Optimization of FOPID controller with hybrid Particle Swarm and Grey Wolf Optimization for
AVR system
Abstract—This paper presents the optimization of the fractional-order proportional-integral-derivative (FOPID) controller with the hybrid particle swarm and grey wolf optimization (HPSGWO) algorithm to control the Automatic Voltage Regulator (AVR) system. To overcome the problem of premature convergence and enhance the efficiency of the proposed HPSGWO algorithm, an improved inertia weight is proposed. The response of the proposed HPSGWO is compared with the PSO and GWO algorithms. An integral-based fitness function namely Integral Time Absolute Error (ITAE) is selected as the fitness function while the rise time, settling time and peak overshoot are considered as the dynamic response analysis. The results of simulation show that the proposed HPSGWO algorithm outperformed the PSO and GWO algorithms with rapid rise time, lesser settling time and the least overshoot in controlling the AVR system.
Keywords—Automatic Voltage Regulator (AVR), optimization, Proportional Integral Derivative (PID) controller, Integral Time Absolute Error (ITAE).
I. INTRODUCTION
Automatic voltage regulators (AVR) are being used largely to achieve effective stability and optimum control for power systems in industries. The major purpose of the AVR is to monitor the voltage fluctuations induced by the system's load supply variations and to keep a steady voltage at the generator's output terminal [1]. In large-scale connections, the maintenance of reliable voltage supplies is inevitable. The control flexibility of the proportional-integral-derivative controller (PID) is one way of reinforcing the efficiency of the AVR system. For years, the PID controller controls several activities in industrial processes [2]–[7]. However, studies have shown that the traditional PID controller has poor noise tolerance and is vulnerable to external interference. The introduction of fractional calculus by Podlubny et al. [8] has lately paved way to migrate from classical PID models to those defined by non-integer-order differential equations [9]. The FOPID controller has gained more popularity in industrial applications due to the advantages of the two additional tuning parameters that can be used to improve control laws when applied to control loops. As presented [10], [11] and [12] the controller’s design has much more versatility in tuning and therefore, has a large area of parameters that govern the controlled system and increases the reliability of the control loop. One way to optimize the FOPID controller parameters is to use metaheuristic algorithms. Metaheuristic algorithms in recent times are one of the very important research areas for scholars, academics and scientists [13]. This is because as
opposed to many conventional techniques, they have core
features and capabilities to solve and generate near-optimal solutions to problems without detailed descriptions of the problem concepts. Since the introduction of the first metaheuristic algorithm, a lot of creativity and technological progress has been made. Several methods such as Bat Algorithm (BA) [14], Flower Pollination Algorithm (FPA) [15], Teaching-Learning-based Optimization (TLBO) [16]
,
PSO [17] and GWO [18] have been used in the past to tune the PID controller in order to control the AVR system.
However, rather than using the conventional PID controller, this work proposes the optimization of the FOPID controller with the hybridization of PSO and GWO algorithms. The hybrid technique is introduced because the mixed algorithm can yield better performance since it selects the right algorithm attributes to boost the weaknesses that exist in individual algorithms. Finally, an improved inertia weight constant is also proposed for the algorithm to ensure effective performance in the optimization process.
The paper is structured as follows. Section II discusses the description of the AVR model. The optimization technique and the hybrid technique are presented in Section III. The results of simulation then are discussed in Section IV.
Finally, the conclusion and discussions are is presented in section V.
