1 Introduction

The electricity generation around the world depends mostly on fossil fuel fired thermal power generation system. But, the rapid increase in the demand of electricity, shortage of fossil fuel supply, and environmental concerns make economic load dispatch (ELD) and emission dispatch problem as the main concerns of electrical power generation system [1]. ELD refers to finding an optimal combination of power generation to minimize the total generation cost while satisfying all other constraints. On the other hand, the goal of emission dispatch is to minimize a number of pollutants from the system. The goal of combined economic emission dispatch (CEED) is to minimize the total generation cost as well as the emission of pollutants by satisfying all other constraints.

Combined economic emission dispatch (CEED) problem is a real-world many-objective optimization problem. Different approaches have been made to formulate or represent CEED problem. The most common way of representing CEED problem is using quadratic function [2,3,4,5,6,7]. However, the nonlinearities of actual power generation system deviate the solution from optimality and thus nullify the rough approximation of quadratic function. It has been found that higher order functions can represent the actual response of power generation system, and thus the solutions can be improved by adopting higher order polynomials [7]. But, the problem is higher order function imposes difficulties in formulating CEED problem and makes it more complex. Thus, to make a trade-off between these two conflicting problems, many researchers have used cubic function to represent CEED problem. It is an industry practice as well. The cubic function representation of CEED problem successfully reduces the growing nonlinearities of modern power generation system [7]. In this research, we have used cubic function to formulate and represent combined economic emission dispatch problem.

Optimization methods based on classical/conventional techniques are the oldest methods used by the researchers to solve CEED problem as reported in [2,3,4,5,6,7]. These methods have been chosen for their advantages like no problem-specific parameters to specify [8], mathematically proven optimality [9], and some of them are computationally fast [10]. However, they are found not to be efficient enough to solve CEED problems as they can immaturely converge into local optimum, show sensitivity to the initial starting points, many of them are not applicable to some types of cost function, i.e., non-smooth, non-convex, non-monotonically increasing cost functions, etc. [11, 12].

In the second phase, artificial intelligence (AI)-based and computational intelligence (CI)-based non-conventional stand-alone techniques have emerged as a replacement for the obsolete classical methods to solve CEED problem. These advanced optimization methods play a pivotal role in alleviating the problems found in the classical approaches in solving CEED problem; for example, they can enable us to solve nonlinear and non-convex cost functions and can achieve nearly global/global solutions. However, some of these methods suffer from many problem-specific parameter selections and high computational time. Majority of these techniques have been inspired by nature and thus are called nature-inspired optimization techniques. Some of the most renowned methods are genetic algorithm (GA) [13], particle swarm optimization (PSO) [14], simulated annealing (SA) [15], bacteria foraging optimization (BFO) [16], differential evolution (DE) [17], firefly algorithm (FFA) [18], gravitational search algorithm (GSA) [19], ant colony optimization (ACO) [20], artificial bee colony (ABC) [21], cuckoo search (CS) [22], bat algorithm (BA) [23], teaching–learning-based optimization (TLBO) [24], flower pollination algorithm [25], and mine blast algorithm (MBA) [26]. Besides, many modified versions of stand-alone CI-based techniques have also emerged to overcome different limitations and drawbacks of stand-alone CI techniques. Some of them are non-dominated sorting genetic algorithm-II (NSGA-II) [27], epsilon-dominance-based genetic algorithm [28], genetic algorithm based on similarity crossover [29], local search integrated PSO [30], quantum-behaved PSO [31], refined PSO [32], fuzzy adaptive modified theta PSO [33], bare-bones multiobjective PSO [34], modulated PSO [35], enhanced PSO [36], gravitational enhanced PSO [37], self-adaptive PSO [38], modified ACO [39], interactive fuzzy satisfying SA [40], and opposition-based GSA [41].

