Combined emission economic dispatch using quantum-inspired particle swarm optimization and its variants

Abstract

The ever-increasing electricity demand, its dependency on fossil fuels, and the consequent environmental degradation are major concerns of this era. The worldwide domination of fossil fuels in bulk electricity generation is rapidly increasing the emissions of CO₂ and other environmentally dangerous gases that are contributing to climate change. The economic and emission dispatch are two important problems in thermal power generation whose combination produces a complex highly constrained nonlinear optimization problem known as combined economic and emission dispatch. The optimization of combined economic and emission dispatch aims to allocate the generation of committed units to minimize fuel cost and emissions, simultaneously while honoring all equality and inequality constraints. Therefore, in this article, we investigate a solution of the combined economic and emission dispatch problem using quantum particle swarm optimization and its two modified versions, that is, enhanced quantum particle swarm optimization and quantum particle swarm optimization integrated with weighted mean personal best and adaptive local attractor. The enhanced quantum particle swarm optimization algorithm achieves particles’ diversification at early stages and shows good performance in local search at later stages. The quantum particle swarm optimization integrated with weighted mean personal best and adaptive local attractor boosts search performance of quantum particle swarm optimization and attains better global optimality. The suggested methods are employed to achieve solution for the combined economic and emission dispatch in four distinct systems, encompassing two scenarios with 6 units each, one with a 10-unit configuration, and another with an 11-unit setup. A comparative analysis with methodologies documented in existing literature reveals that the proposed approach outperforms others, demonstrating superior computational performance and robust efficiency.

Introduction

Electricity is supplied to consumers through a grid comprising generation, transmission and distribution infrastructures. Electrical energy cannot be stored economically and hence requires a proper balance between demand and supply. Under normal conditions, power system operation is termed as a steady-state operation. There are mainly three types of problems encountered in the steady-state mode of operation, hierarchically listed as load flow, optimal economic dispatch (ED) and system control (Kothari, 2012).

Global energy demands are surging due to population growth and the unsustainable use of traditional fossil fuels poses environmental risks. Although, many renewable energy sources (RESs) are available such as wind, solar, etc. that are cleaner and greener than fossil fuels (coal, oil, and natural gas). However, because of the intermittent nature of RES, the bulk electricity is generated by fossil fuels, contributing around $64\mathrm{\%}$ (coal $36\mathrm{\%}$ and gas $20\mathrm{\%}$ ) of worldwide electricity production (International Energy Agency, 2023a,c). In 2022, global coal-fired power generation increased by almost $2\mathrm{\%}$ , marking a slower growth compared to the $8\mathrm{\%}$ seen in $2021$ . Despite this, the year’s growth exceeded the relatively stagnant average seen over the past 5 years. Coal-fired power generation set a new record for the second consecutive year, reaching $\sim$ 10,400 terawatt-hours (TWh) in total. This trend raises concerns about environmental impacts and highlights the ongoing reliance on coal for electricity generation. By 2025, Asia will use half of the world’s electricity, with one-third of it in China. The worldwide electricity demand is expected to increase rapidly by $3\mathrm{\%}$ each year. This growth is mainly in emerging markets and developing countries like China, India, and Southeast Asia. This is the first time in history that Asia will have such a significant impact on global electricity consumption. In $2022$ , the world hit a new record for power sector emissions, reaching about $13.2$ Gt CO₂. This was a $1.3\mathrm{\%}$ increase compared to the previous year. The higher emissions were primarily caused by the increased use of fossil fuels for power generation in the Asia Pacific region (International Energy Agency, 2023b). Countries worldwide are embracing new energy strategies such as the UN’s focus on urban climate action and China’s goals for carbon reduction by 2030 and achieving neutrality by 2060. Studies indicate that China could hit its carbon peak by 2030 and meeting carbon neutrality by 2060 poses challenges, highlighting the importance of technology and reshaping industries and energy systems (Yang et al., 2023).

The term emission dispatch (usually defined as EMD) was introduced after the US Clear Air Act amendment in $1990$ s which declared that electric utilities should decrease emissions of sulphur dioxide (SO₂) by 10 million tons/year and the nitrogen oxides (NOx) by $2$ million tons/year as compared to $1980$ emission level (El-Keib et al., 1994). Moreover, many countries have signed the Kyoto Protocol to commit to a minimum cost of energy production by sending out minimum emissions. Low-cost coal used in thermal power plants is of poor quality with high ash contents which generates huge amounts of hazardous gasses (SO₂ and NOx) (Güvenc et al., 2012). Therefore, another dispatching process was added to the power system to control the emissions of power generating units known as EMD (emission dispatch).

Cost minimization is the basic intention of any good business which is also true in the electric power sector. The priority of the economic procedure of electric power generation is to efficiently encounter power load variation with less costly and pollution-free generation sources, for example, hydro-power. As the load increases beyond the limits of these generating sources, the next tier of efficient plants is initiated, such as fossil fuels based thermal powered plants. In this way, the peak power demand is met by using the most efficient generating sources to the least efficient generating sources. An electric energy management system is used to monitor, control, and optimize the generators of electric power systems.

ED can be defined as the procedure of assigning different levels of generation to the generators within their limits to satisfy the total power demand efficiently at the lowest price. ED is a critical assignment of the power grid with a time scale from $10$ minutes up to a few hours (Sharifian and Abdi, 2022). The electric load on the power system is highly flexible and varies many times in a day. To counter this load, an efficient mechanism is required that could reduce the total cost of generation without violating equality and inequality constraints. In other words, we can say that ED is to distribute the entire load among the generators to reduce the overall cost. In the case of thermal generation, this cost refers to the fuel cost which needs to be minimized (Mahor et al., 2009; Kunya et al., 2023).

The lowest cost condition of ED can lead to the lowest cost with a significant quantity of emissions. On the other hand, the least emissions EMD refers to the production of the least emissions which can deviate from the lowest pricing rate. The nature of both objectives is conflicting and yields a complex optimization problem which is known as combined economic and emission dispatch (CEED). The purpose of multi-objective CEED is to provide optimal load dispatching to the generators with the least fuel price and pollution together while honoring the functioning constraints.

Related work

Researchers have shown a lot of interest in solving the combined form of two different dispatches (ED and EMD) which are conflicting in nature. The aim of solving this multi-objective problem is to reduce the cost and emissions simultaneously. Many optimization techniques have been used to solve the CEED problem. These optimization techniques are categorized as classical or conventional, modern heuristic or non-conventional techniques, and hybrid methods. These methods have their own pros and cons; for instance, conventional methods are mathematically proven optimal and some of these are also recognized for solving problems with less computational time. However, there are some limitations such as premature convergence at local optima and not suitable for non-convex and non-smooth types of cost function, etc. A survey of optimization techniques that are employed to answer the CEED problem over the period ( $2005-2016$ ) shows researchers great interest in finding the optimal solutions (Mahdi et al., 2018). Qu et al. (2018) offered a comprehensive overview of research in economic-emission dispatch (EED), exploring multi-objective evolutionary algorithms’ applications across traditional and dynamic EED problems, along with their integration into wind power, electric vehicles, and micro-grid scenarios. Another recent review article examines various optimization techniques employed in addressing the CEED problem, providing algorithms classification, each with its own set of advantages and disadvantages in tackling this intricate issue, including with and without applications in renewable energy, thereby offering insights for future research (Marouani et al., 2022).

Conventional optimization techniques

Many conventional optimization approaches have been adopted to answer ED and CEED problem such as lambda iteration (LI) (Zhan et al., 2013; Singhal et al., 2014), Newton-Raphson (NR) method (Chen and Chen, 1997, 2003), dynamic programing (DP), quadratic programing (QP) (Fan and Zhang, 1998), interior point method, and Lagrange relaxation (LR) (El-Keib et al., 1994; Shalini and Lakshmi, 2014; Krishnamurthy and Tzoneva, 2012). The environmentally constrained ED (ECED) is one of the initial forms of dispatching problem in which both problems are solved separately. According to Dhillon et al. (1993), LR is used to solve ECED in 1993 and analyzed that environmental constraints are easily accommodated without any major modification. The authors also conclude that their proposed algorithm is reliable, viable for offline and online applications and can generate more revenues for the power sector. They have used the test system consisting of 101 dispatchable generating units and two constraints. A comparison of LR and particle swarm optimization (PSO) to solve CEED is presented by Damodaran and Kumar (2014) and it reveals that LR gives better results than PSO in terms of computational time and fuel cost on IEEE 30 bus, six and 11 generators with power balance, transmission losses, and generator operational constraints. On the other hand, PSO proves itself better than LR with fewer emissions and transmission losses. The authors reach at the conclusion that LR is a better approach than PSO to solve the CEED problem. The solution of EED problem using LR is presented by Shalini and Lakshmi (2014), with an objective function of fuel price and emissions reduction of thermal power plants. The IEEE $30$ node and six generators testing scheme is used with consideration of power balance, transmission losses and generators’ operational constraints to investigate the results of the proposed algorithm. It is also established that the fuel cost and emissions vary with changing weighting factors and LR demonstrates better results. A real-time multi-objective power dispatch optimization problem has been solved by Chen and Chen (1997), by using the NR approach. The QP with incorporation of goal programing is used to solve real-time ED by Fan and Zhang (1998) (Table 1).

Table 1.

Summary of conventional optimization techniques used to resolve CEED problem.

Reference	Method	Units	Constraint
El-Keib et al. (1994)	LR	101	2
Krishnamurthy and Tzoneva (2012)	LR	6,11	3
Shalini and Lakshmi (2014)	LR	6	3
Chen and Chen (1997)	NR	15	3
Fan and Zhang (1998)	QP	5	4

CEED: combined economic and emission dispatch; LR: Lagrange relaxation; NR: Newton–Raphson; QP: quadratic programing.

Table 2.

Summary of non-conventional optimization techniques used to resolve CEED problem.

Authors	Year	Method	Units	Constraints
Secui (2015)	2015	CGBABC	6, 10, 16, 40, 120	4
Sulaiman et al. (2015)	2015	CSA	6, 40	2
Ziane et al. (2015)	2015	SA	6	2
Pandit et al. (2015)	2015	DEFS	16	5
Nguyen and Ho (2016)	2016	BA	6	2
Singh and Dhillon (2016)	2016	OGHS	6, 10, 13, 40	3
Mahdi et al. (2018)	2017	QPSO	3, 5	2
Abbas et al. (2017b)	2017	PSO	6	2
Mason et al. (2017)	2017	PSO AWL	10	3
Chopra et al. (2017)	2017	MPSOSM	10	2
Zou et al. (2017)	2017	NGPSO	6, 10, 11, 40	2
Chellappan and Kavitha (2017)	2017	CSA	6	2
Spea (2017)	2017	FFA	6	2
Mahdi et al. (2019)	2018	QBA	6	2
Khalil et al. (2018)	2018	CSA	6	2
Abdelaziz et al. (2016)	2016	FPA	3, 10, 40	2
Ali and Abd Elazim (2018)	2016	MBA	10, 13	2
Rezaie et al. (2019)	2018	CIHSA	6, 10, 13, 14, 40, 140	3
Zhu et al. (2019)	2019	IMOEA/D-CH	6, 10, 14	3
Bhargava and Yadav (2022)	2020	DE-CSA	3, 6	2
Sakthivel et al. (2021)	2020	MOSSA	6, 10, 40	3
Zaoui and Belmadani (2022)	2021	OWP-OMF	6, 10, 40	2
El-Shorbagy and Mousa (2021)	2021	CMEO	6	3
Chopra et al. (2021)	2021	MPSO_SSM	6, 10, 80	2
Basetti et al. (2021)	2021	QOPO	6, 10, 11, 13, 40	2
Hassan et al. (2022)	2022	MSSA	6, 10, 14	3
Hassan et al. (2022)	2022	MMPA	3, 5, 6, 26	2
Sutar and Jadhav (2023)	2023	ABC-NSGA-II	6, 10, 40	2
Habib et al. (2023)	2023	HBMO	6	5
Singh et al. (2023)	2023	MO-ARQIEA	6	2
Almalaq et al. (2023)	2023	ChMODE	6, 10	3
Luo et al. (2023)	2023	RLADE	6, 11	2
Singh et al. (2023)	2023	NPSO	6	2
Singh et al. (2023)	2023	MPSO	6, 13, 15	2
Peesapati et al. (2023)	2023	EHO	6, 10, 40	3

CEED: combined economic and emission dispatch; CGBABC: chaotic global best artificial bee colony; CSA: cuckoo search algorithm; SA: simulated annealing; DEFS: differential evolution with fuzzy selection; NSGA: non-dominated sorting genetic algorithm; BA: bat algorithm; QPSO: quantum-inspired particle swarm optimization; PSO: particle swarm optimization; PSO AWL: particle swarm optimization avoidance of worst locations; MPSOSM: modified particle swarm optimization using simplex search method; NGPSO: new global particle swarm optimization; CSA: cuckoo search algorithm; FFA: firefly algorithm; QBA: quantum behaved bat algorithm; FPA: flower pollination algorithm; MBA: mine blast algorithm; CIHSA: chaotic improved harmony search algorithm; EHO: elephant herd optimization; ChMODE: hybrid chaotic multi-objective differential evolution; OMF: optimization by morphological filters; MO-ARQIEA: multi-objective adaptive real-coded quantum-inspired evolutionary algorithm; MOSSA: multi-objective squirrel search algorithm.