II. AUTOMATICVOLTAGEREGULATOR(AVR) SYSTEMDESIGN
A. Modelling of AVR system
The AVR comprises of three essential components: exciter, amplifier, sensor and generator which are regarded as linear devices. These essential components are always taken into consideration in the process of developing the mathematical model of the AVR system and are characterized by time constant and gain as are shown in the following Eqs.(1)-(4) [19]:
The transfer function for the amplifier:
𝐺𝐺𝐴𝐴(𝑠𝑠) = 𝐾𝐾𝐴𝐴
1+𝑠𝑠𝑠𝑠𝐴𝐴 (1) The transfer function for the exciter
𝐺𝐺𝐸𝐸(𝑠𝑠) = 𝐾𝐾𝐸𝐸
1+𝑠𝑠𝑠𝑠𝐸𝐸 (2) The transfer function for the generator
𝐺𝐺𝐺𝐺(𝑠𝑠) = 𝐾𝐾𝐺𝐺
1+𝑠𝑠𝑠𝑠𝐺𝐺 (3) The transfer function for the sensor
𝐺𝐺𝑆𝑆(𝑠𝑠) = 𝐾𝐾𝑆𝑆
1+𝑠𝑠𝑠𝑠𝑆𝑆 (4)
where the gains of the amplifier, exciter, generator and sensor are 𝐾𝐾𝐴𝐴, 𝐾𝐾𝐵𝐵, 𝐾𝐾𝐶𝐶 and 𝐾𝐾𝑆𝑆 respectively; 𝑇𝑇𝐴𝐴, 𝑇𝑇𝐵𝐵, 𝑇𝑇𝐶𝐶 and 𝑇𝑇𝑆𝑆
represent their respective time constants. Fig. (1) shows the block diagram of the control system. The time constants and range of gain are given in Table 1 [20]
Fig. 1: Block diagram of the AVR system
Table 1: The components of the AVR transfer function Components Transfer
function
Range of gain and time constants
Parameter values Amplifier 𝐺𝐺𝐴𝐴(𝑠𝑠) = 1+𝑠𝑠𝑠𝑠𝐾𝐾𝐴𝐴
𝐴𝐴
10≤ 𝐾𝐾𝐴𝐴≤400
0.02≤ 𝑇𝑇𝐴𝐴≤0.1 𝐾𝐾𝐴𝐴 = 10 𝑇𝑇𝐴𝐴 = 0.1 Exciter 𝐺𝐺𝐸𝐸(𝑠𝑠) = 1+𝑠𝑠𝑠𝑠𝐾𝐾𝐸𝐸
𝐸𝐸
1≤ 𝐾𝐾𝐸𝐸≤400
0.25≤ 𝑇𝑇𝐸𝐸≤1.0 𝐾𝐾𝐸𝐸 = 1 𝑇𝑇𝐸𝐸 = 0.4 Generator 𝐺𝐺𝐺𝐺(𝑠𝑠) = 𝐾𝐾𝐺𝐺
1+𝑠𝑠𝑠𝑠𝐺𝐺 0.7≤ 𝐾𝐾𝐺𝐺≤400
1.0≤ 𝑇𝑇𝐺𝐺≤2.0 𝐾𝐾𝐺𝐺 = 1 𝑇𝑇𝐺𝐺 = 1 Sensor 𝐺𝐺𝑆𝑆(𝑠𝑠) = 𝐾𝐾𝑆𝑆
1+𝑠𝑠𝑠𝑠𝑆𝑆 1≤ 𝐾𝐾𝑆𝑆≤10
0.001≤ 𝑇𝑇𝑆𝑆≤0.06 𝐾𝐾𝑆𝑆 = 1 𝑇𝑇𝑆𝑆 = 0.01
B. Fractional Order PID controller
The development fractional calculus by Podlubny et al. [8] is a form of improvement on the conventional PID controller which is aimed at improving the controller's tuning versatility, robustness and frequency response. The use of FOPID in industries is ascribed to some more significant benefits obtained from the two extra "tuning knobs" which can be used to improve control laws when implemented to the control loop. Eqs. (5) and (6) describe the output of the PID and FOPID controllers respectively [21].
𝐺𝐺𝑐𝑐(s) = 𝑈𝑈(𝑠𝑠)
𝐸𝐸(𝑠𝑠) = 𝐾𝐾𝑃𝑃 + 𝐾𝐾𝐼𝐼
𝑠𝑠+ 𝐾𝐾𝐷𝐷𝑠𝑠 (5) where u(t) is the PID controller output, the proportional gain is represented by 𝐾𝐾𝑃𝑃, the integral gain is donated by 𝐾𝐾𝐼𝐼 and 𝐾𝐾𝐷𝐷 is the derivative gain. The error is represented as e(t).
