Quasi Oppositional Population Based Global Particle Swarm Optimizer With Inertial Weights (QPGPSO-W) for Solving Economic Load Dispatch Problem

In recent years, power companies have shown increasing interest in making strategic decisions to maintain profitable energy systems. Economic Load Dispatch (ELD) is a complex decision-making process where the output power of the entire power generating units must be set in a way that results in the overall economic operation of the power system. Moreover, it is a constrained multi-objective optimization problem. Now a days, there is a tendency to use metaheuristic methods to deal with the complexity of the ELD problem. Particle swarm optimization (PSO) is a subclass of metaheuristic methods inspired by fish schooling and bird flocking behaviors. However, the optimization performance of the PSO is highly dependent on fitness landscapes and can lead to local optima stagnation and premature convergence. Therefore, in the proposed study, two new variants of the PSO called global particle swarm optimizer with inertial weights (GPSO-w) and quasi-oppositional population based global particle swarm optimizer with inertial weights (QPGPSO-w) are proposed to address the complexity of the ELD problem. The ELD problem is formulated as an optimization problem and validation of the proposed methods is performed on IEEE standards (3, 6, 13, 15, 40 & 140) unit Korean grid ELD test systems under numerous constraints and the obtained results are compared with the several recent techniques presented in the literature. The results obtained with convex systems showed excellent cost-effectiveness, while for non-convex systems sequential quadratic programming (SQP) optimizer was added to discover global minima even more efficiently. The proposed techniques were successful in solving the ELD problem and yielded better results compared to the reported results in the selected cases. It is further inferred that the proposed techniques with less algorithmic parameters reflected improved exploration and convergence characteristics.