Non-conventional optimization techniques

Non-conventional or modern heuristic optimization methods have been extensively used by scholars to answer the CEED problem in recent years (Table 2). To overcome the incompetence of conventional optimization methods, new computational intelligence (CI), nature-inspired advanced CI, and artificial intelligence based techniques have been introduced. Nature-inspired or bio-inspired evolutionary optimization techniques have gained significant popularity and widespread application in engineering and technology over recent decades. They pose a challenge to traditional numerical methods, which rely on assumptions of convexity and continuity and typically use a gradient-based search sensitive to initial solutions. While early bio-inspired techniques had limitations like premature convergence and dependence on control parameters, subsequent improvements and newer variants have effectively tackled these issues (Dubey et al., 2018). Figure 1 shows the various non-conventional optimization methods that have been practiced to solve the CEED problem. Accuracy with high-quality solutions, robustness, powerful computational performance, and fast convergence rate are major features of these optimization techniques (Mahdi et al., 2018). Some of the most important types of optimization techniques that have been used to solve CEED are discussed here.

Figure 1.

Non-conventional optimization techniques used to solve combined economic and emission dispatch (CEED).

The PSO is a stochastic optimization method that consists of a population with self-adaptive features. The idea was formulated by Kennedy and Eberhart (1995). The PSO is one of the most used optimization methods for the solution of the CEED problem. Literature review investigation revealed that Kumar et al. (2003) used PSO to answer the EED problem for the first time. They verified the solution algorithm on a six generators test system with power balance and generator limit constraints and compared the results with classical techniques such as real-coded genetic algorithm (RCGA) and hybrid genetic algorithm. The outcomes show better performance of PSO and global optimality above the mentioned conventional techniques. A review of EED solutions with PSO is presented by Mahor et al. (2009), which concludes that PSO proved to be the best option if it does not undergo trapping into local optima, premature convergence, and dimensionality complications. Moreover, in comparative analysis, PSO has shown better results with less computational time than conventional optimization techniques. Hussain et al. (2019) compared PSO and genetic algorithm (GA) for the CEED of an independent power plant (IPP) in Pakistan across different load demands. Using real IPP data, the study found that PSO performed better than GA in achieving the optimal solution for fuel cost, emission reduction, convergence characteristics, and computational time. Salaria et al. (2021) introduced two versions of global PSO incorporating inertial weights and quasi-opposition-based strategies. These variants were applied to tackle the economic load dispatch (ELD) for the Korean grid using IEEE standards for units 3, 6, 13, 15, 40, and 140. The hybrid PSO with simplex search method (MPSOSM) aimed to counter PSO’s issues of early convergence and stagnation by merging expansive exploration through PSO with localized exploitation via the simplex search technique is suggested by Chopra et al. (2021). This strategy seeks to elevate solution efficacy by harnessing PSO for wide-ranging exploration and employing the simplex search method to fine-tune solutions, steering clear of local minima towards global or nearly optimal points. Singh et al. (2023) presented an improved PSO variant that addressed the limitations of classical PSO. It prioritized retaining particles within the search area, updated positions to maintain global search involvement, dynamically adjusted constriction factors to guide particles towards the global solution and set inertia weight within defined ranges to ensure continuous movement towards the global solution, ultimately enhancing optimization performance. In another study (Singh et al., 2023), a modification was proposed for PSO, integrating a control mechanism for particle velocity through an attraction factor to avoid particle stoppage throughout iterations. This change removed the necessity of updating particle velocities, concentrating solely on adjusting particle positions to enhance computational speed.

A detailed and comprehensive review is presented by Abbas et al. (2017a,b) on CEED solution using PSO suggested many alterations in the elementary assembly of PSO to crack convex and non-convex dispatch problems. Mason et al. (2017) compared standard PSO (SPSO) and a new variant of PSO with avoidance of worst locations (AWLs) and revealed that the new variant performs better than SPSO. According to Chopra et al. (2017), a modified PSO, MPSOSM, is used to solve this problem which has shown better results than classical PSO for quality and convergence.

To increase the local searching capability and to dodge trapping in local optima. Varma et al. (2013) proposed a Gaussian randomizer instead of a traditional uniform randomizer in SPSO with four constraints such as power stability, generating unit output boundaries, ramp rate limit, and forbidden working time zone to achieve more realistic results. Zou et al. (2017) presented a new global PSO (NGPSO) by introducing two key modifications which are a new particle position updating process based on global best (gbest) particle to guide searching and uniform distributor for randomization. Jadoun et al. (2015) proposed two variants which are linear modulated PSO (LMPSO) and sinusoidal modulated PSO (SMPSO) by modifying the particles’ velocity. Both the proposed modified PSO techniques consider power balance, generator limits, and prohibited operating zone constraints and investigate solutions on three test systems in which SMPSO proved better than LMPSO. Hamed et al. (2012) solved the CEED problem using time varying acceleration constants (PSO-TVAC) and illustrated a comparative analysis of different techniques. A variant of PSO, bare-bone PSO and whale swarm optimization algorithms are proposed for combined heat and power economic and emission dispatch problems by Xiong et al. (2022) and Jadoun et al. (2022), respectively. Another variant of PSO, the moth swarm algorithm is used for solving the day-ahead CEED problem in a thermal power system integrated with solar PV units (Ajayi and Heymann, 2021). A grasshopper algorithm with a binary approach is used for solving CEED by Sharifian and Abdi (2022). The effects of valve point loading effect, ramp-rate limits, prohibited zones, and transmission losses are also considered in the study.

The GA is the second most frequent algorithm used to solve the CEED problem. The inspiration for this algorithm was taken from Darwin’s concept of evolution that only tough and fit genes can survive. Many versions of GA have been developed and implemented for EED problems. Song et al. presented a fuzzy logic-controlled GA (FCGA) to solve ED and EMD problems in 1997. Hybrid flower pollination algorithm (HFPA) for optimizing wind-thermal systems, concurrently minimizing cost, emissions, and losses while considering complex constraints, such as valve point loadings, ramp limits, prohibited zones, and spinning reserve. The HFPA method combines flower pollination algorithm (FPA) and differential evolution (DE) to improve solution exploration and prevent premature convergence (Dubey et al., 2015). To adaptively fine-tune crossover possibility and mutation frequency, this algorithm has used two heuristic-based fuzzy logic controllers. The validity of the proposed FCGA is illustrated on six generators with consideration of two constraints for the EED problem (Song et al., 1997). A comparative analysis of evolutionary programing which has been used to answer the CEED problem is presented by Venkatesh et al. (2003). An $\epsilon$ -dominance-based GA for EED is presented by Osman et al. (2009), based on the idea of co-evolution to overhaul algorithm nonlinear constraints. Dhanalakshmi et al. (2011) proposed an improved non-dominated sorting GA (NSGA)-II by presenting two improvements, the first is dynamic crowding distance and the other modification was controlled elitism. Both the improvements are used to encounter the absence of uniform diversity in attained results and the nonappearance of diversity maintaining operator in NSGA.

Artificial bee colony (ABC) is a flock-grounded stochastic search algorithm that replicates the scrounging actions of honeybees. Many researchers have been using ABC to solve the CEED problem. Dixit et al. (2011) used ABC for the CEED problem and implemented it on three and six generating units systems. The comparative analysis shows better performance of ABC than conventional optimization techniques in terms of superiority of results, steady convergence characteristics, and better computational efficiency. ABC with dynamic population size is presented by Aydin et al. (2014), which is an improved version of incremental ABC with local search with the same working mechanism. Both procedures are illustrated on IEEE 30 nodes with considering 40 generators and comparative analysis shows that the proposed algorithm performs better than its predecessor. A chaotic global best ABC (CGBABC) optimization method is presented by Secui (2015) which is based on the global best ABC (GBABC). The CGBABC procedure is verified with five, six, $10$ , $16$ , $40$ , and $120$ generating units systems and a comparison of results shows that the proposed algorithm performed better than classical ABC and GBABC. According to Afandi et al. (2018), two optimization techniques, ABC and GA are used to solve the CEED problem and their comparative analysis reveals that ABC is better than GA in terms of smoothness and fast search. Another comparative analysis is presented by Bhongade and Agarwal (2016) which concludes that ABC is better than GA in terms of generation cost, loss reduction, and less computational time. According to Sutar and Jadhav (2023), a technique combining the ABC optimization algorithm with the NSGA-II is proposed. To gauge its efficiency, the algorithm’s performance was tested on three systems with 6, 10, and 40 coal-based generators. The evaluation included comparing its outcomes on fuel cost and emissions separately with existing techniques documented in the literature. Notably, the comparison did not consider the price penalty factor in assessing the overall fuel cost to ensure a direct comparison among different methodologies.

Nguyen and Ho (2016) presented a new meta-heuristic bat algorithm (BA) to solve the EED problem using the quadratic fuel function. The algorithm is tested on six generators system for two loads (1200 and 1800 MW) and three dispatch cases (fuel, emissions, and both). The authors conclude that BA performs better than other optimization techniques. Niknam et al. (2013) used self-adaptive learning BA with a chaos-based approach to initialize the population. Recently, Mahdi et al. presented a quantum computing idea based on BA known as the quantum behaved BA (QBA) to answer CEED. The proposed algorithm has four major purposes which are ED and three separate emissions dispatches (SO₂, NOx, and CO₂). To compare the results, a unit-wise penalty factor is used and tests are performed in six generators with four different loads. The comparative analysis of different techniques (LR, simulated annealing (SA), and PSO) and QBA displays that the proposed algorithm shows better results, robustness, and computational performance. In future research directions, the authors have mentioned that the addition of QC idea in metaheuristic optimization method delivers valuable and trustworthy means for multi-objective optimization problem (Mahdi et al., 2019). The FPA was designed for addressing ELD and CEED challenges within power systems (Abdelaziz et al., 2016). Its effectiveness was evaluated on 3-units, 10-units, and 40-units systems and compared with other prominent optimization algorithms to affirm its superiority.

