Enhanced Salp Swarm Algorithm for Multimodal Optimization and Fuzzy Based Grid Frequency Controller Design

Abstract

In the present study, an Enhanced SSA (ESSA) has been proposed where the parameter of the SSA technique, which balances the exploration and exploitation phases, has been modified. Additionally, the variable scaling factor is engaged to regulate the salp’s position during the search procedure to minimize the random movement of salps. To demonstrate the effectiveness of the enhanced SSA (ESSA), a set of multimodal test functions are engaged. The statistical outcomes demonstrate that ESSA profits from local optima evasion and quick convergence speed, which aids the proposed ESSA algorithm to outclass the standard SSA and other recent algorithms. The fair analysis displays that ESSA delivers very promising results and outclass current methods. Next, frequency control of power systems is executed by designing a Combined Fuzzy PID (CFPID) controller with the projected ESSA method. Next, a Partially Distributed CFPID (PD-CFPID) controller is designed for a distributed power system (DPS). It is shown that the ESSA method outclasses the SSA method in engineering problems. It is also noted that the ESSA-based PD-CFPID scheme has become more operative in monitoring the frequency than similar structured centralized fuzzy PID (CFPID) as well as PID controller. Finally, the outcomes of the PD-CFPID controller are equated with CFPID and PID for various uncertain situations to validate the benefit of the proposed control approach.

1. Introduction

With the growing count of irregular renewable and distributed energies in the presence of random load changes, today’s power systems are vulnerable to frequency variations [1,2]. Therefore, a suitable controller is required for frequency regulation. The distributed control approach has multiple distributed controllers for each controllable energy source located within the distributed power system (DPS) [3]. However, the distributed control approach requires one-to-one communication between each controllable source. Complete control requirements such as matching collective load and generation or optimum generation combination are not usually achievable with a distributed control scheme. Hence, if system optimization is the objective, a centralized control scheme is generally favored [4]. In a centralized control scheme, all control requirements are handled by a central computer [5]. While centralized approaches frequently suffer if there is a single failure, decentralized schemes are usually associated with difficult practical implementations and costly communication requirements. Considering the shortcomings of both centralized and decentralized control schemes, this paper proposes to use a partially decentralized controller for the frequency regulation of DPS. The proposed partially decentralized control scheme, which compared with the decentralized control can take the coupling information into account, thus enhancing the system performance without making the design process too complicated.

The controller design is again a tough job and numerous methods have been suggested by researchers in this regard. The salp swarm algorithm (SSA) is a recently proposed process encouraged by the swarming conduct of the salps [6]. The advantage of the SSA method over some similar methods such as the firefly algorithm (FA), bat algorithm (BA), state of matter search (SMS), flower pollination algorithm (FPA), particle swarm optimization (PSO), gravitational search algorithm (GSA), and genetic algorithm (GA) has been testified [6]. The SSA algorithm keeps the best results found so far, therefore SSA is unaffected even if the entire populations worsen. In SSA, salps move gradually towards the leading salp, which prevents stagnating in local minima. The parameter c1 is reduced adaptively during iterations, so the algorithm initially explores and then exploits the search area. The above characteristics enable SSA to outperform other algorithms. However, the original SSA method hurts from early convergence, therefore making it inappropriate for multimodal function optimization problems such as controller design problems. In the SSA method, the convergence parameter (c1) is changed gradually in the early phases of iterations and rapidly in the closing phases, which lowers the exploration competence of the process. In the projected enhanced SSA (ESSA) method, the algorithm is altered to conquer the limitations of the SSA method. Additionally, in the SSA, the location of the supporter salp is calculated from its previous location and surrounded by the salp’s location. The equation for finding the location of follower salp is also altered by means of varied scaling factors to avoid the random movement of salps.

Different heuristic-based optimization methods have been used to determine the values of controller parameters for frequency control approaches as shown in

Table 1. Comparative analysis of various frequency control approaches in different test systems.

