Behavior Analysis of the New PSO-CGSA Algorithm in Solving the Combined Economic Emission Dispatch Using Non-parametric Tests

ABSTRACT

This paper proposes a new metahaeuristic algorithm named particle swarm optimization and chaotic gravitational search algorithm (PSO-CGSA) for solving the combined economic and emission dispatch (CEED) problem. First, we determine the efficiency and effectiveness measures of the algorithm and compare it with other well-known algorithms. Then, we analyze the obtained solutions using the statistical procedure proposed in the paper. The proposed procedure contains the following: (i) the behavior analysis of the algorithms when solving the CEED problem, using non-parametric tests, and (ii) the ranking of the algorithms using the PROMETHEE/GAIA multi-criteria decision-making method. The behavior analysis is performed for two cases: (i) when solving individual variants of the CEED problem (single-problem analysis) and (ii) when solving a set of CEED variants (multiple-problem analysis). The results of the applied procedure for the test system with six generators show that PSO-CGSA has (i) the best solution for each tested variant of the CEED problem; (ii) the best standard deviation, mean value, error rate, and behavior for the CEED variant with a bi-objective function that simultaneously minimizes fuel cost and emission, taking into account the valve point effect; and (iii) the best rank when solving a set of CEED variants.

Introduction

A review of the specialized published literature shows that in the past decades, a large number of metaheuristic algorithms have been proposed for solving the combined economic emission dispatch (CEED) problem which is very important in the management and exploitation of power systems (Edwin Selva Rex, Marsaline Beno, and Annrose 2019; Mahdi et al. 2018; Marouani et al. 2022). The CEED is the adjustment of the output power of a number of generators in thermal power plants at a given load and given constraints in the system, minimizing the fuel costs and the emission of toxic gases. The CEED problem is a non-linear and non-convex optimization problem consisting of many variants: with and without valve point effect in power plants, with and without losses in the network, with and without emissions of various pollutants, etc (Zaoui and Belmadani 2022; Hussien et al. 2021, Deb et al. 2021; Jevtić, Jovanović, and Radosavljević 2018b). The authors usually tested their algorithms on standard IEEE test systems after a set of executions on selected variants of the CEED problem, comparing the results with the results of previously proposed algorithms (Nagarajan, Parvathy, and Arul 2019; Radosavljević 2016; Yu, Duan, and Luo 2022). The main criteria used in the published papers to select the best algorithm for solving the CEED problem are robustness, convergence characteristics, computation efficiency, reliability, and suitability of the solution (Arul et al. 2019; Mahdi et al. 2018). Today in the literature can be found a large number of algorithms that their authors propose as the best for solving the CEED problem. It is difficult to determine which ones are truly the best because different authors have used different: variants of the CEED problem, problem dimensions, constraints, methodologies, and algorithms for comparison (Halim, Ismail, and Das 2021; Hooker 1995; Sörensen 2015). The authors in (Jevtić, Jovanović, and Radosavljević 2018a) pointed out that one algorithm does not have to be the best for every variant of the CEED problem. Due to a large number of proposed algorithms for solving the CEED problem and other real-world optimization problems, statistical methodologies have been proposed for a more accurate selection of the best algorithm. Thus, some authors (Barr et al. 1995; Rardin and Uzsoy 2001) proposed statistical methodologies for comparing algorithms based on the application of parametric and non-parametric tests. Other authors (Derrac et al. 2011; Garcia et al. 2009) later applied non-parametric statistical tests to compare the behavior of metaheuristic algorithms over standard benchmark functions. More recently, parametric and non-parametric tests have been used to compare the metaheuristic algorithms on practical optimization problems (Bigham and Gholizadeh 2020; Gholizadeh, Danesh, and Gheyratmand 2020; Jevtić, Jovanović, and Radosavljević 2018a; Nourianfar and Abdi 2022).

In recent years, a large number of hybrid algorithms and improved algorithms have been proposed in the literature to solve the CEED problem. A hybrid algorithm combines two or more algorithms, to obtain better characteristics than individual algorithms. The authors of (Hossein and Hamdi 2019) proposed the TVAC-PSO algorithm combined with the EMA algorithm to solve the different cases of economic emission dispatch with different constraints. In (Mohammad et al. 2019), a hybrid combination of JAYA and TLBO algorithms was proposed for solving different economic dispatch problems with different constraints. In (Murugan et al. 2018), three algorithms are combined: BA and ABC with CSA (CSA-BA-ABC). The proposed CSA-BA-ABC has better capability to escape from local optima with faster convergence rate than the standard BA and ABC. A hybrid algorithm called ACO – ABC – HS that combines ACO, ABC, and HS algorithms achieves excellent performance in solving the CEED problem (Tanuj and Hitesh 2016). The authors of (Xin-Gang et al. 2020)proposed DE-CQPSO algorithm based on fast convergence of differential evolution algorithms and particle diversity of crossover operators of genetic algorithms. DE-CQPSO achieves good effectiveness and robustness in solving environmental economic dispatch problems. Its evaluation index and convergence speed are better than those of the QPSO algorithm. In Hassan et al. (2021), the ISMA algorithm was developed to improve the performance of the SMA algorithm, by updating the solution positions based on operators borrowed from the SCA algorithm. A multiobjective ISMA is developed based on the Pareto dominance concept and fuzzy decision-making. To solve the economic load dispatch problems, the authors of Hassan et al. (2022) proposed the ESCSDO algorithm, which is an improved SDO algorithm using the eagle strategy with 10 chaotic maps. The improvements are as follows: improving population diversity, balancing between local and global search, and avoiding premature convergence of the SDO algorithm. A modified version of the Marine Predators Algorithm (MMPA) is proposed by (Hassan et al. 2022) for solving single- and bi-objective CEED problems. MMPA increases the efficiency of the population in reaching the optimal solution by preventing premature convergence.

