L.C., C.F., N.Z. conceived the study; T.D. proposed the algorithm and wrote the paper; T.D., C.L. and L.C. played important roles in the process Author Contributions: of revising the paper. T.D. and C.L. conceived and designed the experiments; L.C., C.F., N.Z. conceived the study; T.D. proposed the algorithm and wrote the paper; T.D., C.L. and L.C. played important roles in the process Funding: National Key Research and Development Program of China: 2016YFC0401905 the National Natural of revising the paper. of China: 51679095 and 51479076; the Fundamental Research Funds for the Central Science Foundation Universities: HUST:Key 2015MS120. Funding: National Research and Development Program of China: 2016YFC0401905 the National Natural Science Foundation of China: 51679095 and 51479076; the Fundamental Research Funds for the Central Universities: HUST: 2015MS120.

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Acknowledgments: This paper is supported by the National Key Research and Development Program of China (2016YFC0401905), the National Natural Science Foundation of China (No. 51679095, 51479076) and the Fundamental Research Funds for the Central Universities (HUST: 2015MS120). Conflicts of Interest: The authors declare no conflict of interest.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.

Li, C.S.; Zhou, J.Z. Parameters identification of hydraulic turbine governing system using improved gravitational search algorithm. Energy Convers. Manag. 2011, 52, 374–381. [CrossRef] Chen, Z.; Yuan, X.; Tian, H.; Ji, B. Improved gravitational search algorithm for parameter identification of water turbine regulation system. Energy Convers. Manag. 2014, 78, 306–315. [CrossRef] Zhang, H.; Chen, D.; Xu, B.; Wang, F. Nonlinear modeling and dynamic analysis of hydro-turbine governing system in the process of load rejection transient. Energy Convers. Manag. 2015, 90, 128–137. [CrossRef] Li, C.; Zhou, J.; Xiao, J.; Xiao, H. Hydraulic turbine governing system identification using T–S fuzzy model optimized by chaotic gravitational search algorithm. Eng. Appl. Artif. Intell. 2013, 26, 2073–2082. [CrossRef] Fang, H.; Chen, L.; Shen, Z. Application of an improved PSO algorithm to optimal tuning of PID gains for water turbine governor. Energy Convers. Manag. 2011, 52, 1763–1770. [CrossRef] Zheng, Y.; Zhou, J.; Zhu, W.; Zhang, C.; Li, C.; Fu, W. Design of a multi-mode intelligent model predictive control strategy for hydroelectric generating unit. Neurocomputing 2016, 207, 287–299. [CrossRef] Wang, Z.; Li, C.; Lai, X.; Zhang, N.; Xu, Y.; Hou, J. An Integrated Start-Up Method for Pumped Storage Units Based on a Novel Artificial Sheep Algorithm. Energies 2018, 11, 151. [CrossRef] Ma, P.; Ding, F.; Alsaedi, A.; Hayat, T. Recursive least squares identification methods for multivariate pseudo-linear systems using the data filtering. Multidimens. Syst. Signal Proc. 2018, 29, 1135–1152. [CrossRef] Li, J.; Ding, F.; Yang, G. Maximum likelihood least squares identification method for input nonlinear finite impulse response moving average systems. Math. Comput. Model. 2012, 55, 442–450. [CrossRef] Stoica, P.; Li, J. On nonexistence of the maximum likelihood estimate in blind multichannel identification. IEEE Signal Proc. Mag. 2005, 22, 99–101. [CrossRef] Fontan, M.; Breysse, D.; Bos, F. A hierarchy of sources of errors influencing the quality of identification of unknown parameters using a meta-heuristic algorithm. Comput. Struct. 2014, 139, 1–17. [CrossRef] Stefanescu, R.; Schmidt, K.; Hite, J.; Smith, R.C.; Mattingly, J. Hybrid optimization and Bayesian inference techniques for a non-smooth radiation detection problem. Inter. J. Numer. Methods Eng. 2017, 111, 955–982. [CrossRef] Kennedy, J.; Eberhart, R. Particle swarm optimization. In Proceedings of the 1995 IEEE international conference on neural networks, Perth, WA, Australia, 27 November–1 December 1995; pp. 1942–1948. Rashedi, E.; Nezamabadi-Pour, H.; Saryazdi, S. GSA: A gravitational search algorithm. Inf. Sci. 2009, 179, 2232–2248. [CrossRef] Laudani, A.; Fulginei, F.R.; Lozito, G.M.; Salvini, A. Swarm/flock optimization algorithms as continuous dynamic systems. Appl. Math. Comput. 2014, 243, 670–683. [CrossRef] Mirjalili, S. SCA: A sine cosine algorithm for solving optimization problems. Knowl.-Based Syst. 2016, 96, 120–133. [CrossRef] Mirjalili, S. The ant lion optimizer. Adv. Eng. Softw. 2015, 83, 80–98. [CrossRef] Mirjalili, S.; Mirjalili, S.M.; Lewis, A. Grey wolf optimizer. Adv. Eng. Softw. 2014, 69, 46–61. [CrossRef] Lai, X.; Li, C.; Zhang, N.; Zhou, J. A multi-objective artificial sheep algorithm. Neural Comput. Appl. 2018, 1–35. [CrossRef] Wang, W.; Li, C.; Liao, X.; Qin, H. Study on unit commitment problem considering pumped storage and renewable energy via a novel binary artificial sheep algorithm. Appl. Energy 2017, 187, 612–626. [CrossRef] Mirjalili, S.; Lewis, A. The whale optimization algorithm. Adv. Eng. Softw. 2016, 95, 51–67. [CrossRef] Zhang, J.; Xia, P. An improved PSO algorithm for parameter identification of nonlinear dynamic hysteretic models. J. Sound Vib. 2017, 389, 153–167. [CrossRef] Wu, Z.; Yu, D.; Kang, X. Parameter identification of photovoltaic cell model based on improved ant lion optimizer. Energy Convers. Manag. 2017, 151, 107–115. [CrossRef] Ali, M.; El-Hameed, M.A.; Farahat, M.A. Effective parameters’ identification for polymer electrolyte membrane fuel cell models using grey wolf optimizer. Renew. Energy 2017, 111, 455–462. [CrossRef]