Ref. No.	Optimization Technique and Controller	Test System	Remarks
[7]	Quasi oppositional Jaya (QOJAYA) tuned two-degree of freedom (2DOF) PID controller	Two area power systems consist of the thermal and hydropower plants including the nonlinearities.	QOJAYA may not be effective for the systems containing several peaks and trapped in local minima due to its single oppositional-based approach. 2DOF PID controller may not be effective in presence of nonlinearity and uncertainties.
[8]	Sine logistic map based chaotic sine cosine algorithm tuned PID	Islanded microgrid with PV, wind, Fuel Cell, BESS, FESS, DEG, and MT	Time delay and uncertain cases are not considered. The PID controller may not offer effectual control for nonlinear, uncertain systems and in presence of time delay and unstable transfer functions. Centralized PID controller performance may degrade in in presence of nonlinearity and uncertainties.
[9]	Intelligent model predictive control	Microgrid with PV, wind, Fuel Cell, BESS, FESS, DEG, and MT with electric vehicle	Time delay and uncertain cases are not considered. Model predictive control approaches have difficulties with the operation, high maintenance cost, and lack of flexibility resulting in fragile controllers that are not profitable.
[10]	2DOF-tilted integral derivative with filter tuned by bat and harmony search algorithm	Two-area wind-hydro-diesel units with SMES and FACTS devices	Time delay and uncertain cases are not considered. In the 2DOF PID controller, the tuning for the disturbance and the set-point response are not often compatible. Centralized 2DOF-TID performance may degrade in presence of nonlinearity and uncertainties.
[11]	Proportional-derivative with filter cascaded-proportional-integral tuned by the coyote optimization algorithm	Two-area power system with photovoltaic (PV) and wind farm and gas turbine interconnection	The effectiveness of the optimization technique has not been tested in benchmark test functions. Time delay is not considered. Variants of PID controllers are suitable for linear systems and incompatible for nonlinear systems.
[12]	Chaotic atom search optimization based fractional-order PID controller	Multi-area hybrid power system consisting of thermal, hydro, gas, solar-thermal, wind, and aqua electrolyzer-fuel cell	Sever uncertainties such as unavailability of some sources and communication delays are not considered. Decentralized FO PID controllers have difficult practical implementations and costly communication requirements.
[13]	Sailfish optimizer (SFO) optimized fuzzy tilt integral derivative controller	Microgrid containing DEG, PV, FC, AE, BESS, FESS, and FESS	In SHO, the combination may create prey extra prominent to predators and may amplify intra-specific rivalry. Additionally, those in the preferred central locations may have inferior feeding rates. A centralized control scheme may cause a problem in case of a single failure.
[14]	Mayfly optimization-based fuzzy PD-(1 + I) controller	Interconnected microgrid containing Solar–thermal, Wind, Micro-hydro turbine, Biodiesel, and Biogas generators.	In the MO method, if the present locations were away from the best locations, slower convergence may occur. A centralized Fuzzy PD-(1 + I) control scheme is inferior to a decentralized scheme and may cause a problem in case of a single failure.
[15]	Direct synthesis (DS) method based fractional order PID controller	Reheat thermal, hydropower, and non-reheat thermal IEEE 39-bus New England IEEE 39-bus test system along with variable communication delay.	The direct synthesis (DS) approach depends on the process model. The overshoots and undershoots in transient response may be high and fine-tuning may be required. The study is limited to conventional generating units. PID controllers are not suitable for nonlinear and uncertain systems.
[16]	Atom search optimization-based FOPID controller	Two area hybrid power systems containing PEV, WTPG, STPG, and thermal units consider nonlinearities.	In ASO, a small variation in velocities causes inferior exploitation in the later stages of the algorithm. FOPID controllers are not suitable for nonlinear and uncertain systems.
[17]	Grey wolf optimization (GWO) based FOPID controller	A hybrid power system containing PEV, WTG, PV, and DEG with Energy storage elements.	The GWO method calculates the wolves’ positions by the mean locations, ignoring the wolf ladder, which may lead GWO to local convergence. FOPID controllers are not suitable for nonlinear and uncertain systems.
[18]	Ant lion optimizer (ALO) based FOPID controller	Two area systems containing conventional and renewables	The arbitrary walking method in ALO results in a large run time. FOPID controllers are not suitable for nonlinear and uncertain systems.
[19]	Particle swarm optimization-based Fuzzy PI controller	Thermal generator with BESS including the effect of demand response	PSO has poor exploration capability with a large execution time for complex systems. Decentralized FO PID controllers have difficult practical implementations and costly communication requirements.
[20]	Archimedes optimization algorithm tuned integral derivative-tilted (ID-T) controller,	Two area systems containing conventional and renewables units.	AOA is not resistant to the unequal exploration and exploitation stages leading to local optimum. A system with a centralized ID-T controller will collapse in case of a single failure.
[21]	Marine predators algorithm (MPA) tuned PID controller	Two-area system containing conventional and renewable energy sources (wind, PV, and STPP) and energy storing units (SMES and BES)	The disadvantages of the MPA are failure to create a diverse early population, local minima avoidance, and poor exploration. PID controllers are not suitable for nonlinear and uncertain systems.
[22]	Adaptive-neuro-fuzzy inference system (ANFIS) based controller	Single area and two-area hydropower plants.	ANFIS applications in problems with large inputs are computationally expensive. ANFIS controller is complex and requires the expert user to handle.
[23]	Artificial bee colony (ABC) based PID fuzzy logic controller	Fourteen generations Australian test system with wind and battery integration.	ABC suffers from improper exploitation in solving complicated problems. The controller is not adaptive to handle uncertainties.

. The table explains the technique, controller type, test system, and the limitations of the frequency control approach used for various test systems.

The chief contributions of the study are:

i.

To develop an enhanced SSA (ESSA) process and authenticate its effectiveness over original SSA and other similar algorithms [24,25,26,27,28,29,30,31,32,33] using 16 multimodal test functions of different dimensions.

ii.

To design a combined fuzzy PID (CFPID) for frequency control in two area systems and compare its performance with similar approaches reported in the literature.

iii.

To design a partially decentralized combined fuzzy PID (PD-CFPID) controller for frequency regulation of a hybrid power system containing PV, DEG, WTG, with energy storage elements such as FESS, AE, FC, BES, and EV considering the inherent nonlinearities and communication delays.

iv.

To assess the efficacy of ESSA based PD-CFPID under the following uncertain scenarios:

Scenario A: P_WTG, P_PV, and P_D are augmented by 200%;

Scenario B: P_WTG and P_PV are unavailable but the P_D is augmented by 200%;

Scenario C: EV power and load demand pattern are augmented by 100% and 200%, respectively;

Scenario B: P_WTG and P_PV are decreased by 50% but P_EV and P_D are augmented by 100% and 200%, respectively;

Scenario E: The time delay is enhanced to 400 ms.

2. Overview of Enhanced Salp Swarm Algorithm (ESSA)

The main inspiration of the salp swarm method is the spilling over the behavior of the principal swarms on the earth [6]. The whole inhabitants of salp are separated into leader and followers’ groups. The best location of the salp order is taken by the leader and guides the remaining swarms. The remaining salps trail the leader. The location of the salp is categorized by an n-dimensional search area. The food resource T is the aim for the salp in the pursuit search.

The position of the leader is given by Equation (1) as

$$x_j^1=\{\begin{array}{c}{T}_j+c_1((u{b}_j-l{b}_j)c_2+l{b}_j)\begin{array}{cc}& c_3\ge 0.5\end{array}\\ T_j-c_1((u{b}_j-l{b}_j)c_2+l{b}_j)\begin{array}{cc}& c_3\le 0.5\end{array}\end{array}$$

(1)

where $x_j^1$ and $T_j$ represent the locations of the leader and food source in the j-th dimension. The leader changes its place concerning the food resource only. The bounds of j-th dimension are specified by $l{b}_j,u{b}_j$. The parameter $c_2$ and $c_3$ are arbitrary values in [0, 1]. The coefficient $c_1$ equilibriums the exploration and exploitation stages of the SSA method.

