An innovative hybrid algorithm for solving combined economic and emission dispatch problems

As environmental concerns have grown, the combined economic emission dispatch (CEED) problem has gotten a lot of attention. Both the cost of fuel and the emission pollution caused by it must be kept to a minimum. As a result, this paper presents an innovative hybrid approach (ihPSODE) for solving CEED problems. This hybrid technique incorporated novel differential evolution (nDE) and particle swarm optimization (nPSO). Where nDE introduces an improved mutation and crossover approach (to prevent untimely convergence) as well as nPSO introduces a new acceleration coefficient, inertia weight and position improve equation (to alleviate the stagnation). So as to balance among local and global search ability, after ihPSODE population evaluation, the best half individuals are determined and the rest individuals are discarded. Then, nPSO is used in the current population (to sustain exploration and exploitation) and nDE is employed in the nPSO generated population (to improve convergence accuracy). The competence of the proposed algorithms (ihPSODE, nPSO and nDE) is inspected on 23 unconstrained benchmark function and then solved 3 test system (3-, 6- and 40-unit) of economic load dispatch (ELD) and 3 test system (3-, 10- and 40-unit) of CEED problem. The experiments have denoted that the proposed algorithms show competitive results and significant performances.


Introduction
In modern power system operation, economic load dispatch (ELD) is a critical optimization problem (Mansor et al. 2018). Reduce total generation costs are the basic goal of the ELD problem with satisfying equality and inequality limitations and load demand. Apart from the producing capacity limits, the classic ELD primarily analyses the power balance constraint. But, due to real-world restrictions in power system process, ELD essentially consider a multiplicity of real-world limitations like transmission loss, ramp rate limits, multi-fuel options, prohibited operating zones, spinning reserve along with system power demand, etc. It resulted with a non-convex nonlinear ELD problem and finding an optimum solution of this type problem is very challenging and time-consuming. The environmental constraint, which consists of Carbon oxides (Cox), Nitrogen oxides (NOx) and Sulphur oxides (Sox), infects the air, is one of those limitations that is always taken into account. By properly allocating load among available generators, the hazardous environmental pollutants released by fossil fuel power plants can be decreased. However, the power plant's operational costs would rise. As a result, a solution must be found that balances both emissions and fuel costs. It can have been attained by 'combined economic and emission dispatch (CEED)' problem. The key goal of the CEED problem is reduce fuel costs and emissions with satisfying equality and inequality limits plus load demand. Mathematically, the ELD and CEED problem can be expressed as an optimization (minimization) problem, as shown below.
1.1 Mathematical problem formulation 1.1.1 Economic load dispatch (ELD) In ELD the total fuel cost ($/hr) can be mathematical expressed as below: Also, in view of valve-point loadings effect the fuel cost function given as follows.
where F t : total fuel cost of generations ($/hr); a i ; b i andc i : cost factors of a generator i; e i and f i : fuel cost quantities for valve point effects of a generator i; P i : power output of the i th generator. The constraints of ELD problems are listed as follows.
• P min i P i P max i -generator limits • P n i¼1 P i ¼ P D total load demand ð Þ þ P L ðtotal transmission lossÞ -power balance where B ij , B oi and B oo are transmission loss coefficient. • P min i P i P l i;1 : P u i;kÀ1 P i P l i;k : P u i;n i P i P max i ; k ¼ 2; 3; . . .n i -prohibited operating zone (POZ) where n i is the no. of POZ, P u i;k & P l i;k are upper and lower limit of k th POZ for generating unit i: where P t i & P tÀ1 i current & previous output power and UR i & DR i : up & down ramp limit of generating unit i:

