Application of a New Fusion of Flower Pollinated With Pathfinder Algorithm for AGC of Multi-Source Interconnected Power System

As the world’s population grows and energy demand increases, it is necessary to increase the scale of the electrical system, which is more complicated. Consequently, adopting automatic generation control (AGC) scheme to meet the demand becomes inevitable. In this article, the fusion of flower pollinated algorithm (FPA) and pathfinder algorithm (PFA), named hereafter as $h$ FPAPFA, is proposed to achieve maximum control efficiency by combining the exploitation of FPA with the exploration capacity of PFA. The proposed $h$ FPAPFA is meant to regulate two unequal multi-area interconnected power system with different generating units such as thermal, hydro, wind power and diesel plants. The proposed control scheme aims to achieve this by using the new algorithm to optimize the fractional-order set-point weighted PID (FOSWPID) parameters under time domain-based fitness functions namely, integral time square error (ITSE) and integral time absolute error (ITAE) while simultaneously minimizing the power losses. Employing the same interconnected power systems, a comparative study with some recent approaches in renowned journals is conducted. The performance of the proposed method is observed under diverse load conditions scenarios. Moreover, three nonlinearities including boiler dynamics, the governor dead band (GDB) and generation rate constraints (GRC) are further integrated into the system from a pragmatic context. Finally, sensitivity tests involving various parameter changes and the introduction of random step load perturbations are carried out. From the results, the proposed approach outperformed other approaches under different load condition scenarios, incorporation of nonlinearities and random load perturbation, demonstrating the proposed technique’s efficacy and reliability.