It is represented as

$$c_1=K_1\exp \left[-{\left(\frac{4i_c}{I_M}\right)}^2\right]$$

(2)

where $i_c$ and $I_M$ signifies the current and the maximum iterations and $K_1$ is a constant which is set as 2 in SSA.

The location of the remaining salps is restructured by

$$x_j^i=\frac{1}{2}a{t}^2+v_{0}t$$

(3)

Equation (3) corresponds to Newton’s law of motion where $i\ge 2,$ t signifies time, $v_0$ _which means the original speed. These are calculated as: $a=\frac{v_{final}}{v_0}$ and $v_0=\frac{x-x_0}{t}$. Considering $v_0=0$ that variance amid iterations is equal to 1, then Equation (3) is altered as,

$$x_j^i=\frac{(x_j^i+x_j^{i-1})}{2}\begin{array}{cc}& i\ge 2\end{array}$$

(4)

where $x_j^i$ signifies the position of the i-th follower in j-th dimension.

In the proposed ESSA procedure, two SSA parameters were altered for better symmetry between the exploration and exploitation stages. A balance between the two enhances the process. The occasion of the leader is a vital aspect of the SSA process which is eventually guided by parameters $c_1$. The value of $K_1$ is taken as 2 in the SSA process which administrates the movement of the leader salp associated to T. An large $K_1$ may cause salp stirring outside the food source location. In ESSA process $K_1$ is varied to make the spreading of searched arguments near the best than the initial argument. In SSA, $c_1$ changes gradually linearly iterations and quickly in later stages. If $c_1$ is set higher values in the latter stages, it will make the method produce a greater aberration of the leader’s location. These glitches are also quashed in ESSA with the adapted value of $K_1$ by randomly changing to a lesser $c_1$ value in the final phases as:

$$K_1=\left\{\begin{array}{ll}2(1-i_c/I_M)& if randm<0.5\\ 2& if randm\ge 0.5\end{array}\right\}$$

(5)

The position of the follower salp in the original SSA is expressed as:

$$x_j^i=\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(x_j^i+x_j^{i-1}\right)\begin{array}{cc}& i\ge 2\end{array}$$

(6)

where $x_j^i$ and $x_j^{i-1}$ are the positions of i-th and (i − 1)-th salps in j-th dimension. In the early stages, the finest probable search agent is unidentified in the search area. Therefore, the procedure of updating in huge steps firstly will result in stirring salps missing the best locations. Thus, the usage of scaling factors (SF) in ESSA regulates the advancement of search agents during the search procedure. The new equation is formulated as:

$$x_j^i=SF\cdot \left[\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\left(x_j^i+x_j^{i-1}\right)\right]\begin{array}{cc}& i\ge 2\end{array}$$

(7)

where SF is varied linearly from 0.1 to 1 as expressed in Equation (8):

$$SF=1/(10-9\frac{i_c}{I_M})$$

(8)

The inclusion of SF adjusts the drive of saps in the early stages of the method, thus augmenting the search competence of the algorithm. In the advanced phases, as improved solutions are found, salps move in the direction of them at progressively increased speed.

3. Performance Study of ESSA Technique

In this study, the engineering application is a controller design problem. Tuning the controller parameter is a highly nonlinear and complex multimodal optimization problem as many settings of the controller could yield satisfactory performance. As the application area of the present study is the controller design problem, i.e., multimodal optimization problem, therefore benchmark multimodal test functions are used to evaluate the performance of the proposed ESSA technique.

The proposed ESSA is verified on some multimodal functions [34,35] and given in

Table 2. Multimodal functions.