Combined economic emission dispatch (CEED)
When generator units burn fossil fuels, pollutants such as SOx, NOx, and COx are released into the atmosphere. The overall emission of these pollutants, known as emission constrained dispatch (ECD), can be written as.
whereE t : total amount of emissions (lb/hr) and a i ,b i ,c i , w i k i : emission coefficients of the i th unit. Moreover, simultaneously minimizing two objective function F t and E t is the main target of the CEED problem.
The bi-objective problem in Eq. (3) can be altered into a single objective problem, using a h (price penalty factor) method as follows.
where u t ; total cost of the system operation. The price penalty factor h can be calculated by the following procedures for a particular load demand.
(i) Calculate the ratio . . .n$=kg. (ii) Sort the obtained h i values in ascending order. (iii) Add P max i of each unit one at a time starting from the unit with smallest h i until P P max i ! P D . (iv) Catch the last value of h i that attains the previous situation which signifies the price penalty factor for the given load. Equation (5) can be modified as follow (to balance between emission and fuel cost minimization).
where w (specifies type of the optimization problem) is the weight factor and if -(i) w = 1 infers ELD problem (ii) w = 0 implies ECD problem (iii) w = 0.5 indicates CEED problem

Related literature survey
To handle engineering optimization challenges, many traditional optimization techniques such as Newton and quasi-Newton have been created. Moreover, they have some intrinsic limitations, such as high computing cost, local optimal stagnation, and search space derivation (Simpson et al. 1994). It is also challenging to locate the best solution during the problem-solving process. Nowadays, metaheuristics algorithms (MAs) have been established to solve complicated optimization problems. These algorithms may to avoid the limitations of traditional optimization approaches efficiently. As per nature, the MAs are mostly divided into four parts in the literature as follows.
The fundamental drawback of PSO is that it can quickly become trapped in a local optimal solution zone. As a result, in PSO, escaping local optimal solutions and speed of convergence are two essential challenges. To address such concerns, many potential changes to the PSO have been proposed in recent literature. Espitia and Sofrony (2018) proposed VPSO in which particle vortex behaviour and particle circular motions are implemented for improving search capacity and escaping the local minima, respectively. To find the optimum of the current search and gradually explore the search space, Yu et al. (2018) proposed SHPSO by the implementation of social learning PSO. Chen et al. (2018) devised PSOCO where two distinct crossover operations are taken to produce promising exemplars in order to balance diversity. A self-adaptive tool and unique factor (w, c 1 , c 2 ) utilized for enhancing each particle position depending on their fitness in UAPSO which is devised by Isiet and Gadala (2019). Hosseini et al. (2019) developed HAFPSO where fractional-order derivatives and hunter-attack strategy are used to accelerate convergence and avoid stagnation, respectively. Khajeh et al. (2019) proposed MPSO where a novel particle initializing scheme with random distribution is used for uniformly covering the search space. Ang et al. (2020) invented CMPSOWV, in which two diversity maintenance schemes (multi-swarm technique and probabilistic mutation operator) are used to prevent the premature convergence. Lanlan et al. (2020) proposed NOPSO where noninertial velocities update formula, opposition-based learning strategy and adaptive elite mutation strategies are employed to avoid local optimum trapping. Xiong et al. (2020) proposed NMSPSO where three strategies-novel information exchange strategy (for information transfer between sub-swarms), novel leaning strategy (for speed up the convergence) and novel mutation strategy (for better exploration) are incorporated. In the ground of real-world problems, DE also has noteworthy performance and become a great optimizer. Still, it has few concerns like local exploitation ability and convergence rate. So as to reduce its concerns, hordes of effective and robust DE has been intended in the literature. Qiu et al. (2018) proposed MMDE where a novel bottom-boosting mechanism (to maintain the reliability), partial-regeneration strategy (to identify the promising solutions) and mutation operator DE/current/1 (to explore over solution space) are introduced. Zhang and Li (2018) developed DEPS in which a modified parent selection scheme is chosen to use the experience of successful parents while selecting them in mutation operator. Huang et al. (2018) invented hypercubebased NCDE where hypercube neighbourhood-based mutation (to maintain the neighbourhood size in a reasonable range) and self-adaptive techniques (to control the hypercube's radius vector) are used. Yang et al. (2019) developed daDE in which a modified mutation rule is created to utilize the information of the current and the former individual's altogether. It has great benefits on the robustness and convergence speed. Liu et al. (2019) proposed HDEMCO where two layers' top layer (where multi cross operation perform that provides rapid convergence) and bottom layer (where success-history-based adaptive DE is implemented for better global search) are considered. Gui et al. (2019) devised MRDE in which different trial vector generation strategies, regroup scheme and an adaptive strategy are performed to speed up the convergence rate. Li et al. (2020) developed EJADE where a sorting mechanism and a dynamic population reduction strategy are employed to speed up the convergence rate and maintain the diversity, respectively. Hu et al. (2020) invented BADE where an annealing strategy allow the searching space to explore and accelerate the convergence too. Ben (2020) proposed ADE where initial population and a new difference vector (in mutation phase) are created by the knowledge of damage scenario structure and dispersion of individuals, respectively.