Khalil et al. proposed a cuckoo search algorithm (CSA) for CEED and results are illustrated with six generators considering power balance and generators’ limit constraints using the cubic criterion function for the power demand of 150 MW. The comparative analysis reveals that CSA gives better results than LR, SA, and PSO (Chellappan and Kavitha, 2017). Sulaiman et al. proposed CSA for the solution of CEED and to prove effectiveness, the proposed algorithm is tested on 6-units and 40-units generating systems. The comparative analysis of CSA and other recent optimization techniques such as gravitational search algorithm (GSA), NSGA II, and strength pareto evolution algorithm (SPEA) has been presented by Sulaiman et al. (2015). Chellappan et al. presented CSA for CEED problem solution with two constraints and the proposed model is tested on six generators system concluding that the CSA performs better than GA results of CSA (Khalil et al., 2018). Bhargava and Yadav (2022) tackled the CEED problem by combining elements from the CSA and DE into the hybridized model hDECSA. This fusion involved updating solutions from both schemes and integrating them with a random searching model. Notably, this approach leds to reduced computational time compared to PSO, DE, and differential evolution PSO schemes.

The SA is a probabilistic robust optimization method that has been deployed efficaciously in various fields such as very large-scale integration circuit design, image processing, neural networks, economic load dispatch, etc. According to Ziane et al. (2015), SA is used for the solution of the dynamic EED problem. The proposed algorithm is successfully employed with six generators considering two constraints and results are compared with multi-objective modified bacterial foraging optimization (MBFA). The authors conclude that SA provides superior results than the other optimization techniques. The SA with goal attainment technique is proposed by Basu (2005) to solve the EED of a hydro-thermal power plant. The authors conclude that the proposed algorithm almost attains global optimality without imposing any convexity restriction on the generator’s characteristics. The only negative characteristic of the proposed method is that it takes a lengthy execution time. The SA has also been used to solve the CEED problem with new penalty factors by Ziane et al. (2015).

Farhat and El-Hawary (2011) projected a MBFA to solve the EED problem with power transmission losses. The basic BFA shows better results for small constrained problems, however, in larger constrained problems, it shows poor convergence. To overcome this complication along with the high dimensionality problem of EED problem, some modifications have been proposed for BFA. A comparative analysis of BFA and MBFA on six generating units system with consideration of two constraints, the proposed algorithm shows better convergence. An improved BFA is used to solve static and dynamic CEED problems by Pandit et al. (2012). The performance of the proposed improved bacterial foraging algorithm is tested on four different cases and it is revealed that the proposed model shows better and more consistent results.

Abou El Ela et al. (2010) proposed a DE for the EED problem and comparative analysis shows its superiority and effectiveness. An improved DE with fuzzy selection and data-driven surrogate-assisted approach are proposed by Pandit et al. (2015) and Lin et al. (2023), respectively, for the EED problem of multi-area power systems. The proposed algorithm consists of DM preference to direct the examination and to pick the seeds for the next offspring. DEFS is tested on $16$ generators with five constraints to compare with PSO. The firefly algorithm (FFA) is used for the EED problem and tested on six generators system with two constraints. The authors conclude that the proposed algorithm is good in terms of less calculation time, effective, and precise to attain global optimality (Apostolopoulos and Vlachos, 2011). The FFA has many similarities with other swarm intelligence techniques like PSO, BFA, ABC, etc.; indeed its concept and implementation are much simpler. The algorithm is also verified with IEEE 30-bus six generators for the EED problem and outcomes are compared with NSGA, SPEA, NPGA, MBFA, etc. It is concluded that FFA is effective and robust and performs better than other reported optimization techniques (Spea, 2017). A low-carbon ED data-driven model is proposed for urban multi-energy systems by Zhang et al. (2023).

Güvenc et al. (2012) proposed a GSA to solve the CEED problem. The algorithm is tested on four different test systems to demonstrate its effectiveness. A better algorithm opposition-based GSA (OGSA) is proposed by Shaw et al. (2012), to accelerate the efficiency of GSA. The suggested procedure is verified on $23$ complex benchmark test functions and four standard power systems (two cases for $6$ , $40$ , and $140$ units) for the CEED problem. The simulation results confirm the efficiency and sturdiness of the proposed algorithm.

A simplified recursive approach is used by Balamurugan and Subramanian (2007) for the CEED problem and tested on six and 11 generators systems. A comparative analysis of the proposed algorithms is also presented with other optimization techniques to prove its optimality. Basu (2011) has solved economic and environmental dispatch problems by using a multi-objective differential evolution (MODE) optimization approach which is an extension of DE. The algorithm is tested on three test systems and results are compared with pareto differential evolution (PDE), strength pareto evolutionary algorithm 2 (SPEA), and NSGA-II. Another study by Hassan et al. (2022), a modified marine predators algorithm (MMPA) is introduced and employed for determining the optimum solution to the CEED problem. The MMPA approach was used to tackle single and bi-objective CEED problems in diverse power generation systems, spanning 3 to 26 units, across varied load levels while considering only one of three emission types (SO₂, NO_x, and CO₂) as an additional objective (Hassan et al., 2022).

The optimization by morphological filters algorithm was applied to the CEED problem by Zaoui and Belmadani (2022). This implementation took into account two constraints but omitted penalty factors in its approach. A modified honey bee mating optimization (HBMO) algorithm, leveraging collective intelligence, was presented to address the EED challenge within power systems by Habib et al. (2023). This algorithm is specifically tailored to handle nonlinear cost functions and various constraints of power generation. The main objective is to boost the standard algorithm’s effectiveness and balance by enhancing both local and final search approaches, employing an adaptive nonlinear system, and establishing the optimized solution as the termination criterion. According to Zhu et al. (2019), the progression from the multi-objective evolutionary algorithm based on decomposition (MOEA/D) algorithm to constraints handling MOEA/DCH is outlined, targeting the efficient resolution of the dynamic dispatch problem. This development involves the fusion of real-time heuristic constraint adjustments, adaptive threshold punishment mechanisms, and evolutionary control strategies, all designed to navigate complex constraints within the dispatching model. Sakthivel et al. (2021) introduced a multi-objective squirrel search algorithm that integrates the squirrel search algorithm with Pareto-dominance principles for producing non-dominated solutions, preserving diversity via an elitist depository mechanism, and crowding distance sorting. Singh et al. (2023) introduced the multi-objective adaptive real-coded quantum-inspired evolutionary algorithm aiming to tackle premature and slow convergence issues within evolutionary algorithms (EAs). This approach, embedded in the EA framework, adopts a non-dominated sorting technique and integrates quantum principles along with real-coded variables. The quasi-oppositional search-based political optimizer for non-convex CEED problems has been introduced by Basetti et al. (2021). While excelling in faster computation and superior convergence for optimal solutions, it required more iterations as unit numbers increased, constraining its scalability, and computational efficiency in larger applications. The hybrid chaotic multi-objective DE algorithm is presented by Almalaq et al. (2023) which adopted non-domination sorting and crowding distance calculation to accurately derive the Pareto front. To overcome local optima and improve the traditional DE algorithm, it integrates two distinct chaotic maps during initialization, crossover, and mutation stages, replacing the reliance on random numbers for enhanced performance. Luo et al. (2023) introduced reinforcement learning-based adaptive differential evolution (RLADE) to optimize mutation strategy and crossover probability for improved convergence and searchability. It integrated adaptive population size-based state division and fitness-ranking-based rewards in RL to enhance accuracy. Evaluated on quadratic and cubic criterion functions for CEED problems, RLADE outperformed the DE algorithm and its RL-based variants in terms of mean values and standard deviations. Peesapati et al. (2023) introduced the elephant herd optimization algorithm, employing innovative strategies to maintain population diversity and ensure the retention of the fittest individuals in subsequent generations. Comparative analysis demonstrated its superiority over BA and ant lion optimizer algorithms based on the fuel cost function values obtained. Rezaie et al. (2019) presented the chaotic improved harmony search algorithm as an advanced iteration of the harmony search algorithm. By incorporating chaotic patterns for random number generation, dynamically adjusting parameters, utilizing virtual harmony memories, and introducing a new local optimization technique, the algorithm significantly improved robustness, accuracy, and search efficiency while reducing the iterations needed for optimal solutions. A hybrid method merged the exchange market algorithm (EMA) and adaptive inertia weight PSO (AIWPSO) to address the CEED problem. System constraints such as prohibited operating zones and ramp rate limits were handled by the multiple constraint ranking technique and the inclusion of a mutation operator aimed to boost global search capabilities (Nourianfar and Abdi, 2022). The goal behind combining these algorithms is to leverage EMA’s strength in global exploration alongside AIWPSO’s effectiveness in local exploitation. Ali and Abd Elazim (2018) used mine blast algorithm (MBA) to solve ELD and CEEED problems. The authors also studied the effect of valve points on the cost of fuel. The findings of the study demonstrate that the proposed MBA outperforms other optimization techniques in terms of total cost and computational time.

Multiobjective ED with renewable integration

The growing demand for a cleaner environment has sparked attention towards addressing pollution in energy production. Laws in Japan and Europe, along with amendments to the Clean Air Act of 1990, have compelled utilities to modify their strategies, reducing pollution from thermal plants. Renewable energies, like wind turbines and solar panels, are gaining prominence due to their accessibility and ability to produce electricity more sustainably while lessening reliance on resources. Although renewable energy sources offer cleaner electricity production, their natural fluctuations pose challenges. Despite concerns about potential disruptions from large concentrations of wind turbines, they remain crucial in reducing greenhouse gas emissions. Overall, renewable energies present a promising alternative to fossil fuels, offering environmental benefits and energy independence, particularly as fossil fuel reserves diminish, costs rise, and emissions increase.

The changing nature of wind power makes it harder to manage the economic distribution of electricity in power systems. As wind power becomes more prominent, the challenges in scheduling electricity in microgrids increase. This shift requires a new approach to manage the balance between wind power and traditional thermal power in these systems (Alshammari et al., 2021). Balancing electricity production and demand is crucial when dealing with RES. This is because energy production from renewable, like solar panels and wind turbines, can vary and be uncertain over time. For example, solar panels generate electricity based on the available sunlight, which changes throughout the day and across seasons. Similarly, wind energy relies on the varying wind speed, influenced by different factors depending on the location. To accurately predict their energy output, we need to model the unpredictable nature of wind speed and solar radiation (Li et al., 2023).

Motivation and contributions

This study is conducted for a solution to a complex optimization problem, CEED having multiple objectives: optimal load dispatching to the generators with the least fuel price and pollution together while honoring the functioning constraints. The main contributions of the article are listed below:

This study introduces three novel variants of the PSO algorithm tailored for addressing the multiobjective CEED problem: quantum-inspired PSO (QPSO), enhanced QPSO (EQPSO), and QPSO integrated with weighted mean personal best and adaptive local attractor (ALA-QPSO).

The research investigates the potential of these QPSO variants in tackling the CEED problem by evaluating their performance across various parameters. This evaluation involves the application of the algorithms to three distinct systems: 6-unit, 10-unit, and 11-unit systems, each characterized by different load demands.

The competence and effectiveness of the proposed algorithms in terms of robustness, quality of the solution, and computational efficiency are compared with various methods suggested in the literature.

The rest of the article is organized as follows: The “Combined economic and emission dispatch” section introduces the CEED model, outlining the role of the penalty factor, and the considered constraints and the “Optimization techniques” section details the optimization techniques developed to address the CEED problem. A comparative analysis of diverse optimization techniques, applied to three test models, is presented in the “Development of QPSO for CEED problem solution” and “Experimental results and analysis” sections and the “Conclusions” section concludes the article with a discussion of findings and potential future directions.

Combined economic and emission dispatch

The combination of two objectives, fuel cost and emissions minimization of thermal power plants is known as CEED. These two objectives are non-commensurable and conflicting; the reduction of fuel cost leads to excess emissions and vice versa. Therefore, the problem’s solution can only be achieved through some tradeoff between these two objectives.