Recently, researchers show a trend to use hybrid methods to combine the best features of two or more algorithms and thereby to achieve superior performance than stand-alone methods. Some of the most recently introduced hybrid methods to solve CEED problem are backtracking search algorithm with sequential quadratic programming [42], firefly-bat algorithm (FFA-BA) [43], gradient search method and improved Jaya algorithm [44], and differential evolution with simulated annealing technique (DE-SA) [45]. But, the hybrid algorithm usually suffers from long computational time as two or more algorithms operate (in the relay-type hybrid algorithm) to solve CEED problem, where each of the algorithms performs individually into the problem one after another and adds more complexities (in the parallel-type hybrid algorithm).

To alleviate those problems mentioned earlier like immature convergence, non-suitability to certain types of cost function, many problem-specific parameter selections, and long computational time, a new idea is needed to solve CEED problem efficiently and with required optimality. Thus, powerful quantum computing phenomenon is proposed here along with recently developed swarm intelligence (SI)-based bat algorithm to address and overcome the problems found to solve CEED problem. Quantum-behaved bat algorithm (QBA) is thus applied in this research to solve many-objective CEED problem for different loads.

2 Methodology

Quantum-behaved bat algorithm (QBA) is an improved version of bat algorithm (BA). BA was pioneered by Xin-she Yang [46] in 2010 and was inspired by the echolocation or bio-sonar characteristics of bats. It is relatively a new nature-inspired metaheuristic technique that is known for its ability to successfully combine the advantages of many well-known algorithms [47]. BA is easy to implement and more powerful than its predecessor GA and PSO [46]. One of the reasons behind its superiority is that it utilizes some of the major advantages of this algorithm (GA and PSO) in a structured way.

Bat can prey, avoid obstacles, and search food by their advanced echolocation capability as well as its self-adaptive ability to compensate Doppler Effect in echoes. In original BA, Doppler Effect was not considered. Moreover, foraging habitats of bats were not considered; rather, it was considered that bats forage in only one habitat, which was not true and did not reflect the actual behavior of bats [48]. In QBA, both of these phenomena have been considered along with other characteristics of BA. The introduction of quantum behavior in bats diversifies the foraging habitats of bats, which ultimately contributes to the diversification of population. Additionally, it helps to avoid premature convergence in BA.

The basic BA is based on three idealized rules: (1) echolocation technique of bats to sense distance and to calculate a difference between their prey (food) and background barriers, (2) bats vary their wavelength (λ0) and loudness (A0) to search for their prey. They also regulate frequency and rate of their emitted pulses, depending on the distance of their prey; (3) assuming that the loudness is varied from a large (A0) value to a minimum constant value (Amin). The positions (xi) and velocities (vi) of the virtual bats are updated using the following equations:

$$f_{i} = f_{\hbox{min} } + (f_{\hbox{max} } - f_{\hbox{min} } )\alpha$$
(1)
$$v_{i}^{t} = v_{i}^{t - 1} + (x_{i}^{t} - g^{t} )f_{i}$$
(2)
$$x_{i}^{t} = x_{i}^{t - 1} + v_{i}^{t}$$
(3)

where α, fi, fmin, and fmax are random vector in the range of [0, 1], frequency of pulse, minimum frequency, and maximum frequency, respectively. Again,\(v_{i}^{t}\), \(v_{i}^{t - 1}\), \(x_{i}^{t}\), \(x_{i}^{t - 1}\), and \(g^{t}\) stand for the velocity of the ith bat at iteration t, velocity of the ith bat at iteration (t − 1), position of the ith bat at iteration t, position of the ith bat at iteration (t − 1), and current best global location found by the bats, respectively.

