Enhanced Fractional Chaotic Whale Optimization Algorithm for Parameter Identification of Isolated Wind-Diesel Power Systems

Parameters identification of isolated wind-diesel power systems (WDPS) is a significant issue in stability analysis of the power system as well as guaranteeing the power generation through the control system. In this article, enhanced whale optimization algorithms (EWOA) are proposed to deal with the parameter identification problem of a WDPS system. The proposed EWOA effectively tackles the premature convergence problem of WOA by splitting the population into two subpopulations and updating the position of each whale according to the position of the best agent in its current subpopulation, the position of the other subpopulation’s best agent, and the position of the best neighboring agent. Furthermore, fractional chaotic maps are embedded in the search process of EWOA to increase its performance in terms of accuracy. For validation purposes, the proposed algorithms are applied to identify the unknown parameters of WDPS, where different statistical analyzes and comparisons are carried out with other recent state-of-the-art algorithms. Simulation results confirm that the algorithms have less deviation in parameter estimation, more convergence speed, and higher precision in comparison with other algorithms.


I. INTRODUCTION
Renewable energy systems have received extensive attention during the past few decades due to their extensive positive effects on air pollution through decreasing greenhouse-gas emissions to the atmosphere [1]- [3]. With the daily growth of technology throughout the world, the demand for constant and reliable power is increasing significantly. Among all renewable energy resources, wind power generation technology has become the leading power supplying system to deal with this surging power demand [4]- [7]. Isolated wind-diesel power systems (WDPSs) are practical solutions to supply power for the remote facilities, islands, and rural communities where their connection to the central energy supply system is disconnected or somehow limited [6].
Isolated wind-diesel power systems consist of many parts consisting of semiconductors, electronic components, and The associate editor coordinating the review of this manuscript and approving it for publication was Diego Oliva . mechanical gears, which are all inevitably prone to changes in their characteristic variations due to aging and faults. This may lead to reduction in efficiency of the system over time.
Since the isolated WDPS has contributed a small yet critical portion of the world's power production, it is vital to guarantee the power generation through reliable and authentic control strategies [8]. To provide an accurate control system, it is crucial to describe the exact behaviour of the WDPS under operation using an accurate model. The model must closely represent the behaviour of the system, while the accuracy of the model mainly depends on its parameters. Low precision of the system parameters will cause significant error and failure in the control system [9]. Hence, precise parameter identification of the system is a necessity. However, it has found to be a complicated and challenging task due to the highly nonlinear structure of the system.
The parameter identification can be considered as an optimization problem having an appropriate criterion, which is a function of estimation error of the system parameters.
Various optimization techniques have been utilized to handle this problem. Traditional optimization methods, such as Gradient Descent [10] and Newton Raphson [11], seem to be inefficient due to their dependency on initial conditions and differentiating the objective function. Alternatively, meta-heuristic methods have been proven to be practical approaches to deal with different parameter identification problems [12]- [25] as well as practical optimization problems [25]- [28]. A parameter estimation strategy using whale optimization algorithm (WOA) was investigated to develop a precise fuel cell model [12]. A modified bacterial foraging algorithm was also investigated to identify the parameters of fractional-order systems [13]. Aiming at enhancing the exploration and exploitation performances of the basic gravitational search algorithms (GSA), improved GSAs were proposed in [14], [15] for precisely identifying the parameter of water turbine regulation systems. Improved ant lion optimization algorithm augmented with particle swarm optimization [17], and modified GSA [18] were also proposed for parameter identification of hydraulic turbine governing systems. Various optimization algorithms have been introduced and implemented to identify the photovoltaic (PV) parameters [19]- [24]. To name a few, modified cat swarm optimization algorithm [22], improved JAYA optimization algorithm [23], and flexible particle swarm optimization algorithm with an elimination phase [20] for parameters estimation of one-diode and two-diode PV models, as well as sunflower optimization for three-diode PV model [24]. Considering the aforementioned literature, the significant performance of optimization algorithms in parameter identification of industrial systems is apparent, which has attracted an emerging interest among researchers to develop and investigate more efficient methodologies.