I. INTRODUCTION
A. LITERATURE REVIEW The power system is a complex system that consists of interconnected active elements. Meeting the constantly evolving demand from consumers has created huge puzzles for engineers. The top priority of power system industries is to ensure adequate generation, transmission, and electric power distribution. Given the enormous developments in The associate editor coordinating the review of this manuscript and approving it for publication was Siqi Bu . technology, the need for power supply has become essential for individuals and industries [1]. The power systems are becoming more and more complicated since they include multiple sources such as biomass, fossil fuels, solar thermal energy, wind, hydro etc. The renewable energy sources have significantly contributed to the supply of energy to the power grid sectors. However, stable, and good quality power production in the power system still poses a significant challenge due to the continuous change in load. To achieve optimal power distribution to customers, the active power produced must be proportionate to the power demand; VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ otherwise, the frequency of the generating units would be adversely affected [2], [3]. When the power produced falls below the required quantity, the generator units begin to decrease in speed and frequency. The discrepancy between the power generated and the required load demand over a long time leads to an alteration in the system's nominal frequency and voltage profile [4], [5]. To tackle the effect of changes in the system's frequency and tie-line loading, the automatic generation control (AGC) control scheme is employed to regulate the deviations in active power and active power output of each generation unit within the control zone, as well as to maintain the frequency and provide interchange with other fields [6]. Hence, the primary purpose of AGC is to standardize the frequency of the system and tie-line power at a nominal value. To achieve this, several approaches have been explored in the literature, including but not limited to classical control [7], adaptive control [8], optimal control [9], robust control [10] and artificial neural network [11]. However, many modern power systems require an effective controller and more innovative optimization approaches to achieve good frequency and tie-line power stability [12]. The proportional-integral-derivative (PID) controller have been used for many industrial applications over decades [13]- [16]. This is attributable to their simple compositional structure, remarkable capabilities and many other advantages [17]- [19]. Although the PID controller is an admirable preference for many control processes, it becomes ineffective for controlling nonlinear systems, has a low noise immunity and is susceptible to parameter variations. When considering the intrinsic non-linearity of the power systems, the output of a basic proportional-integral controller degrades in terms of the overshoot and setting times of the system's frequency and tie-line power deviation [20]. Subsequently, modern control theories have made significant advances in PID control design by using the principle of fractional calculus to optimize industrial process performance [21]. The fractional-order proportional integral derivative (FOPID) controller, also known as (PI λ D µ ), is an improved version of the regular PID controller. One of the major outstanding performance of the FOPID is found in its two extra tunable parameters (integral order and derivative order) which can be used to enhance the performance of control loops. The FOPID efficacy is evident in its robust performance when dealing with higher order, nonlinearity and systems with delay [22]- [24]. As established in refs. [25]- [28], the FOPID control design has much more adaptability in its fine-tuning scheme and also a large area of parameters that govern the controlled system with increased reliability and stability of the control loop. However, it is essential that FOPID have a clear defining weighting mechanism to present the required optimal control structure. This will invariably lead to a stronger control scheme with clear regulation of the set-point and load control [29]. Hence, one of the motivations for this work is to design a fractional-order set-point weighted PID (FOSWPID) for the AGC system, particularly because it has rarely been applied to the AGC system. Nevertheless, it is worth mentioning that the controllers can only perform optimally if their parameters are well-tuned. Recent approaches such as artificial intelligence (AI) and soft computing (SC) have made important strides when developing the controller architecture to solve technical challenges [30]. Many researchers have proposed the optimization of classical controllers with AI and SC techniques, especially for realizing a balance of the frequency and tie-line power deviations in the AGC system. Mohanty [31] have proposed an AGC using a moth flame optimization algorithm (MFOA) to select the parameters of the proportional-integral-double-derivative (PIDD) controller. The AGC consists of two unequal areas of the thermal system with the inclusion of GRC to reveal its dynamic stability under nonlinearities. A comparative analysis was performed under various scenarios to assess the efficacy of the proposed method. Under time-domain analysis, the proposed MFOA-based PIDD outscored other approaches. From the reports of Debbarma et al. [32], optimal gain control of a two-degree-of-freedom fractional order PID (2-DOF-FOPID) controller was proposed for the AGC of the power system. The gains of the controller and the R parameter selection were tuned with Firefly Algorithm (FA). The results of the simulation showed that the FA-based 2-DOF-FOPID gave superior performance compared with other methods. However, for a more realistic power system, the incorporation of more nonlinear elements such as the GDB and boiler dynamics could further show the efficacy of the proposed technique in refs [31], [32]. In [33] the Harmony Search Algorithm (HSA) was used to stabilize the AGC. To reach optimum equilibrium in frequency and tie-line power deviations, the control strategy involved the optimal configuration of the FO-PID controller's parameters using the suggested algorithm. In comparison to other approaches, the authors indicated that the HSA-based FOPID performed satisfactorily. Fitness dependent optimizer algorithm was employed in reference [34] to select the gains of a modified PID controller namely, integral proportional derivative (I-PD) controller for the AGC of two areas multi-source interconnected power system. The multi-source of the AGC is comprised of gas, hydro and reheat thermal unit. The success of the proposed approach is established with substantial improvement in settling time, overshoot and undershoot in the comparative study. Conversely, the application of the recent metaheuristic algorithm will present a more effective algorithm for selecting the parameters of the I-PD controller compared with the classical fitness dependent optimizer. Furthermore, Nizamuddin et al. [35], examined the performance of bacterial foraging optimization (BFO) algorithm in selecting the ideal gains of the PID controller while monitoring the discrepancy in frequency and tie-line power. The BFO-base PID results in higher efficiency compared to the optimized PID controller of the genetic algorithm (GA). Authors in [36] applied the search group algorithm (SGA) to tune the parameters of the PID controller to regulate the AGC while considering the ITAE-based performance criterion.
The proposed SGA-PID exceeded other techniques in the analysis. Considering the works in references [35], [36], the physical limitation and spontaneous load disturbance of the power system is necessary to show the efficacy of the proposed controllers. Again, Saikia et al. [7] conducted a comparative study between five different configurations of the PID controller. The controllers were optimized using the BFO algorithm. The results of the simulation show that integral-double derivative (IDD)-based BFO outperformed other controllers. The use of heuristic procedures to determine the optimum controller parameters can result in a better control system. Similarly, Nasiruddin [37] et al. designed the AGC for a two-area interconnected power system with multi-energy sources. The BFO algorithm was used to optimize the gains of the PID controller considering 1% step load disturbance in one of the control areas. The output of their study showed that the BFO-based PID surpassed the GA-based PID controller in terms of the system dynamic performance. Singh et al. [38] conducted a performance analysis for the AGC control system using an improved PID controller. The PID controller was enhanced via the incorporation of a filter inform of the derivative term to lessen the impact of noise in the input signal. The proposed controller was optimized using the jaya algorithm (JA). The authors reported an improved performance using the proposed techniques under different scenarios in comparison with differential evolution (DE), particle swarm optimization (PSO), needler-mead simplex (NMS), elephant herding optimization (EHO) and teaching-learning-based optimization (TLBO) algorithms.
In an attempt to regulate the variation in the frequency and tieline power of the multi-area power grid, Jagatheesan et al. [1] investigated the operation of the FPA algorithm for optimal PID controller gains with a 1% step load disturbance in area 1. The findings showed a better performance than with other approaches, namely GA-based PID and PSO-based PID controller under the same condition to ensure a fair comparison. Singh et al. [5], presented an optimal gain of a filtered PID (PIDn) controller using symbiotic organisms search (SOS) algorithm. To investigate the output of a controller, some case studies of various load disruptions were performed in both areas with varying wind penetration levels. Results of simulation showed that the proposed SOS-PIDn gave a superior performance for the change in frequency and tie-line power. In Reference [39], a pathfinder algorithm (PFA) was implemented with a fractional order tilt-integral-derivative (FOTID) controller for regulating the AGC with diverse source power system. Compared to conventional PID controllers and tilt-integral-derivative (TID) controllers using the same algorithm, the proposed PFA-based FOTID showed excellent results with better dynamic response for different sensitivity and robustness measures.