A local random walk is used to generate new solution for each bat once a solution is selected from the current best solutions. The new position or solution can be described as below:

$$x_{\text{new}} = x_{\text{old}} + \varepsilon A^{t} ,$$
(4)

where ε is a random number in the range of [− 1,1] and At is the average loudness of all bats at iteration t. In QBA, new position or solution is achieved using the following equations:

$$\begin{aligned} x_{id}^{t + 1} & = g_{d}^{t} \times [1 + j(0,\sigma^{2} )] \\ \sigma^{2} & = \left| {A_{i}^{t} - A^{t} } \right| + \varepsilon \\ \end{aligned}$$
(5)

where \(j(0,\sigma^{2} )\) is a Gaussian distribution with mean 0 and standard deviation \(\sigma^{2}\). \(x_{id}^{t + 1}\) and \(g_{d}^{t}\) are the position of the ith bat at iteration t + 1 and current best global location found by the bats at dimension d. \(A_{i}^{t}\) is the loudness of ith bat at iteration t. ε is integrated here to ensure the standard deviation \(\sigma^{2}\) remains positive. The loudness Ai and pulse emission rate ri are updated in each iteration using the following equations:

$$\begin{aligned} A_{i}^{t + 1} = \delta A_{i}^{t} \hfill \\ r_{i}^{t + 1} = r_{i}^{0} [1 - \exp ( - \gamma t)] \hfill \\ \end{aligned}$$
(6)

where \(A_{i}^{t}\)\(A_{i}^{t + 1}\), \(r_{i}^{0}\), and \(r_{i}^{t + 1}\) refer to the loudness of ith bat at iteration t, the loudness of ith bat at iteration t + 1, initial pulse emission rate of ith bat, and pulse emission rate of the ith bat at iteration t + 1, respectively. δ and γ are constants whose range are [0,1] and greater than 0 (γ > 0), respectively.

To make the algorithm more similar to the actual scenario of bats and thus make it more efficient, two more idealized rules have been considered along with the three idealized rules [46] found in the original BA. They are: (1) Bats have different foraging habitats rather than one single foraging habitat that depends on a stochastic selection and (2) bats have the self-adaptive capability to compensate for Doppler Effect in echoes. In QBA, quantum-behaving virtual bats position can be defined using the equation below:

$$x_{id}^{t} = g_{d}^{t} + \beta \left| {mbest_{d} - x_{id}^{t} } \right|\ln \left( {\frac{1}{u}} \right),\quad u(0,1) < 0.5$$
(7a)
$$x_{id}^{t} = g_{d}^{t} - \beta \left| {mbest_{d} - x_{id}^{t} } \right|\ln \left( {\frac{1}{u}} \right),\quad u(0,1) \ge 0.5$$
(7b)

where \(x_{id}^{t}\) refers to the position of an ith bat in dimension d at iteration t.

Consideration of bats self-adaptive compensation for Doppler Effect changes the updating formulas as mentioned in Eqs. (1) and (2). The equations can be rewritten as below:

$$f_{id} = \frac{{(340 + v_{i}^{t - 1} )}}{{(340 + v_{g}^{t - 1} )}} \times f_{id} \times \left[ {1 + C_{i} \times \frac{{(g_{d}^{t} - x_{id}^{t} )}}{{\left| {g_{d}^{t} - x_{id}^{t} } \right| + \varepsilon }}} \right]$$
(8)
$$v_{id}^{t} = (w \times v_{id}^{t - 1} ) + (g_{d}^{t} - x_{id}^{t} )f_{id}$$
(9)
$$x_{id}^{t} = x_{id}^{t - 1} + v_{id}^{t}$$
(10)

where \(f_{id}\) refers to the frequency of ith bat in dimension d; \(v_{g}^{t - 1}\) refers to the velocity of the global best position at iteration t − 1, and \(C_{i}\) is a positive number of ith bat in the range of [0, 1]. For simplicity, we can assume if the value of C is 0, then bat cannot compensate for Doppler Effect in echoes and if C = 1, it means bat can fully compensate for Doppler Effect in echoes. Inertia weight w is introduced here to update the velocity and has similar characteristics like the inertia weight found in PSO [49]. The pseudocode of QBA is given below in Algorithm 1.

figure a

QBA generates solution by solving the cost function (see below Eq. 17). For different values of P, within its prescribed range, the algorithm provides different solutions. With each step, the solutions are updated according to the formula given in Eqs. (7a–10). The best solution achieved by the bats is known as gbest. With each iterating steps, gbest is updated if the current best solution is smaller than previous best solution. The algorithm stops updating after it reaches its maximum number of iterations. The gbest value after maximum number of iterations becomes the final value of the algorithm.

