PSO_α: A Fragmented Swarm Optimisation for Improved Load Frequency Control of a Hybrid Power System Using FOPID

Abstract

Particle swarm optimisation (PSO) is one of the widely adopted meta-heuristic methods for solving real-life problems. Its practical utility can be further enhanced by improving its performance. In order to acheive this, academics have presented several variants of the original PSO over the past few years, including the quantum PSO (QPSO), bare-bones PSO (BB-PSO), hybrid PSO, fuzzy PSO, etc. In this paper, the performance of PSO is improved by proposing a fragmented swarm optimisation approach known as the PSO_α. The PSO_α is tested and compared with PSOs over 14 different benchmarking cost functions to validate its efficacy. The analysis is also carried out to see the impact of α on its performance. It is observed that the average value of the cost function over 50 simulations obtained using the fragmented swarm approach is lower than that obtained using the standard PSO in 12 out of 14 benchmark functions. Similarly, the fragmented approach outperforms the standard PSO in 13 out of 14 benchmark functions when compared with the best fitness value achieved out of 50 simulations. Finally, the proposed approach is applied to solve the well-known real-life optimisation problem of load frequency control (LFC) in power systems. A test system comprising both renewable and traditional power sources is considered to evaluate the efficacy of the proposed technique. A fractional order proportional-integral-differential (FOPID) controller is used, whose parameters are optimised using the proposed PSO for achieving the LFC. The proposed fragmentation approach can be applied with other optimisation techniques to improve their performance.

1. Introduction

In 1995, Kennedy and Eberhart proposed a meta-heuristic method known as particle swarm optimisation (PSO) [1]. Since then, the widespread use of PSO has been found in solving real-life problems [2,3]. The convergence of the algorithms to a local minim is still a challenge that has been reported in [4]. Several methods and variants of the original PSO were proposed in the past several years to overcome this problem and improve its performance. These approaches are classified into four categories: proper selection of algorithm parameters [5,6,7,8], design of proper evolution strategies [9,10,11], hybrid PSO algorithms [12,13,14,15], and modified versions of the original PSO [16]. The choice of the PSO parameters plays an important role in its performance [5,6]. Researchers developed methods for the proper selection of these parameters, thereby resulting in improved performance. Fuzzy logic-based selection, use of control theory, selection of inertia weights, etc., are some of these methods [17]. These methods are specific to a particular application. Their performance in solving other real-life problems is yet to be validated. However, it is generally the case with any optimisation method or its variant that it is practically not feasible for a single algorithm to work effectively on all real-life optimisation problems.

The second category of improvement techniques is based on modifying evolution rules to achieve enhanced performance. Some of the examples are noted in [18], such as bare bones PSO (BB-PSO), QPSO, and APSO. The literature reveals that the conventional PSO can be combined with different algorithms to increase the PSO’s performance. To mention a few examples, in [19], fuzzy logic is applied in PSO to improve its performance. Similarly, chaotic PSO, discussed in [20], and opposition-based PSO, discussed in [21], are some other examples where PSO is used in conjunction with other algorithms. These techniques enhance the PSO’s efficiency but also make it more complicated. Some scientists attempted to reduce the computational complexity of the PSO while trying to improve its efficiency simultaneously. Simplified PSO (SPSO) and many optimising liaisons (MOL) are some of these techniques that enhance the standard PSO’s performance and also decrease its computational complexity [22]. The fragmented approach has shown a significant improvement in the performance of the GA [23], which opened a new paradigm in performance improvement of the PSO. In this paper, a similar approach is applied for PSO to propose a fragmented particle swarm optimisation, which is known as the PSO_α, for improving the performance of the standard PSO. The performance of the proposed PSO is compared by rigorously analysing it on 14 standard benchmark test functions. Further, the efficacy of the proposed PSO_α is validated through its implementation in the power system from the perspective of the load frequency control (LFC) analysis.