Function Name	Expression	Range	Dim.	Opt.
Generalized Schwefel	$mm{f}_1\left(y\right)={\displaystyle {\displaystyle \sum }_{i=1}^n} -y_{i}sin\left(\sqrt{\left|y_i\right|}\right)$	[−500, 500]	10	−2094.9145
Generalized Rastrigin	$mm{f}_2\left(y\right)={\displaystyle {\displaystyle \sum }_{i=1}^n} \left[y_i^{2 }-10coscos \left(2\pi y_i\right)+10\right]$	[−5.12, 5.12]	10	0
Ackley	$\begin{array}{c}mm{f}_3\left(y\right)=-20expexp \left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^n} y_i^2}\right)\\ -exp\left(\frac{1}{n}{\displaystyle {\displaystyle \sum }_{i=1}^n} cos\left(2\pi y_i\right)\right)+20\end{array}$	[−32, 32]	10	0
Generalized Griewank	$mm{f}_4\left(y\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum }_{i=1}^n} y_i^2-{\displaystyle {\displaystyle \prod }_{j=1}^n} cos\left(\frac{y_i}{\sqrt{i}}\right)+1$	[−600, 600]	10	0
Generalized Penalized Function 1	$\begin{array}{c}mm{f}_5\left(y\right)=\frac{\pi }{n}\{10sinsin \left(\pi x_i\right)\hfill \\ +{\displaystyle {\displaystyle \sum }_{i=1}^{n-1}} {\left(y_i^{-}1\right)}^2\left[1+10si{n}^2\left(\pi x_{i+1}\right)\right]\}\\ +{\left(x_n-1\right)}^2+{\displaystyle {\displaystyle \sum }_{i=1}^n} u(y_i,10,100,4)\\ x_i=1+\frac{y_i+1}{4}\\ u\left(y_i,a,k.m\right)=\{k{\left(y_i-a\right)}^m{y}_i>a 0-a<y_i\\ <a k{\left(-y_i-a\right)}^m{y}_i<a \end{array}$	[−50, 50]	10	0
Generalized Penalized Function 2	$\begin{array}{l}mm{f}_6\left(y\right)=0.1\{si{n}^2+{\displaystyle {\displaystyle \sum }_{i=1}^n} {\left(y_i-1\right)}^2\left[1+si{n}^2\left(3\pi y_i+1\right)\right]\\ +{\left(y_n-1\right)}^2\left[1+si{n}^2\left(2\pi y_n\right)\right] \}\\ +{\displaystyle {\displaystyle \sum }_{i=1}^n} u\left(y_i,5,100.4\right)\end{array}$	[−50, 50]	10	0
Shekel’s Foxholes	$mm{f}_7\left(y\right)={\left(\frac{1}{500}+{\displaystyle {\displaystyle \sum }_{j=1}^{25}} \frac{1}{j+{{\displaystyle \sum }}_{i=1}^2{\left(y_i-a_{ij}\right)}^2}\right)}^{-1}$	[−65.536, −65.536]	2	1
Kowalik	$mm{f}_8\left(y\right)={\displaystyle {\displaystyle \sum }_{i=1}^{11}} {\left[a_i-\frac{y_1\left(b_i^2+b_i{y}_2\right)}{b_i^2+b_i{y}_3+y_4}\right]}^2$	[−5, 5]	4	0.0003
Six-Hump Camel-Back	$mm{f}_9\left(y\right)=4y_1^2-2.1y_1^4=\frac{1}{3}{y}_1^6+y_1{y}_2-4y_2^2+4y_2^4$	[−5, 5]	2	−1.0316
Branin	$mm{f}_{10}\left(y\right)={\left(y_2-\frac{5.1}{4y^2}{y}_1^2+\frac{5}{\pi }{y}_1-6\right)}^2+10\left(1-\frac{1}{8\pi }\right)cos\left(y_1\right)+10$	[−5, 10] [0, 15]	2	0.398
Goldstein–Price	$\begin{array}{l}mm{f}_{11}\left(y\right)=[1+{\left(y_1+y_2+1\right)}^2(19-14y_1+3y_1^2-14y_2\\ +6y_1{y}_2+3y_2^2)]\\ \times [30+{\left(2y_1-3y_2\right)}^2\times (18-32y_1+12y_1^2\\ +48y_2-36y_1{y}_2+27y_2^2)]\end{array}$	[−2, 2]	2	3
Hartman’s 1	$mm{f}_{12}\left(y\right)=-{\displaystyle {\displaystyle \sum }_{i=1}^4} c_{i}exp\left(-{\displaystyle {\displaystyle \sum }_{j=1}^3} {\mathsf{a}}_{ij}{\left(y_j-p_{ij}\right)}^2\right)$	[0, 1]	3	−3.86
Hartman’s 2	$mm{f}_{13}\left(y\right)=-{\displaystyle {\displaystyle \sum }_{i=1}^4} c_{i}exp\left(-{\displaystyle {\displaystyle \sum }_{j=1}^6} {\mathsf{a}}_{ij}{\left(y_j-p_{ij}\right)}^2\right)$	[0, 1]	6	−3.32
Shekel’s 1	$mm{f}_{14}\left(y\right)=-{\displaystyle {\displaystyle \sum }_{i=1}^5} {[\left(Y-{\mathsf{a}}_i\right){\left(Y-{\mathsf{a}}_i\right)}^T+c_i]}^{-1}$	[0, 10]	4	−10.1532
Shekel’s 2	$mm{f}_{15}\left(y\right)=-{\displaystyle {\displaystyle \sum }_{i=1}^7} {[\left(Y-{\mathsf{a}}_i\right){\left(Y-{\mathsf{a}}_i\right)}^T+c_i]}^{-1}$	[0, 10]	4	−10.4028
Shekel’s 3	$mm{f}_{16}\left(y\right)=-{\displaystyle {\displaystyle \sum }_{i=1}^{10}} {[\left(Y-{\mathsf{a}}_i\right){\left(Y-{\mathsf{a}}_i\right)}^T+c_i]}^{-1}$	[0, 10]	4	−10.5363

. Multi-modal functions (mmf) 1 to 6 are of 10 dimensions and 7 to 16 are of different fixed dimensions. To authenticate the efficacy of the suggested ESSA algorithm it is equated with SSA as well as with WCMFO [24], ABC [25], GA [26], CamWOA [27], PSO [28], GSA [29], PSOGSA [30], WCA [31], MFO algorithm [32] and DA [33] as given in [24,27]. The no. of function estimations is set as 100,000 for all approaches for an unbiased assessment. In ESSA, 1000 iterations with 100 no. of search agents are considered. The parameters of all the techniques are taken from reference [24,27]. The statical outcomes found in 30 runs are assembled. For the appropriate selection of SF, several equations are tried and evaluated. It is realized that for the many functions best outcomes are found when SF is found by Equation (8) as related to others.

The statistical result of ESSA and SSA obtained along with the outcomes of the other 10 methods available in reference [24,27] are presented in

Table 3. Statistical results for 10 dimension functions.