The whale optimization algorithm as one of the most recent evolutionary algorithms motivated by the hunting behaviour of humpback whales [29], has been utilized for distinct optimization problems in recent years [30], [31]. The spiralling mechanism represents the exploitation phase consisting of encircling the prey and the spiral bubble-net feeding manoeuvre; while, the exploration phase is carried out by a random search for prey. Although WOA is proven to yield superior performance over many optimization algorithms in terms of solving complex optimization problems [32], [33], it still suffers from premature convergence when it comes to solving large-scale problems, which defects its performance. Consequently, various modifications have been reported in the literature to enhance its exploration and exploitation capabilities and achieve better performance. Authors in [34] developed the WOA based on Pareto dominance to solve multiobjective optimization problems. This algorithm used an exterior archive as a storage for the non-dominated solutions detected during the optimization mechanism. Taking advantage of a cosine function-based nonlinear dynamic strategy, a modified WOA was proposed in [35] to equilibrate the exploitation and exploration abilities to enhance the algorithm's efficiency for solving large-scale optimization problems. In [36], a refraction-learning-based WOA augmented with a modified Logistic-model-based conversion parameter update rule was developed to make a trade-off between diversity and convergence during the search process of WOA when solving high-dimensional problems. In this context, in order to solve the premature convergence of WOA and modify the exploration process, an enhanced WOA based on quadratic interpolation was proposed [37]. In addition, modified versions of WOA have achieved remarkable results in other applications, such as water resources demand estimation [38], maximizing the power capture of variable-speed wind turbines [39], task allocation [40], parameter identification of solar cell diode model [41], quadratic assignment problem [42], terminal voltage control of fuel cells [43], and short-term natural gas consumption prediction [44]. Although these modifications have yielded some performance enhancements to the conventional WOA, they still suffer from other drawbacks such as lack of exploitation accuracy [36]- [39], [40], lack of exploration accuracy and getting stuck in local optima [34], [39], [41], [42], [44], and low convergence rate [34], [35], [38]- [41], that need to be adequately addressed.
In this article, modified versions of WOA algorithm with/without fractional chaotic map, namely, the enhanced WOA (EWOA) and fractional chaotic EWOA (FC-EWOA), are proposed. In the proposed algorithms, to enhance the convergence of WOA, the population is efficiently divided into two equivalent subpopulations, and each agent updates its position in respect of the position of the best agent, so-called the leader, in its current subpopulation, the position of the other subpopulation's leader, and the position of the best neighboring agent. Due to the ergodic and nonrepetition behaviours of chaotic maps, the combination of integer-order chaotic maps and meta-heuristic algorithms has proved to deliver significant improvements to the performance of algorithms [45], [46]. On the other hand, incorporation of fractional calculus with chaotic maps have enriched the dynamical behaviour of maps by demonstrating different distributions in comparison with integer-order counterparts [47]- [52]. According to the investigation results reported in the literature, the main superiorities of fractionalorder chaotic maps compared with the integer-order chaotic maps can be summarized as (i) wider chaotic regions can be achieved due to the addition of fractional-order [50], (ii) more random chaotic sequences, more stability, and higher level of security are guaranteed [50], [51], and (iii) better ergodicity and distribution characteristic are illustrated [52]. In this context, researchers have deployed fractional chaotic maps to enhance the performance of optimization algorithms, such as fractional chaotic ensemble particle swarm optimizer [50] and fractional flower pollination algorithm [53]. Thus, in this study, incorporation of fractional-order variant of the chaos maps into the proposed EWOA is developed to boost its performance. Comparative investigations, along with different statistical analyzes are performed to testify the behaviour of the proposed algorithms through parameter identification of VOLUME 8, 2020 isolated WDPS. According to the results achieved, the proposed strategies not only enhance the exploitation ability, but also expedite the convergence speed of the basic WOA.
This article is established into the following sections. The problem of WDPS identification is described in Section 2. The new enhanced fractional chaotic whale optimization algorithms are proposed in Section 3. Section 4 illustrates the comparative behaviour evaluation of the proposed algorithms through parameter identification of WDPS. Section 5 provides conclusions.