In the power system, the overall power generation must match the entire load plus losses for multi-area power systems to operate efficiently. The frequency must be within its nominal range to maintain the system’s stability. This has drastically attracted the attention of researchers toward LFC analysis. The frequency is kept at its average value with the aid of various controllers [24]. One vital aspect of the controller is to optimize its parameters using several optimisation techniques, such as PSO, GA, FA, BPSO, etc. One of the most comprehensive surveys on LFC is reported in [25]. In order to operate and regulate a power system effectively and reliably, LFC is crucial in that the frequency is ensured to be within the nominal range for the reliable operation of the power system. The significant attempts are found in the literature where various controllers are used to enhance the power system’s dynamic performance throughout various operating shifts. The state-of-the-art review on different controllers is presented by Latif et al., in [26], for integrated power systems comprising both traditional and renewable energy sources. The various controllers, in conjunction with optimisation techniques, are comprehensively reviewed by Tungadio et al., in [27]. Within such context, a review on some of the preliminary works is subsequently follows.

A firefly algorithm is suggested in [28] for the multi-area system’s load frequency regulation. The reported approach is used to optimise the gains of the proportional integral/proportional integral derivative controller for two- and three-area power systems. The LFC for a multi-source power system using different intelligent optimisation techniques is reported in [29]. R.K. Sahu et al., reported the tilt integral derivative controller using a filter constant to deal the issue of LFC for a multi-area system [30]. In this, for optimally designing the controller, differential evolution (DE) heuristic techniques are used to minimise integral time absolute error (ITAE) performance index. C.K. Shiva proposed a novel quasi-oppositional harmony search algorithm in [31] to deal the issue of LFC for a three-area power system under a deregulated regime. Authors in [32] reported the use of PIDA controller, where the parameters of the controllers are tuned with the TLBO algorithm. The firefly algorithm is used for LFC for a single area power system in [33]. Paliwal, N. [34] claimed that the implementation of the grey wolf optimiser (GWO) method has been used to estimate the best PID controller settings for LFC in multi-source networks. A novel deep reinforcement learning technique to achieve coordinated control and improve the performance of DIC-AGC in the performance-based frequency regulation market is proposed in [35]. A novel improved gravitational search algorithm-binary particle swarm optimisation (IGSA-BPSO) is introduced in [36] to address problems in the automatic generation control of interconnected deregulated power systems. For the load frequency management in the interconnected power system, the tilted integral derivative controller is modified and is known as an integral derivative-tilted (ID-T) controller whose parameters are tuned using archimedes optimisation algorithm [37]. In [38], fractional-order calculus and interval type-2 fuzzy inference systems are used to design the primary control system for the LFC problem. In order to enhance the performance of the system, authors in [39] use a novel improved squirrel search algorithm for the controller design where PID controllers with varying degrees of freedom are utilised. The power system’s automated generation control (AGC) problem is addressed in [40,41] using a novel fuzzy PID controller with filtered derivative action and fractional order integrator controller. In [42], a two area hydrothermal power system is considered for LFC analysis where gravitational FA is used for the optimal tunning of the controller parameters. In [43], the fractional order PI^λ D^µ controller is used for the LFC analysis of a interconnected power system. Contrary to these works, we propose a fragmented PSO optimisation technique, and its efficacy is evaluated with reference to the power system within the context of LFC.

The major contributions of the paper are summarized as follows:

• A fragmented swarm optimisation approach, known as the PSO_α, is proposed to improve the performance of the conventional PSO;
• The 14 different benchmarks are considered to test the performance of PSO_α;
• The analysis is also carried out to see the impact of α on its performance;
• The proposed approach is applied to solve the well-known real-life optimisation problem of LFC in power systems where the test system is considered with both renewable and traditional power sources;
• A fractional order proportional-integral-differential (FOPID) controller is used, whose parameters are optimised using the proposed PSO for the LFC of the hybrid power system.

The article is organised into five different sections. The description of the proposed fragmented swarm optimisation approach in Section 2 follows the introduction. The performance of the proposed PSO is analysed in the third section with a comparative benchmark analysis. In Section 4, the multi-source multi-area power system is designed with a FOPID controller as a test system. The efficacy of the proposed system is evaluated through simulations. The simulation results and discussion are reported in Section 5. The last section presents the conclusion of the reported research work.