The term operating cost of ED refers to the cost of fuel used in a thermal power plant. There are sets of valves in the fuel inlet of each generating unit which are used to control the power production. For instance, the set of valves is opened in sequence to raise the steam inlet of fuel to increase the production. Throttling losses are high during the opening time of the valve and low when it is fully opened. This phenomenon is known as the valve point loading effect. The entire expenditure of a fossil fuel-based thermal plant mainly depends on fuel cost and the other expenditures (maintenance costs, labor costs, and cost of fuel supplies) are generally fixed percentages of fuel cost. The formulation of an ED optimization is usually represented by the quadratic cost function given in equation (1). The first objective function is fuel cost $F_c$ is a sum of N generating units’ fuel cost which needs to be minimized given by equations (1) and (2) (Kothari and Dhillon, 2004).

$$F_c=\sum _{i=1}^N{F}_i(P_{gi})=\sum _{i=1}^N(a_i{P}_{gi}^2+b_i+P_{gi}+c_i)$$

(1)

By including the valve point loading effect the quadratic equation can be illustrated as (Güvenc et al., 2012)

$$F_c=\sum _{i=1}^N[a_i{P}_{gi}^2+b_i+P_{gi}+c_i+|d_i\sin e_i(P_{gi}^{min}-P_{gi})]$$

(2)

where “ $F_i$ ” is the fuel cost function of the $i\mathrm{th}$ generating unit, “ $F_c$ ” is the total fuel cost function of N generating units, “ $P_{gi}$ ” is the actual power generation of the $i\mathrm{th}$ generator, ( $a_i$ , $b_i$ , $c_i$ , $d_i$ , $e_i$ ) are the cost coefficients of the $i\mathrm{th}$ generator and “N” is the number of generating units.

There are many pollutants released by thermal power plants such as sulphur dioxide, carbon dioxide, nitrogen oxides, etc. Emissions of these gases need to be minimized for every generating unit to reduce pollution. Some researchers take each pollutant separately as a separate objective function, however, in this article, we have considered all those emissions together as a collective emission dispatch. The construction of EMD problem is represented by a quadratic function of emissions in equation (3) and with valve point loading effect represented by $E_T$ is given by equation (4).

$$E_T=\sum _{i=1}^N({\alpha }_i{P}_{gi}^2+{\beta }_i+P_{gi}+{\gamma }_i)$$

(3)

$$E_T=\sum _{i=1}^N[{\alpha }_i{P}_{gi}^2+{\beta }_i+P_{gi}+{\gamma }_i+\eta \exp ({\delta }_i{P}_{gi})]$$

(4)

where $E$ and $E_T$ are the emission function of the $i\mathrm{th}$ generating unit and total emission function of $N$ generating units. The symbols ${\alpha }_i$ , ${\beta }_i$ , ${\gamma }_i$ , ${\delta }_i$ , and ${\eta }_i$ show the emission constants of the $i\mathrm{th}$ generating unit.

Role of penalty factor

To change a multi-objective problem to a single objective, a penalty factor has been introduced denoted by “h.” The penalty factor plays a role in changing the physical meaning of emissions (kg/h or lb/h or ton/h) to fuel cost ($/h), to equalize the units of both functions. In this way, the difference of dimensions is eliminated and two objectives are transformed into one objective. Hence the total fuel cost CEED formulation represented by $F_T$ is given by equation (7).

$$F_T=\sum _{i=1}^N[(a_i{P}_{gi}^2+b_i{P}_{gi}+c_i)+h_i(d_i{P}_{gi}^2+e_i{P}_{gi}+f_i)]$$

(5)

There are six types of penalty factors reported in literature (Ziane et al., 2015; Krishnamurthy and Tzoneva, 2011; Afandi et al., 2016; Meziane et al., 2020), however, the most commonly used type is the max–max penalty factor.

$$h_i=\frac{{(a}_i{P}_{gi,max}^2+b_i{P}_{gi,max}+c_i)}{(d_i{P}_{gi,max}^2+e_i{P}_{gi,max}+f_i)}$$

(6)

$$h_i=\frac{a_i{P}_{gi,max}^2+b_i{P}_{gi,max}+c_i+|d_i\sin (e_i(P_{gi}^{min}-P_{gi,max}))|}{{\alpha }_i{P}_{gi,max}^2+{\beta }_i{P}_{gi,max}+{\gamma }_i+{\eta }_i\exp ({\delta }_i{P}_{gi,max})}$$

(7)

Two types of constraints

Power balance constraint

The power balance can be explained as the total active power generation of all the generators is essential to meet total power requirement. Therefore, total power production should be equivalent to the summation of power demand and power losses as shown in equation (8).

$$P=\sum _{i=1}^N{P}_{gi}=P_D+P_L$$

(8)

where $P$ , $P_{gi}$ , $P_D$ , and $P_L$ stand for total power generation of all the generating units, the power production of individual generator i, entire power requirement, and total power losses, respectively. For example, if power demand is $250$ and $3$ MW losses then a system consisting of six generating units is scheduled in such a way as to provide a total of $253$ MW power.

The losses that occurred during the power transmission to the grid such as power dissipation in conductors of transmission lines, transformers, and corona losses are recognized as transmission losses. The simple formula used to calculate power transmission loss is given in equation (9) and is known as George’s method (George, 1960).

$$P_L=\sum _{i=1}^N\sum _{j=1}^N{(P}_{gi}*B_{ij}*P_{gj})$$

(9)

where $B_{ij}$ is the transmission losses coefficient of George’s formula, $P_{gi}$ and $P_{gj}$ are the actual power inserted at the $i\mathrm{th}$ bus and the $j\mathrm{th}$ bus.

Generator operational constraint

A power generating unit must work between its lower and upper limits of generation. Operating a generator below its lower limit is uneconomical or technically infeasible and the upper limit is the maximum output that can be taken from it. Generator operational constraint, also known as generation limit constraint, can be defined as $P_{i,min}\le P_i\le P_{i,max}$ where

$P_{i,min}$ lower limits or minimum value of power output of the $i\mathrm{th}$ unit.

$P_{i,max}$ upper limits or maximum value of power output of the $i\mathrm{th}$ unit.

Optimization techniques

Quantum particle swarm optimization

QPSO is an improved version of PSO inspired by the quantum computing phenomenon. In QPSO, a quantum bit (Qubit) and spin gate are presented to enhance the range of the population. Moreover, to avoid local optima trapping, self-adaptive, and sequence mutation have been introduced. The particle’s state is depicted by the wave function rather than the position and velocity. Hence, QPSO shows more efficiency, stronger search capabilities, and faster convergence than basic PSO.

A survey presented by Manju and Nigam (2014), reveals that hundreds of papers reported successful implementation of quantum-inspired computational intelligence (QCI) and this combination of quantum mechanics (QM) and CI has yielded promising results for real-world applications. The first idea of QPSO was presented by Sun et al. (2004) with the model named quantum delta potential well model for PSO (QDPSO).

Later on, a cooperative approach to QPSO was introduced by Gao et al. (2007), to solve high-dimensional problems and maintain stability between global and local search. Another modification was presented by Su et al. (2009) in which a crossover operator was introduced to escape from local optimal. The testing of this model on different functions such as the sphere, Rosenbrock, Rastrigrin’s, Griewank, and Shaffer’s proves its performance better than basic PSO and original QPSO.

The next improvement was presented by Sun et al. (2012), in which search strategies were analyzed along with an investigation of the contraction-expansion (CE) coefficient’s modifications. Two methods of controlling CE were also proposed which have fixed and time-varying values. However, the authors also concluded that further improvements are required to control and determine the value of parameters for QPSO. The final form of QPSO was presented by Sun et al. (2012, 2011) which experimentally tested different variants of PSO on 12 different CEC 2005 benchmark functions and the comparative ranking showed that QPSO stands at second level.

QPSO is a new version of the PSO technique with the same working principles, based on population and self-adaptive features. For an optimal solution in multidimensional search space, particles are not updating their position by adding velocity instead each particle has quantum behavior which is formulated by a wave function (Mahdi et al., 2018, 2017). The main modification can be summarized as the particles’ state is depicted by the wave function which consists of angle and quantum bit (Qubit) instead of the position and velocity of PSO. The Qubit can simultaneously stay in two different quantum states which is its major difference with classical bit. The wave function $\psi$ can be explained as

$$\psi =\alpha |0>+\beta |1>$$

(10)

where $\alpha$ and $\beta$ are two complex numbers which satisfy

$${\left|\alpha \right|}^2+{|\beta |}^2=1$$

(11)

The $|0>$ represents the spin up state, while $|1>$ represents spin down state. In this way, Qubit simultaneously stay in up and down state. This superposition can be explained by equation (13)

$$\psi =\sin \theta |0>+\cos \theta |1>$$

(12)

where $\theta$ represents the phase of Qubit,

$$\theta =\arctan \left(\frac{\beta }{\alpha }\right)$$

(13)

The initialization procedure of QPSO is almost the same as PSO, random particle positions are calculated within the domain. These best values are selected as personal best “pbest” and the best fitness among all is selected as “gbest.” The QPSO introduces the local attractor which means that convergence is achieved if each particle converges to that local attractor. The local attractor is calculated according to equations (15) or (16)

$$P_i^t=[P_{i,1}^t,P_{i,2}^t,\dots ,P_{i,j}^t,\dots ,P_{i,D}^t]$$

(14)

$$P_{i,j}^t=\frac{c_1{rand1}_{i,j}^t{P}_{ij}^t+c_2{rand2}_{i,j}^t{G}_j^t}{c_1{rand1}_{i,j}^t+c_2{R}_{i,j}^t}$$

(15)

$$P_{i,j}^t=\phi P_{i,j}^t+(1-\phi )G_j^t$$

(16)

where $i,j,t$ are the particle, dimension, and iteration, respectively, $c_1$ and $c_2$ are the learning factors also known acceleration constants, $P$ is the local attractor, $P_{i,j}^t$ is the personal best (Pbest), $G_j^t$ is the global best (Gbest), and ${rand}_1$ and ${rand}_2$ are the random numbers between (0 and 1), and $\phi$ are the uniformly distributed random numbers

$$\phi =\frac{c_1{rand1}_{i,j}^t}{c_1{rand1}_{i,j}^t+c_2{R}_{i,j}^t}$$

(17)

The equation of local attractor can be written as equation (18)

$$P=\phi Pbest+(1-\phi )Gbest$$

(18)

The CE coefficient is denoted by $\alpha$ , which can be either fixed or time-varying

$$\alpha =(w2-w1)\left(\frac{t_{max}-t}{t_{max}}\right)+w1$$

(19)

where t is the present iteration and $t_{max}$ is the maximum iteration. Finally, the particle position is updated using equation (20)

$$X_{i,j}^{t+1}=P_{i,j}^t\pm \alpha |C_j^t-X_{i,j}^t|\ln \left(\frac{1}{u_{i,j}^{t+1}}\right)$$

(20)

where $P_{i,j}^t$ is the personal best position of particle i, $\alpha$ is the CE coefficient, $X_{i,j}^{t+1}$ is the new position at iteration $t+1$ , $X_{i,j}^t$ is the position at iteration $t$ , $C_j^t$ is the mean best of positions of all particles is known as mbest, and $u_{i,j}^{t+1}$ is the random no. evenly spread over “0” and “1.”

The parameter $\alpha$ (CE) is the only parameter of QPSO that needs to be tuned to control the convergence swiftness of the process and its value, $\alpha <1.781$ is used to guarantee the convergence. There are two methods to set CE, the first is to fix the value of $\alpha$ in the range $0.5-0.8$ and the best practice is to set at $0.75$ . However, fixing the value of $\alpha$ is sensitive to the size of the population and number of iterations, if those parameters are changed then the $\alpha$ needs to be adjusted accordingly to get the desirable performance. The second option is time-varying CE which can be calculated by equation (19). The researchers suggest that setting the value of $w2$ between 0.8 and 1.2 and $w1<0.6$ generates acceptable performance. The linearly decreasing $w1$ for QPSO can generate better performance (Sun et al., 2012).