3 Problem formulation

Real-world optimization problems usually consist of two or more objectives. The objectives are usually incommensurable and conflicting in nature. This conflicting behavior gives rise to a set of solutions, instead of a single solution. It is because no single solution can be considered better than other solutions as there is no combination of decisions that can be considered to be better than any other decision on all other objective functions. This set of solutions is known as the Pareto-optimal solution or Pareto-optimal front.

CEED is a multiobjective optimization problem that usually refers to the minimization of fuel cost and emission of hazardous gases and particulates while satisfying total load demand and all other equality and inequality constraints. In this research, we consider the minimization of SO2, NOx, and CO2 as separate three objectives. Thus, in our case, by considering ELD along with emission objectives, CEED becomes a four-objective optimization problem.

ELD is represented here using cubic criterion function, where total fuel cost F(P) in ($/h) can be expressed as

$$F(P) = \sum\limits_{i = 1}^{n} {a_{i} P_{i}^{3} + b_{i} P_{i}^{2} + c_{i} } P_{i} + d_{i} ,$$
(11)

where n is the total number of generating units, Pi is the real output power of generating unit i; ai, bi, ci, and di are the fuel cost coefficients of the generating unit i.

Emission dispatch is divided into separate three objectives and is also represented using cubic function as given below:

$$E_{\text{SO2}} (P) = \sum\limits_{i = 1}^{n} {e_{\text{SO2i}} P_{i}^{3} + f_{{{\text{SO}}2i}} P_{i}^{2} + g_{{{\text{SO}}2i}} } P_{i} + h_{{{\text{SO}}2i}} ,$$
(12)
$$E_{{{\text{NO}}x}} (P) = \sum\limits_{i = 1}^{n} {e_{{{\text{NO}}xi}} P_{i}^{3} + f_{{{\text{NO}}xi}} P_{i}^{2} + g_{{{\text{NO}}xi}} } P_{i} + h_{{{\text{NO}}xi}} ,$$
(13)
$$E_{\text{CO2}} (P) = \sum\limits_{i = 1}^{n} {e_{{{\text{CO}}2i}} P_{i}^{3} + f_{{{\text{CO}}2i}} P_{i}^{2} + g_{{{\text{CO}}2i}} } P_{i} + h_{{{\text{CO}}2i}} ,$$
(14)

where ESO2(P), ENOx(P), and ECO2(P) in (kg/h) are the emission functions of SO2, NOx, and CO2, respectively. eSO2i, fSO2i, gSO2i, hSO2i, eNOxi, fNOx2i, gNOxi, hNOxi, eCO2i, fCO2i, gCO2i, and hCO2i are coefficients of SO2 emission, NOx emission, and CO2 emission of ith generating unit, respectively.

Power balance and generation limit constraints are considered in this research. Total output power generation PT (in MW) must satisfy total load demand (in MW), i.e., it must be equal to the summation of total load demand and total power loss (in MW). Power balance constraint can be defined as below:

$$P_{\text{T}} = \sum\limits_{i = 1}^{n} {P_{i} } = P_{\text{D}} + P_{\text{L}} ,$$
(15)

where PD and PL are total load demand (in MW) and real power transmission loss (in MW), respectively. Each generating unit has its minimum and the maximum limit of power generation. Within this limit, a generator can operate satisfactorily. Generator limit constraint can be defined as below:

$$P_{{i,{ \hbox{min} }}} \le P_{i} \le P_{{i,{ \hbox{max} }}} ,$$
(16)

where Pi,min and Pi,max refer to the minimum and maximum output power of ith generating unit, respectively.