2. Fragmented Particle Swarm Optimisation

The movement of particles in a swarm is the foundation for the conventional PSO. Every particle in a swarm is identified by its velocity and position in the search space. The particles’ location, updated with each subsequent iteration, defines the answer to the objective function. Let X_i and V_i represent the position and velocity of the ith particle. Then, we have Equations (1) and (2) which represent X_i and V_i, respectively. N is the swarm size for an m-dimensional solution.

$$X_i=\left(x_{i1},x_{i2},\dots \dots ,x_{im}\right)$$

(1)

$$V_i=\left(v_{i1},v_{i2},\dots \dots ,v_{im}\right)$$

(2)

P_i and P_g represent the personal best position for the ith particle, and the global best position for the swarm. At the kth iteration, Equations (3) and (4) can be used to update the position of the ith particle and its velocity [16].

$$x_{id}\left(k\right)=x_{id}\left(k-1\right)+v_{id}\left(k\right)$$

(3)

$$v_{id}\left(k\right)=v_{id}\left(k-1\right)+a_1{r}_1\left\{p_{id}\left(k-1\right)-x_{id}\left(k-1\right)\right\}+a_2{r}_2\left\{p_{gd}\left(k-1\right)-x_{id}\left(k-1\right)\right\}$$

(4)

where, r₁ and r₂ are randomly generated numbers, and a₁ and a₂ are constants. r₁ and r₂ lie in the range of [0, 1]. With the fitness function F, Equation (5) can be used to update P_i [16].

$$P_i\left(k\right)=\left\{\begin{array}{cc}{X}_i\left(k\right) \hfill & \mathrm{if} F\left(X_i\left(k\right)\right)\le F\left(P_i\left(k-1\right)\right)\hfill \\ P_i\left(k-1\right) \hfill & \mathrm{if} F\left(X_i\left(k\right)\right)>F\left(P_i\left(k-1\right)\right)\hfill \end{array}\right\}$$

(5)

To comprehensively review conventional PSO, its steps are depicted in Figure 1.

In the present work, the PSO_α technique is proposed to enhance the performance of the conventional PSO, which is based on a fragmented approach. The fundamental notion underlying the approach is the sub-particle division of each solution, which fragments the swarm into two, as given by Equation (6).

$$X_i=\left\{\left[X_{i1}\right],\left[X_{i2}\right]\right\}$$

(6)

where X_i1 contains the first m/2 sub-particles, and X_i2 contains the next m/2 sub-particles in the solution. In the following iterations, these sub-particles are updated in α fragments sequentially, rather than simultaneously, as seen in Figure 2.

In the first iteration, only the first sub-particle of the ith particle in the swarm (X_i1) is updated. The next sub-particle X_i2 is updated in the next iteration. In the similar way, all sub-particles are updated such that only one sub-particle is updated at any iteration. Consequently, in the αth iteration, the X_iα sub-particle is updated. As long as the termination criteria are not met, this cycle is repeated.

3. Performance Analysis

For the performance analysis, 14 different benchmark functions summarised in [44,45] are utilised. The dimension of a particle and the maximum number of iterations are considered as 20 and 100, respectively, in the simulation. The efficacy of the PSO_α depends on the number of sub-particles (i.e., the value of α), which will also be analysed in this section. However, there has not been any attempt to adjust the parameters of either the conventional PSO or the suggested PSO. The different values of α are considered to perform the simulations reported in

Table 1. Average fitness values of the cost functions for 50 simulations.