This paper presents a new hybrid algorithm called particle swarm optimization and chaotic gravitational search algorithm (PSO-CGSA) for solving the CEED problem. We obtained the PSO-CGSA by improving the PSOGSA (Mirjalili and Hashim 2010; Mirjalili, Hashim, and Sardroudi 2012). In the published literature, PSOGSA has been proposed as the best algorithm for solving the CEED problem (Jiang, Ji, and Shen 2014; Radosavljević 2016). In the PSOGSA algorithm, the gravitational search algorithm (GSA) (Rashedi, Nezamabadi-Pour, and Saryazdi 2009) was combined with particle swarm optimization (PSO) (Kennedy and Eberhart 1995) to obtain a better exploration compared to PSO (Khan and Ling 2021). We embed the Gauss/mouse chaotic map into the formula for the gravitational constant and thus obtain PSO-CGSA, which has better exploration than PSOGSA. We solve four variants of the CEED problem with different objective functions using PSO-CGSA, PSOGSA, and group of algorithms (PSOS-CGSA (Ullah et al. 2020), PPSO (Ghasemi et al. 2019) and PPSOGSA (Ullah et al. 2019) with performance similar to theirs. The PSO-CGSA outperforms the other tested algorithms in terms of best solution, but some performance measures of the algorithms (mean, standard deviation (SD) and error rate values, and convergence rates) are close to each other and vary depending on the CEED variant being solved. Therefore, we analyze the behavior of the algorithms using parametric and non-parametric tests. Analysis of the behavior of the algorithms allowed us to determine which conclusions from concrete sets of solutions can be generalized to the entire population of solutions, i.e., whether the differences between the algorithms are real or random. We conduct the analysis for each individual variant of the CEED problem (single-problem analysis) and all four tested variants simultaneously (multiple-problem analysis). As a performance measure in behavior analysis, we use the error rate for each algorithm. In our analysis, the algorithm’s error rate (for a concrete CEED variant) is the average value of the percentage differences between the values of individual results obtained for 30 runs and the best value obtained. The single-problem analysis shows that different algorithms have different behavior in solving different variants, and that PSO-CGSA has the best behavior in solving the variant with a bi-objective function that simultaneously minimizes fuel cost and emission, taking into account the valve point effect. The multiple-problem analysis showed no differences in the behavior of the tested algorithms when solving all variants of the CEED problem simultaneously, i.e., in this case, algorithms have the same behavior. At the end of the procedure, we propose in this paper, we rank the algorithms according to different performance measures using the PROMETHEE/GAIA multi-criteria decision-making (MCDM) method (Brans 1982; Brans and De Smet 2016). The PROMETHEE/GAIA method showed that PSO-CGSA is the best-ranked algorithm for solving all variants of the CEED problem simultaneously.

The main contributions of this paper are as follows: (i) Developing an algorithm for solving the CEED problem, called PSO-CGSA, to improve the performance of PSOGSA; (ii) Appling the proposed PSO-CGSA to solve individual variants of the CEED problem (which can be single-objective or bi-objective) and multiple variants simultaneously; (iii) analysis of the behavior of algorithms when solving the CEED problem, using nonparametric tests, for cases: a) solving individual variants (single-problem analysis), and b) solving a set of CEED variants simultaneously (multiple-problem analysis). (iv) ranking algorithms using the PROMETHEE/GAIA multi-criteria decision-making method, for solving a set of CEED variants.

CEED Problem

The fuel cost function for an individual generator g in a thermal power plant usually has a quadratic shape,

(1) $F_g^{\mathit{\hspace{0.17em}}}\left(P_g\right)=a_g+b_g{P}_g+c_g{P}_g^2$(1)

where F_g ($/h) is the fuel cost of the generator g; P_g (MW) is the power output of the generator g; a_g, b_g, and c_g are the fuel cost coefficients of the generator g.

If the valve point effect in a power plant is taken into account, the fuel cost function takes the form:

(2) $\mathit{F}{{\mathit{ }}^{\mathrm{\prime }}}_{\mathit{g}}^{\mathit{\hspace{0.17em}}}\left(P_g\right)=a_g+b_g{P}_g+c_g{P}_g^2+\left|d_{g}sin\left(e_g\left(P_g^{min}-P_g\right)\right)\right|$(2)

where d_g and e_g are the coefficients of impact due to the movement of the valve and $P_g^{min}$ is the minimum power of the generator g.

The function that models the emission of gases in a thermal power plant is the sum of the quadratic and exponential functions of the generator output power,

(3) $E_g\left(P_g\right)={\alpha }_g+{\beta }_g{P}_g+{\eta }_g{P}_g^2+{\xi }_{g}exp\left({\lambda }_g{P}_g\right)$(3)

where E_g (t/h) is the weight of gases emitted during the operation of the generator g; P_g (MW) is the output power of the generator g; ${\alpha }_g$, ${\beta }_g$, ${\eta }_g$, ${\xi }_g$ and ${\lambda }_g$ are the emission coefficients of the generator g.

The objective function (OF), applied in solving the CEED problem, minimizes the sums of costs and/or emissions of all generators. OFs usually have the following forms, depending on the features of the power system under consideration:

(4) $\begin{array}{rl}& \mathit{O}{F}_1={\sum }_{\mathit{g}\in \mathit{G}}\mathit{F}(P_g)+\mathit{\gamma }{\sum }_{\mathit{g}\in \mathit{G}}\mathit{E}(P_g)+{\lambda }_p{\left({\mathit{P}}_{\mathit{sl}}-P_{\mathit{sl}}^{lim}\right)}^2,\mathit{g}=\mathit{\hspace{0.17em}}1,\mathit{\hspace{0.17em}}2,\dots ,\mathit{G}\\ & \mathit{O}{F}_2={\sum }_{\mathit{g}\in \mathit{G}}\mathit{F}(P_g)+{\sum }_{\mathit{g}\in \mathit{G}}\left|d_{g}sin\left(e_g\left(P_g^{min}-P_g\right)\right)\right|+{\lambda }_p{\left({\mathit{P}}_{\mathit{sl}}-P_{\mathit{sl}}^{lim}\right)}^2,\mathit{g}=\mathit{\hspace{0.17em}}1,\mathit{\hspace{0.17em}}2,\dots ,\mathit{G}\text{break}\\ & \mathit{O}{F}_3={\sum }_{\mathit{g}\in \mathit{G}}\mathit{E}(P_g)+{\lambda }_p{\left({\mathit{P}}_{\mathit{sl}}-P_{\mathit{sl}}^{lim}\right)}^2,\mathit{g}=\mathit{\hspace{0.17em}}1,\mathit{\hspace{0.17em}}2,\dots ,\mathit{G}\text{break}\\ & \mathit{O}{F}_4={\sum }_{\mathit{g}\in \mathit{G}}\mathit{F}(P_g)+\mathit{\gamma }{\sum }_{\mathit{g}\in \mathit{G}}\mathit{E}(P_g)+{\sum }_{\mathit{g}\in \mathit{G}}\left|d_{g}sin\left(e_g\left(P_g^{min}-P_g\right)\right)\right|+{\lambda }_p{\left({\mathit{P}}_{\mathit{sl}}-P_{\mathit{sl}}^{lim}\right)}^2,\mathit{g}=\mathit{\hspace{0.17em}}1,\mathit{\hspace{0.17em}}2,\dots ,\mathit{G}\end{array}$(4)

OF₁ minimizes fuel costs and emissions simultaneously; OF₂ minimizes fuel costs taking into account the valve point effect in thermal power plants; OF₃ is used when only the emission of gases is minimized; and OF₄ includes costs and emissions simultaneously, taking into account the valve point effect. The quadratic penalty term ${\lambda }_p{\left({\mathit{P}}_{\mathit{sl}}-P_{\mathit{sl}}^{lim}\right)}^2$ is added to each OF (4) to satisfy the power balance constraint in the system during the optimization process. λ_p is the penalty factor. P_sl and P_sl^lim are defined by (10). G is the number of generators in the system. Other symbols in (4) are explained in (1), (2), and (3). The power balance constraint is

(5) ${\sum }_{\mathit{g}\in \mathit{G}}^{\mathit{\hspace{0.17em}}}{P}_g-P_D^{\mathit{\hspace{0.17em}}}-P_{\mathit{loss}}=0,$(5)

where P_D is the total power of the consumer, P_loss is the power of loss in the transmission system.