II. WDPS MODEL
A standalone hybrid WDPS is considered, consisting of static VAR compensator (SVC) which provides the required reactive power, isolated load, synchronous generator (SG), and induction generator (IG) driven by diesel engine and wind turbine (WT), respectively. As Fig. 1 depicts, the SG connected to the diesel generator (DG) form the diesel generator set, where it acts as a local grid for the wind system, which is formed by a connection of the IG and the WT. The reactive-power balance of the system under steadystate condition can be expressed as follows [8], [54]: where Q SG and Q SVC denote the reactive powers generated by DG and SVC, respectively; Q L is the demanded reactivepower-load, and Q IG is the required reactive power by the generator. It is assumed that the WDPS has a reactive power load change Q L . Thanks to the impact of the SVC and AVR controllers, due to the change in the system terminal voltage V t , the required reactive power will change, and also the reactive power generation system will increase as Q SVC + Q SG . Therefore, the net reactive power surplus in the system can be represented as [8]: The reactive-power control transfer function diagram for the WDPS is depicted in Fig. 2. Accordingly, by assuming the existence of small perturbations, the complete linearized model of the system can be achieved as follows (see [55] for more details).
where The WOA is an interesting nature-inspired algorithm motivated by the hunting behavior of humpback whales [29]. This algorithm mimics the attacking mechanism of a swarm of whale individuals to solve optimization problems, where the position of each whale indicates the feasible solution.
The WOA consists of three stages: (a) encircling the prey, (b) spiral bubble-net feeding maneuver, (c) search for the prey, where the first two phases represent the exploitation process, and the last phase represents the exploration. As the whales recognize the location of the prey, they dive deep and start creating bubble-nets in a spiral pattern around the prey and simultaneously swim-up to the surface. During the prey encircling phase, each whale encircles the target prey and maintains a candidate solution, and the best solution is determined. Correspondingly, other individuals update their positions with respect to the best search agent. This behavior at iteration t is represented as: where the symbol |·| denotes the absolute value, ζ and η represent the coefficient parameters, and * denote the position vector of the current agent and the best solution acquired so far, respectively. The parameters η and ζ are also calculated as: where κ is the damping coefficient, which linearly decreases from 2 to 0 with respect to iteration number, q is a constant value, which is set to 2, and υ ∈ [0, 1] is a random value. The bubble-net behavior consists of two strategies, namely, the shrinking encircling mechanism and the spiral position update. In the first approach, the coefficient parameter ζ varies in the range [−κ, κ] and κ ∈ [0, 2], thus, by setting random values for ζ ∈ [−1, 1] the new search agent can be located between its original position and the position of the current best agent. In the second approach, the agent's new position is updated regarding the position of the prey and the agent according to the following spiral equation: where the distance between the ith agent and the prey is computed by = * (t) − (t) , the constant δ defines the shape of the logarithmic spiral, and ε ∈ [−1, 1] is a random value. In order to simultaneously perform the shrinking circling together with the spiral-shape movement, a random probability coefficient is adopted that updates the position of agents during the optimization process by choosing between the following two mechanisms: where ψ ∈ [0, 1] is a random value. In order to perform the exploration process, the search agents need to adequately far-off the best agent and try to search randomly through the search space. To this end, ζ is randomly chosen either greater than 1 or less than −1, and the agents update their positions regarding randomly chosen agents rather than the best agent so far. The mathematical behavior of the exploration process is expressed by: where rand refers to a randomly selected agent from the current population.

B. FRACTIONAL CHAOTIC MAPS
Chaos theory studies the behavior of iterated functions that return random values in each iteration. Chaotic functions are highly sensitive to initial conditions, such that any small difference in initial conditions yield to divergent in the generated sequence of values. The integration of chaotic functions into optimization algorithms have demonstrated promising improvements in the speed of convergence of the algorithms and the solutions diversity. Thus, they have been widely studied for global optimization and engineering problems [45], [46]. Different chaotic maps are introduced in the literature [56], due to their interesting characteristics such as a) the chaotic maps are generated by a deterministic dynamic rule, b) the time series is bounded between upper and lower limits, c) they act aperiodic, and d) the sequence is dependant to the initial condition.