	Standard PSO	PSOα
		α = 2	α = 5	α = 10	α = 20
Fletcher-Powell	461,103.9	2,604,529	2,721,716	3,022,818	2,514,629
Griewank	0.049925	0.065658	0.082329	0.080676	0.117824
Penalty #1	0.622927	0.009717	0.002073	2.25 $\times {10}^{-15}$	2.68 $\times {10}^{-12}$
Penalty #2	0.550521	0.04043	0.001538	0.00022	8.59 $\times {10}^{-13}$
Quartic	8.67 $\times {10}^{-13}$	2.98 $\times {10}^{-15}$	5.1 $\times {10}^{-37}$	1.06 $\times {10}^{-27}$	1.42 $\times {10}^{-23}$
Rastrigin	49.17073	24.93886	19.34199	17.97034	15.94734
Rosenbrock	16.78169	17.46503	11.3112	12.12923	16.18238
Schwefel 1.2	−6452.71	−6603.55	−6647.61	−6696.6	−6672.26
Schwefel 2.21	7.881805	23.08167	15.24715	10.32565	4.047511
Schwefel 2.22	16.15846	3.12 $\times {10}^{-11}$	5.18 $\times {10}^{-11}$	5.56 $\times {10}^{-8}$	1.29 $\times {10}^{-7}$
Schwefel 2.26	0.862617	0.861845	0.642069	0.46599	0.242072
Sphere	3.39 $\times {10}^{-17}$	1.72 $\times {10}^{-15}$	1.23 $\times {10}^{-23}$	7.58 $\times {10}^{-17}$	3.76 $\times {10}^{-15}$
Step	183.86	5.06	0.28	0.04	0
Ackley	12.03881	0.636369	0.239076	0.05167	0.000602

, regarding the average cost function. Likewise,

Table 2. Best fitness values of the cost functions for 50 simulations.

	Standard PSO	PSOα
		α = 2	α = 5	α = 10	α = 20
Fletcher-Powell	310,355.6	828,986.5	1,190,779	1,157,620	1,221,396
Griewank	1.11 $\times {10}^{-16}$	0	0	5.55 $\times {10}^{-16}$	1.44 $\times {10}^{-14}$
Penalty #1	1.28 $\times {10}^{-18}$	1.65 $\times {10}^{-16}$	6.01 $\times {10}^{-28}$	4.36 $\times {10}^{-20}$	1.13 $\times {10}^{-17}$
Penalty #2	1.8 $\times {10}^{-18}$	3.91 $\times {10}^{-17}$	2.09 $\times {10}^{-25}$	4.6 $\times {10}^{-18}$	6.8 $\times {10}^{-16}$
Quartic	3.01 $\times {10}^{-34}$	1.62 $\times {10}^{-16}$	4.26 $\times {10}^{-44}$	1.43 $\times {10}^{-35}$	3.58 $\times {10}^{-29}$
Rastrigin	23.87898	9.949591	6.964713	7.959672	6.964713
Rosenbrock	2.315889	1.251602	3.410537	2.001622	0.007852
Schwefel 1.2	−7113.69	−7457.95	−7113.69	−7110	−7228.44
Schwefel 2.21	0.211203	7.728768	6.977263	2.658325	0.149143
Schwefel 2.22	3.44 $\times {10}^{-11}$	2.97 $\times {10}^{-15}$	8.69 $\times {10}^{-16}$	1.93 $\times {10}^{-10}$	5.5 $\times {10}^{-9}$
Schwefel 2.26	0.192718	0.547877	0.376101	0.266043	0.128634
Sphere	3.46 $\times {10}^{-23}$	1.41 $\times {10}^{-16}$	1.04 $\times {10}^{-27}$	1.62 $\times {10}^{-19}$	6.02 $\times {10}^{-18}$
Step	2	0	0	0	0
Ackley	1.48 $\times {10}^{-10}$	1.05 $\times {10}^{-8}$	1.11 $\times {10}^{-12}$	1.03 $\times {10}^{-9}$	1.14 $\times {10}^{-8}$

summarises the best fitness value for the cost functions. The best results for each cost function are highlighted in boldface. It should also be noted that for α = 1, the fragmented PSO behaves as the standard PSO.

It is essential to proceed with some caution when analysing the results in Table 1 and Table 2. A single optimisation strategy cannot effectively handle every test bench function. However, the choice of the number of fragments are a critical parameter in the performance of the proposed PSO, as observed from the results reported in Table 1 and Table 2. But overall, fragmentation of the swarm does result in improved optimisation. Table 1 shows that the average value of the cost function over 50 simulations, obtained using the fragmented swarm approach, is lower than that obtained using the standard PSO in 12 out of 14 benchmark functions. Similarly, the fragmented approach outperforms the standard PSO in 13 out of 14 benchmark functions when compared w.r.t to the best fitness value achieved out of 50 simulations.