Function	ESSA		SSA		GA		PSO
	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev
mmf1y	−2796.049	3349.21	−2930.15	3099.72	−3692.39	182.42	−2742.78	274.7175
mmf2y	0	0	11.8731	3.7767	3.8E−4	3.2E−4	1.757	1.1592
mmf3y	8.88E−16	0	0.2089	0.4824	8.88E−16	1.0E−31	8.88E−16	1.00E−31
mmf4y	3.29E−14	1.897E−14	0.268906	0.14071	5.6E−2	3E−2	0.1244	8.04E−2
mmf5y	3.501E−16	2.437E−16	0.09329	0.16637	5.73E−05	1.4E−4	4.71E−32	1.67E−47
mmf6y	1.829E−15	1.456E−15	1.831E−3	4.164E−3	6.21E−05	1.1E−4	1.34E−32	5.56E−48
Function	DA		WCA		GSA		MFO
	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev
mmf1y	−3213.66	431.748	−3422.55	304.572	−1694.53	190.6721	−3329.13	288.317
mmf2y	11.561	10.177	20.993	10.524	1.392	1.214	12.8372	7.352
mmf3y	3.14E−05	1.7E−04	2.42E−15	1.79E−15	1.28E−10	6.71E−11	8.88E−16	1.00E−31
mmf4y	0.3846	0.3826	0.1502	9.44E−2	1.67E−2	2.79E−2	1.78E−01	8.43E−02
mmf5y	0.5296	0.6912	1.036E−2	5.67E−2	7.95E−21	3.23E−21	3.11E−02	9.487E2
mmf6y	0.5292	0.7173	7.3E−4	2.7E−3	5.67E−20	1.88E−20	1.10E−3	3.33E−3
Function	CS		PSOGSA		ABC		WCMFO
	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev
mmf1y	−3712.01	167.4447	−3271.6	278.08	−3922.73	88.61857	−3729.7	96.325
mmf2y	6.574	1.367	23.281	12.968	3.677	1.0365	2.089	1.508
mmf3y	1.24E−15	1.08E−15	4.94E−12	2.26E−12	1.21E−06	9.37E−07	8.88E−16	1.00E−31
mmf4y	3.96E−02	8.8E−3	0.2004	0.1141	0.281	0.1086	9.91E−02	5.31E−2
mmf5y	9.77E−05	1.3E−4	0.2491	0.581	1.9E−3	1.3E−3	2.00E−29	6.44E−29
$mm{f}_6\left(y\right)$	1.31E−09	1.39E−09	3.11E−21	1.06E−21	8.3E−3	5.1E−3	4.49E−22	2.06E−21

and

Table 4. Statistical results for fixed dimensional functions.

Function	ESSA		SSA		GA		PSO
	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev
mmf7y	0.998	0	0.998	0	0.998	8.83E−14	1.56	0.959
mmf8y	3.421E−4	1.678E−4	8.863E−4	3.397E−4	8.4E−4	2.9E−4	7E−4	3.2E−4
mmf9y	−1.0316	0	−1.0316	0	−1.03	5.02E−10	−1.03	3
mmf10y	3.98E−1	8E−17	3.98E−1	4.0E−16	3.98E−1	4.73E−7	3.98E−1	1.13E−16
mmf11y	3	4.054E−14	3	4.3E−14	3	1.21E−8	3	4.52E−16
mmf12y	−3.86	2.14E−15	−3.86	5E−15	−3.86	2.203E−3	−3.86	2.7E−15
mmf13y	−3.2504	3.017E−2	−3.25	6.001E−3	−3.32	2.170E−2	−3.26	6.04E−2
mmf14y	−10.1532	5E−14	−9.7342	1.6244	−10.2	0.00048	−9.31	1.9255
mmf15y	−10.4029	2.14E−11	−10.2271	0.9629	−9.93	1.822	−9.52	2.00228
mmf16y	−10.5364	4E−15	−10.5364	4E−15	−9.61	2.405	−10	1.6357
Function	DA		WCA		GSA		MFO
	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev
mmf7y	1.1	0.303	0.998	3.39E−16	3.4	2.578637	1.03	0.181483
mmf8y	1.34E−3	5.11E−4	3.69E−4	2.32E−4	1.8E−3	4.9E−4	8.37E−4	2.54E−4
mmf9y	−1.03	2.55E−11	−1.03	0	−1.03	0	−1.03	0
mmf10y	3.98E−1	7.6E−13	3.98E−1	3.79E−16	3.98E−1	1.13E−16	3.98E−1	1.13E−16
mmf11y	3	1.38E−6	3	1.79E−14	3	4.02E−15	3	1.95E−15
mmf12y	−3.86	1.587E−03	−3.86	2.71E−15	−3.86	2.71E−15	−3.86	2.71E−15
mmf13y	−3.25	6.72E−02	−3.26	6.04E−2	−3.32	1.36E−15	−3.22	4.5066E−2
mmf14y	−9.81	1.28	−8.31	2.718	−7.45	3.381188	−7.56	3.323037
mmf15y	−10.4	0.192	−9.52	2.002	−10.4	0	−9.35	2.423664
mmf16y	−10.3	1.06	−9.82	2.235	−10.5	9.03E−15	−10.3	1.39948
Function	CS		PSOGSA		ABC		WCMFO
	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev	Av.	St.Dev
mmf7y	0.998	3.39E−16	1.06	0.252	0.998	1.02E−13	0.998	5.36E−16
mmf8y	3E−4	4.23E−9	3.79E−3	7.5E−3	7E−4	1.3E−4	3.0E−4	1.07E−15
mmf9y	−1.03	0	−1.03	0	−1.03	7.36E−11	−1.03	0
mmf10y	3.98E−1	1.13E−16	3.98E−1	1.13E−16	3.98E−1	5.68E−09	3.98E−1	1.13E−16
mmf11y	3	4.52E−16	3	4.52E−16	3	8.64E−05	3	9.57E−15
mmf12y	−3.86	2.71E−15	−3.86	2.71E−15	−3.86	7.89E−11	−3.86	2.71E−15
mmf13y	−3.32	1.26E−13	−3.26	6.032E−2	−3.32	4.82E−06	−3.25	6.027E−2
mmf14y	−10.2	1.81E−15	−5.9	3.421068	−10.1	6.784E−3	−8.89	2.361515
mmf15y	−10.4	6.18E−14	−5.76	3.454976	−10.4	2.886E−3	−10.4	1.59E−12
$mm{f}_{16}\left(y\right)$	−10.5	2.15E−12	−6.99	3.890193	−10.5	4.074E−3	−10.5	4.09E−14

for 10 dimensions and fixed dimensions mmfs’, respectively. It can be noticed from Table 3 and Table 4 that suggested ESSA outperforms original SSA in 15 out of 16 mmfs’ (except $mm{f}_6\left(y\right)).$ It can also be realized from Table 3 that ESSA outperforms all other 11 techniques in the majority of functions (in 3 out of 6 mmfs’). For fixed dimension multimodal functions, the proposed ESSA provides an optimum of equally best results in 9 out of 10 mmfs’ as evident from Table 4.