Considering the above-mentioned characteristics, chaotic maps can effectively increase the exploration power of the stochastic search processes, which make them perfect replacements for the existing random generators. Thus, in this article, three fractional chaotic maps are embedded to adjust κ in the proposed EWOA algorithm, namely FC-EWOA. The fractional chaotic maps including the Fractional Sine map [57], the Fractional Logistic map [57], and the Fractional Tent map [58], together with their relationships and distributions are listed in Table 1.

C. PROPOSED FC-EWOA
Although the WOA is an efficient optimization algorithm in dealing with global optimization and engineering problems [30]- [33], but yet, it has the main shortcoming that is premature convergence, which leads the algorithm to  [44]. In this article, enhanced whale optimization algorithms, namely EWOA and FC-EWOA, are proposed.
The EWOA consists of modifications to the position update procedure, which will be discussed, and the FC-EWOAs are modified versions of EWOA, which incorporate fractional chaotic maps into the search process of EWOA. Here, the proposed fractional chaotic EWOA algorithms are denoted by FC1-EWOA (with logistic map), FC2-EWOA (with sine map), and FC3-EWOA (with tent map). Through the exploitation process of WOA, each agent updates its position based on the position of the best agent, which reduces the diversity of the population, leading to a low convergence rate. The population is efficiently divided into two semiindependent subpopulations with the same number of agents, where the best agent in each subpopulation is determined. To address the convergence problem in WOA, the position of each agent is updated in terms of three factors, namely the best agent's position in its current subpopulation, the best agent's position in the other subpopulation, and the neighboring agent's position denoted by nbest that have better fitness than itself. This behavior can be expressed as follows: where * 1 and * 2 represent the position of the best agents in the current and the other subpopulation, respectively. nbest denotes the position of the neighboring agent with better fitness, and η i = q i υ are the coefficient parameters, where {q 1 , q 2 , q 3 } = {2, 1, 0.5}. Therefore, we have: where the coefficient parameterζ i is computed as: Remark 1: The convergence speed of the algorithm through the exploitation process depends on the parameters q 1 , q 2 and q 3 . Higher values of q 1 result in faster movement of the agents towards * 1 and less exploitation accuracy. On the other hand, although lower values of q 1 leads to better exploitation accuracy, the convergence rate would be reduced. Higher values of q 2 attracts the agents to move towards the other subpopulation ( * 2 specifically), while the current subpopulation has not been well-exploited. Therefore, lower values of q 2 decreases the impact of * 2 on the agents' position update procedure, which reduces the diversity of the solutions. Similar to q 2 , lower values of q 3 decreases the diversity of the solutions due to the lower impact of nbest on the agents' update procedure. Alternatively, higher values of q 3 increase the diversity, but the agents get misled, which reduces the convergence rate and the accuracy of the algorithm.
As it is apparent, the basic WOA and the proposed EWOA consist of several arbitrary selected variables that notably affect the algorithm performance. The most key variable is the linearly decreasing damping coefficient κ ∈ [0, 2], which is a crucial factor in the convergence rate of FC-EWOA. Thus, here, this variable is chaotically tuned. In this regard, κ is chaotically varied between 0 and 2, instead of being linearly decreased with respect to the iteration number as follows: where (Q − κ) f represents the chaos damping coefficient that follows the distribution of the FC-maps stated in Table 1 with index f ∈ {1, 2, 3}, denoting the fractional chaotic maps, T and t are the number of the total and the current iterations, respectively. On the other hand, κ h and κ l are the upper and the lower values of κ, and NC f is defined as: where F − Ch f is the considered fractional chaotic map, min(F − Ch f ) and max(F − Ch f ) represent the minimum and the maximum distribution range of the fractional chaotic maps, respectively, and ρ = 0.6 and ϑ = 0.9 are the normalization interval ranges.
In FC-EWOA, the position of each agent through the bubble-net procedure is calculated as: Similar to the basic WOA, in order to calculate the new position of the agents, the random probability coefficient ψ ∈ [0, 1] is chosen between the shrinking circling and the spiral-shape movement.