Computational complexity is an important parameter that has to be considered while comparing different algorithms. The computational burden is dependent on the number of cost function evaluations. The proposed fragmentation technique only fragments the solution space and it requires the same number of cost function evaluations as the standard PSO. Thus, the proposed PSO has a similar computational complexity as the standard PSO. To further understand the proposed approach, boxplots of some of the test functions are shown in Figure 3.

The boxplots further corroborate the results shown in Table 1 and Table 2. It can be clearly seen that, depending on the cost function, the fragment size has to be optimally chosen to achieve better results.

4. Design of a Test System for Performance Analysis of the PSO_α in LFC

In this section, the multi-source, multi-area test system is described, followed by design aspects of the FOPID controller for the LFC of the test system.

4.1. Multi-Source Multi-Area Power System

As a test system for LFC analysis using PSO_α, a multi-source multi-area power system is considered. Such a test system is shown in Figure 4, which comprises two areas, referred to as area-1 and area-2. The transfer function of the test system is shown in Figure 4a whereas, the transfer function of individual power plants are shown in Figure 4b. Where the different constants used are as follows: ${\mathrm{k}}_1=0.543478$, ${\mathrm{k}}_2=0.326084$, ${\mathrm{k}}_3=0.130438$, ${\mathrm{k}}_4=0.4312$, and ${\mathrm{k}}_5=1/2.4$. Both of these areas are of a multi-source hybrid in nature from the generation’s perspective. Particularly, the thermal plant, hydro plant, and gas turbine plant are considered to infuse the multi-source characteristics in both areas. The frequencies in both the areas, and tie-line power flow between area-1 and area-2 are key parameters of interest.

4.2. FOPID Controller

The FOPID controller is proposed for the LFC with the objective to enhance the system performance compared to the traditional PID-based controller. Compared to the PID controller, the FOPID controller has two additional parameters, namely $\lambda$ and $\mu$, other than $K_p$, $K_i$ and $K_d$. Here, $\lambda$ defines integrator order and $\mu$ defines derivative order, whereas proportional gain, integral gain, and derivative gain are defined by $K_p$, $K_i$ and $K_d$, respectively. The block schematic representation of the FOPID controller for the proposed system is represented in Figure 5. The transfer function of the FOPID controller is characterised by Equation (7).

$$G_c(s)=\frac{U(s)}{R(s)}=K_p+\frac{K_i}{s^{\lambda }}+K_d{s}^{\mu }$$

(7)

The approach for using the proposed PSO_α optimisation techniques for the LFC of the test system is schematically represented in Figure 6. The iterative methodology is used to obtain the performance of the test system with optimised FOPID controller parameters. The parameters of the FOPID controller, i.e.,$K_p$, $K_i$, $K_d$, $\lambda$ and $\mu$, are required to be optimised for the LFC analysis of the test system. For their optimisation, the PSO_α optimisation techniques are used for the optimization constraints, and initial values of optimization parameters work as inputs. The inputs to the FOPID controller are frequency variation in the corresponding area, and tie-line power between area-1 and area-2. For the tuning of $K_p$, $K_i$, $K_d$, $\lambda$ and $\mu$, the integral time absolute error (ITAE) is chosen as an objective (fitness) function. The objective function includes the tie-line power and frequency of the proposed test system as defined using (8).

$$ITAE={\displaystyle \underset{0}{\overset{t}{\int }}\left(\left|\Delta F_1\right|+\left|\Delta F_2\right|+\left|\Delta P_{tie}\right|\right)} t dt$$

(8)

where $\Delta F_1$, $\Delta F_2$, and $\Delta P_{tie}$ represent the frequency variation in area-1, area-2, and tie-line power between area-1 and area-2, respectively.