Moreover, to increase the performance of a single algorithm, one of the primary research advices is the hybrid strategy. Because of diverse optimization methods have dissimilar search behaviours and benefits. Hence, to reduce their individual weaknesses (like untimely convergence, particle stagnation, etc.), hybrid methods are more preferred nowadays to solve complex optimization problems. As a result, many hybrid algorithms for DE and PSO are presented in the literature in order to improve their performance. Seyedmahmoudian et al. (2015) proposed DEPSO, where DE is employed to adds diversity on traditional PSO. However, it may not appropriate for multimodal optimization problems. Parouha and  devised DPD in which DE is executed in the inferior and superior groups, while PSO is employed in the mid-group. But, for solving complex real-world problems it may need some moderations. Tang et al. (2016) proposed HNTVPSO-RBSADE, which employs a nonlinear time varying PSO and a ranking-based self-adaptive DE to result in dynamic exploration and exploitation. Parouha and Das (2016a) developed MBDE in which swarm mutation and swarm crossover for DE is used to direct knowledge and improve the solution quality. Parouha and Das (2016b) proposed DE-PSO-DE in which the population is divided into three groups (A, B, & C) and executed in parallel manner. Famelis et al. (2017) devised DE-PSO where DE is merged with a velocity-update rule of PSO to enhance diversity. Mao et al. (2018) proposed DEMPSO in which DE is added first to lessen the search space and then acquired populations used modified PSO (MPSO) as an initial population to speed up the convergence rate. Tang et al. (2018) developed SAPSO-mSADE in which selfadaptive PSO (SAPSO) and modified self-adaptive DE (mSADE) are evolved to balance diversity and reduce potential stagnation, respectively. Too et al. (2019) invented BPSODE where the strength of binary PSO (BPSO) and binary DE (BDE) are inherited and computed in sequence. Dash et al. (2020) proposed HDEPSO in which three DE operations (mutation, modified crossover and selection) are fused with the best particles of PSO for enhancing global searching ability. Parouha and Verma (2021) proposed innovative hybrid algorithm of DE and PSO for bound-unconstrained optimization and ELD problem with or without valve point effects. It combined with novel DE (to avoid premature convergence) and PSO (to escape stagnation). Further, Verma and Parouha (2021) applied the innovative hybrid algorithm to solve non-convex dynamic economic dispatch problem and numerical, graphical as well as comparative results shows its capability to solve considered optimization problems.
Furthermore, a related recent review of DE, PSO and their hybrids as well as other MAs variants for solving CEED problem are mentioned as further. Mahdi et al. (2019) proposed QBA, in which a cubic criterion function is employed to represent this problem to reduce the nonlinearities of the system. The quantum behaviour of bats donates population divergence and diversifies the foraging habitats. In addition, early convergence in BA can be prevented. Jiang et al. (2019) devised GPSOA, where it integrates PSO with gravitation laws of GSA. Here, the particle's velocity is reorganized by random support of PSO and GSA. Additionally, Weibull-based probability density function is also used, to designate the stochastic individualities of wind speed and output power. Rezaie et al. (2018) proposed CIHSA, which is the combination of IHSA and CHSA. Where IHSA has a suitable convergence rate with high accuracy and CHSA has a high strength to find the best solutions in altered runs. Moreover, to dynamically tuning the parameters, employing virtual harmony memories and generate random numbers it uses chaotic patterns. Goudarzi et al. (2020) proposed MGAIPSO, it utilizes three operators, an arithmetic crossover and a mutation operator from GA to produce elite off-springs and maintain diversity; a nonlinear timevarying double-weighted technique from PSO to obtain a substantial balance between exploration and exploitation. Edwin Selva Rex et al. (2019) proposed a novel hybrid algorithm (GA-WOA) using GA and Whale optimization for solving CEED problem. This method verified on four different test systems and it is superior to other heuristic methods with slight increase in the CPU execution time. Rashid et al. (2020) invented MIW-PSO, in which a multiple inertia weight is incorporated in PSO to improve its convergence characteristics for minimizing fuel cost and pollutant emission in the uncertain energy production expense and random load. Bibi et al. (2020) developed GOA, where the comfort zone operator of GOA assists in extracting stupendous simulations results of minimized fitness of multi objective functions that shows the efficiency of GOA in term of highly optimal and feasible solution satisfying all the system equality and inequality operational constraint. Khatsu et al. (2020) proposed PPSO in which a linear and nonlinear time varying weight inertia factor (LWF and NLWF) are introduced in PPSO to enhance its searching ability. Goyal et al. (2020) proposed BBO, where an optimum generator scheduling has been achieved by employing BBO with all system constraints. Sakthivel et al. (2021) proposed MOSSA, it integrates squirrel search algorithm along with Pareto-dominance principle to generate non-dominated solutions. Also, it employed outward elitist depository tool with flocking distance categorization (to retain the distribution diversity) and utilized fuzzy decision-making strategy (to select the finest cooperated solution). Ajayi and Heymann, (2021) devised MSA. It is encouraged by the effort of moths the moonlight towards. Also, to provide the essential spinning reserve capacity, it slated thermal generators as solar energy is intermittent in nature. Hassan et al. (2021) proposed CAEO, which uses the chaotic maps which enhance a variety of the solution spaces and improve the convergence capabilities to achieve the optimum solutions as well as avoid the local minima.