Energies 2018, 11, 2367

25. 26. 27.

28. 29.

30.

31. 32. 33. 34. 35.

36. 37.

38. 39. 40. 41. 42.

29 of 29

Tsai, S.H.; Chen, Y.W. A novel fuzzy identification method based on ant colony optimization algorithm. IEEE Access 2016, 4, 3747–3756. [CrossRef] Zou, W.; Li, C.; Zhang, N. AT-S Fuzzy Model Identification Approach based on a Modified Inter Type-2FRCM Algorithm. IEEE Trans. Fuzzy Syst. 2018, 26, 1104–1113. [CrossRef] Li, C.; Chang, L.; Huang, Z.; Liu, Y.; Zhang, N. Parameter identification of a nonlinear model of hydraulic turbine governing system with an elastic water hammer based on a modified gravitational search algorithm. Eng. Appl. Artif. Intell. 2016, 50, 177–191. [CrossRef] Zhang, N.; Li, C.; Li, R.; Lai, X.; Zhang, Y. A mixed-strategy based gravitational search algorithm for parameter identification of hydraulic turbine governing system. Knowl.-Based Syst. 2016, 109, 218–237. [CrossRef] Li, C.; Mao, Y.; Zhou, J.; Zhang, N.; An, X. Design of a fuzzy-PID controller for a nonlinear hydraulic turbine governing system by using a novel gravitational search algorithm based on Cauchy mutation and mass weighting. Appl. Soft Comput. 2017, 52, 290–305. [CrossRef] Li, C.; Zhang, N.; Lai, X.; Zhou, J.; Xu, Y. Design of a fractional-order PID controller for a pumped storage unit using a gravitational search algorithm based on the Cauchy and Gaussian mutation. Inf. Sci. 2017, 396, 162–181. [CrossRef] Li, C.; Xiao, Z.; Xia, X.; Zou, W.; Zhang, C. A hybrid model based on synchronous optimisation for multi-step short-term wind speed forecasting. Appl. Energy 2018, 215, 131–144. [CrossRef] Majdi, M.; Seyedali, M. Whale optimization approaches for wrapper feature selection. Appl. Soft Comput. 2018, 62, 441–453. Majdi, M.; Seyedali, M. Hybrid whale optimization algorithm with simulated annealing for feature selection. Neurocomputing 2017, 260, 302–312. Ling, Y.; Zhou, Y.; Luo, Q. Lévy Flight Trajectory-Based Whale Optimization Algorithm for Global Optimization. IEEE Access 2017, 5, 6168–6186. [CrossRef] Abdel-Basset, M.; Manogaran, G.; El-Shahat, D.; Mirjalili, S. A hybrid whale optimization algorithm based on local search strategy for the permutation flow shop scheduling problem. Future Gener. Comput. Syst. 2018, 85, 129–145. [CrossRef] Kaveh, A.; Ghazaan, M.I. Enhanced whale optimization algorithm for sizing optimization of skeletal structures. Mech. Based Des. Struct. Mach. 2017, 45, 345–362. [CrossRef] Emary, E.; Zawbaa, H.M.; Salam, M.A. A proposed whale search algorithm with adaptive random walk. In Proceedings of the IEEE 13th International Conference on Intelligent Computer Communication and Processing, Cluj-Napoca, Romania, 7–9 September 2017; pp. 171–177. Oliva, D.; El Aziz, M.A.; Hassanien, A.E. Parameter estimation of photovoltaic cells using an improved chaotic whale optimization algorithm. Appl. Energy 2017, 200, 141–154. [CrossRef] Rao, R.V. Jaya: A simple and new optimization algorithm for solving constrained and unconstrained optimization problems. Inter. J. Ind. Eng. Comput. 2016, 7, 19–34. Heikki, H.; Marko, L.; Antonietta, M.; Eero, S. DRAM: Efficient adaptive MCMC. Stat. Comput. 2006, 16, 339–354. Mirjalili, S.; Gandomi, A.H. Chaotic gravitational constants for the gravitational search algorithm. Appl. Soft Comput. 2017, 53, 407–419. [CrossRef] Cheng, X.; Zhang, J. Water Turbine Automatic Regulation; Waterpower Press: Beijing, China, 2010. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).