5. Simulation Results and Discussion

The proposed PSO_α optimisation techniques are applied to tune the FOPID parameters to achieve the LFC of multi-source multi-area power system. The system configurations used for the implementation and simulations are as follows: MATLAB (R2016a) software with i5-6200 CPU@ 2.30 GHz with 16 GB RAM. The performance analysis of the proposed optimisation technique is carried out for two different cases. In the first case study, the load variation in area-1 and area-2 is considered as 1% and 2%, respectively. In the second case study, area-1 and area-2 are subjected to the 3% and 2% load variation, respectively. For these case studies, ITAE as an objective function is chosen to optimise all five parameters of the FOPID controllers. For optimisation, the standard PSO and PSO_α optimisation techniques are considered. The different parameters for the proposed case studies are included in the Appendix A. The simulation results and discussion for both the case studies are followed in subsequent sections.

5.1. Case Study-I

In case study-I, the load variation in area-1 and area-2 is considered as 1% and 2%, respectively. The best FOPID controller parameters are selected using both standard PSO and the proposed PSO_α optimisation techniques. Further, to observe the efficacy of the proposed optimisation techniques, the dynamic responses are captured through the simulation. The simulation results for the perturbed responses in frequency deviations of area-1 ($\Delta F_1$) and area-2 ($\Delta F_2$) are plotted in Figure 7a,b, respectively. The plots for simulation results depicting dynamic response in the tie-line power flow ($\Delta P_{12}$) are shown in Figure 7c. The analysis of Figure 7 reveals that the proposed PSO_α optimisation techniques outperform the standard PSO in terms of dynamic perturbed response of $\Delta F_1$, $\Delta F_2$ and $\Delta P_{12}$. Nevertheless, the number of sub-particles α plays a critical role in performance enhancements, which can be validated from the perturbed responses shown in Figure 7 for $\Delta F_1$, $\Delta F_2$ and $\Delta P_{12}$ with ${\mathrm{PSO}}_2, \mathrm{PSO}$₅ and ${\mathrm{PSO}}_{10}$ Optimally tuned parameters of FOPID controllers with different optimisation techniques are shown in

Table 3. Optimally tuned parameters of FOPID controllers with different optimisation techniques for case study-I.

FOPID Parameters	PSO	PSO2	PSO5	PSO10
Kp1	0.01	5.1816	2.3064	10
KI1	8.8499	5.6965	7.2508	8.6551
KD1	5.3573	3.0801	3.3643	5.1749
µ1	0.791	1	0.8454	1
λ1	0.5	0.9949	0.5	0.6662
Kp2	9.6153	10	6.0612	4.6897
KI2	10	10	8.381	10
KD2	5.7386	3.8028	2.5067	3.5886
µ2	0.9967	1	0.9709	1
λ2	0.8292	0.5	0.9338	0.7885
ITAE	0.0411	0.0575	0.0314	0.0291

.

To further augment the performance analysis, the perturbed response of the test system for case study-I can be analysed in terms of the settling time, undershoot, and overshoot. Thus, for evaluating the efficacy of the proposed PSO_α-driven FOPID controller for LFC, the settling time, undershoot/overshoot results for different parameters, i.e., $\Delta F_1$, $\Delta F_2$ and $\Delta P_{12}$ are summarised in

Table 4. Key performance parameters for the LFC of the test system pertaining to case study-I with PSO, PSO₂, PSO_5, and PSO₁₀.

	PSO	PSO2	PSO5	PSO10
ITAE	0.0411	0.0575	0.0314	0.0291
$\mathrm{Peak} \mathrm{Overshoot} (\Delta F_1$)	1.68%	1.46%	1.47%	1.14%
$\mathrm{Peak} \mathrm{Overshoot} (\Delta F_2$)	2.91%	2.4%	2.15%	2.51%
$\mathrm{Peak} \mathrm{Overshoot} (\Delta P_{12}$)	0.158%	0.081%	0.055%	0.168%
$\mathrm{Settling} \mathrm{Time} (\Delta F_1$)	5.98 s	9.23 s	4.99 s	3.5 s
$\mathrm{Settling} \mathrm{Time} (\Delta F_2$)	4.82 s	5.57 s	4.57 s	4.82 s
$\mathrm{Settling} \mathrm{Time} (\Delta P_{12}$)	18 s	12 s	9.5 s	9.5 s