I. INTRODUCTION
The world is currently going through uncertain times. The corona pandemic has halted economic progress worldwide. The energy sector, which was a backbone of economic progress, also had serious consequences [1]. The year 2020 saw a huge energy drop of 5%, which a few years ago could be considered unrealistic or even desecrated [2]. Although the consumption of household electricity is increasing due to the recession in industry and the situation in economic sector has proved that the overall demand has The associate editor coordinating the review of this manuscript and approving it for publication was Jagdish Chand Bansal.
decreased [3]. The impact of this decline was felt so severely by some entities in the power sector that only in America, 19 companies went bankrupt [4]. The current situation seems bleak, but all is not lost, as the post-corona scenario could give an impetus to the global energy sector and revive it to its former glory. The disruption in development will significantly motivate investment firms to strengthen their strategy and use people's panic-buying psychology due to Covid-19 to make profits, thus stimulating the demand for energy in the process. Such positive trends for energy demand can be seen in the automotive sector which has increased in demand after corona due to people's preferences to avoid public transport [5], [6]. To meet the growing demand for energy in future VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ and create a sustainable energy infrastructure for our future generations, renewable energy sources must be vigorously integrated into the energy infrastructure. The penetration of renewable energy sources has seen a rising trend, even during the Covid-19 situation, and it is expected to retain its trajectory. The current positive trends of renewable integration are largely attributed to the incentives and policies of governments around the world. In 2022, the future for renewable energy seems a bit uncertain, as most of these incentives and policies are likely to end. The post-Covid-19 energy situation and uncertainty over global energy policy, coupled with a very volatile energy demand, are forcing the need for a backup plan. Global energy generation is still largely attributed to fossil fuels that have been in service for decades and have a proven track record of resilience, reliability, and efficient grid integration. To ensure energy supply, these thermal generation sources cannot be overlooked and must be utilized in an efficient and cost-effective manner. Economic load dispatch (ELD) is a well-documented optimization problem related to the economy of the power system, which involves the control of thermal resources so that maximum power can be extracted at the lowest possible fuel cost [7]- [9]. Economic load dispatch of thermal resources deals with non-linear, non-convex fuel cost curves under constraints such as energy balance, valve point effect, generators limits and prohibited operating zones. All these limitations make the ELD problem extremely challenging for optimization engineers, making it ideal for research. Simply put, ELD is the transmission of power in an economical way for a given load demand, so that no restriction is violated. ELD has been the subject of research for decades. The ongoing literature in ELD proves its necessity and importance in the overall energy mix. The ELD problem was initially addressed by traditional solution techniques such as lambda iteration method, quadratic programming, linear programming, gradient method, Newton's method and similar mathematical methods to find gradient or iterative search techniques. These approaches can solve the ELD problem, but with increasing dimensions and system size, these methods face severe performance degradation. It has also been noted in several studies that conventional approaches are lacking in adequate global exploration and are more prone to local exploitation. This tendency of trapping in local optima severely hampered their overall effectiveness. When conventional approaches did not yield the desired results, evolutionary and swarm-based approaches took the lead to solve the complex real world optimization problems [10]- [15]. Among these categories, swarm intelligence-based approaches have been found to be very effective in achieving the optimal solution of the ELD problem. These swarm-based techniques are a subclass of metaheuristic methods and can be further divided into two groups, such as global swarm optimizers (GSOs) and local swarm optimizers (LSOs). Among them the GSOs are more popular; the GSO techniques mimic the behavior of fish, insects, animals, or birds [16]- [19].
These techniques start with some random initial population and achieve the best position in the transition by learning from the personal experience of each solution as well as the experience of best solution in a coordinated manner. GSO techniques are widely recognized as the black-box problem solvers; many of them have few control parameters and are easy to implement in a computer code. Although most of these methods show promising results on unimodal optimization problems, many of them are challenged by multimodal and other complex fitness landscapes of the real-world optimization problems. To overcome the aforesaid shortcomings several new variants of swarm-based techniques were proposed like modified particle swarm (MPSO) [20], stochastic modified particle swarm (SM-PSO) [21], [22], hybrid mutated particle swarm (HMPSO) [23], improved particle swarm (IPSO) [24] and particle swarm with random drift (PSORD) [25], an expanded particle swarm (XPSO) [26], triple archives particle swarm (TAPSO) [27] and U-based particle swarm (U-PSO) [28].
One such global PSO variant namely global inertial PSO (GPSO-w) has been proposed in [29]. GPSO-w is an enhanced and advanced PSO algorithm with better exploration, exploitation, and convergence capabilities that can handle many fitness landscapes. In this study, to explore broader regions of the search space and to find the best solutions for the selected ELD cases, GPSO-w augmented with a quasi-oppositional population called the QPGPSO-w is used on IEEE standards (3, 6, 13, 15, 40 & 140) unit ELD test systems and the results obtained are compared with the results of several other meta heuristic techniques reported in recent literature.
The core contributions of the study are as follows: 1) The study seeks to identify two new meta heuristic methods to solve the ELD problem under various constraints. 2) A new quasi-oppositional global particle swarm optimizer with inertial weights (QPGPSO-w) is being developed by incorporating the quasi-oppositional population in the global particle swarm optimizer with inertial weights (GPSO-w).
3) The validation of the proposed methods for the solution of IEEE standards (3, 6, 13, 15, 40 & 140 )unit Korean grid ELD test systems under different constraints have been carried out.
The paper is structured into the following sections. Section II describes problem formulation and explains the solution techniques. Section III explains the overview of the Quasi-Oppositional GPSO-w. Section IV provides details of simulation and discussion of results. And section V draws some useful conclusions.

II. PROBLEM FORMULATION
ELD is a well-known optimization problem in power system with the aim of assigning optimal load to thermal units to reduce the total fuel cost, subject to the operational limitations of power system. Mathematically, ELD contains non-convexity and non-linearity, also ELD is a challenging mathematical problem due to hard binding constraints such as power balance and soft constraints such as generator limits, prohibited operating zones (POZs) and valve point effect. The computation of transmission losses at each dispatch level assigned to thermal units can also be part of the ELD problem. Numerically, the principal goal is to minimize the operating cost of generation units that can be modeled as formulated in Eq. 1 and Eq. 2. minimize The quadratic approach of cost curves for thermal unit fuel without valve point effect is described by Eq. 1. Detailed costs including the impact of valve point are described by Eq. 2. In the above equations, a, b, c, e and f represent cost coefficients; Nx indicates the total number of generating units available for scheduling, P i represents the ith power output of the generating unit and P il shows the minimum power generating limit of the ith generating unit [9].
These objective functions are subjected to following equality and inequality constraints. Equality constraints include the power generation balance as described by Eq. 3.
In Eq. 3 P generated represents the total scheduled power, P required indicates the total power demand of the system in megawatts (MW) and P loss shows loss of the transmission network of the system in MW.
Inequality constraints include generation limits and prohibited operating zones (POZs) described by Eq. 4 and Eq. 5.
where P il and P ih are the minimum and maximum power generation limits of the ith generator, P i represents power scheduled on the ith generation unit and P i1 to P in represent the workable operating zones of the ith generation unit. The transmission losses can be calculated from loss coefficient matrix B using following Eq. 6.
where B ik , B i0 and B 00 are transmission loss coefficients. The overall fitness function including equality constraints and objective can be defined as: Pi − P required − P Loss + objective (7) Penalty in the above mentioned equation is a constant value that is usually excess than 100.