B. RESEARCH GAP AND MOTIVATION
Most of the above-mentioned approaches used a single algorithm to optimize the parameters of the different variants of the PID controllers for the AGC system. However, maintaining a balance between diversification (exploration of the search space) and intensification (exploitation of the search space) is a major prerequisite in presenting an effective algorithm [40]. To achieve this, two or more algorithms are hybridized to form a memetic algorithm that can identify the suitable algorithm features and mitigate the flaws of each. In this work, the flower pollinated algorithm is combined with the pathfinder algorithm to harness the global search capability and local convergence abilities of both algorithms to optimize the parameters of the FOSWPID controller for the AGC system. The flower pollinated algorithm (FPA) is a population-based algorithm that is characterized by its simplicity and flexibility. FPA tends to attain a good exploitation process. This is attributable to its unique parameter referred to as switching probability. In addition, the levy distribution facilitates the exploration of the global search towards the best solution. However, according to the operation of the FPA, the solution to the optimization problem is reliant upon interaction with pollen individuals. This has a detrimental consequence of being conveniently stuck at a local minimum [41]. Secondly, due to the application of population diversity especially with multimodal problems, FPA is highly pruned to being stuck in local optimum [42]- [45]. On the other hand, the pathfinder algorithm (PFA) is a recently proposed algorithm that mimics the characteristics of animal group social movements and the leadership system of swarms to find the best food area or prey (global optimum). By avoiding local optima, the PFA performance is competent when used for high-dimensional, dynamic, multimodal problems. However, the PFA has a weak exploitative capability. Given their respective strengths, FPA and PFA are suitable for hybridization. The mixed algorithm outperforms the individual algorithms because it identifies the appropriate algorithm features to mitigate the flaws of each one attaining a good equilibrium between intensification and diversification. This paper aims to present a hybrid of optimization-based algorithms which comprises the integration of FPA and pathfinder algorithm PFA for addressing the problems of load changes and frequency deviation in the AGC system.

C. CONTRIBUTION AND PAPER ORGANIZATION
The main contributions of this work can be explained as follows: 1) A fusion of flower pollinated algorithm (FPA) and pathfinder algorithm (PFA) is proposed for the automatic generation control system with a multi-source interconnected power system. 2) The proposed approach is applied to optimally design the fractional-order set-point weighted PID (FOSW-PID) controller for a two-area multi-source interconnected AGC consisting of thermal, hydro, wind power and diesel generating units.

3) Performance comparison between the h FPAPFA-based
FOSWPID controller over DE-PID [46], PSO-PIDD, The organization of the rest of this paper is as follows: Section 2 describes the material and methods, followed by the control structure and the overview of the optimization techniques; Section 3 discusses the proposed algorithm; Section 4 presents the results and discussion. Finally, the conclusion is summarized in section 5.