. These results are recorded to strengthen the comprehensive analysis of the overall performance of the test system with standard PSO-driven FOPID and the proposed PSO_α-driven FOPID controller. From the summarised results, it is inferred that the proposed PSO_α-driven FOPID controller provides better system stability and robust performance with an optimum choice of sub-particles α, when compared to the standard PSO-based FOPID controller. Specifically, with α = 10, the proposed PSO₁₀-based FOPID controller provides a more stable and robust performance. However, this is the maximum value of α that can be used, as the maximum iterations are 10. Nevertheless, a trade-off can be observed between the system LFC performance parameters and value α, which can be corroborated from the results summarized for α = 5 and α = 10. Moreover, the ITAE = 0.0291, is observed to be the minimum in the case of PSO₁₀-based FOPID controller which further corroborates the efficacy of the proposed PSO_α-driven FOPID controller and the significance of sub-particles α on its performance.

5.2. Case Study-II

Under case study-II, the load fluctuation in area 1 is considered as 3%, and in area-2 is considered as 2%. The best FOPID controller parameters are selected using both the standard PSO and the proposed PSO_α optimization techniques for the LFC analysis.

The simulation results for the perturbed responses in frequency deviations of area-1 ($\Delta F_1$) and area-2 ($\Delta F_2$) are plotted in Figure 8a,b, respectively. The plots for the simulation results depicting dynamic response in the tie-line power flow ($\Delta P_{12}$) are shown in Figure 8c. Optimally tuned parameters of FOPID controllers with different optimisation techniques are shown in

Table 5. Optimally tuned parameters of FOPID controllers with different optimisation techniques for case study-II.

FOPID Parameters	PSO	PSO2	PSO5	PSO10
Kp1	8.7717	3.3908	10	5.5647
KI1	7.6618	3.5364	8.1693	8.4972
KD1	3.0907	2.7922	6.0918	5.7859
µ1	0.9748	0.8824	1	1
λ1	0.8058	1	1	0.71
Kp2	3.3765	10	6.8539	10
KI2	4.8148	0.01	7.205	10
KD2	2.2872	5.1184	6.8575	6.513
µ2	1	1	0.9217	1
λ2	0.9901	0.5	0.6523	0.7604
ITAE	0.1225	0.3152	0.0788	0.0433

. From Figure 8, it is inferred that the proposed PSO_α optimisation techniques outperform the standard PSO in terms of the dynamic perturbed response of $\Delta F_1$, $\Delta F_2$, and $\Delta P_{12}$. Nevertheless, the number of sub-particles α play critical role in performance enhancements that can be validated from the perturbed responses shown in Figure 8 for $\Delta F_1$, $\Delta F_2$ and $\Delta P_{12}$, with ${\mathrm{PSO}}_2$ and ${\mathrm{PSO}}_5$.

To evaluate the efficacy of the proposed technique, the perturbed response of the test system for case study-II is analysed in terms of the settling time, undershoot, and overshoot pertaining to the different parameters, such as $\Delta F_1$, $\Delta F_2$, and $\Delta P_{12}$. The results are comprehensively summarised in

Table 6. Key performance parameters for the LFC of the test system, pertaining to case study-II, with PSO, PSO₂, PSO₅ and PSO₁₀.

	PSO	PSO2	PSO5	PSO10
ITAE	0.1225	0.3152	0.0788	0.0433
$\mathrm{Peak} \mathrm{Overshoot} (\Delta F_1$)	4%	4.27%	3.2%	3.2%
$\mathrm{Peak} \mathrm{Overshoot} (\Delta F_2$)	3.37%	2.59%	2.24%	2.08%
$\mathrm{Peak} \mathrm{Overshoot} (\Delta P_{12}$)	0.214%	0.367%	0.063%	0.138%
$\mathrm{Settling} \mathrm{Time} (\Delta F_1$)	9.4 s	20 s	5.9 s	4.59 s
$\mathrm{Settling} \mathrm{Time} (\Delta F_2$)	6.55 s	15 s	8.02 s	5.29 s
$\mathrm{Settling} \mathrm{Time} (\Delta P_{12}$)	9.956 s	50 s	12 s	4.84 s