A. OVERVIEW OF GPSO-W
The authors in [29] introduced GPSO-w by augmenting the working of PSO. In GPSO-w, N distinct solutions are initialized having N distinct velocity vectors. The initial values are randomly selected within the range of search space. The solutions then progressively iterate to optimal solution by updating their location and velocity vectors based on following equations.

18:
Update G B (j) corresponding to fit(G B (j)) 19: end for 20: Print best solution G B (maxiter) and best fitness fit(G B (maxiter)) simultaneously for obtaining an optimum candidate solution. This optimization technique depends on quasi opposition initialization and quasi opposition generation jumping, by which optimum initial candidate solutions may be achieved by making use of opposite points, even without the availability of previous information about the solutions. A similar methodology is reported in [31]. The quasi population has been extensively integrated to improve the exploitation capabilities of many algorithms. The quasi opposite population progresses as follows:  For each solution in the population an opposite number is defined as where X i is the ith d-dimensional solution at a particular iteration. X 0 is the opposite number of this ith solution and X qoi is the respective opposite population. Quasi population is integrated with GPSO-w, as illustrated by the pseudo code provided in Algorithm 2.
The effect of proposed QPGPSO-w can also be seen graphically in the distribution of the solution obtained by applying it to solve the following equation: f (x, y) = 9 + (x − 2) 2 + (y + 3) 2 (11) Eq. 11 is solved for 100 iterations by both GPSO-w and QPGPSO-w. From Figure. 1 it can be seen that QPGPSO-w performs better than GPSO-w by achieving optimal results in 32 iterations compared to 60 iterations taken by the latter technique. Figure. 2 shows the solution transition in QPSPSO-w during iteration 2 and iteration 4 respectively. We can see that the solution space is thoroughly utilized by a quasi population leading to the improvement of convergence and local optima stagnation.

IV. SIMULATION RESULTS
Both GPSO-w and proposed QPGPSO-w were used to implement standard ELD-IEEE test systems. The included test systems consist of: 1) 3 thermal unit convex system at a load demand of 150 MW proposed by A.J. Wood in [32].
2) IEEE standard 6 thermal unit test convex system having POZs constraint at a load demand of 1263 MW including transmission losses. The data is taken from [7]. 3) IEEE standard 15 thermal unit convex system having POZs constraint at a load demand of 2630 MW including transmission losses. The data is taken from [33]. 4) IEEE standard 13 thermal unit non-convex system having valve point constraint at a load demand of 1800 MW. The data is taken from [34]. 5) IEEE standard 40 thermal unit non-convex system having valve point constraint at a load demand of 10500 MW. The data is taken from [35]. 6) Korean 140 Unit convex test system at a load demand of 49342 MW taken from [24].     MATLAB 2016 software was used to perform simulations. The framework used had 8 GB RAM and Intel core i5 processor. The results achieved were compared with other similar results in the literature and are presented below.

A. SMALL SCALE CONVEX TEST SYSTEMS
The convex systems of (3, 6 & 15) thermal units were simulated for 20 runs and the iterations were kept at 500 per run for each system. The most optimal results obtained by GPSO-w and QPGPSO-w are shown in Tables. 3, 4 and 5. Table. 1 shows a comparison of the results obtained with other techniques available in the literature. It is clear from Table. 1 that for 3 unit test system QPGPSO-w has achieved better results in terms of cost ranging from 0.0602 to 0.0472 0.0472 $/hr compared to the lambda iteration method (LI), teaching learning based optimization (TLBO), disruption based symbiotic organism search (DSOS), and the GPSO-w methods.    FCEP respectively. Convergence characteristics of GPSO-w and QPGPSO-w are shown in Figure. 3. From Figure. 3 it can be noted that QPGPSO-w initially has a faster convergence rate and can later achieve more optimal solutions in the iterations. This search trend confirms the benefit of using a quasi population strategy.