The second constraint that must be met during the optimization process is:

(6) $P_g^{min}\le P_g\le P_g^{max}$(6)

where $P_g^{min}$ and $P_g^{max}$ are the minimum and maximum power of generator g, respectively.

Power loss in the power system, P_loss, is expressed as a quadratic function of generators’ powers, i.e., from the Kron’s formula for transmission loss:

(7) $P_{\mathit{loss}}={\sum }_{\mathit{g}\in \mathit{G}}^{\mathit{\hspace{0.17em}}}{\sum }_{\mathit{j}\in \mathit{G}}{P}_g{B}_{\mathit{gj}}{P}_j+{\sum }_{\mathit{g}\in \mathit{G}}{B}_{0\mathit{g}}{P}_g+B_{00}$(7)

where B_gj and B_0g are the coefficients of the B-loss matrix and B₀₀ is a constant.

Calculations of the P_loss, slack generator power P_sl, and $P_{\mathit{sl}}^{lim}$ are performed in each step of the iterative process as follows.

One of the generators is selected as a dependent generator (slack generator). For this generator, the value of the output power, P_sl, is calculated from (5) as follows:

(8) $P_{\mathit{sl}}=P_D+P_{\mathit{loss}}-{\sum }_{\genfrac{}{}{0ex}{}{g=1}{g\ne sl}}^{\mathit{G}}{P}_g$(8)

The power loss, P_loss, and P_sl are obtained from the following steps: 1) setting the initial value $P_{\mathit{loss}}=P_{\mathit{loss}}^{\left(0\right)}=0$ in (8); 2) obtaining the value $P_{\mathit{sl}}^{\left(0\right)}$ from (8) for the initial value $P_{\mathit{loss}}^{\left(0\right)}=0$; 3) calculating the new value $P_{\mathit{loss}}^{\left(1\right)}$ from (7); 4) checking whether the error value ε is less than the given error tolerance value δ, i.e.,

(9) $\mathit{\epsilon }=\left|P_{\mathit{loss}}^{\left(1\right)}-P_{\mathit{loss}}^{\left(0\right)}\right|,\begin{array}{cc}\mathit{\hspace{0.17em}}& \mathit{\epsilon }\le \mathit{\delta }\end{array}$(9)

and 5) obtaining the value of $P_{\mathit{sl}}^{\left(1\right)}$from (8) for $P_{\mathit{loss}}=P_{\mathit{loss}}^{\left(1\right)}$. Constraint (5) (power balance) is met if condition (9) is satisfied. Otherwise, the procedure is repeated.

When the value of P_sl is calculated, it is necessary to check whether the value of P_sl satisfies the constraint (6). After that, the variable P_sl^lim is defined as follows:

(10) $P_{\mathit{sl}}^{lim}=\left\{\begin{array}{ccc}{P}_{\mathit{sl}}^{max}& \mathit{if}& P_{\mathit{sl}}>P_{\mathit{sl}}^{max}\\ P_{\mathit{sl}}^{min}& \mathit{if}& P_{\mathit{sl}}<P_{\mathit{sl}}^{min}\\ P_{\mathit{sl}}& \mathit{if}& P_{\mathit{sl}}^{min}\le P_{\mathit{sl}}\le P_{\mathit{sl}}^{max}\end{array}\right.$(10)

Thereafter, the obtained values of P_sl and P_sl^lim are inserted into the quadratic penalty term of the objective function OF (4).

PSO-CGSA

We obtained the PSO-CGSA by improving the PSOGSA. The PSOGSA combines the PSO algorithm and the GSA. The main novelty of PSO-CGSA is the embedded Gauss/mouse chaotic map in the gravitational constant formula, which resulted in improved exploration and convergence rate of PSO-CGSA compared to PSOGSA. Mirjalili and Gandomi (2017) have previously shown that introducing different chaotic maps into the GSA gravitational constant formula yields chaos-based GSA (CGSA) with improved performance compared to GSA (trapping in local minima is reduced and convergence rate is increased).

The equations in PSO-CGSA for updating the current velocity, v_i(t), and current position (solution candidate), x_i (t), of search agent i, during the iterative process, are as follows:

(11) ${\mathbf{v}}_i\left(\mathit{t}+1\right)=r_0\cdot {\mathbf{v}}_i\left(\mathit{t}\right)+C_1\cdot r_1\cdot {\mathbf{a}}_i\left(\mathit{t}\right)+C_2\cdot r_2\cdot \left(\mathbf{gbest}\left(\mathit{t}\right)-{\mathbf{x}}_i\left(\mathit{t}\right)\right)$(11)

(12) ${\mathbf{x}}_i\left(\mathit{t}+1\right)={\mathbf{x}}_i\left(\mathit{t}\right)+{\mathbf{v}}_i\left(\mathit{t}+1\right)$(12)

where gbest (t) is the best position of all search agents so far; r₀ is the inertia weight which plays the role of balancing the global search and local search; C₁ and C₂ are the acceleration coefficients associated with the agent’s own best position and the best positions of any agent in the whole swarm, respectively; r₁, r₂ and r₀, are uniformly distributed random numbers in [0,1]; a_i (t) is the acceleration of ith search agent in current iteration t, which is defined in (Mirjalili and Gandomi 2017). a_i (t) depends on the gravitational constant Gr with the embedded Gauss/mouse chaotic map, as explained in (Mirjalili and Gandomi 2017). The final form of a_i (t) is: $a_{\mathrm{i}}\left(\mathit{t}\right)=F_{\mathrm{i}}\left(\mathit{t}\right)/M_i\left(\mathit{t}\right)$, where $F_{\mathrm{i}}\left(\mathit{t}\right)$ is the total force that acts on the ith agent at iteration t, which depends on Gr; M_i (t) is the mass of each agent.

The gravitational constant defines the intensity of gravitational forces between search agents (Rashedi, Nezamabadi-Pour, and Saryazdi 2009). Gr is larger in the initial iterations and smaller in the later iterations. Therefore, Gr pushes search agents to larger steps in the initial iterations and less in the final iterations. In this way, Gr balances exploration and exploitation. The Gauss/mouse chaotic map changes Gr chaotically during each iteration, which improves exploration and convergence rate.