. These results are comparatively analysed with not only the standard PSO-driven FOPID but also with proposed PSO_α-driven FOPID controllers for the different values of sub-particles α. From the summarised results, it can be concluded that the proposed PSO_α-driven FOPID controller provides better system stability and robust performance with an optimum choice of sub-particles α, when compared to the standard PSO-based FOPID controller. The optimum results are obtained for the highest possible number of sub-particles, i.e., α = 10. Specifically, with α = 10, the proposed PSO₁₀-based FOPID controller provides a more stable and robust performance as the following key parameters are recorded: peak overshoot ($\Delta F_1$) = 3.2%, peak overshoot ($\Delta F_2$) = 2.08%, peak overshoot ($\Delta P_{12}$) = 0.138%, settling time ($\Delta F_1$) = 4.59 s, settling time ($\Delta F_2$) = 5.29 s, and settling time ($\Delta P_{12}$) = 4.84 s. Moreover, the ITAE = 0.0433 is observed to be the minimum in the case of the PSO₁₀-based FOPID controller. Nevertheless, a trade-off between the system performance and number of sub-particles α can be observed which can be corroborated from the reults summarized for PSO₅ and PSO₁₀. In a nutshell, the proposed PSO_α-driven FOPID controller perfoms better than the standard PSO, and the significant impact of sub-particles α is observed on its performance.

Moreover, the convergence plot for the objective function is compared for both standard PSO, PSO₂, PSO₅, and PSO₁₀ at different values of the iteration, which is as shown in Figure 9.

The convergence plot shows that the convergence performance is even better for the proposed PSO_α optimisation techniques. In addition, it is important to note that all the optimisation techniques have the same initial parameters, which is reflected in the same ITAE value at the beginning of optimisation. Nevertheless, the choice of α is critical in enhancing the performance of the proposed optimisation techniques. The results reveal that fragmentation improves the performance of the PSO for α = 5 and α = 10. However, for α = 2, the performance degrades. Thus, the parameters of the FOPID controller must be tuned with PSO₅ and PSO₁₀ for a better LFC of the multi-source multi-area test system.

6. Conclusions

To summarize, the proposed optimisation technique, i.e., PSO_α, uses a segmentation approach to improve the performance of the conventional PSO. The segmentation methodology is a useful method for enhancing the PSO’s performance. The recommended PSO outperforms the conventional PSO, not only in terms of average fitness value but also in terms of achieving the lowest fitness value when applied to 14 benchmark functions and studied in over 50 simulations of 100 iterations each. Apart from the performance improvement, the proposed technique’s efficacy is validated for a multi-area multi-source power system for the LFC analysis. In the test system, a FOPID-based controller is used for the LFC analysis. The parameters of the FOPID controller are tuned using the proposed PSO_α techniques. The comparative LFC analysis is performed under the dynamic perturbed response of $\Delta F_1$, $\Delta F_2$ and $\Delta P_{12}$ for two case studies to study the efficacy of the proposed PSO_α techniques. It was found that the proposed PSO_α works better than the conventional PSO. Moreover, with α = 5 and α = 10, the proposed PSO₁₀-based FOPID controller provides a more stable and robust performance as the following key parameters are recorded: ITAE = 0.0411 with PSO, ITAE = 0.0575 with PSO₂, ITAE = 0.0314 with PSO₅, and ITAE = 0.0291 with PSO₁₀ under case study-I. Similarly, the following key parameters are recorded under case study-II: ITAE = 0.1225 with PSO, ITAE = 0.3152 with PSO₂, ITAE = 0.0788 with PSO₅, and ITAE = 0.0433 with PSO₁₀. The choice of α is critical in the performance enhancement of the proposed optimisation techniques. However, the optimum value of α for which the performance of PSO improves depends on the nature of the fitness function and needs to be explored further. In addition, the fragmentation approach can be used by other optimisation techniques or other variants of PSO to achieve improved performance, which is left as an open research problem.