A. TWO AREA POWER SYSTEM MODEL
The fundamental objective of the AGC system is to always preserve the power system at a required nominal value in the event of load disturbance. The transfer function model of the multi-source interconnected power system is shown in Fig. 1 [46]- [48]. The system under consideration consists of two areas with different energy sources such as thermal (reheat), hydro and gas and diesel plant units. At a nominal frequency of 60Hz, each control area has a rating of 2000MW and a nominal loading of 1000MW. The two multi-source areas are connected via a tie-line. As shown in Fig. 1, the area control errors are denoted by ACE 1 and ACE 2 ; T G1 and T G2 are the time constants of the speed governor in seconds; T T 1 and T T 2 are the turbine time constant for area 1 and 2; R H 1 and R G1 are the governor speed regulation parameters in p.u for area 1; the frequency bias parameters are represented by B 1 and B 2 ; the change in load are given as PD 1 and PD 2 ; the thermal constants are given as K th1 and K th2 respectively; P 12 is the synchronization power coefficient, K GD is the diesel turbine speed governor constant, T D is the diesel turbine time constant; F 1 and F 2 are the frequency deviations in area 1 and 2 whereas P Tie is the marginal shift in line power (p.u). The details of all the parameters are presented in the Appendix. For a comprehensive detailed equations, readers are suggested to consult. [34], [49].

B. STRUCTURE OF THE CONTROLLER AND OBJECTIVE FUNCTION 1) FRACTIONAL-ORDER CALCULUS PRELIMINARIES
Fractional-order calculus (FOC) is a branch of mathematics that focuses on differentiation and integration of real or complex order [50]. Compared with several conventional integer techniques, the FOC has proven more suitable for analyzing and modelling real-time systems [51]. The application of FOC is expanding in many areas, including quantum mechanics, control theory, heat-flux exchange, electronics, bioengineering, chemical analysis of solutions [52]. The FOC system is also referred to as the generalized integration and differentiation for non-integer orders and they are mathematically represented by fractional-order differential equations. The fractional-order integral-differential operator can be defined as Eq. (1) [53] where αD r t , a, t and r are the fundamental operators, lower limit, upper limit, and the order of the operator. There is no unified definition of the fractional-order derivative, but there have been three generally accepted definitions, including Caputo, Grunwald-Letnikov (GL) and Riemann-Liouville (RL) [41]. Each of the three definitions has its individual properties. The Caputo definition is given in Eq. (2).
where (n − 1) ≤ α ≤ n, n is an integer, α is the real number and (.) is an Euler function. Furthermore, the Grunwald-Letnikov definition can be represented as Eq. (3) αD where α and t are the lower and upper limit of the integration respectively, r j represents the binomial coefficient.
Depending on the integration or differentiation, the value of r can either be negative or positive. The Reimann-Liouville (RL) is the simplest and most basic form of fractional-order derivative definition. It is defined as Eq. (4) where (n − 1) ≤ α ≤ n, n is an integer, α is the real number and (.) is an Euler function. Numerous approximation approaches can directly provide the approximate s-transfer function of the fractional-order differentiator and integrator. A well-known approximation was given by Oustaloup [54] which utilizes a recursive distribution of zeros and poles. The simplified form of the Oustaloup approximation definition for the fractional-order differentiator s α is presented in Eq. (5) As presented in Eq. (5), K is the feature gain that is retained to provide the unit gain at 1 rad/sec. The approximate frequencies of poles and zeros are ω k and ω k , respectively. N is a function of the number of poles and zeros. It should be noted that ω k and ω k are the nominal pole and zero frequencies at the n th instant whose values are true within the lower frequency (ω b ) and the high frequency (ω h ) of the system The poles, zeros and gain of the filter can be recursively evaluated as: where α is the order of the differ-integration, the order of the filter is represented as (2N + 1). The output of a signal transiting the filter (as seen in Eq. (1)) is an approximation to the fractionally differentiated or integrated signal αD r t f (t) [55]. In this study, the order of the filter is selected to be 5 th order Oustaloup's recursive approximation, and the chosen frequency range is 10 −3 , 10 3 rad/sec. In addition, Podlubny [53], proposed the most frequently used version of a fractional order PID controller which has both an integrator order (λ) and a differentiator order (µ). The transfer function in time domain is given as: where k p is the proportional gain; k i is the integral gain, k d is the derivative gain; λ is the integral order and µ is the derivative order. Taking the Laplace transform of Eq. (9) gives the continuous transfer function of the FOPID controller as described in Eq. (10) When using the FOPID controller, it is apparent that besides the standard parameters K P , K I and K D the additional two parameters namely, integral order λ and derivative-order µ should also be considered. To realize the complex FOPID controller, the FOMCON toolbox in MATLAB is used in this study [56], [57].