The gravitational constant is updated as follows:

(13) $\mathit{Gr}\left(\mathit{t}\right)=C_{\mathit{g}/m}^{\mathit{norm}}\left(\mathit{t}\right)+\mathit{G}{r}_0\cdot exp(-\mathit{\alpha }\cdot \mathit{t}/t_{max})$(13)

where $C_{\mathit{g}/m}^{\mathit{norm}}\left(\mathit{t}\right)$ is the normalized Gauss/mouse chaotic map in iteration t, Gr₀ is the initial gravitational constant; $\mathit{\alpha }$ is descending coefficient, and; t_max is the maximum number of iterations.

The procedure for solving the CEED problem using PSO-CGSA is as follows: First, the initial population of N randomly selected search agents is generated. The position of ith search agent is defined as follows:

(14) ${\mathbf{x}}_i=\left[P_i^1,\dots ,P_i^k,\dots ,P_i^n\right]\begin{array}{cc}\mathit{\hspace{0.17em}}& \mathit{i}=1,2,\dots ,\mathit{N}\begin{array}{cc}\mathit{\hspace{0.17em}}& \mathit{n}=\mathit{G}-1\end{array}\end{array}$(14)

where $P_i^k$ is the power output of generator k; G is the number of all generators; n is the number of generators in search space without the slack generator. The power of the slack generator is calculated in each iteration using the equilibrium condition in the system (5). After initialization, in the iterative procedure, each search agent’s fitness is calculated using the objective function (4). Then the update Gr (t), a_i (t), gbest (t), v_i (t) and x_i (t) is performed in each iteration. This procedure is repeated until the end criterion is met. Figure 1 shows the flowchart of the PSO-CGSA for solving the CEED problem.

Figure 1. Flowchart of the PSO-CGSA algorithm for solving the CEED problem.

Testing the Algorithms

In this paper, we focus on the experimental analysis of the behavior of the proposed PSO-CGSA algorithm and four algorithms with similar performance in solving the CEED problem. We perform this analysis for the cases of solving individual variants of the CEED problem and in the case of solving four variants of CEED problems, simultaneously. Before starting the analysis in the next section, we test the proposed PSO-CGSA on three test systems and compare it with the algorithms proposed in the published literature. The algorithms are implemented on a platform of 1.6 GHz with 3 GB RAM running MATLAB R2017a. All the results shown herein are the best values obtained over 30 runs. The coefficient values of the algorithms were determined by trial-and-error technique and are shown in Table 1. PPSO has no coefficients, i.e., it is a self-adaptive algorithm. The specified error tolerance value is δ = 10⁻⁶ MW.

Table 1. Adjusted coefficients of algorithms applied to test systems.

Algorithm	Test systems 1 and 2	Test system 3
PSOGSA	N=50; T = 200; C1 = 2; C2 = 2; Gr0 = 1; α = 20	N=200; T = 1500; C1 = 2; C2 = 2; Gr0 = 100; α = 10
PSO-CGSA	N=50; T = 200; C1 = 2; C2 = 2; Gr0 = 1; α = 20; Gauss/mouse chaotic map	N = 200; T = 1500; C1 = 2; C2 = 2; Gr0 = 100; α = 10 Gauss/mouse chaotic map
PSOS-CGSA	N=50; T = 200; C1i = .5; C1f = 2.5; λ = .0001; Gr0 = 1; α = 20	N=200; T = 1500; C1i = .5; C1f = 2.5; λ = .0001; Gr0 = 100; α = 10
PPSOGSA	N=50; T = 200; Gr0 = 1; α = 20	N=200; T = 1500; Gr0 = 100; α = 10

Test system 1 consists of three generators with a total demand of 850 MW. The operating limits, fuel cost coefficients, and emission coefficients for this system are taken from (Benasla, Belmadani, and Rahli 2014). In this test system, the expression for transmission-line losses is: $P_{\mathit{loss}}=0.00003P_1^2+0.00009P_2^2+0.00012P_3^2$ where P₁, P₂ and P₃ are the power outputs of generators 1, 2 and 3, respectively. The scaling factors appearing in (4) are taken from (Bhattacharya and Chattopadhyay 2011) and they are as follows: γ = 147582.78814 ($/ton). Test system 1 in this case is considered with P_loss. Test system 2 is standard IEEE 30-bus 6-generator system with total load demand of 283.4 MW, and with NO_x emission. Fuel cost coefficients, emission coefficients, and B-loss matrices are taken from (Aydin et al. 2014). Test system 3 consists of 40 generating units with valve point effects and NO_x emission. The total load demand is 10,500 MW and no transmission losses are considered. The input data for this test system are taken from (Aydin et al. 2014). The best solutions obtained using PSO-CGSA for all three systems are given in the Tables 2–4. Table 5 shows the best, mean, and SD values obtained using PSO-CGSA for three test systems. The best solutions are compared with solutions of other proposed algorithms in the published literature (Table 6). From Table 6, it can be seen that the best solutions obtained using PSO-CGSA are the same as those obtained by other algorithms in the case of test system 1 and test system 2 or are very close to other algorithms in the case of test system 3.

Table 2. Best solutions obtained using PSO-CGSA for test system 3 with P_loss..

Generation	Fuel cost minimization	NOx emission minimization
P1 (MW)	435.19842	508.58043
P2 (MW)	299.96996	250.44250
P3 (MW)	130.66058	105.72290
Fuel cost ($/h)	8344.59272	8365.11413
NOx (ton/h)	0.098686	0.095924
Ploss (MW)	15.82897	14.74583
T (s)	0.25630	0.26254

Table 3. Best solutions obtained using PSO-CGSA for test system 2 with P_loss.

CEED variant→ ↓Generation	OF1	OF2	OF3	OF4
P1 (MW)	22.55457	5.00074	41.09248	5.00000
P2 (MW)	35.45556	12.52558	46.36680	18.61974
P3 (MW)	57.00574	83.53917	54.44145	80.01452
P4 (MW)	74.53916	74.81549	39.03722	74.81330
P5 (MW)	54.82134	79.79939	54.44575	78.16257
P6 (MW)	41.55631	29.60127	51.54931	28.74480
Fuel cost ($/h)	612.25304	619.88285	646.20738	616.88086
NOx (ton/h)	0.203570	0.226581	0.194179	0.222882
Ploss (MW)	2.53269	1.88164	3.53302	1.95493

Table 4. Best solutions obtained using PSO-CGSA for test system 3 with VPE.