𝐺𝐺𝑐𝑐(s) = 𝑈𝑈(𝑠𝑠)
𝐸𝐸(𝑠𝑠) = 𝐾𝐾𝑃𝑃 + 𝐾𝐾𝐼𝐼𝑠𝑠−𝜆𝜆+ 𝐾𝐾𝐷𝐷𝑠𝑠𝜇𝜇 (6) where 𝐾𝐾𝑃𝑃, 𝐾𝐾𝐼𝐼, 𝐾𝐾𝐷𝐷 are proportional, integral and derivative gains respectively. The integral order is represented by λ while μ is the derivative order. The block diagram of the AVR is shown in Fig. (1)
III. OPTIMIZATION TECHNIQUES A. Particle Swarm Optimization (PSO)
The PSO is a population-driven evolutionary algorithm that focuses on the social features of fish schooling and bird flocking. The algorithm is among the most effective strategies used in addressing optimization problems and problems with nonlinearity, multiple optima and high dimensionality [22]. The simplicity, stable convergence characteristics and good computational competence of the PSO algorithm have given it more preference over many other algorithms. The PSO contains a population of candidate solution referred to as a Swarm which moves in a D dimensional search region where certain measurement of efficiency and fitness are being optimized. Each particle has a position and velocity that are denoted in vectors as position vector 𝑋𝑋𝑖𝑖 = (𝑥𝑥𝑖𝑖1,𝑥𝑥𝑖𝑖2…𝑥𝑥𝑖𝑖𝐷𝐷) and velocity vector 𝑉𝑉𝑖𝑖 = (𝑣𝑣𝑖𝑖1,𝑣𝑣𝑖𝑖2…𝑣𝑣𝑖𝑖𝐷𝐷) respectively. Given that 𝑖𝑖 is the index of a particular particle, such particle recalls its best position in a vector 𝑃𝑃𝑖𝑖 = (𝑝𝑝𝑖𝑖1,𝑝𝑝𝑖𝑖2…𝑝𝑝𝑖𝑖𝐷𝐷) while the best vector position among the surrounding neighbours is then stored as a vector 𝑃𝑃𝑔𝑔 = �𝑝𝑝𝑔𝑔1,𝑝𝑝𝑔𝑔2…𝑝𝑝𝑔𝑔𝐷𝐷�. The modified velocity and position of each particle can be calculated according to the following equations [22]:
𝑣𝑣𝑖𝑖𝑖𝑖𝑡𝑡+1 = w𝑣𝑣𝑖𝑖𝑖𝑖(𝑡𝑡) + 𝐶𝐶1𝑟𝑟1 (𝑝𝑝𝑖𝑖𝑖𝑖 - 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡)) + 𝐶𝐶2𝑟𝑟2(𝑝𝑝𝑔𝑔𝑖𝑖 - 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡))
(7)
𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡+1) = 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡) + 𝑣𝑣𝑖𝑖𝑖𝑖(𝑡𝑡+1) (8) where:
𝑝𝑝𝑖𝑖𝑖𝑖: particle best position of agent i
𝑝𝑝𝑔𝑔𝑖𝑖: global particle best position of the group 𝑣𝑣𝑖𝑖𝑖𝑖(𝑡𝑡): velocity of agent i at iteration t,
𝐶𝐶𝑗𝑗: correction factor,
r: random number between 0 and 1 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡): current position of agent i at iteration t
w is referred to as the weighting function represented as given in Eq. (10)
w = 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 - 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚− 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚
𝑖𝑖𝑡𝑡𝑖𝑖𝑖𝑖𝑚𝑚𝑚𝑚𝑚𝑚 × iter (10) where the final and initial weights are 𝑤𝑤𝑚𝑚𝑚𝑚𝑚𝑚 and 𝑤𝑤𝑚𝑚𝑖𝑖𝑚𝑚 respectively; 𝑖𝑖𝑖𝑖𝑖𝑖𝑟𝑟𝑚𝑚𝑚𝑚𝑚𝑚 and iter symbolizes the maximum and current iteration numbers respectively.
A.B. Grey Wolf Optimization
Mirjalili et al. [23] in 2014 developed the Grey Wolf Optimizer (GWO) by replicating the leadership and hunting structure of the grey wolves. The approach is a reflection of the social chain of command and hunting actions in the community of grey wolves [24]. As shown in Fig. 2, there are different modes of imitations that are used in the grey wolf order. The four simulations are Alpha (α), Beta (β), Delta (δ) and Omega (ω). The group leader who is responsible for decision making is the Alpha (α). Decisions such as the place to sleep, the waking and hunting time are being made by Alpha (α). The second rank in the hierarchy consists of Beta (β). Beta (β) wolf assists Alpha (α) in decision making on a variety of activities such as hunting. As shown in Fig. 2, Omega (ω) is the lowermost ranking which follows Alphas
(α) and Betas (β). Omega (ω) wolves submit to the various predominant wolves. The Delta (δ) is subordinate wolf and does not have a place with alpha, beta or omega. Exploration in grey wolf optimization starts with random generation of wolves’ population (solutions). Such wolves determine the site of the prey (optimum) by an iterative technique throughout the hunting (optimization) phase [25]. Fig. 2 shows the Grey wolf hierarchy.