The global search capability of QPSO makes it effective in exploring complex solution spaces, which is important for determining optimal solutions to complex CEED problems. It can balance exploration and exploitation which helps optimize both emissions levels and ED needs. Furthermore, lower computational effort and parallelization potential increase its efficiency, especially in large energy systems with numerous variables and constraints. Additionally, its ability to escape local optima is vital for finding globally optimal solutions.

Enhanced quantum particle swarm optimization

The combination of PSO and quantum mechanics breeds a new global optimization method named QPSO. It shows encouraging performance in terms of convergence, searchability, solution accuracy, and robustness. Jia et al. (2015) presented EQPSO to overcome the global searching ability limitation of QPSO for a limited number of iterations. A novel way of calculating the local attractor for QPSO is to improve the performance of QPSO’s global search. The author tried to achieve diversification of particles at the early iterations stage and have a good performance in local search at later iteration stages. The proposed local attractor found by equation (21)

$$P_{i,j}^t=\frac{L_c-C_c}{L_c}\beta P_{i,j}^t+\frac{C_c}{L_c}(1-\beta )G_j^t,\beta U(0,1)$$

(21)

where $L_c$ is the total number of iterations and $C_c$ is the current number of iteration.

QPSO with weighted mean personal best and adaptive local attractor

The most recent version of QPSO is presented by Chen (2019) with weighted pbest and adaptive local attractor and termed as ALA-QPSO. The authors claim that the proposed algorithm can simultaneously boost the search performance of QPSO and attain better global optimality. The weighted mean, pbest, was obtained by measuring difference in effect of particles with different fitness values. The other improvement in adaptive local attractor is calculated by using a sum of squares of deviation of particles fitness as the coefficient of linear combination by utilizing the sum of particles fitness value deviation square. Weighted mean personal best position can be calculated using equations (22) and (23)

$$r_i\left(t\right)=\{\begin{array}{c}\frac{1}{S-1}(1-f_{obj}^i(t)/\sum _{k=1}^S{f}_{obj}^k(t))\\ \frac{1}{S},others\end{array},if\sum _{k=1}^S{f}_{obj}^k(t)\ne 0\}$$

(22)

$$\begin{array}{cc}{mpbest}^t& ={\displaystyle \sum _{i=1}^s{r}_i(t){pbest}_i^t}\\ & =\left({\displaystyle \sum _{i=1}^s{r}_i(t){pbest}_{i1}^t,\dots ,{\displaystyle \sum _{i=1}^s{r}_i\left(t\right){pbest}_{id}^t,\dots ,{\displaystyle \sum _{i=1}^s{r}_i(t){pbest}_{iD}^t}}}\right)\end{array}$$

(23)

where $t$ is the current number of iterations, $f_{obj}^k$ is the objective function value to the $i\mathrm{th}$ particle at iteration $t$ , $S$ is the number of particles or swarm sizes, and $r_i(t)$ is the coefficient of best position of the $i\mathrm{th}$ particle.

To enhance the efficiency of population based optimization techniques, an individual must wander through the entire search space rather than gathering around local optima and in later stages it is necessary to converge toward the global optima. The novel way of computing adaptive local attractors is presented in (24) and (25)

$${Ala}_{id}^t=\phi \frac{{\sigma }^2(t)}{S}{pbest}_{id}^t+(1-\phi )(1-\frac{{\sigma }^2\left(t\right)}{S}){gbest}_d^t,\phi U(0,1),d=1,2,\dots ,D$$

(24)

$$x_{id}^{t+1}=\left\{\begin{array}{c}{Ala}_{id}^t+\alpha |{mpbest}_d^t-x_{id}^t|\ln \left(\frac{1}{u}\right),ifrand\ge 0.5\\ {Ala}_{id}^t-\alpha |{mpbest}_d^t-x_{id}^t|\ln \left(\frac{1}{u}\right),otherwise\end{array}\right\}$$

(25)

where ${\sigma }^2(t)$ is the sum of squares of deviation of particle’s fitness values used to dynamically adjust the algorithm for best path exploration calculated as

$${\sigma }^2\left(t\right)=\sum _{i=}^S{\left(\frac{f_i^{(t)}-f_{avg}^{(t)}}{F}\right)}^2,f_{avg}^{(t)}=\frac{1}{S}\sum _{i=1}^S{f}_i^{(t)}$$

(26)

where $f_i^{(t)}$ is the fitness of the $i\mathrm{th}$ particle at iteration t, $f_{avg}^{(t)}$ is the average fitness of swarm at iteration t, and $F$ is the normalized calibration factor calculated as

$$F=\left\{\begin{array}{c}max|f_i^{\left(t\right)}-f_{avg}^{\left(t\right)}|,ifmax|f_i^{\left(t\right)}-f_{avg}^{\left(t\right)}|>1\\ 1,otherwise\end{array}\right\}$$

(27)

Development of QPSO for CEED problem solution

The step-wise procedure of QPSO development is as follows: $N_P$ is the number of particles in a swarm and $n$ is the number of members in one particle. The first step of the process starts with iteration $I=1$

Initialize all QPSO parameters such as acceleration constants ( $c_1$ and $c_2$ ), inertia weights ( $w_{max}$ and $w_{min}$ ), uniform random quantities ( ${rand}_1$ and ${rand}_2$ ), and maximum number of iterations ${iter}_{max}$ . At this stage, the algorithm also reads CEED data (coefficients of fuel and emission).

Calculation of random initial position of particles using $P_{g_i,max},P_{g_i,min}$ , which are predefined for each generator.

$$P_{g_i}=P_{g_i,min}+rand\left(1\right)(P_{g_i,max}-P_{g_i,min})g=\overline{1,N_P},i=\overline{1,n-1}$$

(28)

Checking generators limits according to equations:

For lower limit violation $P_{g_i}=P_{g_i,min},P_{g_i}\le P_{g_i,min}$ .

For upper limit violation $P_{g_i}=P_{g_i,max},P_{g_i}\ge P_{g_i,max}$ .

For staying between limits $P_{g_i}=P_{g_i},P_{g_i,min}\le P_{g_i}\le P_{g_i,max}$ .

Initial power of slack bus generator is calculated based on power balance constraint equation (8) and stated in equation (29)

$$\begin{array}{r}{P}_{gd}+\sum _{i=1,i\ne d}^N(P_{gi})=\sum _{i=1,i\ne d}^N\sum _{j=1,j\ne d}^N{P}_{gi}{B}_{ij}{P}_{gj}\\ \sum _{j=1,j\ne d}^N{P}_{gi}(B_{jd}{B}_{dj}{P}_{gd}+B_{dd}{P}_{gd}^2+P_D,g=\overline{1,N_P}\end{array}$$

(29)

where $P_{gd}$ is the power generated by slack generator, $P_D$ is the total power demand of the system, and $\sum _{i=1,i\ne d}^N(P_{gi})$ total active power provided by all generators except slack generator. Now, the whole actual power vector is $P_g=[P_{gd}{P}_{gi},i=1,n,i\ne d].$

Calculation of penalty factor according to equations (6) or (7) for all generators.

Calculate fuel cost, emission and total CEED fuel cost as below

$$F_c=\sum _{i=1}^n{(a}_i{P}_{gi}^2+b_i{P}_{gi}+c_i)$$

(30)

$$E_T=\sum _{i=1}^n{(d}_i{P}_{gi}^2+e_i{P}_{gi}+f_i)$$

(31)

$$F_T=\sum _{i=1}^N[(a_i{P}_{gi}^2+b_i{P}_{gi}+c_i)+h_i(d_i{P}_{gi}^2+e_i{P}_{gi}+f_i)]$$

(32)

pbest and gbest are defined as follows:

$$P_g^{best}=min{P}_{gi}^{best},g=\overline{1,N_P,}i=1,n$$

(33)

$$G^{best}=min{P}_g^{best},g=\overline{1,N_P}$$

(34)

The second phase starts here $I=1$ .

Calculate CE coefficient ( $\alpha$ ) according to equation (19) and mbest( $C_g^t$ ) value as equation (36)

$$\alpha =(w2-w1)\left(\frac{I_{max}-I}{I_{max}}\right)+w1$$

(35)

$$C_g^I=\frac{1}{N_P}\sum _1^{N_P}{P}_g^{best}$$

(36)

Calculate local attractor, gbest and pbest values using the following equation:

$$P_{i,j}^I=\phi P_{g,i}^{best}+(1-\phi ){G_i}^{best},g=\overline{1,N_P,},i=1,n-1$$

(37)

The position of particle is updated using equation (20)

$$P_{gi}^{new}=P_{i,j}^I\pm \alpha |C_g^I-P_{gi}^{best}|\ln \left(\frac{1}{u_{i,g}^{I+1}}\right),g=\overline{1,N_P,}i=1,n-1$$

(38)

Check the limits of new positions according to step 3.

Calculate new slack power as per step 4 and new total real power vector is designed as $P_g^{new}=[{P_{gd}}^{new}{P_{gi}}^{new},i=1,n,i\ne d].$

New values of objective function are calculated as per step 5.

Update the new objective function $F_T^{new}$ as follows if

$$F_T^{new}<F_T^{{best}^{I-1}}then{F}_T^{best}=F_T^{new}and{P}_{Tgi}^{{best}^{new}}=P_{Tgi}^{{best}^{new}}$$

(39)

else

$$F_T^{best}=F_T^{{best}^{I-1}}and{P}_{Tgi}^{{best}^{new}}=P_{Tgi}^{{best}^{I-1}}$$

(40)

$$G^{best}=P_g^{best},g=\overline{1,N_P}$$

(41)

where. $I$ is the iteration number, $G^{best}$ is the one best solution for the whole swarm, $P_g^{best}$ is the one best solution per particle best solution, for whole system $P_g^{best}=$ $[P_1^{best},P_2^{best},P_3^{best},\dots ,\dots P_N^{best}]$ , and $G^{best}=min{P}_g^{best}$ for $g=\overline{1,N_P}$ .

The flowchart of QPSO is presented in Figures 2 and 3 which summarize the procedure.

Figure 2.

Quantum particle swarm optimization (QPSO) algorithm.

Figure 3.

Development of algorithm for combined economic and emission dispatch (CEED) problem.

Experimental results and analysis

This section elaborates on the experimental results and discussion. The simulations are performed for four different CEED test systems using QPSO and its variants EQPSO and ALA-QPSO to test and validate their performances. These test systems have been used by many researchers as benchmarks in power systems to solve CEED problems, listed as (1) IEEE standard 30 bus system with six generators, (2) six generating units system, (3) 10 generating unit system, and (4) IEEE 69 bus 11 generators system. The findings of this article are also compared with different studies reported in the literature that use the same criteria. Table 3 shows different test systems considered in this study.

Table 3.

Different test systems considered in this study.

Test case	Units	$P_D$ (MW)	$P_L$ (MW)	Valve point loading	hi
Case 1	6	1000	No	No	Equation (6)
Case 2	6	1200	Yes	No	Equation (6)
Case 3	10	2000	Yes	Yes	Equation (7)
Case 4	11	2500	No	No	Equation (6)

The parameter $\alpha$ (CE coefficient) is the only parameter that needs to be tuned to control the convergence swiftness of the process and its value, $\alpha <1.781$ is used to guarantee the convergence. There are two methods to set CE, the first one is to fix the value of $\alpha$ ranges ( $0.5-0.8$ ). However, fixing the value of $\alpha$ is sensitive to the size of the population and the number of iterations, if those parameters are changed then the $\alpha$ needs to be adjusted accordingly to get the desirable performance. The second option is time-varying CE which can be calculated by equation (19). For all the algorithms, parameters $w1$ and $w2$ are set as 0.4 and 0.9, respectively, while $\alpha$ is taken as time-varying. Similarly, for all the test systems, $w_{eco}$ and $w_{emi}$ are set as 1. All the results are an average of a total of $50$ number of runs simulated in MATLAB $2016a$ and executed in core i7-10510U, CPU @ $2.30$ GHz with $16$ GB RAM.