Fuel cost minimization
P1–8 (MW)	110.89695	111.32738	120.00000	179.73392	93.67359	139.99917	299.99927	284.63084
P9–16 (MW)	284.69897	130.00168	168.79440	168.79786	125.00000	304.52518	304.51976	394.28007
P17–24 (MW)	489.28182	489.27973	511.27543	511.28493	523.28781	523.29415	523.29343	523.28990
P25–32 (MW)	523.27868	523.28276	10.00313	10.00000	10.00000	96.99873	190.00000	190.00000
P33–40 (MW)	190.00000	199.99271	200.00000	200.00000	109.99851	110.00000	110.00000	511.27922
Fuel cost($/h)		Best = 121655.40100		Mean = 122324.58390		SD = 489.8573
NOx emission minimization
P1–8 (MW)	113.53950	111.60843	118.13958	170.27477	96.89723	120.18281	297.16624	295.87542
P9–16 (MW)	292.72350	145.78011	301.05393	293.40114	436.07887	427.36530	428.48672	422.89642
P17–24 (MW)	438.84124	439.73230	436.37343	437.97616	438.12872	443.46073	445.26797	442.62456
P25–32 (MW)	439.71646	437.06493	27.61079	32.06760	28.20013	96.58466	169.13982	171.69657
P33–40 (MW)	171.89704	199.89915	199.99691	199.76626	98.19233	98.10074	101.81123	434.38033
NOx ($/h)		Best = 176683.58330		Mean = 177223.217642		SD = 480.35663

Table 5. Best, mean, and SD values and computation time obtained using PSO-CGSA for: test system 1 with P_loss; test system 2 with VPE and P_loss; and test system 3 with VPE.

	Value	Test system 1	Test system 2	Test system 3
Minimization of fuel cost	Best	8344.59272	635.82117349	121655.40100
	Mean	8344.59272	641.85381911	122324.58390
	SD	4.68039e-012	10.639425531	489.8573
	T (s)	0.25630	7.6848	332.9290
Minimization of emission	Best	0.095924	0.19417851	176683.5833
	Mean	0.095924	0.19417851	177223.217642
	SD	8.62335e-016	2.605893E–11	480.35663
	T (s)	0.26254	9.5886	322.7140

Table 6. Comparison of the best solutions for test systems 1, 2 and 3.

Algorithms	Test system 1 with Ploss		Test system 2 with Ploss		Test system 3 with VPE
	Fuel cost ($/h)	Emission (t/h)	Fuel cost ($/h)	Emission (t/h)	Fuel cost ($/h)	Emission (t/h)
PSO-CGSA	8344.59272	0.095924	605.99837	0.1941785	121655.401	176683.5833
PSOGSA (Radosavljević 2016)	8344.59272	0.095924	605.99837	0.194179	121461.429	176678.0193
SOA (Benasla, Belmadani, and Rahli 2014)	8342.658	0.09586
BBO (Bhattacharya and Chattopadhyay 2011)	8344.5927	0.095923
ISMA (Hassan et al. 2021)			605.998	0.19418	121546.89	176682.67
MSA (Jevtic et al. 2017)			605.99837	0.194179
NGPSO (Zou et al. 2017)			605.998	0.19418
ABC (Aydin et al. 2014)					121414.80	176682.25
DE-HS (Sayah, Hamouda, and Bekrar 2014)					121414.937	176679.684

The best, mean and SD values for the cases of applying PSO-CGSA and similar algorithms, PSOGSA, PSOS-CGSA, PPSO, and PPSOGSA, to the test system 2 are shown in Table 7. From the results shown in Table 7, it is evident that PSO-CGSA provides the best minimum value for each OF. The SD of the results obtained using PSO-CGSA are the smallest in the cases of OF₃ and OF₄. The best solutions for the power outputs, emission, and fuel cost obtained using PSO-CGSA for the test system 2 with P_loss are given in Table 3. The convergence characteristics of the proposed PSO-CGSA and those of PSOGSA, PSOS-CGSA, PPSO, and PPSOGSA applied to the OF₄ are illustrated in Figure 2. From Figure 2, it can be seen that ascending speeds are high at the beginning for all 5 algorithms, which shows the high convergence. However, PSO-CGSA can achieve an optimal solution after a smaller number of iterations than the other 3 algorithms. Thus, PSO-CGSA is demonstrated to have a better convergence property in comparison with the PSOGSA, PSOS-CGSA, PPSO, and PPSOGSA.

Figure 2. Convergence curves of five algorithms applied to the OF₄ function.

Table 7. The best, mean, and SD values and error rates (ER) obtained using PSOGSA, PSO-CGSA, PSOS-CGSA, PPSO, PPSOGSA for the test system 2 with VPE.

		Without Ploss					With Ploss
OF	Value	PSOGSA	PSO-CGSA	PSOS-CGSA	PPSO	PPSOGSA	PSOGSA	PSO-CGSA	PSOS-CGSA	PPSO	PPSOGSA
OF1	Best	810.08691677	810.08691677	810.08691718	810.08691677	810.08691678	815.82291063	815.82291064	815.82291132	815.82291063	815.82291065
	Mean	810.08691677	810.08691685	810.08691898	810.08691678	810.08692413	838.12323528	818.33434955	817.35615238	815.82291071	816.53572124
	SD	1.1563E–13	4.22315E–08	1.64546E–06	3.32276E–08	1.02976E–05	19.281139171	9.247815964	4.164067717	1.943017E–07	1.796448603
	ER	1.440175E–14	1.005208E–08	2.728771E–07	2.047276E–09	9.090171E–07	0.704750930	3.078412E–01	1.879381E–01	9.799457E–09	8.737320E–02
	T (s)	2.2225	2.27019	2.21458	2.52444	2.11191	4.7059	7.1313	7.3582	10.3494	9.0215
OF2	Best	631.34081604	631.33158822	631.33792228	631.34584785	631.33645979	635.82371948	635.82117349	635.82370386	635.82662371	635.87618352
	Mean	635.20705028	634.46861916	631.85135595	633.88554529	631.55245423	655.39554491	641.85381911	636.80361571	637.20528044	639.52205046
	SD	6.091202511	6.255671017	2.500370943	4.836767964	0.177679698	14.598860248	10.639425531	5.122108726	2.982804554	6.687647727
	ER	6.138552E–01	4.968912E–01	8.232880E–02	4.045350E–01	3.498415E–02	3.078596976	9.487960E–01	1.545155E–01	2.176881E–01	5.820626E–01
	T (s)	2.20407	2.26435	2.26070	2.58963	2.17785	4.9228	7.6848	6.9434	10.6581	10.3837
OF3	Best	0.1942029389	0.1942029389	0.1942029390	0.1942029389	0.1942029389	0.19417851	0.19417851	0.19417851	0.19417851	0.19417851
	Mean	0.1942029389	0.1942029389	0.1942029397	0.1942029389	0.1942029419	0.20306643	0.19417851	0.19531448	0.19417851	0.19489220
	SD	5.08925E–16	1.87538E–11	5.31515E–10	2.40828E–11	5.82718E–09	0.008337065	2.605893E–11	0.005514249	7.607846E–11	0.002362168
	ER	2.572568E–13	1.467562E–08	4.319794E–07	6.554731E–09	1.551293E–06	4.577187324	2.366824E–08	5.850120E–01	1.438471E–08	3.675418E–01
	T (s)	2.22695	2.23544	2.24769	2.51482	2.15401	4.9735	9.5886	9.2108	11.9797	8.366
OF4	Best	857.51164134	857.51106243	857.51131632	857.51628304	857.51416291	861.70732533	861.70293668	861.70310011	861.70710405	861.70952875
	Mean	857.53210434	857.51466806	857.51654466	857.58482752	857.56840932	888.77777902	861.70704653	861.70717144	861.75652704	863.04344517
	SD	0.015613473	0.003481224	0.004877423	0.080117864	0.050989199	21.739694419	0.003512251	0.003532636	0.061506758	3.383061494
	ER	2.453836E–03	4.204765E–04	6.393188E–04	8.602232E–03	6.687598E–03	3.142015790	4.769452E–04	4.914416E–04	6.219122E–03	1.555650E–01
	T (s)	2.21250	2.25667	2.24374	2.49056	2.16292	4.4224	7.9009	7.3845	10.471	10.1775