2) SETPOINT WEIGHTED FRACTIONAL ORDER PID CONTROLLER
The use of fractional order PID (PI λ D µ ) controller in previous works have shown that it has the potential to independently increase the system's efficiency in terms of sensitivity and responsiveness [58], [59]. However, relying solely on FOPID does not guarantee optimum results. As a result, the addition of set-point weighting parameters allows for a more stable control system [60]. In the same vein, fractional integral and differentiation offer a way to increase the likelihood of fine-tuning the controller's parameters to improve proficiency [61]- [63]. The synthesis of these two frameworks offers a basis to address the problems of the AGC system. Consequently, the FOSWPID controller is chosen in this paper for regulating the frequency deviation and tie-line power deviation in the AGC system. Following the past works on the AGC system which entails the application of PID-based controllers, the FOSWPID controller is similarly considered in this study for two unequal areas with VOLUME 9, 2021 the multi-source power system. The block diagram of the FOSWPID controller is shown in Fig. 2. The design of the FOSWPID requires the determination of seven parameters: b and c which are the proportional and derivative set-point weighting parameters respectively, proportional gain (K p ), integral gain (K i ), derivative gain (K d ), integral order (λ) and derivative order (µ). The control signal u(t) of a standard PID is a sum of P, I and D actions as described in Eq. (11) where K p , K i and K d are proportional, integral, and derivative gains, respectively. The three control actions P, I and D are functions of error signal e (t) given as where r (t) is the reference or setpoint signal and y (t) can be defined as the output signal or control signal. From Eq. (11), the control action of SWPID can be obtained by weighting proportional action with b and derivations action with c as given in Eq. (13) [29]. with where b, c ∈ [0, 1] are the proportional and derivative setpoint weights, respectively. The error associated with integral action is not weighted to avoid steady-state control error [29]. Therefore, e (t) = e i (t) as described in Eqs. (12) and (15). The Laplace transform of the control signal from Eq. (16) is given as: The control action of SWPI λ D µ can be obtained by replacing the first-order integral and derivative actions with fractional order λ, µ as described below [29]: The error inputs entering the controllers in area 1 and area 2 are expressed as: By considering P r1 and P r2 as the rated power of the areas, Then, where F 1 and F 2 are the frequency deviation in area 1 and area 2, respectively; P tie is the change in tie-line power; B 1 and B 2 correspond to the frequency bias parameters.
The performance criteria can be described as a quantitative measure used to show the effectiveness of a control system [64]. A well-defined objective function reveals the device proficiency and illustrations if it satisfies the requirements of the controller's design. Findings have revealed that the performance of a typical PID controller has long been established by using some performance criterion namely, integral squared error (ISE), integral of time-weighted absolute error (ITAE), integral of time multiply squared error (ITSE) and integral of absolute error (IAE) [65]. The minimization of the fitness function is an essential procedure in attaining optimized parameters for the parameters. The ITAE has been reported to be an effective objective function for the AGC system [66], [67]. This is because of its simplicity of use, reliability and improved efficiency in outputs [68]- [70]. The ITAE criterion exceeds IAE and ISE in its ability to minimize the settling time. Nevertheless, the ITSE is well-known for its capability to impose a higher penalty on large oscillations due to its squared error, and hence successfully help to minimize large oscillations in frequency and tie-line power deviations. Hence, the proposed algorithm uses the ITSE to display the performance of controllers. The mathematical description of ITSE is given in Eq. (25) and other cost functions are defined by Eqs. (27)- (28).
94154 VOLUME 9, 2021 where t is the simulation time, F i is the frequency deviation in area i, P Tie−i−k is the change in tie-line power connected in-between area i and area k. The parameters of the FOSW-PID controller are optimized by the proposed h FPAPFA algorithm by minimizing J according to the following constraints:

