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 PSOα, is proposed to improve the performance of the conventional PSO;
The 14 different benchmarks are considered to test the performance of PSOα;
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
Xi and
Vi represent the position and velocity of the
ith particle. Then, we have Equations (1) and (2) which represent
Xi and
Vi, respectively.
N is the swarm size for an
m-dimensional solution.
Pi and
Pg 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].
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
Pi [
16].
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).
where
Xi1 contains the first
m/2 sub-particles, and
Xi2 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 (Xi1) is updated. The next sub-particle Xi2 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 Xiα 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, regarding the average cost function. Likewise,
Table 2 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.
6. Conclusions
To summarize, the proposed optimisation technique, i.e., PSOα, 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 PSOα techniques. The comparative LFC analysis is performed under the dynamic perturbed response of , and for two case studies to study the efficacy of the proposed PSOα techniques. It was found that the proposed PSOα 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.