B. NON CONVEX TEST SYSTEMS
Similarly, (13 & 40) unit non-convex thermal systems were also simulated for 20 runs and the iterations were kept at 500 per run for each system. The most optimal results obtained by GPSO-w and QPGPSO-w are shown in Tables. 6 and 7. From Tables. 6 and 7, it can be seen that both GPSO-w and QPGPSO-w were able to achieve an optimal solution of given test systems at a reasonable solution cost. The results obtained with QPGPSO-w demonstrate its efficacy compared to GPSO-w. Although QPGPSO-w was able to outperform GPSO-w, the results obtained are only a marginal improvement. To ensure that QPGPSO-w reaches the global optimum, the results were further optimized using the MATLAB SQP optimizer. The best results obtained after QPGPSO-w-SQP are shown in Table. 8. Table. 2  Convergence properties of the best solution of GPSO-w and QPGPSO-w are presented in Figure. 5. Figure. 4 shows    convergence properties of QPGPSO-w-SQP. From Table. 2 we can also observe a significant improvement in execution times, with the proposed techniques showing an improvement of up to 45 seconds in the total execution time per run.

C. LARGE SCALE CONVEX TEST SYSTEMS
The 140 unit Korean convex ELD test system consists of a combination of coal, oil fuel, LNG, and nuclear units. Twenty runs of 500 iterations each were performed for both GPSO-w and QPGPSO-w. The best results obtained with GPSO-w and QPGPSO-w are shown in Table. 8. The results obtained show that the QPGPSO-w compared to GPSO-w was not only able to obtain an optimal solution but also achieved a cost comparable to the global solution available in the literature.
QPGPSO-w achieved the best cost of 1665379 $/hr at an average cost of 1735099 $/hr in a run time of 40 seconds per run. The optimum cost achieved by QPGPSO-w is only 0.59% higher than the current global minimum.

V. CONCLUSION AND FUTURE WORK
In this study, the global particle swarm optimizer with inertial weights algorithm (GPSO-w) and quasi-oppositionbased global particle swarm optimizer with inertial weights (QPGPSO-w), which are new variants of swarm-inspired (SI) metaheuristic algorithms, were used to solve the IEEE standards (3, 6, 13, 15, 40 & 140) unit Korean grid ELD test systems under various constraints. The optimization potential of the proposed metaheuristic techniques has been validated VOLUME 9, 2021 by comparing their searching performance with several recent approaches reported in the literature. From the simulation results it could be seen that the energy generation costs of the systems have decreased significantly by augmenting the GPSO-w with quasi-oppositional population QPGPSO-w in the selected cases. In addition, the convergence characteristics and optimization potential of the QPGPSO-w algorithm are significantly better than the GPSO-w algorithm. Results for small-scale IEEE standards (3, 6 & 15) thermal unit convex system show that QPGPSO-w achieved better results in terms of cost up to 0.000735%, 0.093% and 1.735% for IEEE standards (3, 6 & 15) units respectively. The proposed approaches have also reduced the likelihood of a premature convergence on the ELD problem due to improved searching properties. Furthermore, for non-convex systems, sequential quadratic programming (SQP) optimizer searched the global minimum even more effectively. For non-convex system, the operation of QPGPSO-w was enhanced by passing the results through the SQP optimizer, and the resulting QPGPSO-w -SQP achieved better results in terms of cost up to 0.81% and 6.11%, for (13 & 40) thermal unit non-convex test systems respectively. For 140-units, the Korean ELD test system, the QPGPSO-w, outperformed the GPSO-w and found better results in relatively less iterations. In the future, this improved modification of the GPSO-w may be tested on other practical ELD issues with more operational constraints, for example prohibited operating zones, different fuel options and transmission losses, as well as with renewable energy sources. In addition, emerging metaheuristic methods can also be validated in the selected ELD cases.

APPENDIX A RESULT TABLES
See Tables 3-9.

APPENDIX B CONVERGENCE CHARACTERISTICS
See Figure 5.