C. OVERVIEW OF FLOWER POLLINATED ALGORITHM
The Flower Pollinated Algorithm (FPA) is an effective segment of bio-inspired metaheuristic algorithm which replicates the pollination mechanism of flowers. Pollination can be simply put as the process of reproduction in plants.
It involves the transportation of pollen grains to the receptive area of the female reproductive organ [71]. The movement is carried out by a variety of agents called pollinators. Examples of pollinators include bats, butterflies, bees, wind, water etc. Biotic pollination happens when pollination is carried out by insects or livestock. On the other hand, if the pollination agent is inanimate things such as wind and water, this is called abiotic pollination. Moreover, the process of pollination can be achieved without involving pollinators. This is called selfpollination. Self-pollination involves the transportation of pollen from the male to the female part of the same plant.
One of the peculiar characteristics of the process of floral pollination is ''floral constancy.'' This is a concept that describes pollinators' preference for only visiting specific flora species while ignoring others [42]. The major significance of the concept is its guarantee of optimum pollen transfer which then stimulates and enhances flower reproduction [72].

D. MATHEMATICAL DESCRIPTION OF FPA
The FPA simulates the pollination and flower constancy functionality described above and has been modelled mathematically using four distinct rules by Yang [73]: R1: Biotic and cross-pollination replicate the global search process which is carried out by pollinators in the similitude of Lévy flight.
R2: The local search simulates the Abiotic and selfpollination phenomenon.
R3: The flower constancy is viewed as a reproductive rate associated with the resemblance of two flowers.
R3: The operation of global pollination and local pollination are premised on switch probability p [0, 1].
At the start of the algorithm's execution, the initial population is generated randomly to determine the current best solution. The pollination class should be determined with respect to a fixed probability p which is a random number r [0, 1] to evaluate the new solution. If rand < p, global pollination is actualized and the expression for the flower constancy regarding R1 and R2 is given as: where x t i stands for solution i at time t, γ is the scaling factor, gbest is the current best solution, L is a step size which shows the tenacity of pollination, and it can be expressed as: where is the standard gamma function, and the distribution is effective for large steps s > 0. Considering the second rule i.e., when rand > p, the local pollination process is executed. The local pollination and flower constancy can be described as: where x t i and x t k are pollen from separate flowers of the same plant species, ∈ is a random number between 0 and 1. The main purpose of flower pollination is reproduction. To achieve this, the movement of the pollinators to the favourable region is a key factor. This region of the optimal solution is referred to as the global optimum and is depicted by gbest. Consequently, to enhance the pollination process and attain the global optimum, the basic FPA algorithm is improved by infusing the exploratory competence of the PFA algorithm. The PFA algorithm is discussed in the next session.

E. PATHFINDER ALGORITHM (PFA)
The Pathfinder Algorithm is a new algorithm that was proposed by Yapaci and Cetinkaya [74]. The PFA simulates the features of social movements of the animal group and imitates the leadership structure of the swarms to locate the best food region or prey (global optimum). The PFA consists of a population of candidate solutions known as a swarm. The proposed approach begins by arbitrarily initializing herd VOLUME 9, 2021 members' locations. Afterwards, the fitness of each member is determined and the member's position with the best fitness is chosen as a pathfinder. The pathfinder leads the course of the remaining swarm members and navigates the search space using Eq. (36) while concurrently generating the vector of fluctuation rate (A) using Eq. (37) in every iteration. The process continues until the maximum iteration is reached. The pathfinder's function is not contingent on the group operation and this differentiates the PFA from the rest of the swarm-based algorithms and makes it exceptional [75]. To look for prey or feeding area and for following the pathfinder, the model below was proposed in [74]: where the current iteration is represented by K, the position vector of the ith member is denoted as x i , the position vector of the jth member is denoted as x j , R 1 and R 2 are the random vectors and can be expressed as demonstrated in Eq. (34) Here, r 1 and r 2 are random variables uniformly generated in the range [0, 1], α is the interaction coefficient that determines the stability of a member along with their neighbour, and β is the attraction coefficient that sets the random distance between each member and the leader. Moreover, ε is the vector of vibration and is given as: where D ij is the distance between two members and K max is the maximum iterations. The position of the pathfinder is updated as described in Eq.(36) [74]: where u 1 and u 2 are the random vectors in the range [−1, 1], and r 3 denotes the random vector uniformly generated in the range of [0, 1]. Choosing the optimal values of A and ε ensures that all the members have adequate random motion which enhances exploitation and exploration. Similarly, both α and β also enhance the exploration phase by providing a random movement of the members to get a close pathfinder in the process of finding the feeding area or the hunt.