Convergence figures for ESSA and SSA are related in Figure 1 for some mmfs’ from which it can be noticed that the convergence feature of ESSA is improved. This is because of modifying the value K₁ by randomly changing to a lower value C₁ during the latter stages of iteration and the use of SF to restrict the drive of saps in the initial periods of the search process. It is worthwhile to mention here that for mmf₁, typical ESSA converges near to the best value of −2094.9145. Similarly, for mmf₃ and mmf₁₄ even though both ESSA and SSA converge to the same value, ESSA converges faster compared to SSA.

4. Proposed Frequency Control Approach

The proposed ESSA method is applied tried for frequency regulation of hybrid power system (HPS) as shown in Figure 2. It contains PV, WTG FC, DEG, AE, BESS, FESS, and electric vehicles, etc. [4,36,37,38,39]. The performance of HPS in the presence of wind and solar generation is extremely erratic and therefore the power balance is a complex task. Hence this problem can be resolved by appropriate management among stability and controllability. In this study, a partially decentralized controller is engaged, which makes it a modest control and requires less maintenance as well as dealing with fewer controllable parameters. Limiters are involved in each component to match fluctuations in the electromechanical features of the specific subsystem. In addition, time delays associated with input/output to the controller (10 ms) are considered. In Figure 2, the control participation factors (CPF) of controllable sources such as DEG, FESS, BESS, and EV are represented by corresponding CPFs.

4.1. Modelling of Parts

4.1.1. Wind Turbine Generator (WTG)

The WTG power output is assumed as [5]:

$$P_{WP}=\frac{1}{2}\rho A_R{C}_P{V}_W^3$$

(9)

The transfer function (TF) of WTG is

$$G_{WT{G}_N}(s)=\frac{K_{WTG}}{1+s{T}_{WTG}}=\frac{\Delta P_{WTG}}{\Delta P_{WP}}$$

(10)

where N = 1, 2, 3.

4.1.2. Photovoltaic Cell (PV)

PV power output is assumed as [5]

$$P_{PV}=\eta \cdot S\cdot \varphi [1-0.005(T_a+25)]$$

(11)

where $\eta$ is the conversion efficiency of PV cells and taken as 10%, S is the area of PV array taken as 4084 m² and $\varphi$ is the solar irradiation in kW/m². A PV TF is assumed as [5]

$$G_{PV}(s)=\frac{K_{PV}}{1+s{T}_{PV}}=\frac{\Delta P_{PV}}{\Delta \varphi }$$

(12)

4.1.3. Aqua Electrolyzer

Aqua electrolyzer receives a majority of the total power from renewable sources. The AETF is stated as [5]

$$G_{AE}(s)=\frac{K_{AE}}{1+s{T}_{AE}}=\frac{\Delta P_{AE}}{U_2}$$

(13)

4.1.4. Fuel Cell

The TF of FC is assumed as

$$G_{F{C}_N}(s)=\frac{K_{FC}}{1+s{T}_{FC}}=\frac{\Delta P_{FC}}{\Delta P_{AE}}$$

(14)

where N = 1, 2.

4.1.5. Diesel Engine Generator (DEG)

The TF of DEG is assumed as

$$G_{DEG}(s)=\frac{K_{DEG}}{1+s{T}_{DEG}}=\frac{\Delta P_{DEG}}{\Delta U}$$

(15)

4.1.6. FESS and BESS

The TF of FESS and BESS are [5]

$$G_{FESS}(s)=\frac{K_{FESS}}{1+s{T}_{FESS}}=\frac{\Delta P_{FESS}}{\Delta U}$$

(16)

$$G_{BESS}(s)=\frac{K_{BESS}}{1+s{T}_{BESS}}=\frac{\Delta P_{BESS}}{\Delta U}$$

(17)

4.1.7. Limiter and Saturation

The energy storage devices are included in the control loop and are regulated by the controller. They have constraints that permit the devices to operate in the nonlinear region. These limitations avert the mechanical shudder because of sudden frequency variations. The constraints are given by

$$\begin{array}{c}|\stackrel{•}{P_{FESS}}<0.09|, |\stackrel{•}{P_{BESS}}<0.09|, |\stackrel{•}{P_{DEG}}<0.01|, |\stackrel{•}{P_{EV}}<0.01|, |P_{FESS}<0.9|,\\ |P_{BESS}<0.2|, |P_{EV}<0.8| \mathrm{and} 0\le P_{DEG}\le 0.45\end{array}$$

4.1.8. Electric Vehicle

The figure presentation modeling of EV for frequency control is demonstrated in Figure 3 [40]. The LFC signal $\Delta U$ is supplied to the EV for discharging/charging. Parameters $\pm BkW$ signify the battery capacity. The existing battery energy is signified by E that is kept inside the restrictions $E_{\max }$ and $E_{\min }$ presumed as 90% and 60%. $K_1$ and $K_2$ are found as $K_1=E-E_{\max }$, $K_2=E-E_{\min }$. The stored energy part in Figure 3 computes the remaining stored energy [40].

4.1.9. Power System

The input and the output to the power system are the difference in power (ΔP) and (Δf) and is given as:

$$G(s)=\frac{\Delta f}{\Delta P}=\frac{1}{D+sM}$$

(18)

Here D characterizes the damping parameter and M means inertia value.