Research gaps (Inspiration/motivation)
Despite the fact that a large number of MAs have been introduced in the literature, they have not been able to solve a wide range of problems (Wolpert and Macready 1997). Particularly, for some problems an algorithm can produce satisfactory outcomes but not for others. As a result, for solving a variety of optimization problems, there is a need to develop some competent algorithms. Furthermore, over separate efforts hybrid algorithms are now favoured more to solve complex real-world problems. Hence, from the inspiration of above-mentioned facts and literature investigation motivation of this study is to design an effective and innovative hybrid algorithm with the combination of novel variants of MAs. Between popular MAs, PSO and DE as well as their hybrids successfully applied in diverse optimization parts due to their leading search ability and simple structure. Consequently, after a wide analysis of the literature on different variants of DE and PSO with their hybrids the subsequent resulting opinions is evaluated and encouraged form them.
(i) The PSO is largely dependent on its parameters (which direct particles to the optimum) and position update (to balance diversity). As a result, numerous investigators have attempted to improve the accuracy and speed of PSO by modifying its control parameters and position update equation. (ii)

Contribution
Inspired by above remarks and literature study planned the following for solving CEED problems.
(i) nPSO (novel PSO): It includes novel gradually decreasing and/or increasing control factors and new equation for position update of the particle. (ii) novel DE (nDE): It includes combination of novel operators (mutation and crossover) and new related control parameters. (iii) ihPSODE (innovative hybrid PSODE): It integrated with advised novel PSO (to increase the population diversity) and novel DE (it produce perturbations to avoid the algorithm trapping into local optima).
The proposed algorithms have justified on 23 typical basic benchmark functions then used to solve 3 test systems (3, 6 and 40-unit test system) of ELD and 3 test systems (3, 10 and 40-unit test system) of CEED problem. Comparative experiments endorse the efficiency and superiority of the presented methods.
The rest part of this paper is presented in the following ways. Section 2 presents the proposed algorithms in details. Proposed methods are validated on 23 basic benchmark functions in Sect. 3. In Sect. 4, the proposed algorithms applied on 3 different test systems of ELD and CEED problems. Conclusions would be drawn in Sect. 5 with future work.