Fig. 2: Grey Wolf Hierarchy [26]
The hunting behavior of wolves is generally categorized in stages as follows [24]:
1. Encircling prey
Eqs. (11) - (12) are the proposed equations in demonstrating the encircling behaviour of GWO.
𝐷𝐷��⃗ = |(𝐶𝐶⃗𝑋𝑋⃗𝑃𝑃 - 𝑋𝑋⃗(𝑖𝑖))| (11)
𝑋𝑋⃗(𝑖𝑖+ 1) = 𝑋𝑋⃗𝑃𝑃(𝑖𝑖) - 𝐴𝐴⃗𝐷𝐷��⃗ (12)
where 𝐶𝐶⃗ and 𝐴𝐴⃗ symbolizes the coefficient vectors, the current iteration is given as t, 𝑋𝑋⃗𝑃𝑃(𝑖𝑖) is the vector position of the victim. 𝑋𝑋⃗ is the position vector of a grey wolf.
Considering the encircling process of the grey wolves around the prey, vectors 𝐴𝐴⃗ and 𝐶𝐶⃗ are:
𝐴𝐴⃗ = 2 𝑎𝑎.���⃗ 𝑟𝑟���⃗1 - 𝑎𝑎⃗ (13) 𝐶𝐶⃗ = 2𝑟𝑟���⃗2 (14) where 𝑟𝑟���⃗1 and 𝑟𝑟���⃗2 are random vectors in the range [0,1], 𝑎𝑎⃗
declined linearly from 2 to 0 throughout iterations.
2. Hunting process
In GWO, the initial three best solutions acquired are retained while the other hunting agents are forced to change their positions because of the best search agents location [25].
Therefore, the following mathematical expressions are proposed.
𝐷𝐷��⃗𝛼𝛼= |𝐶𝐶����⃗ 1𝑋𝑋⃗𝛼𝛼 - 𝑋𝑋⃗|,
𝐷𝐷��⃗𝛽𝛽= |𝐶𝐶����⃗ 2𝑋𝑋⃗𝛽𝛽 - 𝑋𝑋⃗|, (15) 𝐷𝐷��⃗𝛿𝛿= |𝐶𝐶����⃗ 3𝑋𝑋⃗𝛿𝛿 - 𝑋𝑋⃗|
𝑋𝑋⃗1= 𝑋𝑋⃗𝛼𝛼 - 𝐴𝐴����⃗ (1 𝐷𝐷�����⃗𝛼𝛼),
𝑋𝑋⃗2= 𝑋𝑋⃗𝛽𝛽 - 𝐴𝐴����⃗ (2 𝐷𝐷����⃗𝛽𝛽), (16) 𝑋𝑋⃗3= 𝑋𝑋⃗𝛿𝛿 - 𝐴𝐴����⃗ (3 𝐷𝐷����⃗𝛿𝛿),
From Eqs. (15) and (16) the location of the prey signifies the best fitness obtained for α, (𝑋𝑋∝), β, (𝑋𝑋𝛽𝛽), and δ(𝑋𝑋𝛿𝛿) wolfs.
The wolfs’ positions are updated using equation (17).
𝑋𝑋⃗1 (𝑖𝑖+ 1) = 𝑋𝑋�⃗1 (𝑡𝑡) + 𝑋𝑋�⃗2(𝑡𝑡)+ 𝑋𝑋�⃗3(𝑡𝑡)
3 (17) 3. Attacking Prey (Exploitation)
Eq. (15) – (17) present the mathematical description of how grey wolves complete hunting by attacking the prey after they stop moving. Two parameters are tested to describe the concept of approaching the prey in a mathematical manner. 𝑎𝑎⃗
which linearly decreases from 2 to 0 and fluctuations of 𝐴𝐴⃗
which similarly decreases with 𝑎𝑎⃗ [23].
4. Search for Prey (Exploration)
The positions of ω, β and δ determine optimal search in grey wolf algorithm. Wolfs diverge from one another in pursuit of prey and congregate while attacking. Mathematically, search agent diverges towards prey when 𝐴𝐴⃗>1 or 𝐴𝐴⃗<-1.
Consequently, GWO demonstrates more random behaviour all through the optimization and thereby favouring exploration and avoidance of local optima [26].