4.2. Combined Fuzzy PID Structure

The fuzzy logic control (FLC) based PID controller can be designed by tuning scaling factors and PID parameters [41,42,43,44,45]. In the present work, a combined fuzzy PID (CFPID) is projected as revealed in Figure 4 [45]. To ensure that the controller is adaptive, the input signal is accepted by fuzzy and also straight to the PID as revealed in Figure 5. The membership functions are allocated linguistic PoBg (Positive Big), PoSm (Positive Small), Ze (Zero), NeSm (Negative Small), and NeBg (Negative Big) for inputs/output variables as shown in

Table 5. Fuzzy rule foundation for CFPID.

ede	NeBg	NeSm	Ze	PoSm	PoBg
NeBg	NeBg	NeBg	NeSm	NeSm	Ze
NeSm	NeBg	NS	NeSm	PoSm	PoSm
Ze	NeSm	NeSm	Ze	PoSm	PoSm
PoSm	NeSm	Ze	PoSm	PoSm	PoBg
PoBg	Ze	PoSm	PoSm	PoBg	PoBg

. A Mamdani fuzzy inference engine is used in the present study.

An integral square error (ISE) criterion is taken as an objective function as:

$$J={\displaystyle \underset{0}{\overset{T}{\int }}\left[{(\Delta f)}^2+{(\Delta U)}^2/kn\right]}\cdot dt$$

(19)

To ensure that both terms in Equation (19) participate in the search technique, kn is allocated 5.

5. Results and Discussions

5.1. Performance Examination of Proposed Frequency Control Approach

The effectiveness of the recommended ESSA tuned CFPID structure for frequency control is examined in two widely employed 2-area power systems [46,47,48,49,50,51,52]. The first test system is a non-reheat system and the second test system is a reheat system. For a fair comparison with published frequency control approaches, the same load disturbance (10% SLP for the first test system and 1% SLP for the second test system) and objective function (ITAE) are considered. In addition, two similar CFPID controllers are assumed in each area. The ESSA optimized CFPID controller parameters are provided in

Table 6. Optimized parameters for two test systems.

1st test system	K1 = 1.9847, K2 = 0.5802, KP1 = 1.7138, KI1 = 1.9742, KD1 = 0.2796, KP2 = 1.4912, KI2 = 1.9966, KD2 = 0.2249
2nd test system	K1 = 1.9847, K2 = 1.9867, KP1 = 1.9959, KI1 = 1.9742, KD1 = 0.002, KP2 = 1.9965, KI2 = 1.9966, KD2 = 0.0999

. The results of the proposed controller are related to traditional as well as numerous new optimization techniques, for example, Ziegler Nichols (ZN) [46], GA [46], BFOA [46], PSO [47], hybrid BFOA-PSO [47], NSGA-II based PI [48], NSGA-II optimized PIDF [48], hybrid PSO-PS optimized fuzzy PI [43] as well as mMSA Fuzzy PD-PI [37] for the first test system and shown in

Table 7. Performance with different controllers for 1st test system.

Method/Controller	ITAE	Settling Time (Ts) S			Undershoot (Us)-ve
		Δf1	Δf2	ΔPtie	Δf1	Δf2	ΔPtie
GA: PI [46]	2.7475	10.3	10.3	9.3	0.23	0.19	0.07
BFOA: PI [46]	1.8379	7.1	5.5	6.35	0.27	0.23	0.08
hBFOA-PSO-PI [47]	1.1865	6.6	6.2	5.73	0.24	0.21	0.071
NSGA-II: PIDF [48]	0.387	4.86	3.03	4.34	0.103	0.052	0.023
hPSO-PS-Fuzzy PI [43]	0.1438	5.25	4.07	4.01	0.07	0.035	0.012
mMSA: Fuzzy PD-PI [37]	0.1783	2.71	4.29	3.95	0.052	0.038	0.011
Proposed ESSA based CFPID	0.0262	1.21	1.05	1.02	0.0483	0.0167	0.0058

and Figure 6 for the first test system.

It is obvious from Table 7 and Figure 6 that the least ITAE value and best system response are attained with an ESSA-based CFPID controller compared to other approaches usually used in solar electricity control [49]. For the second test system, the results are compared FA [50], SOSA [51], ABC [52] tuned PID and SGWO tuned AFPID [39]. The outcomes are gathered in

Table 8. Objective function values for the different controllers for 2nd test system.

Technique: Control Structure	ITAE
PID/FA [50]	10.12 × 10−2
PID/SOSA [51]	9.96 × 10−2
PID/ABC [52]	5.72 × 10−2
SGWO: AFPID [53]	2.79 × 10−2
Proposed ESSA based CFPID	1.34 × 10−2

and Figure 7. The minimum ITAE value and best system response are attained with ESSA-based CFPID controller compared to other approaches as evident from Table 8 and Figure 7. This justifies the usefulness of the proposed CFPID controller structure.

5.2. Frequency Control of DPGS by PD-CFPID

In the next step, DPS is considered and the controllers are optimized by the ESSA technique. The load demand (P_D) pattern is revealed in Figure 8a. The renewable outputs are characterized by Equations (17)–(19) and adopted from [5,36,37,38,39] as presented in Figure 8b,c. Table 8 provides the ESSA optimized PID, CFPID, and PD-CFPID controller parameters. For comparison, the results of MFO, PSO, GA, DA, GSA, and SSA optimized PID and J values are also specified in

Table 9. Optimized values.