III. THE PROPOSED HYBRID APPROACH
One of the major tasks needed for the optimal operation of metaheuristic algorithms, especially in control systems, is the achievement of adequate equilibrium between exploration (intensification) and exploitation (diversification). As mentioned earlier, each algorithm has its distinctive capabilities, which can favour exploration or exploitation [76]. As a result, when two algorithms are combined, the mixed algorithm performs better as it chooses the right algorithm attributes to improve the stability between exploration and exploitation, minimizing local minima trapping problems. Exploration ensures that the algorithm reaches the different preferred areas of the search space, while exploitation ensures the search for optimal solutions within the region. An important feature of the proposed h FPAPFA is the capability to attain a balance between exploration and exploitation. The fusion of these algorithms is essential to find the optimal solution, primarily to optimize the parameters of the FOSWPID controllers for the multi-area interconnected AGC system. Whereas FPA has the potential to exploit the search space, it falls short when it comes to exploration. The PFA, on the other hand, is a recent algorithm that is more versatile in exploration. In PFA, there is a pathfinder (global best) which leads the whole population. Two core parameters of PFA, A and ε as defined in Eqs. (34) and (37), play a key role in achieving optimum exploration by maintaining a multidirectional and random movement [74]. The PFA parameters keep on changing in each iteration and incline each member to move towards the leader. Consequently, it enables a good equilibrium between exploration and exploitation. By merging both techniques, the pollination process in FPA is combined with the pathfinder technique in such a way that the position of the FPA is updated with the equations of PFA. The description of the hFPAPFA is shown in Fig. (3) while the flowchart is described in Fig. (4).

IV. RESULTS AND DISCUSSION
The transfer function model of the AGC of the multi-source interconnected power system under consideration is designed and simulated in MATLAB version 9.4 (R2018a) software. Thermal, hydro, wind power and diesel power plants are considered as the power sources with load demand in each area as described in Fig. 1 [34], [77]. The parameters of the proposed algorithm and other algorithms used in this study are presented in Table 1 while the optimal values of the proposed controller in comparison with others are presented in Table 2.
The algorithm is executed for 50 iterations with 10 runs. The fractional-order modelling and control (FOMCON) toolbox of MATLAB [78] was employed for the fractional-order (FO) identification and controller design and optimization. The system is first subjected to different load conditions before undergoing sensitivity tests involving a variety of parameter variations. Moreover, boiler dynamics, the GDB and the GRC have been incorporated into the model to demonstrate the competence of the system towards nonlinearities. The system is further subjected to a random step load perturbation while the nonlinearities are still present to further reveal its efficacy.
The performances indices for the dynamic response considered are the ITSE and ITAE indexes, settling time and peak undershoot. The population size (NP) and the total number of iterations are 30 and 50 respectively. The parameters of the FOSWPID structured AGC system are optimized with the proposed hFPAPFA and the performance is compared with the performance achieved using the techniques such as DE-PID [46], PSO-PIDD, SOS-PID [47], SSA-TID and TLBO-PID [48].

A. LOAD CHANGE SCENARIOS
The subsection describes the response of the power system under different load conditions. At the initial stage, a step load increase of 0.1 p.u is applied in area 1 with no load change in area 2. The performance of the system is observed and compared with other methods. Figs. 5(a)-(c) show the response curve obtained from the results of simulation for frequency deviations and tie-line power in both areas. As depicted in Fig. 5(a), the proposed approach yielded a more desirable response with lesser undershoot and preferable curve smoothness. Also, it is glaring from the results presented in Table 3 Table 5.

B. SENSITIVITY TEST
The fundamental aim of conducting a sensitivity analysis is to determine the system's resilience to large variations in working conditions [79]. Hence, this session is dedicated to the sensitivity test analysis of the controllers which involves varying T G , T T , T R and R at the rate of ±25% from their nominal values without interfering with their optimal values. The results obtained as shown in Table 6 and Figs. (8)- (11) show a little change in the dynamic response of the controllers, within acceptable ranges which is also very close to the results obtained previously for the proposed hFPAPFA-FOSWPID. From a close observation of the response curve as displayed in Figs. (8)- (11), it can be inferred that the impact of the variation in the time constants has a marginal effect on the system. Consequently, it can be said that the proposed control method offers a control system that is sufficiently reliable and robust.