B.C. The Proposed Hybrid Approach (HPSGWO) Several hybrid nature-inspired techniques have lately been developed by researchers to boost the performance of the exploration and exploitation of existing algorithms. Based on [27] and according to [28] when two algorithms are hybridized at either high level or low level they are referred to as co-evolutionary techniques of hybridization. The hybridization technique of PSO and GWO algorithms proposed in this paper is said to be a low evolutionary mixed hybrid. Because the functionality of both PSO and GWO are combined, then it is low. It is also co-evolutionary because the two algorithms are not implemented one after another, i.e.
they run parallel. It is mixed because the two algorithms are involved in the final solution of the problem. According to [28] instead of using usual mathematical equations, inertia constant can be used to control the exploration and exploitation of the grey wolf in the search space as demonstrated in Eq. (18).
𝐷𝐷��⃗𝛼𝛼= |𝐶𝐶����⃗ 1𝑋𝑋⃗𝛼𝛼 - 𝑤𝑤 ∗ 𝑋𝑋⃗|
𝐷𝐷��⃗𝛽𝛽 = |𝐶𝐶����⃗ 2𝑋𝑋⃗𝛽𝛽 - 𝑤𝑤 ∗ 𝑋𝑋⃗| (18) 𝐷𝐷��⃗𝛿𝛿 = |𝐶𝐶����⃗ 3𝑋𝑋⃗𝛿𝛿 - 𝑤𝑤 ∗ 𝑋𝑋⃗|
The velocity and position of the modified equations are proposed as follows to combine the PSO and GWO variants:
𝑣𝑣𝑖𝑖𝑖𝑖𝑡𝑡+1 = w*(𝑣𝑣𝑖𝑖𝑖𝑖(𝑡𝑡) + 𝐶𝐶1 𝑟𝑟1 (𝑝𝑝𝑖𝑖𝑖𝑖 - 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡)) + 𝐶𝐶2𝑟𝑟2(𝑝𝑝𝑔𝑔𝑖𝑖 - 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡)))
(19)
𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡+1) = 𝑥𝑥𝑖𝑖𝑖𝑖(𝑡𝑡) + 𝑣𝑣𝑖𝑖𝑖𝑖(𝑡𝑡+1) (20)
To find optimum solution in a population-based optimization technique, the control of global exploration and local exploitation must be adequately considered [29]. As a result, a particle with superior performance suggests that it has more
capturing ability than others and that it is easier to achieve the optimal globally than others [30], [31]. The concept of inertia weight (𝑤𝑤) was presented by Shi and Eberhart [32] to improve the performance of the conventional PSO algorithm by harmonizing the local and global search throughout the optimization process. The traditional PSO algorithm has a fixed inertia weight factor. When the inertia weight is overly large, the convergence speed slows down; when the value is too small, it quickly comes down to an optimum local solution [33]. There have been many works on the modification of inertia weight, but it is not clear how this parameter could be modified to boost PSO algorithm efficiency [34]. Nevertheless, the dynamic modification of the inertia weight during iteration will improve the global exploration and local exploitation during the search process.
The linear decreasing inertia weight particle swarm optimization (LDIWPSO) was proposed by Shi and Eberhart [29]. The inertia weight (𝑤𝑤) decreases linearly over the iteration. The mathematical expression is represented in Eq.
(21) [35]:
𝑤𝑤𝑡𝑡= (𝑤𝑤𝑠𝑠𝑡𝑡𝑚𝑚𝑖𝑖𝑡𝑡 − 𝑤𝑤𝑖𝑖𝑚𝑚𝑖𝑖) �𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠 − 𝑡𝑡
𝑚𝑚𝑚𝑚𝑚𝑚 � + 𝑤𝑤𝑖𝑖𝑚𝑚𝑖𝑖 (21) where the initial and final values of the inertia weight are denoted as 𝑤𝑤𝑠𝑠𝑡𝑡𝑚𝑚𝑖𝑖𝑡𝑡 and 𝑤𝑤𝑖𝑖𝑚𝑚𝑖𝑖 in sequence; 𝑖𝑖 represents the present iteration; 𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚 is the maximum iteration number, and 𝑤𝑤𝑡𝑡 is the inertia weight value in the 𝑖𝑖th iteration. An exponential term is incorporated into the conventional LDIWPSO to overcome the problem of premature convergence and enhance its performance. Given the rapid and decreasing nature of the exponential functions, these functions have gained a great deal of attention as an option for decreasing inertia weight strategies [29]. The proposed improved LDIWPSO version is presented in Eq. (22):
𝑤𝑤𝑡𝑡= (𝑤𝑤𝑠𝑠𝑡𝑡𝑚𝑚𝑖𝑖𝑡𝑡 − 𝑤𝑤𝑖𝑖𝑚𝑚𝑖𝑖) �𝑠𝑠𝑚𝑚𝑚𝑚𝑚𝑚𝑠𝑠 − 𝑡𝑡
𝑚𝑚𝑚𝑚𝑚𝑚 � + 𝑤𝑤𝑖𝑖𝑚𝑚𝑖𝑖 ×𝑖𝑖−(
𝑇𝑇𝑚𝑚𝑚𝑚𝑚𝑚𝑡𝑡 4
)2
(22)
where 𝑤𝑤𝑠𝑠𝑡𝑡𝑚𝑚𝑖𝑖𝑡𝑡 and 𝑤𝑤𝑖𝑖𝑚𝑚𝑖𝑖 are given as 0.4 and 0.9 respectively.