Behavior Analysis

In this paper, we perform an experimental analysis of the behavior of five algorithms over the four most common variants (4) of the CEED problem. All five algorithms are hybrid or self-adaptive based on the PSO and provide the same or approximately the same solutions to the CEED problem. These are the following algorithms: PSOGSA, PPSO, PSOS-CGSA, PPSO-GSA and the new PSO-CGSA algorithm, which we propose in this paper. As pointed out above, the PSOGSA algorithm was presented in the published literature as the best compared to other previously proposed algorithms for solving the CEED problem. Therefore, we choose PSOGSA and algorithms with similar performance to compare with the new PSO-CGSA algorithm. We solve the CEED problem within the standard test system 2.

Table 7 shows the best values (best), mean best values (mean), SD, and error rates obtained by the tested algorithms for the objective functions (OF₁ – OF₄). The algorithm’s error rate is calculated as the average value of the percentage differences between the values of the individual results obtained for 30 runs and the best value obtained for a concrete OF. Table 7 shows (in bold) that PSO-CGSA gives the lowest best value for all OFs compared to other algorithms. PSO-CGSA gives the best all other obtained values (mean, SD, and error rate) for bi-objective OF₄ function, which is the most complicated among the four tested functions. Other tested algorithms give similar and in some cases the same values as PSO-CGSA. The PSOGSA algorithm has the closest values to the PSO-CGSA algorithm.

Based on the values shown in Table 7, it can be concluded that PSO-CGSA shows the best results, good robustness, and good powerful optimization ability which increases by increasing the complexity of the objective function. Convergence curves of all algorithms applied to the OF₄ function are shown in Figure 2. From Figure 2 follows that the convergence curve of PSO-CGSA is the best. The best solutions for the power outputs, emission, fuel cost, and P_loss obtained by using PSO-CGSA are given in Table 3.

Below we analyze the algorithm’s behavior in the optimization problem after applying statistical tests to the obtained results. In this way, we get a more accurate assessment of which algorithm is the best. We use normality tests of Lilliefors, Shapiro–Wilk and D’Agostino-Pearson, Levene’s test of homoscedasticity, paired t-test, Wilcoxon’s non-parametric test, and Friedman’s non-parametric test. For testing and the analyses, we use the SPSS and MATLAB platforms.

Single-problem Analysis

We first analyze the behavior of the algorithms over each of the five variants of the CEED problem separately (single-problem analysis). Table 8 gives the probability values (p-values) of the applied tests on the results of each algorithm for each variant of the problem. These values show the extent to which the distribution of results in the sample deviates from the normality distribution. We choose a p-value of .05 as the level of significance (α). From Table 8, we conclude that the results of most algorithms are not subject to the normal distribution law (p-values are less than .05) except in two cases: PSO-CGSA for OF₁ and PSOS-CGSA for OF₃.

Table 8. p-values of normality tests: KS Lilliefors Modification, Shapiro–Wilk, and D’Agostino–Pearson.

Algorithm	KS Lilliefors Modification				Shapiro-Wilk				D’Agostino-Pearson
	OF1	OF2	OF3	OF4	OF1	OF2	OF3	OF4	OF1	OF2	OF3	OF4
PSOGSA	0.0000	0.0000	0.0000	0.0851	0.0000	0.0000	0.0000	0.0297	0.0000	0.0240	0.0000	0.0689
PSO-CGSA	0.2000	0.0000	0.0158	0.0072	0.2983	0.0000	0.0485	0.0005	0.0966	0.0009	0.1503	0.0384
PSOS-CGSA	0.0042	0.0000	0.2000	0.0544	0.0011	0.0000	0.0625	0.0014	0.0025	0.0000	0.1678	0.0028
PPSO	0.0000	0.0000	0.0000	0.0007	0.0000	0.0000	0.0000	0.0001	0.0000	0.0000	0.0000	0.0011
PPSOGSA	0.0000	0.1517	0.0000	0.0002	0.0000	0.0037	0.0000	0.0001	0.0012	0.0027	0.0000	0.0030

Figure 3 shows histograms and plots of the quantiles from the results and the normal distribution (Q–Q plots) for all algorithms over the OF₄ function. The Q–Q plots confirm the results of applied normality tests.

Figure 3. Histograms and Q-Q plots for all algorithms applied to the OF₄ function.

Below we apply Levene’s test of homoscedasticity, which shows that sample variances of the results of each algorithm for each OF are not homogeneous (all p-values are zero), i.e., they are different. From the results of the tests of normality and homoscedasticity, we conclude that parametric tests cannot be applied to analyze the behavior of algorithms. Therefore, we use the nonparametric Wilcoxon test for pairwise comparisons of algorithms, suggested by Garcia et al. (2009). Wilcoxon test is based on the ranking of algorithms for each OF. Table 9 gives the p-values of the Wilcoxon test for mean ranks differences of algorithms in the pair. For comparison, p-values of the parametric paired t-test are also offered. Also, Table 9 shows the differences in error rates of algorithms in pairs. Table 9 shows that the p-values obtained by the Wilcoxon test and the p-values obtained by the t-test are similar except in 5 cases (out of a total of 50 cases). Since we have previously determined that only non-parametric tests are acceptable for further analysis, we use only the p-values of the Wilcoxon test as relevant in these 5 cases. Table 9 shows that the vast majority of pairs (41 pairs) of algorithms have a lower p-value than the significance level (.05), and in the case of 9 pairs, p-values are greater than .05. This means that in 41 pairs, there is a difference in the behavior of the algorithms and in 9 pairs there is no difference. The error rate differences determine which of the algorithms in the pair is better (has a lower error rate). A minus sign in front of the average rate difference indicates that the first algorithm in the pair is better than the second and vice versa. Table 9 shows that the PSOGSA algorithm has the best behavior in solving the OF₁ and OF₃ functions, the PSOS-CGSA behaves best for the OF₂ function, and the PSO-CGSA algorithm has the best behavior for the OF₄ function.