C. INTEGRATION OF NONLINEARITIES
Investigating the system's behavior in the presence of physical limitations, also known as nonlinear components, is a necessary step to provide a reliable result from a realistic control, rendering the system undulating [79]. The dead band in the steam turbine is caused by the linkage that connects the piston of the servo to the camshaft. This occurrence is referred to as backlash. According to reference [48], the aforementioned operation occurs in the rack and pinion used to rotate the camshaft that is responsible for controlling the valves. The speed of the GDB has a substantial impact on the dynamic performance of the AGC system. The backlash nonlinearity effect produces a continuous sinusoidal oscillation with a natural period of about 2s [48]. As per [46], the backlash nonlinearity of 0.005% is considered at the thermal unit while 0.02% is considered in the hydro system. Furthermore, in this  analysis, a GRC of 3% per minute is used for thermal units [46], [80]. The GRC for hydro unit is 270 per cent for raising generation per minute and 360 per cent for lowering generation per minute [81]. Fig. 12 shows the model system with the inclusion of boiler dynamics, GRC and GDB. Fig. 13 shows the boiler dynamics while Figs. 14 (a) and (b) describe the GRC in the thermal and hydro systems, respectively. The nominal parameters of the nonlinear elements are presented in the Appendix. The dynamic responses of the nonlinear system are depicted in Figs. 15(a)-(c) representing the frequency deviation in area 1, frequency deviation in area 2 and the tie-line deviation, respectively. The proposed hFPAPFA-FOSWPID retained its dominance by having the smallest performance index of (ITSE = 0.01217, ITAE = 0.6150) over for PSO   In the same vein, its performance is evident with the least value of peak undershoot.

D. RANDOM SYSTEM LOADING
To elucidate the system's robustness the system is further subjected to random step load perturbation in all areas, as seen in Fig. 16 (a). As shown in Fig.16(b), the proposed method overtook others in terms of frequency deviation in area 1. This is apparent in the smoothness of the curve relative to others. That being said, a similar superior response is obtained in area 2. The proposed approach kept its curve smoother over time, while the others encountered 94164 VOLUME 9, 2021 more dampening as disclosed in Fig.16(c). Alike, the stability of the proposed approach for the tie-line power deviation is further divulged in the pattern of its curve by exhibiting the least deviant curve as shown in Fig. 16(d) compared with other methods. The responses confirmed the robustness of the proposed approach under different conditions, and this time, considering random load step perturbation. Consequently, it can be concluded that the controller is robust and effective.

V. CONCLUSION
In this work, a new algorithm based on the merger of the flower pollinated algorithm and the pathfinder algorithm is used to optimize the setpoint weighted fractional-order PID (FOSWPID) controller for the automatic generation control of a multi-source interconnected power system with renewable energy sources (thermal, hydro, wind power and VOLUME 9, 2021 diesel plants). The system under study was examined under three load changes scenarios. For a fair comparison, the proposed approach's efficiency was examined by comparing its dynamic response with some recently published modern metaheuristic algorithms such as DE, PSO, SSA, SOS, and TLBO while subjecting them to the same multi-source interconnected power system. The results indicated that the proposed controller had a successful dynamic response compared to other controllers with minimal error criterion values, settling time and peak undershoots. Additionally, the robustness of the proposed controlling scheme is examined under different parameter variations in the range of ±25. Moreover, to further test the efficacy of the proposed algorithm three physical constraints such as boiler dynamics, GDB and GRC are incorporated into the system to be more pragmatic.
The proposed approach maintained its superiority in the presence of physical constraints, confirming its efficiency. In addition, a random step load perturbation was introduced into the system under nonlinearities. By having the minimum value of the indexes namely, ITSE, ITAE, settling time and peak undershoot, the system's aggregate performance explicitly showed that the proposed algorithm generates a high-reliability response under various conditions, parameter variance, and nonlinearity effects.
In this work, we focused on a two-area power system. However, future work could be expanded to a more complex multi-source system comprised of several areas and incorporation of plug-in hybrid electric vehicles (PHEVs) for frequency stability improvement and renewable energy sources (RES). Furthermore, the proposed controller or some other control structures such as fractional order setpoint weighted cascaded controller can be designed for the study systems while employing some of the most recent optimization techniques and energy storage devices (ESS).