Test system 1

The test system contains six generators to meet a total power demand of $1000$ MW. This test system is considered as lossless and hence loss coefficients are not taken into account. The unit data consisting of generator limits with cost and emission function coefficients are taken from Basu (2011) and listed in Appendix Table 16. Table 4 presents the results of optimization techniques considered in this study. The comparison of best results of cost and emissions obtained in Table 4 and other population-based optimization techniques from literature are presented in Table 5.

Table 4.

Comparative analysis of best solutions achieved for test case 1 ( $P_D=1000$ MW).

Units	QPSO	EQPSO	ALA-QPSO
$P_1$ (MW)	250.36085	250.2247	250.34
$P_2$ (MW)	257.94356	257.83301	257.938
$P_3$ (MW)	159.54525	159.6278	159.511
$P_4$ (MW)	169.81692	169.35099	169.822
$P_5$ (MW)	83.715946	84.181681	83.7834
$P_6$ (MW)	78.617469	78.781812	78.6059
$F_c$ ($/h)	51264.475	51274.242	51265.4
$E_T$ (kg/h)	827.08566	826.64628	827.09
$F_T$ ($/h)	90933.22284	90936.443	90933.7
Best iteration	385	872	842

QPSO: quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor; EQPSO: enhanced quantum particle swarm optimization.

Table 5.

Best fuel cost and emission of test case 1 ( $P_D=1000$ MW).

Methods	Cost ($/h)	Emission (kg/h)
$\lambda$ -iteration (Balamurugan and Subramanian, 2007)	51264.6	828.72
Recursive (Balamurugan and Subramanian, 2007)	51264.5	828.715
PSO (Balamurugan and Subramanian, 2007)	51269.6	828.863
DE (Balamurugan and Subramanian, 2007)	51264.6	828.715
Simplified recursive (Balamurugan and Subramanian, 2007)	51264.6	828.715
GA similarity crossover (Güvenç, 2010)	51262.31	827.2612
GSA (Güvenc et al., 2012)	51255.78	827.138
QPSO	51264.475	827.0856
EQPSO	51274.242	826.64628
ALA-QPSO	51265.4	827.09

PSO: particle swarm optimization; DE: differential evolution; GA: genetic algorithm; GSA: gravitational search algorithm QPSO: quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor; EQPSO: enhanced quantum particle swarm optimization.

The results obtained in case 1 are shown in Table 4 with the generation level of each unit. The results are an average of a total of 50 runs considering the population size of 100 and 1000 iterations. The fuel cost of QPSO is minimum with moderate emissions to provide minimum overall CEED fuel cost. The EQPSO provides a lesser amount of emissions, however, with higher fuel cost, the overall CEED fuel cost by this method is higher than others. The comparative analysis also shows that QPSO is fast and provides early convergence with a lesser number of iterations. The next table presents a comparison of these techniques with other conventional and non-conventional optimization techniques. As results reported in the literature have no overall fuel cost mentioned in their papers, we cannot conclude which algorithm performs better than others. Furthermore, in this study, only an average of 50 runs is considered, whereas the other techniques reported in the literature did not mention this criterion in their study. Thus, we cannot necessarily conclude about which individual algorithm performs the best. However, only looking at the table, it can be said that the minimum fuel cost is achieved by GSA (Güvenc et al., 2012) and minimum emissions are attained by QPSO.

Test system 2

In this test system, six generating units are used to meet the total power demand ( $P_D=1200$ MW). Transmission losses are taken into account for this case and calculated using equation (15). To ensure power balance constraint, the system needed to supply extra power according to equation (14). The test system data and loss coefficients are taken from Basu (2011) and presented in Appendix Tables 17 and 18.

Table 6 provides the comparison of results for test system 2 with all the generation levels, power loss, total power generation, fuel cost, emissions, and overall CEED fuel cost. The new global PSO (NGPSO) provides the best results for this case with minimum emissions and overall fuel cost. However, they did not mention whether their provided results are from a single run or an average of a certain number of runs. The QPSO achieved second position while EQPSO stands at third position among all these techniques by providing a balancing solution for fuel cost and emissions.

Table 6.

Comparative analysis of best solutions achieved for test case 2 ( $P_D=1200$ MW).

	MODE	PDE	NSGA-II	SPEA 2	QOTLBO	TLBO	$\epsilon$ v-MOGA	OGHS	NGPSO
Units	Basu (2011)	Basu (2011)	Basu (2011)	Basu (2011)	Roy and Bhui (2013)	Roy and Bhui (2013)	Afzalan and Joorabian (2013)	Singh and Dhillon (2016)	Zou et al. (2017)	QPSO	EQPSO	ALA-QPSO
$P_1$	108.6284	107.3965	113.1259	104.1573	107.3101	107.8651	108.9318	105.7331	144.0425	107.7397	107.35109	111.511
$P_2$	115.9456	122.1418	116.4488	122.9807	121.497	121.5676	123.1808	119.0825	150	123.8299	125.42433	125.808
$P_3$	206.7969	206.7536	217.4191	214.9553	206.501	206.1771	205.1513	205.2976	190.5069	196.8892	196.07995	194.521
$P_4$	210	203.7047	207.9492	203.1387	206.5826	205.1879	206.67	204.7772	192.9284	207.8262	207.27872	207.946
$P_5$	301.8884	308.1045	304.6641	316.0302	304.5555	306.5555	304.8553	305.804	284.9082	308.7873	309.08047	304.973
$P_6$	308.4127	303.3797	291.5969	289.9396	304.6036	304.1423	302.6093	308.9128	288.0455	306.572	306.39911	306.826
$P_L$	51.672	51.4808	51.204	51.2018	51.4781	51.4955	51.3985	49.6072	50.4315	51.6445	51.613	51.6446
$P_T$	1251.672	1251.4808	1251.204	1251.2018	1251.4781	1251.4955	1251.3985	1249.6072	1250.4315	1251.645	1251.613	1251.6446
$F_c$	64843	64920	64962	64884	64912	64922	64838.57	64722.74	66538.34	64954.76	64979.842	65095.1
$E_T$	1286	1281	1281	1285	1281	1281	1285.4896	1281.349	1228.3654	1280.09	1279.3177	1276.18
$F_T$	126347.98	126191.64	126242.04	126357.35	126181.04	126191.04	126103.55	126008.47	125289.9	125772.53	125786.68	125890

MODE: multi-objective differential evolution; PDE: pareto differential evolution; NSGA-II: non-dominated sorting genetic algorithm; SPEA: strength pareto evolution algorithm.

Test system 3

In this test system, 10 generating units are used to fulfill a power demand of $2000$ MW. This case also includes the power transmission losses as well as the valve point loading effect. Power losses are calculated using the B matrix formula and all input data are taken from Basu (2011) and provided in Appendix Tables 19 and 20.

Table 7 gives comparative results of different non-conventional optimization techniques reported in the literature for test system 3. The CEED fuel cost of these optimization techniques is taken from Zou et al. (2017) to compare the overall efficiency of optimization techniques for providing the best compromise solution. In this test case, QPSO gives higher emission but lower fuel cost and transmission losses which yield the best overall fuel cost results than all the other techniques.

Table 7.

Comparative analysis of best solutions achieved for test case 3 ( $P_D=2000$ MW).

	MODE	PDE	NSGA-II	SPEA 2	GSA	$\epsilon$ v-MOGA	QOTLBO	TLBO	OGHS	NGPSO
Units	Basu (2011)	Basu (2011)	Basu (2011)	Basu (2011)	Güvenc et al. (2012)	Afzalan and Joorabian (2013)	Roy and Bhui (2013)	Roy and Bhui (2013)	Singh and Dhillon (2016)	Zou et al. (2017)	QPSO	EQPSO	ALA-QPSO
$P_1$	54.9487	54.9853	51.9515	52.9761	54.9992	54.1807	55	55	55	55	55	53.7143	54.3244
$P_2$	74.5821	79.3803	67.2584	72.813	79.9586	78.4981	80	80	79.9998	80	80	76.8147	79.1997
$P_3$	79.4294	83.9842	73.6879	78.1128	79.4341	84.7653	84.8457	83.9202	85.2236	81.2398	119.319	111.151	113.877
$P_4$	80.6875	86.5942	91.3554	83.6088	85	81.3502	83.4993	82.8342	84.3022	80.8334	119.805	122.895	120.556
$P_5$	136.8551	144.4386	134.0522	137.2432	142.1063	138.0526	142.921	132.0131	137.1243	160	125.338	125.439	133.998
$P_6$	172.6393	165.7756	174.9504	172.9188	166.567	166.2667	163.2711	173.988	155.8936	235.0087	153.09	159.842	158.765
$P_7$	283.8233	283.2122	289.435	287.2023	292.8749	295.466	299.8066	299.7099	299.9981	289.3507	270.761	279.413	271.263
$P_8$	316.3407	312.7709	314.0556	326.4023	313.2387	326.7642	315.4388	317.9684	315.726	297.4542	308.135	320.996	305.949
$P_9$	448.5923	440.1135	455.6978	448.8814	441.1775	428.9338	428.5084	427.0166	434.9409	401.5072	425.189	413.188	423.802
$P_{10}$	436.4287	432.6783	431.8054	423.9025	428.6306	429.6309	430.5524	431.3955	436.0067	401.4275	426.345	419.215	421.008
$P_L$	84.3271	83.9331	84.2496	84.0612	83.9869	83.9085	83.8433	83.8459	84.2152	81.8215	82.9847	82.6708	82.7436
$P_T$	2084.32	2083.93	2084.24	2084.06	2083.98	2083.9	2083.84	2083.84	2084.21	2081.82	2082.98	2082.67	2082.74
$F_c{F}^5$	1.1348	1.1351	1.1354	1.1352	1.1349	1.13422	1.1346	1.13471	1.1314	1.16179	1.1319	1.13449	1.13553
$E_T$	4124.9	4111.4	4130.2	4109.1	4111.4	4120.52	4110.2	4113.5	4144.4	3939.2	4303.77	4285.13	4268.72
$F_T$	218181.5	217867.6	218908.7	218306.1	217853.2	218015	217790.7	217885.5	218345.1	216170.5	215716.3	216580.4	216305.9

MODE: multi-objective differential evolution; PDE: pareto differential evolution; NSGA-II: non-dominated sorting genetic algorithm; SPEA: strength pareto evolution algorithm; NGPSO: new global particle swarm optimization; GSA: gravitational search algorithm; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Test system 4

This system comprises 11 generating units with a total power demand of $2500$ MW. This system is considered without transmission losses. In this system, the valve point loading effect is not included and all the input data are taken from Balamurugan and Subramanian (2007) given in Appendix Table 21.

The results obtained in test case 4 are presented in Table 8. The system is considered lossless and all the generating units fulfill a total power demand of $2500$ MW. The comparative analysis shows QPSO provides a fuel cost of 12410.2, which is the lowest among all the other techniques. The NGPSO achieves a minimum value of emission of 1661.7118 with the highest value of fuel cost of 13025.9072. The QPSO stands first among all the optimization techniques by providing a minimum value of overall CEED fuel cost. The other variants EQPSO and ALA-QPSO also provide minimum results than other techniques.

Table 8.

Comparative analysis of best solutions achieved for test case 4 ( $P_D=2500$ MW).