Proposed methodology
This section briefs the basics PSO and DE then discussed and the proposed methodology in detail.
2.1 Particle swarm optimization (PSO) (Kennedy and Eberhart 1995) In a D-dimensional optimization space x i;j ¼ x i;1 ; x i;2 ; . . .; x i;D À Á and v i;j ¼ v i;1 ; v i;2 ; . . .; v i;D À Á denote the position and velocity of the i th particle, respectively. During the evolutionary process, each individual constantly adjusts its velocity and position by following the personal best experience pbest i;j ¼ pbest i;1 ; pbest i;2 ; . . .; pbest i;D À Á and the population best experience gbest j ¼ gbest 1 ; gbest 2 ; . . .; gbest D ð Þ . The specific mathematical formulations are presented as below.
v i;j ðt þ 1Þ ¼ wv i;j ðtÞ þ c 1 r 1 pbest i;j ðtÞ À x i;j ðtÞ À Á þ c 2 r 2 gbest j ðtÞ À x i;j ðtÞ À Á ð6Þ where c 1 (cognitive) and c 2 (social) are acceleration coefficients, t stand for the number of iterations, r 1 and r 2 are two randomly selected numbers within the range [0,1] and w is the inertia weight.
2.2 Differential Evolution (DE) (Storn and Price 1997) In DE algorithm, an initial population which includes np individuals are randomly generated in the D-dimensional solution space. In the searching space, each individual represents a candidate solution. Three main operations on DE given as follows.
Mutation: A mutant vector v i;j ðtÞ generated as follows, for every target vector x i;j ðtÞ.
where x r 1 , x r 2 and x r 3 are randomly chosen vectors from the target population as r 1 6 ¼; r 2 6 ¼ r 3 6 ¼ i 2 ½1; np. The control parameter F is a scaling factor, which amplifies the difference vector. Crossover: By combining v i;j ðt þ 1Þ & x i;j ðtÞ, a new vector say trail vector u i;j ðt þ 1Þ is generated as follows.
where j = 1, …,D. randðjÞ 2 a random number lies between 0 and 1. C r is a control factor called crossover rate. Selection: Between the target and trail vector a comparison made on the basis of their fitness function value. After this, a vector which has better fitness function value is selected for the next generation as follows.

New particle swarm optimization (nPSO)
The steps of nPSO for optimization given as follows-1st step-Initialize position (x i;j t ð Þ, randomly) and velocity (v i;j t ð Þ) for each particle. 2nd step-Evaluate fitness function value f ðx i;j t ð ÞÞ. 3rd step-Choose and update pbest (personal best) and gbest (global best). 4th step-For each particle, calculate velocity by Eq. (11) and position by Eq. (12) as follows- In velocity update Eq. (6) a linearly decreased included and restructured by Eq. (11) and in velocity update Eq. (7) a nonliner decreasing factor n t ¼ e 1À tmaxþt tmaxÀt ð Þ involved and updated by Eq. (12), where t max &t-maximum number & current iterations. These presented parameter has the following feature- (i). large and small values of w maintain exploitation and exploration in earlier and latter stages of the nPSO evolution process. (ii). increasing value of c 2 and decreasing value of c 1 helps individually for global and personal best results. (iii). if particles get stuck nearby the global best result then n t may profited for local search in later stage, i.e. it oscillate particles for next iterations to achieve best positions or results. The behaviour of presented factor w; c 1 ; c 2 shown in Fig. 1 and n t shown in Fig. 2.
5th step-Repeat above from step 2 to step 4 until termination condition met.

New differential evolution (nDE)
The steps of nDE for optimization given as follows-1st step-Initialization.
Initially np individuals population x i;j t ð Þ (target vectors) generated randomly in D-dimensional solution space. 2nd step-Calculation.
Calculate the fitness function value of the generated population. 3rd step Evolution.
Genetic operators (novel mutation, novel crossover and selection) are performed in evolution.

(I) Novel mutation.
A mutant vector v i;j ðt þ 1Þ is generated by following formula- where best i;j is best fittest vector and s is convergence factor. During the evolution process of nDE & ihP-SODE (defined below), the presented convergence factor s has the following feature-(i) Early phase-choice of mutation (possibly s ! 1:5) probability is maximum which is helpful for maintained population diversity efficiently. Therefore, nDE and ihPSODE possibly adept the promising regions. (ii) Middle phase-choice of mutation (possibly s\2:5) probability is minor which is benefited for faster convergence. Thus, it accelerate search behaviour nDE and ihPSODE in this stage and helpful for global exploration capability. (iii) Later phase-choice of mutation (possibly s 2:2) probability need to improve. Hence, it provided more of convergence precision and local exploitation facility, speed to nDE and ihPSODE. (iv) Hence, larger s and smaller s is preserve population diversity and helpful for global convergence ability, respectively, in nDE and ihPSODE evolution process. The behaviour of presented factor s shown in Fig. 3 during the evolution process of nDE and ihPSODE.
tmax Àtþ1 ð Þ Â 0:7 is use to generate trial vectoru i;j ðt þ 1Þ. The novel C r has the following feature-(i) When C r small, it maintains diversity of the population and accelerates the convergence speed of the nDE. (ii) When C r is equal to 0.9 then 90% trial individuals can be formed by mutation which increases/preserve global search ability of nDE.
is used for to selection operation.
4th step-Repeat above from step 2 to step 3 until termination condition met.