Table 9. The p-values of the paired t-test and the Wilcoxon test and the differences of the error rates for the analysis of single functions.

Function	Algorithm pair	t-test	Wilcoxon test	Difference of the error rates
OF1	PSOGSA-PSOCGSA	0.000	0.000	−1.01E–08
	PSOGSA-PSOSCGSA	0.000	0.000	−2.73E–07
	PSOGSA-PPSO	0.011	0.000	−2.05E–09
	PSOGSA-PPSOGSA	0.001	0.000	−9.09E–07
	PSOCGSA-PSOSCGSA	0.000	0.000	−2.63E–07
	PSOCGSA-PPSO	0.000	0.000	8.00E–09
	PSOCGSA-PPSOGSA	0.001	0.000	−8.99E–07
	PSOSCGSA-PPSO	0.000	0.000	2.71E–07
	PSOSCGSA-PPSOGSA	0.012	0.098	−6.36E–07
	PPSO-PPSOGSA	0.001	0.000	−9.07E–07
OF2	PSOGSA-PSOCGSA	0.682	0.530	1.17E–01
	PSOGSA-PSOSCGSA	0.012	0.001	5.32E–01
	PSOGSA-PPSO	0.316	0.975	2.09E–01
	PSOGSA-PPSOGSA	0.003	0.153	5.79E–01
	PSOCGSA-PSOSCGSA	0.048	0.008	4.15E–01
	PSOCGSA-PPSO	0.711	0.329	9.24E–02
	PSOCGSA-PPSOGSA	0.016	0.877	4.62E–01
	PSOSCGSA-PPSO	0.059	0.000	−3.22E–01
	PSOSCGSA-PPSOGSA	0.500	0.000	4.73E–02
	PPSO-PPSOGSA	0.013	0.000	3.70E–01
OF3	PSOGSA-PSOCGSA	0.000	0.000	−1.47E–08
	PSOGSA-PSOSCGSA	0.000	0.000	−4.32E–07
	PSOGSA-PPSO	0.027	0.000	−6.55E–09
	PSOGSA-PPSOGSA	0.008	0.000	−1.55E–06
	PSOCGSA-PSOSCGSA	0.000	0.000	−4.17E–07
	PSOCGSA-PPSO	0.016	0.006	8.12E–09
	PSOCGSA-PPSOGSA	0.009	0.000	−1.54E–06
	PSOSCGSA-PPSO	0.000	0.000	4.25E–07
	PSOSCGSA-PPSOGSA	0.055	0.054	−1,12E–06
	PPSO-PPSOGSA	0.009	0.000	−1.54E–06
OF4	PSOGSA-PSOCGSA	0.000	0.000	2.03E–03
	PSOGSA-PSOSCGSA	0.000	0.000	1.81E–03
	PSOGSA-PPSO	0.001	0.001	−6.15E–03
	PSOGSA-PPSOGSA	0.002	0.002	−4.23E–03
	PSOCGSA-PSOSCGSA	0,141	0.185	−2.19E–04
	PSOCGSA-PPSO	0.000	0.000	−8.18E–03
	PSOCGSA-PPSOGSA	0.000	0.000	−6.27E–03
	PSOSCGSA-PPSO	0.000	0.000	−7.96E–03
	PSOSCGSA-PPSOGSA	0.000	0.000	−6.05E–03
	PPSO-PPSOGSA	0.404	0.393	1.91E–03

Each of the three best algorithms has different p-values in each pair with other algorithms. Therefore, we determine one p-value, the so-called true p-value, for each of the best algorithms in the pair, according to the following formula (Garcia et al. 2009; Kaveh 2021):

(15) $\mathit{p}=1-\prod _{\mathit{i}=1}^k\left(1-p_i\right)$(15)

where p_i is the p-value of the algorithm in the ith pair in which the algorithm is better; k is the number of pairs. Table 10 gives the true probability values of the best algorithms in pairs calculated according to (15). Table 10 also shows the corresponding values of confidence intervals calculated from the obtained p-values (15), as 100 (1−α), where we assume the significance level α = p.

Table 10. True values of p and confidence intervals for algorithms with better performance in pairs.

Function	Algorithm with better performance	True value of p	Confidence interval, %
OF1	PSOGSA	0.000	100
OF2	PSOS-CGSA	0.008992	99.1
OF3	PSOGSA	0.000	100
OF4	PSO-CGSA	0.000	100

From the results of testing algorithms on individual variants of the CEED problem, the following can be concluded: (1) the results obtained are not subject to normal distribution except in the case of PSO-CGSA for the OF₁ function and PSOS-CGSA for the OF₃ function (Table 8); (2) The homoscedasticity test shows that the sample variances of the algorithms are not homogeneous, so parametric tests cannot be used to analyze the behavior of the algorithms, but non-parametric ones; (3) Wilcoxon’s test of pairs of algorithms shows that different algorithms have the best behavior for different functions: PSOGSA for OF₁ and OF₃, PSOS-CGSA for OF₂, and PSO-CGSA for OF₄; (4) The new PSO-CGSA algorithm has the best behavior in solving the most complex variant of the CEED problem that has the OF₄ objective function (which minimizes fuel cost and emission simultaneously, taking into account the valve point effect).

Multiple-problem Analysis

We perform multiple-problem analysis by analyzing the behavior of each algorithm over all functions of the CEED problem simultaneously. Since there are dissimilarities in the results for different OFs, we use the error rates of the given results as a performance measure of the algorithms (Table 7). We apply statistical tests to each sample consisting of the error rates for the algorithm. We first use normality tests to determine whether error rates in the sample are subject to normal law. Table 11 shows the p-values of normality tests for each algorithm. Table 11 shows that the p-values are less than the significance level (.05), which means that the error rates do not follow a normal law in either sample. Histograms of error rates and Q–Q diagrams for algorithms confirm the results in Table 11. Therefore, we apply non-parametric tests to analyze the behavior of each algorithm in solving all OFs simultaneously. We use Wilcoxon’s and Friedman’s non-parametric tests to compare algorithms in pairs. We determine whether the error rates of two algorithms in a pair represent two populations with different median values. Table 12 gives the p-values obtained using Wilcoxon’s and Friedman’s tests for all pairs of algorithms. Obviously, all p-values in Table 12 are greater than the significance level (α = .05), which means that there is no difference in the behavior of the algorithms when solving the CEED problem as a multiple-problem.

Table 11. P-values of normality tests to analyze problems with a group of functions.

Algorithm	KS Lilliefors Modification	Shapiro-Wilk	D’Agostino-Pearson
PSOGSA	0.001	0.001	0.0035
PSOCGSA	0.001	0.001	0.0035
PSOSCGSA	0.001	0.002	0.0035
PPSO	0.001	0.002	0.0035
PPSOGSA	0.055	0.046	0.0097

Table 12. p-values of Wilcoxon and Friedman tests to analyze problems with a group of functions.