	GSA	GA-SC	SRA	PSO	DE	$\lambda$ iteration	RA	NGPSO
Units	Güvenc et al. (2012)	Güvenç (2010)	Balamurugan and Subramanian (2007)	Balamurugan and Subramanian (2007)	Balamurugan and Subramanian (2007)	Balamurugan and Subramanian (2007)	Balamurugan and Subramanian (2007)	Zou et al. (2017)	QPSO	EQPSO	ALA-QPSO
$P_1$	138.9382	138.8618	139.672	NA	NA	NA	NA	243.334	139.5409	143.151	139.308
$P_2$	110.2728	112.1312	112.781	NA	NA	NA	NA	210	112.693	119.094	115.419
$P_3$	147.9728	146.7169	145.802	NA	NA	NA	NA	250	145.6597	147.59138	148.72896
$P_4$	221.1072	222.1041	221.527	NA	NA	NA	NA	169.033	221.1789	216.75	219.986
$P_5$	137.7986	137.1962	136.774	NA	NA	NA	NA	142.615	140.2077	132.766	138.661
$P_6$	217.9015	217.3208	218.578	NA	NA	NA	NA	168.843	218.2389	216.486	219.05
$P_7$	141.3801	140.4711	140.261	NA	NA	NA	NA	142.592	140.0407	136.283	139.918
$P_8$	349.6497	348.9008	345.046	NA	NA	NA	NA	317.289	344.522	343.534	344.018
$P_9$	327.3178	326.5188	329.484	NA	NA	NA	NA	276.543	328.9797	337.357	332.072
$P_{10}$	363.4766	363.5275	363.645	NA	NA	NA	NA	303.229	363.0512	357.984	355.135
$P_{11}$	344.1847	346.2508	346.43	NA	NA	NA	NA	276.519	345.8874	348.999	347.7
$F_c$	12422.6626	12,423.77	12424.94	12428.63	12425.06	12424.94	12424.94	13025.9072	12410.2	12430.7	12421.6
$E_T$	2002.9499	2003.0304	2003.3	2009.72	2003.35	2003.301	2003.301	1661.7118	2002.17	1992.47	2008.36
$F_T$	19939.8006	19938.3912	19994.9822	19969.7622	19942.2898	19941.986	19941.986	19261.1978	18930.8	18967.1	18990.4

PSO: particle swarm optimization; GSA: gravitational search algorithm; DE: differential evolution; NGPSO: new global particle swarm optimization; GSA: gravitational search algorithm; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Comparison of three optimization techniques for fuel cost minimization of four CEED cases

A comparison of the three optimization techniques used to solve the CEED problem is provided in this section. To analyze their performances, all the algorithms are tested in four different test cases and two case studies.

Case study 1

In the first case study, the number of iterations is set to 200, the population size to 40 and results are taken after averaging a total of 50 runs. The results of three algorithms in terms of minimum, maximum, mean, median, and standard deviation are provided in Tables 9 to 11. In both case studies, $F_c$ , $E_c$ , and $F_T$ are fuel cost, emissions, and overall CEED fuel cost, respectively. The values of fuel cost and emissions are taken simultaneously by solving combined economic and emission dispatch problems. Many researchers solve ED and EMD dispatch separately, however, in this article, we have solved the combination of ED and EMD. The QPSO shows lesser variations of results with minimum standard deviation and provides the best output in terms of fuel cost, emissions, and overall CEED fuel cost in all the test cases.

Table 9.

Fuel cost comparative analysis of CEED.

Test case	Algorithm	$F_c$ min	$F_c$ max	$F_c$ mean	$F_c$ median	$F_c$ std
Case 1	QPSO	51264.4751	51264.4754	51264.4753	51264.4753	$4.28\times {10}^{-5}$
	EQPSO	51168.0621	51635.5002	51379.0551	51381.8809	102.193952
	ALA-QPSO	51129.479	51443.4928	51279.7353	51280.3444	68.7656895
Case 2	QPSO	64954.7625	64954.7628	64954.7627	64954.7627	$8.08\times {10}^{-5}$
	EQPSO	64850.4402	65093.7017	64979.8423	64974.6472	45.9346593
	ALA-QPSO	64745.372	65643.0563	65095.1024	65036.8232	185.769937
Case 3	QPSO	113193.205	113193.264	113193.245	113193.248	$9.32\times {10}^{-3}$
	EQPSO	112736.62	114608.32	113449.124	113364.729	448.497377
	ALA-QPSO	112839.867	114935.377	113553.315	113495.549	440.731128
Case 4	QPSO	12410.2039	12410.2039	12410.2039	12410.2039	$8.32\times {10}^{-6}$
	EQPSO	12396.3714	12478.6553	12430.7255	12429.9383	14.5688465
	ALA-QPSO	12380.7227	12468.1318	12421.658	12422.3322	20.6276144

CEED: combined economic and emission dispatch; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Table 10.

Emission comparative analysis of CEED.

Test case	Algorithm	$E_T$ min	$E_T$ max	$E_T$ mean	$E_T$ median	$E_T$ std
Case 1	QPSO	827.08565	827.085661	827.085655	827.085655	$2.17\times {10}^{-6}$
	EQPSO	812.693842	833.32157	823.266673	822.499392	4.17690644
	ALA-QPSO	820.272927	836.562264	826.851767	826.520308	3.41864389
Case 2	QPSO	1280.09047	1280.09049	1280.09048	1280.09048	$4.27\times {10}^{-6}$
	EQPSO	1273.55847	1285.53464	1279.31771	1279.13887	2.28632533
	ALA-QPSO	1254.43346	1292.17944	1276.17509	1276.90305	7.60588854
Case 3	QPSO	4303.77233	4303.78296	4303.77796	4303.778	$1.72\times {10}^{-3}$
	EQPSO	4143.62663	4378.0764	4285.13056	4289.78456	56.3950647
	ALA-QPSO	4117.49316	4366.47193	4268.72435	4267.21154	60.3787444
Case 4	QPSO	2002.16991	2002.16998	2002.16994	2002.16994	$1.48\times {10}^{-5}$
	EQPSO	1938.33805	2017.06875	1992.47406	1995.10721	16.0040768
	ALA-QPSO	1936.13938	2074.04941	2008.36598	2007.62619	32.9186313

CEED: combined economic and emission dispatch; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Table 11.

CEED overall fuel cost comparative analysis.

Test case	Algorithm	$F_T$ min	$F_T$ max	$F_T$ mean	$F_T$ median	$F_T$ std
Case 1	QPSO	90933.2228	90933.2228	90933.2228	90933.2228	$4.69\times {10}^{-11}$
	EQPSO	90941.2928	91141.4162	91002.2942	90998.7956	45.8202332
	ALA-QPSO	90935.495	91023.2369	90954.9445	90948.4498	21.4627022
Case 2	QPSO	125772.526	125772.526	125772.526	125772.526	$3.81\times {10}^{-11}$
	EQPSO	125775.908	125806.929	125786.676	125784.342	8.19738153
	ALA-QPSO	125773.005	126216.676	125890.369	125837.834	125.127418
Case 3	QPSO	215716.273	215716.273	215716.273	215716.273	$8.17\times {10}^{-7}$
	EQPSO	215906.748	218007.796	216580.404	216531.843	391.512307
	ALA-QPSO	215814.265	218185.209	216305.981	216108.07	522.779213
Case 4	QPSO	18930.8015	18930.8015	18930.8015	18930.8015	$1.77\times {10}^{-11}$
	EQPSO	18935.6647	19033.6408	18967.1403	18964.0929	17.9137005
	ALA-QPSO	18940.4448	19082.1865	18990.4243	18984.1297	36.456813

CEED: combined economic and emission dispatch; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Case study 2

In the second case study, the number of iterations is set to 1000, the population size is 200 and results are taken after averaging a total of 50 runs. The comparative results of case study 2 are provided in Tables 12 to 14. This case study also shows that QPSO achieved first place in most of the cases. However, in Table 14, case 3, the average results achieved by EQPSO are better than QPSO. It shows that for a larger number of iterations, EQPSO can find better results but larger standard deviation is its demerit. These results are also shown in later convergence Figure 5.

Figure 5.

Convergence characteristics of all three algorithms (50 runs and 200 iterations).

Table 12.

Fuel cost comparative analysis of CEED.

Test case	Algorithm	$F_c$ min	$F_c$ max	$F_c$ mean	$F_c$ median	$F_c$ std
Case 1	QPSO	51264.4752	51264.4753	51264.4752	51264.4752	$3.07\times {10}^{-5}$
	EQPSO	51237.2916	51308.509	51274.2422	51277.2049	15.381993
	ALA-QPSO	51256.2144	51275.1223	51265.3709	51265.0912	3.97065014
Case 2	QPSO	64954.7625	64954.7628	64954.7627	64954.7627	$6.29\times {10}^{-5}$
	EQPSO	64930.2573	64997.9261	64960.4749	64961.4673	15.1135829
	ALA-QPSO	64743.94	65510.1507	65026.5907	64957.5975	170.380692
Case 3	QPSO	113193.248	113193.249	113193.248	113193.248	$7.66\times {10}^{-5}$
	EQPSO	112804.615	114380.035	113372.245	113263.09	393.450155
	ALA-QPSO	113048.696	113905.039	113192.655	113156.533	187.177547
Case 4	QPSO	12410.2039	12410.2039	12410.2039	12410.2039	$4.49\times {10}^{-6}$
	EQPSO	12406.1372	12431.8745	12417.4252	12416.3147	6.29345901
	ALA-QPSO	12404.4535	12418.6382	12410.7234	12410.858	3.00684575

CEED: combined economic and emission dispatch; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Table 13.

Emission comparative analysis of CEED.

Test case	Algorithm	$E_T$ min	$E_T$ max	$E_T$ mean	$E_T$ median	$E_T$ std
Case 1	QPSO	827.085652	827.085659	827.085655	827.085655	$1.60\times {10}^{-6}$
	EQPSO	824.959261	829.262933	826.646285	826.470657	0.92633037
	ALA-QPSO	826.647614	827.559116	827.050461	827.064338	0.20144688
Case 2	QPSO	1280.09048	1280.09049	1280.09048	1280.09048	$3.16\times {10}^{-6}$
	EQPSO	1278.16744	1281.82275	1279.9715	1279.92167	0.76271581
	ALA-QPSO	1259.03497	1294.07236	1278.00315	1280.07938	7.8273936
Case 3	QPSO	4303.77805	4303.7781	4303.77807	4303.77807	$1.48\times {10}^{-5}$
	EQPSO	4114.13642	4486.1596	4333.60578	4345.42923	75.1572472
	ALA-QPSO	4227.56351	4373.98046	4316.53134	4308.74727	35.6742923
Case 4	QPSO	2002.16992	2002.16995	2002.16994	2002.16994	$5.95\times {10}^{-6}$
	EQPSO	1988.62729	2011.05768	2000.95781	2000.92992	5.28172868
	ALA-QPSO	1989.79505	2009.83672	2001.52417	2001.91092	4.40992373

CEED: combined economic and emission dispatch; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Table 14.

CEED overall fuel cost comparative analysis.