The objectives of minimizing fuel cost and emissions of CO2, SO2, and NOx can be converted into a single objective using a price penalty factor. A unit-wise max/max penalty factor [7] is considered in this research to solve CEED problem. The objective function (OF) will then be to minimize the total cost FT in ($/h) and can be described as

$$\begin{aligned} {\text{OF}} & = \hbox{min} (F_{\text{T}} ) \\ F_{\text{T}} & = \sum\limits_{i = 1}^{n} {\{ F(P_{i} ) + h_{\text{Si}} E_{\text{SO2}} (P_{i} )} + h_{\text{Ni}} E_{{{\text{NO}}x}} (P_{i} ) + h_{\text{Ci}} E_{\text{CO2}} (P_{i} )\} \\ h_{\text{Si}} & = \sum\limits_{i = 1}^{n} {\frac{{F(P_{i,\hbox{max} } )}}{{E_{\text{SO2}} (P_{i,\hbox{max} } )}}} \\ h_{\text{Ni}} & = \sum\limits_{i = 1}^{n} {\frac{{F(P_{i,\hbox{max} } )}}{{E_{{{\text{NO}}x}} (P_{i,\hbox{max} } )}}} \\ h_{\text{Ci}} & = \sum\limits_{i = 1}^{n} {\frac{{F(P_{i,\hbox{max} } )}}{{E_{\text{CO2}} (P_{i,\hbox{max} } )}}} \\ \end{aligned}$$
(17)

where FT ($/h) is the total cost of the power generation system; F(Pi,max), ESO2(Pi,max), ENOx(Pi,max), and ECO2(Pi,max) are total fuel cost ($/h), total SO2 emission (kg/h), NOx emission (kg/h), and CO2 emission (kg/h) for maximum output power of ith generating unit, respectively; hSi, hNi, and hCi are the max/max penalty factors of SO2, NOx, and CO2 emissions of generating unit i, respectively.

4 Results and discussion

This section presents simulation results with comprehensive performance analysis of QBA for solving many-objective CEED problem. All the simulations are done using MATLAB R2015a and executed with i5-3470 CPU @ 3.20 GHz (4 CPUs), ~ 3.2 GHz and 4 GB RAM PC. QBA is applied to solve many-objective CEED problem for four different loads (150 MW, 175 MW, 200 MW, and 225 MW), where CEED problem is formulated using cubic criterion function as shown in Eq. (17). QBA is implemented in six-unit power generation system. All the objectives are converted into a single objective, and the final result, i.e., total cost ($/h), is presented to compare it with other methods found in the literature. Table 1 shows the parameter settings for QBA to solve many-objective CEED problem.

Table 1 Parameter settings of QBA for solving CEED problem
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QBA has many parameters to tune. Population size and number of iterations are taken 2000 and 200, respectively, to get the optimal value. In our investigation, we have found that same value of gamma and delta (δ = γ = 0.9) gives more accurate and robust results. Total 30 number of runs are considered here as a fair test of robustness, and the average of the outcomes of these runs is reported in this section. All the data are taken from [7] as shown in Tables 2, 3, 4, and 5, and the penalty factors are collected from Eq. (17).

Table 2 Fuel cost coefficients and generator power limits for six-unit system
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Table 3 Emission coefficients with max/max penalty factors of SO2 for six-unit system
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Table 4 Emission coefficients with max/max penalty factors of NOx for six-unit system
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Table 5 Emission coefficients with max/max penalty factors of CO2 for six-unit system
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In order to test and verify the effectiveness of QBA, the obtained results for four different loads (150, 175, 200, and 225 MW) are compared with LR, PSO, and SA that have been used to solve many-objective CEED problem, where many-objective CEED problem is defined using cubic criterion function and solved considering unit-wise max/max price penalty factors. Table 6 shows overall results of QBA for the six-unit power generation system considering different load demands by mentioning the fuel cost ($/h), emission of SO2 (kg/h), emission of NOx (kg/h), emission of CO2 (kg/h), and the total cost ($/h) after using unit-wise max/max price penalty factor. The standard deviation shown in Table 6 demonstrates that the results provided by QBA for solving many-objective CEED problem are very much reliable and robust for different load demands. QBA needs less population size and a number of iterations than other methods like LR [7], PSO [50], and SA [51].