The improved LDIWPSO can be considered as an extended version of LDIW which incorporates the natural (base e) inertia weight strategy [36]. The proposed enhanced LDIWPSO introduces the improved efficiency of the PSO algorithm in the hybrid technique.
IV. SIMULATIONRESULTSANDDISCUSSION The simulation was implemented in MATLAB/Simulink environment. Fig. 3 shows the Simulink model of the proposed control system. Fig. 4 shows the terminal voltage step response of the AVR system without using the FOPID controller. Results of simulation obtained shows that the 𝑖𝑖𝑖𝑖= 0.24755(s), 𝑖𝑖𝑠𝑠=6.9676(s) and 𝑀𝑀𝑝𝑝=61.6058%. The result indicates an instability in the power system which is evident in the voltage oscillation and large overshoot.
Fig. 3: Simulink model of the proposed control for the AVR system
Fig. 4: Output response of AVR without FOPID controller.
The characteristics of the convergence curve for PSO- FOPID, GWO-FOPID and HPSGWO-FOPID are shown in Fig. 4, Fig. 5 and Fig. 6 respectively. It can be deduced from the three figures that the proposed HPSGWO-FOPID controller delivers a swift convergence at a more accurate evaluation value. This means that as opposed to other controllers, the proposed controller requires lesser execution time to obtain the optimal values of the FOPID controller parameters.
Fig. 4: Convergence Curve of PSO-FOPID controller
Fig. 5: Convergence Curve of GWO-FOPID controller
Fig. 6: Convergence Curve of HPSGWO-FOPID controller
The step response of the terminal voltage of the AVR with, PSOFOPID, GWOFOPID and HPSGWO-FOPID controllers are shown in Fig. 7 while the results of simulation is presented in Table 2. The proposed controller exhibits a better performance than other controllers with 𝑖𝑖𝑖𝑖=0.24993(s), 𝑖𝑖𝑠𝑠=4.1993(s) and 𝑀𝑀𝑝𝑝=42.7654%. The proposed controller displays lesser oscillation and attains rapid stability at the shortest time with a fast response and better smoothness.
Fig. 7: Output response with AVR for HPSGWO-FOPID, PSOFOPID and GWOFOPID controllers
Table 2
Parameters FOPID PSO- FOPID
GWO- FOPID
HPSGWO- FOPID
𝐾𝐾𝑝𝑝 1.214 1.053 1.2 0.9
𝐾𝐾𝑖𝑖 0.48 0.58103 0.69704 0.55
𝐾𝐾𝑖𝑖 0.32021 0.40622 0.30693 0.45
𝜆𝜆 0.65 0.83077 0.92921 0.7
𝜇𝜇 0.60 0.64171 0.66347 0.65
𝑖𝑖𝑖𝑖(s) 0.24755 0.2495 0.25543 0.24993 𝑖𝑖𝑠𝑠(s) 6.9676 4.2148 4.4102 4.1993 𝑀𝑀𝑝𝑝 (%) 61.6058 47.9594 54.8138 42.7654
V. CONCLUSION
The main aim of this paper is to determine the optimum values of the FOPID controller using the HPSGWO algorithm in order to improve the step response of the terminal voltage of the AVR system within the shortest time and minimum overshoot. The key idea behind the combination is to improve the exploitive ability in PSO with the exploration capability in GWO. Furthermore, an improved inertia weight was also proposed to enhance the global exploration and local exploitation during the search process thereby aiding the proposed algorithm find optimal values of the FOPID controller. The results of simulation show that the proposed HPSGWO algorithm surpassed both PSO and GWO algorithms in terms of the rise time, settling time and overshoot. It can be deduced from the results that the proposed HPSOGWO-based FOPID is a suitable controller for regulating the terminal voltage of AVR system.
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