Algorithm pair	Wilcoxon test	Friedman test
PSOGSA-PSOCGSA	0.465	1.000
PSOGSA-PSOSCGSA	0.465	1.000
PSOGSA-PPSO	0.715	0.317
PSOGSA-PPSOGSA	0.715	0.317
PSOCGSA-PSOSCGSA	0.715	0.317
PSOCGSA-PPSO	0.465	0.317
PSOCGSA-PPSOGSA	0.715	0.317
PSOSCGSA-PPSO	0.465	1.000
PSOSCGSA-PPSOGSA	0.715	0.317
PPSO-PPSOGSA	0.465	1.000

Ranking Algorithms Using the PROMETHEE/GAIA Method

In the previous section, we found that there was no significant difference in the behavior of the five tested algorithms when solving CEED problems with all variants (functions) simultaneously. In this section, we determine the best-ranked algorithm for solving the CEED problem according to the best results, SD, mean values, and error rates of individual variants of the problem. To solve this problem, we use the PROMETHEE method. This MCDM method is widely used to solve various decision problems (Aherwar et al. 2019; Arsić, Nikolić, and Jevtić 2021; Brans and De Smet 2016). Additionally, we use the GAIA (Geometrical Analysis for Interactive Assistance) plane for the graphical representation of the obtained results. The initial data of a multicriteria problem in Table 7.

Using the steps for calculating of the PROMETHEE method (Arsić, Nikolić, and Jevtić 2021; Brans 1982), the calculation of “outranking flows“was performed, i.e., positive (Ф+) and negative (Ф-) flow and complete net flow Ф(a). The final rank of the analyzed algorithms was shown for all scenarios (best results, SD, mean values, and error rates) in Table 13 and Figure 4.

Figure 4. GAIA plane of selection of the best-ranked algorithm according to: a) best results, b) SD, c) mean values, d) error rates.

Table 13. Results of the complete ranking of the algorithm according to the functions.

Scenario	Best				SD
↓Alternatives	Φ^+	Φ^−	Φ	Rank	Φ^+	Φ^−	Φ	Rank
PSOGSA	0.2500	0.3125	−0.0625	3	0.6875	0.3125	0.3750	1
PSO-CGSA	0.5625	0.000	0.5625	1	0.5625	0.4375	0.1250	2
PSOSCGSA	0.3125	0.4375	−0.1250	4	0.4375	0.5625	−0.1250	3
PPSO	0.0625	0.5000	−0.4375	5	0.4375	0.5625	−0.1250	3
PPSOGSA	0.3125	0.2500	0.0625	2	0.3750	0.6250	−0.2500	4
Scenario	Mean				Error rate
↓Alternatives	Φ^+	Φ^−	Φ	Rank	Φ^+	Φ^−	Φ	Rank
PSOGSA	0.1875	0.3750	−0.1875	3	0.6250	0.3750	0.2500	1
PSO-CGSA	0.3750	0.1875	0.1875	1	0.5625	0.4375	0.1250	2
PSOSCGSA	0.3750	0.2500	0.1250	2	0.5000	0.5000	0.0000	3
PPSO	0.1250	0.3750	−0.2500	4	0.5000	0.5000	0.0000	3
PPSOGSA	0.3125	0.1875	0.1250	2	0.3125	0.6875	−0.3750	4

The best-ranked algorithm applied in four different functions according to the best results and mean is PSO-CGSA, followed by PPSOGSA. According to the SD and error rates in the considered functions, the obtained results indicate that the best-ranked algorithm is PSOGSA, while PSO-CGSA is in second place.

Figures 4(a–d) show the projection of functions in the three-dimensional plan, where they are saved with 92.3%, 93.7%, 96.5%, and 99.4% of the total information, respectively. All the values obtained are above the recommended 70% (Aherwar et al. 2019), which means that all the information provided by the GAIA plan is very reliable.

Discussion

The new PSO-CGSA algorithm was obtained by improving the hybrid PSOGSA proposed in the published literature to solve the CEED problem. The improvement was made by embedding the chaotic Gauss-mouse map into the formula for the gravitational constant. This allowed better exploration and better avoidance of trapping in local minima. PSO-CGSA was tested on the following four variants of the CEED problem: (1) minimization of fuel costs and emissions simultaneously (function OF₁) without the valve point effect in power plants; (2) minimization of NO_x gas emission (OF₂); (3) minimization of fuel costs with the valve point effect (OF₃); and (4) minimization of costs and emissions simultaneously with the valve point effect (OF₄). In all four variants, the VPE is taken into account. The IEEE 30-bus system with 6 generators was used for testing. Simultaneously with the PSO-CGSA algorithm, PSOGSA and three algorithms (PSOS-CGSA, PPSO and PPSOGSA) with similar performance were tested. The test results showed that the new PSO-CGSA gives the best value of OF compared to other algorithms in all tested variants of the CEED problem. PSO-CGSA showed the best robustness and the fastest convergence in solving the most complicated function OF₄. To reliably evaluate the performance of the algorithms, we analyzed the behavior of the algorithms using parametric and non-parametric tests. We performed the analysis for each OF individually (single-problem analysis) and for all OFs simultaneously (multiple-problem analysis). In the analysis, we used error rate as a performance measure. Single-problem analysis showed that PSO-CGSA has the best behavior for OF₄, PSOGSA for OF₁ and OF₃, and PSOS-CGSA for OF₂. The behavior analysis confirmed that PSO-CGSA has the best performance when solving the most complex, function (OF₄) and that PSOGSA has the best performance when solving the simpler OF₁ and OF₃ functions. Multiple-problem analysis showed that all tested algorithms have approximately the same behavior when solving all variants of the CEED problem simultaneously. Bearing in mind that multiple-problem analysis does not favor one algorithm, we applied the PROMETEE/GAIA MCDM method to select the most suitable algorithm for simultaneously solving all variants of the CEED problem. Using this method, we ranked the tested algorithms based on their performance measures obtained when solving individual variants. PSO-CGSA is the best-ranked algorithm based on best and mean values and ranks second (after PSOGSA) when ranking is based on SD and error rate. This means that based on the results of the PROMETEE/GAIA method, PSO-CGSA can be chosen as the most acceptable algorithm for simultaneously solving all variants of the CEED problem.

Conclusion

The results of this paper show that the new metaheuristic algorithm, PSO-CGSA, can be successfully applied to solve the CEED problem. The procedure used in the paper to select the best algorithm is based on the application of non-parametric tests and MCDM methods, so it can be applied to solve other optimization multiple-problems. This procedure is particularly suitable in cases where a large number of algorithms with similar performance have been proposed to solve a specific multiple-problem consisting of many individual problems.

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Ministry of Education, Science and Technological Development of Serbia.