Table 6 Results of CEED problem for six-unit system using QBA for different loads
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Table 7 shows comparison of fuel cost ($/h) of the six-unit system for different load demands. It can be seen from Table 7 that QBA gives better fuel cost result than other methods such as classical mathematical-based LR, and CI-based PSO and SA methods as a reduction of 0.24 $/h for 150 MW, 32.41 $/h for 175 MW, 9.65 $/h for 200 MW, and 6.89 $/h for 225 MW is observed from its nearest method SA. Figure 1 depicts the comparison graph of different methods for different loads. We can conclude from Table 7 and Fig. 1 that QBA provides the minimum fuel costs, SA closely follows QBA, whereas PSO and LR methods provide the maximum fuel cost. It also demonstrates that quantum computing-based CI algorithm is the most efficient algorithm, while CI-based SA and PSO come after that and classical LR is the least efficient method. Tables 8, 9, and 10 and Figs. 2, 3, and 4 show the comparison of emission for different gases like SO2, NOx, and CO2 (kg/h) considering different load demands in six-unit power generation system. For SO2 and NOx emissions, SA provides the best result for all the load demands, whereas QBA, PSO, and LR come in the second, third, and fourth positions, respectively. On the other hand, for CO2 emission, QBA performs better for 150 and 225 MW loads, while SA performs better for 175 and 200 MW loads. Finally, Table 11 summarizes our main objective, i.e., comparison of minimization of the total cost ($/h), which is depicted in Fig. 5. QBA is found to provide the best overall results for all load demands in six-unit power generation system. It reflexes the power of quantum computing-integrated CI methods over classical and stand-alone CI methods.

Table 7 Comparison of results of fuel cost for six-unit system
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Fig. 1
figure 1

Flowchart of quantum-behaved bat algorithm (QBA)

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Table 8 Comparison of results of SO2 emission for six-unit system
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Table 9 Comparison of results of NOx emission for six-unit system
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Table 10 Comparison of results of CO2 emission for 6-unit system
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Fig. 2
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Fuel cost variations of methods for different load demands in six-unit system

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Fig. 3
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Emission variations of SO2 of methods for different load demands in six-unit system

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Fig. 4
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Emission variations of NOx of methods for different load demands in six-unit system

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Table 11 Comparison of results of total cost for six-unit system
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Fig. 5
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Emission variations of CO2 of methods for different load demands in six-unit system

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Figure 6 shows the convergence curves of QBA for solving many-objective CEED problem using different loads. These curves are tallied with Table 6. It can be seen from the figure that the curves tend to converge very fast, converging within 20 iterations, whereas LR [7] and SA [51] need 1500 and 10,000 iterations, respectively, to get to the optimal point. It demonstrates the computational prowess of the used QBA technique. The curves presented from the simulation results also show that they are smooth and reliable for all the four cases. It has also been found that the exploited QBA technique is quite robust against trapping into the local optima. Finally, it can be said that QBA is found to provide robust, reliable, and accurate solution with powerful computational efficiency, which is better than SA, PSO, and LR.

Fig. 6
figure 6

Total cost variations of methods for different load demands in six-unit system

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Pareto-optimal fronts for different pairs of objectives are shown in Figs. 7, 8, and 9. Although, we get a set of solutions for different objectives in each run, here, we have only shown the significant sets of solutions for load 225 MW. The Pareto-optimal front of fuel cost and emission of SO2 shows good diversity characteristics of non-dominated solutions, whereas Pareto-optimal fronts of fuel cost and emission of NOx, and fuel cost and emission of CO2 show less diversity and tend to concentrate on certain areas in the middle. It is understandable as more objectives increase the probability of having any two arbitrary solutions to be non-dominated to each other [52, 53]. Again, in many-objective problem, there are many objectives in which a trade-off (one is better in one objective, while worse in any other objective) can occur. Furthermore, the proportion of non-dominated solutions in the population increase when we deal with a finite-sized population-based approach [52, 54]. However, we have just taken the average of final values, i.e., total cost after using a unit-wise price penalty factor among the objectives to convert all the objectives into a single objective. It can be said clearly that the Pareto-optimal fronts obtained by using quantum computing idea based BA show good diversity characteristics of the non-dominated solutions. In short, it can be said that the many-objective CEED problem is solved efficiently by QBA (Fig. 10).