As the exploitation process completes, the two subpopulations are merged into one population, and the best agent is evaluated. Consequently, the position of every agent can be updated in terms of the whole population, which leads to more random movements through the exploration process. The mathematical behavior of the exploration process is as stated in (10) and (11), with the chaos damping coefficient given in (15) instead of κ ∈ [0, 2]. In summary, the pseudo-code for the proposed FC-EWOA can be presented in Algorithm 1. Moreover, the flowchart of the FC-EWOA is depicted in Fig. 3.

IV. SIMULATION RESULTS AND DISCUSSIONS
In this section, comparative performance evaluations of the new algorithms are conducted through mathematical benchmark functions and parameter identification of isolated WDPS. In this regard, the comparisons are carried out with eight well-established meta-heuristic algorithms, including the GSA [58], the firefly algorithm (FA) [59], the grey wolf optimizer (GWO) [60], the bacterial foraging optimization (BFO) [61], the bat algorithm (BA) [62], the flower pollination algorithm (FPA) [63], the dragonfly algorithm (DA) [64], and the basic WOA [29]. The parameters settings of the optimization algorithms under comparison are tabulated in Table 2. For each algorithm, 100 independent runs are performed with a population size of 50. In the performance evaluation section, the dimension of all test problems is set to D = 50. Besides, the maximum number of function evaluations (NFEs) of 10, 000 × D and 20,000 are set as the stopping criteria for benchmark functions optimization and parameter identification evaluations, respectively.

A. PERFORMANCE EVALUATION
To testify the performance of the proposed algorithms, 8 problems adopted from the CEC2017 benchmark test for every search agent 9: Update the chaos damping coefficient using (15); 10: Update ζ , η, ε, and ψ; 11: Evaluate using (12); 12: if-1 ψ < 0.5 13: if-2 |ζ | < 1 14: Update the current search agent's position using (13); 15: NFE=NFE+1; 16: elseif-2 |ζ | ≥ 1 17: Merge the subpopulations into one population; 18: Update the current search agent's position using (10) and (11) with chaos damping coefficient (15); 19: NFE=NFE+1; 20: Rank the whale agents and determine the current best agent ( * ); 21: Define the subpopulations and assign the leaders and members; 22: end if-2 23: elseif-1 ψ ≥ 0.5 24: Update the position of the current search agent using (17); 25: NFE=NFE+1; 26: end if-1 27: end for 28: Rank the whale agents; 29: Determine the current best agent ( * ) in each subpopulation; 30: end while suite are used as objective functions. The test problems are considered with various difficulty levels and categorized as unimodal functions, simple multimodal functions, hybrid functions, and composition functions with the search range of [−100, 100], as illustrated in Table 3. The experimental results obtained by the algorithms are shown in Table 4, where standard deviation (S.D) represents the standard deviation, and the best minimum mean value achieved for each function is highlighted in bold.
The test results in Table 4 show that the proposed algorithms exhibit the best performance on all functions. FC1-EWOA performs significantly better than all other algorithms with 6 best results on (f 2 , f 3 , f 4 , f 6 , f 7 , f 8 ), followed by  FC2-EWOA with 2 best results on (f 1 , f 5 ). From Table 4 and based on the overall performance ranking of all algorithms, FC1-EWOA is the best algorithm, followed successively by FC2-EWOA, FC3-EWOA, EWOA, GWO, WOA, GSA, FA, BFO, BA, FPA, and DA.
For better demonstration of performance comparisons, the convergence curves of average best solutions of the proposed algorithms and other algorithms on the 8 selected benchmark problems are depicted in Fig. 4. According to Fig. 4, the proposed EWOA and its fractional versions demonstrate the fastest rates of convergence throughout the early stages in comparison to other methods, which illustrates one of the superiorities of the proposed algorithms. Another superiority is their capability to deal with multimodal, hybrid, and composition functions. As is clearly shown in Fig. 4, the proposed algorithms maintain excellent search abilities both globally and locally, while other algorithms are mostly gotten trapped into local optimum prematurely. Besides, although in the first stages, the convergence of FC1-EWOA is relatively slower than the other three proposed algorithms, in 6 problems it surpasses them in the middle stages.