Test case	Algorithm	$F_T$ min	$F_T$ max	$F_T$ mean	$F_T$ median	$F_T$ std
Case 1	QPSO	90933.2228	90933.2228	90933.2228	90933.2228	$3.45\times {10}^{-11}$
	EQPSO	90933.3792	90943.1149	90936.4429	90935.8074	2.40174496
	ALA-QPSO	90933.2242	90933.8634	90933.3155	90933.2872	0.10300555
Case 2	QPSO	125772.526	125772.5256	125772.526	125772.526	$6.51\times {10}^{-11}$
	EQPSO	125772.737	125778.1654	125774.471	125774.089	1.33500593
	ALA-QPSO	125772.54	126030.9003	125844.985	125825.658	82.9858889
Case 3	QPSO	215716.273	215716.273	215716.273	215716.273	$1.41\times {10}^{-10}$
	EQPSO	214143.153	216313.69	215617.457	215773.17	545.518673
	ALA-QPSO	215718.21	216287.042	215793.893	215758.423	130.23932
Case 4	QPSO	18930.8015	18930.8015	18930.8015	18930.8015	$3.67\times {10}^{-12}$
	EQPSO	18932.288	18950.1548	18938.1092	18937.6018	4.15483513
	ALA-QPSO	18930.969	18935.2811	18931.8175	18931.543	0.94942998

CEED: combined economic and emission dispatch; QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Convergence efficiency of three algorithms

Figure 4 depicts the random single-run behavior of algorithms for four test cases with a population size of 40 and 200 iterations. It shows that QPSO converges faster than the other two variants. The EQPSO spreads its population wider than the other two optimization approaches to achieve better local search. The convergence behavior of ALA-QPSO is in between these two techniques.

Figure 4.

Convergence characteristics of all three algorithms (single run and 200 iterations).

For better analysis of convergence, all algorithms are tested for 50 runs and their average results are plotted as shown in Figure 5. The graphs show the same results as discussed earlier. Both the variants of QPSO are modified in such a way that they scatter their population in a wider area for better local search. However, ALA-QPSO’s adaptive local attractor is set in such a manner that the population wanders in the whole search space at early stages and converges at a later stage. To investigate further, the number of iterations is increased to 1000 and then it can be seen in Figure 6 that EQPSO achieves better results than QPSO for case 3.

Figure 6.

Convergence characteristics of test case 3 (50 runs and 1000 iterations).

The computational complexity of CEED algorithms arises from the intricate nature of power system optimization problems involving economic and emission considerations. Researchers continue to explore strategies to mitigate this complexity and enhance the performance of these algorithms in real-world applications. The computational complexity of a PSO algorithm is typically assessed in terms of its time complexity, which represents the computational resources required to find a solution. The execution time of the three algorithms is given in Appendix Table 16 under the conditions of 40 population size, 200 iterations, and 50 run average result. It is evident from Table 15 that the best time is achieved for QPSO, Test Case 1 and comparatively EQPSO elapsed the maximum time because of its feature of spreading population in wider range.

Table 15.

Comparison of computational time of three cases.

Test case no.	Algorithm	Total elapsed time (S)	Best iteration time (S)	Best iteration
Test case 1	QPSO	0.5826	0.3571	121
Test case 1	ALA-QPSO	0.8752	0.8465	190
Test case 1	EQPSO	1.1143	0.9781	176
Test case 2	QPSO	1.329	1.107	166
Test case 2	ALA-QPSO	1.3039	1.1658	179
Test case 2	EQPSO	1.6813	1.6811	200
Test case 3	QPSO	0.6448	0.5853	182
Test case 3	ALA-QPSO	0.6801	0.6626	194
Test case 3	EQPSO	0.5978	0.5133	172

QPSO: quantum particle swarm optimization; EQPSO: enhanced quantum particle swarm optimization; ALA-QPSO: QPSO with weighted mean personal best and adaptive local attractor.

Table 16.

Six-units generator characteristics.

	$a_i$	$b_i$	$c_i(\mathrm{\$}/h)$	$d_i$	$e_i$	$f_i$
Unit	($/(MW)2h)	($/MWh)		(kg/(MW)2h)	(kg/MWh)	(kg/h)
1	0.1525	38.54	756.8	0.0042	0.33	13.86
2	0.106	46.16	451.325	0.0042	0.33	13.86
3	0.028	40.4	1050	0.00683	$-$ 0.5455	40.267
4	0.0355	38.31	1243.53	0.00683	$-$ 0.5455	40.267
5	0.0211	36.328	1658.57	0.0046	$-$ 0.5112	42.9
6	0.018	38.27	1356.66	0.0046	$-$ 0.5112	42.9

Table 17.

Six-units generator characteristics.

Unit	$a_i$	$b_i$	$c_i$	$d_i$	$e_i$	$f_i$	$P_{min}$	$P_{max}$
	($/(MW)2h)	($/MWh)	($/h)	(kg/(MW)2h)	(kg/MWh)	(kg/h)	(MW)	(MW)
1	0.15247	38.539	756.7988	0.00419	0.32767	13.8593	10	125
2	0.10587	46.1591	451.3251	0.00419	0.32767	13.8593	10	150
3	0.03546	38.3055	1243.5311	0.00683	$-$ 0.54551	40.2669	35	210
4	0.02803	40.3965	1049.9977	0.00683	$-$ 0.54551	40.2669	35	225
5	0.01799	38.2704	1356.6592	0.00461	$-$ 0.51116	42.8955	125	325
6	0.02111	36.3278	1658.5696	0.00461	$-$ 0.51116	42.8955	130	325

Table 18.

Transmission loss B coefficient.

0.00014	0.000017	0.000015	0.000019	0.000026	0.000022
0.000017	0.00006	0.000013	0.000016	0.000015	0.00002
0.000015	0.000013	0.000065	0.000017	0.000024	0.000019
0.000019	0.000016	0.000017	0.000071	0.00003	0.000025
0.000026	0.000015	0.000024	0.00003	0.000069	0.000032
0.000022	0.00002	0.000019	0.000025	0.000032	0.000085

Table 19.

Ten-units generator characteristics.

No.	$a_i$	$b_i$	$c_i$	$d_i$	$e_i$	${\alpha }_i$	${\beta }_i$	${\gamma }_i$	${\eta }_i$	${\delta }_i$	$P_{min}$	$P_{max}$
	($/(MW)2h)	($/MWh)	($/h)	($/h)	(rad/MW)	(lb/(MW)2h)	(lb/MWh)	(lb/h)	(lb/h)	(1/MW)	(MW)	(MW)
1	0.12951	40.5407	1000.403	33	0.0174	0.04702	$-$ 3.9864	360.0012	0.25475	0.01234	10	55
2	0.10908	39.5804	950.606	25	0.0178	0.04652	$-$ 3.9524	350.0056	0.25475	0.01234	20	80
3	0.12511	36.5104	900.705	32	0.0162	0.04652	$-$ 3.9023	330.0056	0.25163	0.01215	47	120
4	0.12111	39.5104	800.705	30	0.0168	0.04652	$-$ 3.9023	330.0056	0.25163	0.01215	20	130
5	0.15247	38.539	756.799	30	0.0148	0.0042	0.3277	13.8593	0.2497	0.012	50	160
6	0.10587	46.1592	451.325	20	0.0163	0.0042	0.3277	13.8593	0.2497	0.012	70	240
7	0.03546	38.3055	1243.531	20	0.0152	0.0068	$-$ 0.5455	40.2669	0.248	0.0129	60	300
8	0.02803	40.3965	1049.998	30	0.0128	0.0068	$-$ 0.5455	40.2669	0.2499	0.01203	70	340
9	0.02111	36.3278	1658.569	60	0.0136	0.0046	$-$ 0.5112	42.8955	0.2547	0.01234	135	470
10	0.01799	38.2704	1356.659	40	0.0141	0.0046	$-$ 0.5112	42.8955	0.2547	0.01234	150	470

Table 20.

Transmission loss B coefficient.

0.000049	0.000014	0.000015	0.000015	0.000016	0.000017	0.000017	0.000018	0.000019	0.00002
0.000014	0.000045	0.000016	0.000016	0.000017	0.000015	0.000015	0.000016	0.000018	0.000018
0.000015	0.000016	0.000039	0.00001	0.000012	0.000012	0.000014	0.000014	0.000016	0.000016
0.000015	0.000016	0.00001	0.00004	0.000014	0.00001	0.000011	0.000012	0.000014	0.000015
0.000016	0.000017	0.000012	0.000014	0.000035	0.000011	0.000013	0.000013	0.000015	0.000016
0.000017	0.000015	0.000012	0.00001	0.000011	0.000036	0.000012	0.000012	0.000014	0.000015
0.000017	0.000015	0.000014	0.000011	0.000013	0.000012	0.000038	0.000016	0.000016	0.000018
0.000018	0.000016	0.000014	0.000012	0.000013	0.000012	0.000016	0.00004	0.000015	0.000016
0.000019	0.000018	0.000016	0.000014	0.000015	0.000014	0.000016	0.000015	0.000042	0.000019
0.00002	0.000018	0.000016	0.000015	0.000016	0.000015	0.000018	0.000016	0.000019	0.000044

Table 21.

Eleven-units generator characteristics.

	$a_i$	$b_i$	$c_i$	$d_i$	$e_i$	$f_i$	$P_{min}$	$P_{max}$
Unit	($/(MW)2h)	($/MWh)	($/h)	(kg/(MW)2h)	(kg/MWh)	(kg/h)	(MW)	(MW)
1	0.00762	1.92699	387.85	0.00419	$-$ 0.67767	33.93	20	250
2	0.00838	2.11969	441.62	0.00461	$-$ 0.69044	24.62	20	210
3	0.00523	2.19196	422.57	0.00419	$-$ 0.67767	33.93	20	250
4	0.0014	2.01983	552.5	0.00683	$-$ 0.54551	27.14	60	300
5	0.00154	2.12181	557.75	0.00751	$-$ 0.4006	24.15	20	210
6	0.00177	1.91528	562.18	0.00683	$-$ 0.54551	27.14	60	300
7	0.00195	2.10681	568.39	0.00751	$-$ 0.40006	24.15	20	215
8	0.00106	1.99138	682.93	0.00355	$-$ 0.51116	30.45	100	455
9	0.00117	1.99802	741.22	0.00417	$-$ 0.56228	25.59	100	455
10	0.00089	2.12352	617.83	0.00355	$-$ 0.41116	30.45	110	460
11	0.00098	2.10487	674.61	0.00417	$-$ 0.56228	25.59	110	465

Conclusions

In this study, we have used QPSO and its two modified versions, EQPSO and ALAQPSO, to solve the CEED problem. These algorithms have been implemented on IEEE standard 6-, 10-, and 11-units systems. The proposed algorithms are investigated on four different test cases, that is, Test case 1 PD = 1000 MW, Test case 2 PD = 1200 MW, Test case 3 PD = 2000 MW, and Test case 4 PD = 2500 MW and two case studies have been considered. The efficacy of the proposed metaheuristic techniques is confirmed through a comparison of their performance with methodologies documented in the literature. From simulation results, key findings could be summarized as follows:

In case study 1, where the number of iterations is set to 200 with 40 population size. In all the test cases, QPSO showed fewer variations in results with minimum standard deviation and provided mean best output in terms of fuel cost: 51264.4753 for test system 1, 64954.7627 for test system 2, 113193.245 for test system 3, and 12410.2039 for test system 4; emissions: 827.085655 for test case 1, 1280.09048 for test case 2, 4303.77796 for test case 3, and 2002.16994 for test case 4; and overall CEED fuel cost: 90933.2228 for test case 1, 125772.526 for test case 2, 215716.273 for test case 3, and 18930.8015 for test case 4.

In the second case study, the number of iterations has been set to 1000, with 200 population size and results are taken after averaging a total of 50 runs. This case study also showed that the QPSO achieved first place in all the cases except test case 3 where the average results for overall CEED fuel cost achieved by EQPSO (215617.457) were better than QPSO (215716.273).

The convergence characteristics of QPSO also confirmed its efficiency in terms of robustness as it required fewer iterations to achieve the solution than the other two variants. The EQPSO and ALA-QPSO also produced some quality results as compared to other optimization techniques. However, the convergence efficiency of these two algorithms is lower and requires a larger number of iterations.

In the future, QPSO and its variants need to be tested on larger testing units integrated with renewable energy sources with more practical constraints such as prohibited operating zones and ramp rate limits. The efficiency of QPSO along with its enhancements also required to be tested on multi-objective dynamic economic and emission dispatch with demand side management.