Technique	Optimized Parameters			J Value (×10−2)
PID
	KP	KI	KD
MFO	0.5919	0.3137	1.1244	789.15
PSO	0.6370	0.3719	1.3587	789.05
GA	0.6016	0.3106	1.0199	788.89
DA	0.6716	0.3685	1.2448	788.97
GSA	0.6484	0.3550	1.5325	787.62
SSA	0.7106	0.2211	1.0787	783.20
ESSA	0.7172	0.2489	1.0371	782.75
CFPID
ESSA: CFPID	K1 = 1.3967, K2 = 1.5397 KP1 = 1.5495, KI1 = 1.5744, KD1 = 0.8160 KP2= 0.2288, KI2 = 0.7581, KD2 = 1.3056			781.99
ESSA: PD-CFPID	K1= 1.4319, K2 = 0.8438 KP1 = 1.5183, KI1 = 1.9570, KD1 = 1.9764 KP2 = 0.1992, KI2 = 0.6637, KD2 = 1.9570 CPF-FESS = 1.9564; CPF-BESS = 1.7787 CPF-DEG = 1.9987; CPF-EV = 1.7965			368.31

from which it can be seen that, with the same PID controller, a minimum objective function value of 782.75 × 10⁻² is obtained with the proposed ESSA technique. Hence, it is established that in the controller design problem, the proposed ESSA outperforms MFO, PSO, GA, DA, GSA, and SSA techniques. The CFPID controller optimized by ESSA reduces the J value to 781.99 × 10⁻². This validates the dominance of the CFPID over the PID. It can also be understood from Table 9 that the suggested PD-CFPID controller significantly reduces the J value to 368.31 × 10⁻².

To assess the controller’s effectiveness, the following cases are taken.

• Case 1: Normal operation:

Here, solar and wind generation and load outlines as specified in Figure 8 are presumed. The dynamic response with ESSA-based PID, CFPID, and PD-CFPID is presented in Figure 9. It is clear from Figure 9a that the performance with PD-CFPID is slightly better than PID and CFPID. The power outputs with the proposed PD-CFPID controller are shown in Figure 9b,c. In the time range t = 50 to 100 s it can be seen from Figure 8a that P_D is 0.6 pu. The P_WTG is around 0.4 p.u., P_PV is 0.2 pu which is clear from Figure 8b,c. Aqua electrolyzer takes 0.24 p.u. from the generation of renewable sources used by a fuel cell and the fuel cell gives a power that is equal to 0.0001 pu. DEG also becomes coupled to DPS at t = 50 s and supplies a power equal to 0.007 p.u. which is revealed in Figure 9b. The whole generation is obtained by totaling the power generation from the renewable sources less the aqua electrolyzer power, and by adding the power from DEG the fuel cell becomes 0.367 p.u. This is 0.23 p.u. less than P_D so BESS, FESS, and electric vehicles (EV) deliver less power to meet P_D. P_FESS = 0.0075 p.u., P_BESS = 0.0022 p.u., P_PEV = 0.23 p.u. which are exhibited in Figure 9b,c. When the power is released by the storage devices and power discharged by Pev is added to the generated power, the supply power becomes 0.6 p.u., which is the same as P_D. Hence, the power balance is maintained and the system is frequently stable.

• Case 2: Uncertain Cases

The renewable sources are associated with uncertainty and so is the case with load demand and EV. Keeping in view of some realistic variation/uncertainties, different cases are considered. To prove the supremacy of suggested PD-CFPID over CFPID and PID, the responses with PD-CFPID are compared with CFPID and PID under various uncertain conditions as:

Scenario A: P_WTG, P_PV, and P_D are augmented by 200%;

Scenario B: P_WTG and P_PV are unavailable but the P_D is augmented by 200%;

Scenario C: EV power and load demand pattern are augmented by 100% and 200%, respectively;

Scenario B: P_WTG and P_PV are decreased by 50% but P_EV and P_D are augmented by 100% and 200%, respectively;

Scenario E: The time delay is enhanced to 400 ms.

The responses for the above situations are presented in Figure 10a–e. It can be comprehended from Figure 10a that, for scenario A, when solar, wind powers, and load demand patterns are increased by 200%, the system is unstable with the PID. System stability is preserved with both CFPID and PD-CFPID. However, the response degrades with CFPID under scenario A whereas the response remains unaffected with the proposed PD-CFPID controller. For scenario B, when P_WTG and P_PV are unavailable but the P_D is augmented by 200%, the system is unstable with PID. The response further degrades with the CFPID controller in scenario B compared to the same for scenario A. with the proposed PD-CFPID controller, the response remains more or less the same. For scenarios C, D, and E, the system is unstable with both PID and CFPID controllers as evident from Figure 10c–e. However, the system stability is retained with the suggested PD-CFPID controller and frequency deviations lie inside the acceptable boundary.

In the present study, only multimodal functions have been tested as the controller design problem is a multimodal task. The usefulness of the proposed MSSA for unimodal functions and other benchmark functions needs further investigation.

6. Conclusions

In this paper, the enhanced salp swarm algorithm (ESSA) method was presented for solving multimodal optimization and controller design. In the suggested ESSA method, the convergence parameter is varied suitably for an improved equilibrium among the exploration and exploitation stages of the technique. Additionally, variable scaling factors are engaged to alter the salp’s position in the search procedure to minimize the random movement of salps. The effectiveness of ESSA is confirmed in some multimodal test functions. It is seen that for 10-dimensional multimodal functions (mmfs’), the ESSA algorithm provides improved results more than SSA, GA, ABC, CamWOA, MFO, PSO, GSA, WCA, PSOGSA, DA, and WCMFO in four out of six mmfs’. For fixed dimension mmfs’, proposed ESSA provides optimum or equally best outcomes in nine out of ten mmfs’. The projected control method is evaluated in two widely used systems. It is detected that the J value is reduced by 81.78% and 76.57% related to the best-published results for the two test systems, respectively.

The proposed ESSA method is then used to optimize a partially decentralized combined fuzzy PID (PD-CFPID) controller for frequency control of DPS. It is perceived that the ESSA-based PD-CFPID controller is better than PID and CFPID controllers and maintains frequency stability under various uncertain scenarios. The objective function value with the PD-CFPID controller is reduced by 52.94% and 52.90% compared to PID and CFPID controllers.

As a future direction, experimental verification of the present work can be performed and the state of charge of EVs can be included in the system.