To sum up, as stated in Table 4 and can be seen in Fig. 4, the proposed algorithms demonstrate the best performance in terms of convergence rate and optimization, followed by GWO, WOA, GSA, and FA, which deliver better results in comparison with other methods under study.

B. PARAMETER IDENTIFICATION OF WDPS
The parameter identification of isolated WDPS is conducted in this sub-section. This problem is transformed as an optimization problem, where the goal is to identify the parameters such that the difference between the actual and current data is minimized. Therefore, we consider the following objective functions, which are the mean square error (MSE), the mean absolute error (MAE), the root mean square error (RMSE), respectively.
In Table 5, we list the actual parameters values and the search ranges of unknown parameters for the WDPS. The comparative performance evaluation results of the proposed methods for the WDPS parameter identification problem with respect to other algorithms are presented in Tables 6 and 7. According to Table 6, the proposed EWOA and FC-EWOA algorithms can obtain more accurate parameters for the WDPS system. Table 7 shows the applied error metrics together with their corresponding standard deviation, such that the best results indicating the minimum error is emphasized in bold. From Table 7, we observe that the error values obtained by the proposed EWOA algorithms are remarkably less than other methods.
Besides, the FC1-EWOA demonstrates the best performance among all algorithms by achieving the least possible identification error value in terms of MAE, RMSE, and MSE. Results manifest the superiority of the proposed EWOA and FC-EWOA algorithms, yielding outstanding parameter identification performance with 16 accurate identified parameters achieved out of 16 parameters, followed by GWO and WOA with 3 and 2 accurate parameters, respectively.
It is worth pointing out that the results accuracy is computed and presented to eight and four decimal places, respectively. To intuitively compare and analyze the quality of the VOLUME 8, 2020   solutions from a statistical point of view, Table 8 provides the comparative Friedman test [65] results, where the column ''Rank'' denotes the performance order of the algorithms. The results regarding the Friedman test show the superiority of the proposed algorithms in comparison with others, which verifies the above conclusion again. In order to test the conformity between the real and the estimated WDPS parameters, the transient responses of Q SG , Q SVC , and Q IG for a 5% step increment in the reactive power load are illustrated in In addition, although the GWO and WOA algorithms demonstrate better fitting results compared with BA, yet, the accuracy of their results is markedly less than the proposed EWOA and FC-EWOA, which again, indicates  the remarkable performance of the proposed algorithms. Fig. 8 depicts the average convergence profile of the objective function for the proposed algorithms in comparison with other methods. From Fig. 8, it can be observed that the proposed EWOA and FC-EWOA algorithms demonstrate significantly higher convergence speed than others.
Accordingly, the proposed FC-EWA algorithms converge to the minimum value of MSE within 2000 function evaluations, followed by the proposed EWOA with 3600 function evaluations, indicating the exceptional performance of the proposed algorithms. On the other hand, although other methods could not achieve the minimum MSE value, still GWO, WOA, GSA, and FA demonstrated more desirable performance compared to BFO, FPA, BA, and DA.
The convergence rates of the WDPS parameters are depicted in Fig. 9, yielding more searching accuracy and faster convergence speed of the proposed algorithms in contrast with other algorithms. The aforementioned comparisons demonstrate that utilizing the proposed strategy presented   in subsection 3.3 and embedding the FC-maps with the proposed algorithm affects the accuracy and enhances the performance of the basic WOA.

V. CONCLUSION
Accurate modelling and appropriate controller design for the isolated WDPS require accurate identification of the model parameters. For this purpose and aiming at the main disadvantage of the whale optimization algorithm, which is premature convergence, four novel enhanced WOA algorithms are introduced. The proposed EWOA develops the position update procedure of WOA such that more accurate exploitation behavior is achieved. Besides, to enhance the performance of EWOA in terms of accuracy, three different fractional chaotic maps are taken into account in the search process of WOA. Efficiency and superiority of the proposed algorithms are validated via parameter identification of WDPS in comparison with other algorithms. Results confirm the remarkable performance of the proposed algorithms with respect to solution precision and speed of convergence.