Fig. 7
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Convergence curves of QBA for different load demands in six-unit power system

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Fig. 8
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Pareto-optimal front for fuel cost and emission of SO2 (load 225 MW)

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Fig. 9
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Pareto-optimal front for fuel cost and emission of NOx (load 225 MW)

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Fig. 10
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Pareto-optimal front for fuel cost and emission of CO2 (load 225 MW)

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5 Conclusion and future research directions

In this research, quantum computing idea integrated BA (QBA) is successfully applied to six-unit power generation system to solve CEED problem for different load demands (150, 175, 200, and 225 MW). Cubic criterion function is used in this research to represent many-objective CEED problem. According to the best of our knowledge, this is the first time QBA has been exploited to solve CEED problem, regardless of the number of objectives. The four objectives considered in this paper are converted into a single objective using unit-wise max/max price penalty factor. The obtained results for different load demands are compared among LR, PSO, SA, and QBA. The comparison shows that QBA performs better than other methods in terms of providing high quality, robust, and stable solutions. The feasibility of cubic function with QBA in such many-objective optimization problem is also proved from this research. Another advantage of QBA over these methods is that it takes less number of iterations to converge into the optimum point, which demonstrates its computational superiority and prowess. Pareto-optimal solutions for QBA is also shown here. After running the algorithm several times (30 runs), it can be stated that the QBA algorithm successfully avoids trapping into the local optima. The standard deviation also confirms that the results are robust, suitable, and stable. Although QBA successfully provides the minimum result for the total cost which is our main objective and for emission of CO2 for some load demands but some other cases like for emission of SO2, NOx, and CO2, it is found that often SA and in few cases LR provide better results than QBA. The simulation results also demonstrate and justify that the integration of quantum computing idea with advanced metaheuristic algorithm provides a reliable and useful tool to solve such real-world many-objective optimization problem.

By this research, we have identified some aspects that will help to secure our objectives, i.e., minimize both fuel cost and emission of pollutants more accurately and realistically. It has two definite directions: One is to further improve the optimization methods to have computationally more powerful, robust, and reliable tool for solving CEED problem and second is to better formulate and represent the CEED problem with necessary constraints, so that it represents the actual condition of real-word power generation system.

One of the major problems in QBA is it has many parameters. Measures should be taken to reduce the parameters so that the operation of QBA becomes more easy, flexible but effective. Few works have already done to reduce or make BA without any parameter as reported in [55, 56].

Additional real-world constraints like transmission loss, generator ramp rate limit, prohibited operating zones, tie-line limit should be considered. Along with that, reliability level, load adjusting time, reserve capacity, load demand forecast, unit commitment, and even renewable technology can be integrated with the power generation system. These will obviously make the modeling much more complex but will be very effective and implement friendly for real-world problem. One important thing is that we need to have/collect/acquire necessary data to simulate and run such model.

Furthermore, some kind of selection mechanism should be adopted along with this research to choose the optimal solution from many sets of optimal solutions. Many-objective optimization problem like CEED makes it difficult for the decision maker to choose an optimal solution from a multidimensional Pareto-optimal solution. The implementation of fuzzy set theory along with ANN could be a suitable solution for this problem.

Finally, other advanced metaheuristic algorithms like CS algorithm might be a good choice to integrate with quantum computing phenomenon to solve non-convex CEED problem. We are suggesting CS algorithm because it has been proved that this technique utilizes some nature-inspired features like brood parasitism of some cuckoo species and Levy flight behavior of some birds and insects that enable them to perform better than most of the techniques like PSO and GA. Lastly, a hybrid version of QCI methods might also bring some fruitful changes to handle/solve many-objective CEED problem for large systems.