Innovative hybrid PSODE (ihPSODE)
The steps of nPSO for optimization given as follows-1st step-Initialization.
Initially np individuals population x i;j t ð Þ generated randomly in D-dimensional solution space. 2nd step-Calculation.
Calculate the fitness function value of the generated population. 3rd step-Categorization.
As per fitness function value categorize population. 4th step-Application and elimination. 1. Apply novel particle swarm optimization (nPSO) in best half population and eliminate rest worst population 2. Apply novel differential evolution (nDE) in nPSO generated population 5th step-Combination. Combine population generated by nPSO and nDE. 6th step-Repetition.
The evolution process is repeated till the termination condition is not satisfied.

Validation of presented algorithms
To confirm the performance of the presented algorithms, experiments are investigated on twenty-one basic benchmark functions (Bbf s ) specified as f 1 $ f 7 -unimodal, f 8 $ f 13 -multimodal and f 14 $ f 23 -fixed-dimension. The data of these Bbf s are listed in Table 1. All experiments are conducted on a computer with a Win10 operating system and Intel(R) Core (TM) i5-2350H GHz with 4 GB RAM, and simulated in C language (C-free Standard 4.0). All data analysed or generated throughout this study are involved in this paper. Similarly, availability of data and materials are cited in references of the paper. Additionally, the data of this study are available from the corresponding author upon reasonable request which support the findings. No additional data archiving is required. Furthermore, to handle constraint, a penalty term is added to the objective function. Because of its higher efficiency, the bracket operator penalty (Deb 1995) was chosen for this study. Besides after several tryouts fine-tuning value of R=1 Â e 03 is recommended for presented algorithms.
In each table, whole best values are stained by bold with corresponding methods. Termination criteria, population size and independent run of presented methods are taken same as of the comparative algorithms, for fair comparison. The comparative methods results are taken directly from the original papers.

Case-1
In this case, to evaluate the efficiency of presented algorithms obtained results on 23 Bbfs compared with the nine well-known algorithms GA (Goldberg and Holland 1988), PSO (Kennedy and Eberhart 1995), GSA (Rashedi et al. 2009), TLBO (Rao et al. 2011), GWO (Mirjalili et al. 2014), MVO (Mirjalili et al. 2016), WOA (Mirjalili and Lewis 2016), TSA (Kaur et al. 2020) and RSA (Abualigah et al. 2022). The parameters of proposed and all compared methods are listed in Table 2. The relative simulation results over 30 runs are reported in Table 3, in terms of average objective function values (A vg ) and standard deviation (S td ). Simulation results (in terms A vg. ) of the proposed algorithms (nPSO, nDE and ihPSODE) is noted in Table 3 and illustrated as follows-(i) for unimodal functions (f 1 $ f 7 Þ: ihPSODE is superior than all compared algorithms, nDE is better than all compared algorithms but lesser in case off 4 and f 5 with ihPSODE and nPSO produce better results than all compared algorithms but lesser in case off 4 ; f 5 , f 6 and f 7 with ihPSODE and nDE. (ii) for high-dimensional multimodal functions (f 8 $ f 13 Þ:  -5.12,5.12] 0 -5,5] 0.00030 -5,5] 0.398 -3.86 -3.32 -10.1532 -10.4028 An innovative hybrid algorithm for solving combined economic and emission dispatch problems ihPSODE is superior than all compared algorithms and equal with RSA, WOA and TLBO in case of f 9 and f 11 , nDE is better than all compared algorithms and equal with RSA, WOA and TLBO in case of f 9 and f 11 but lesser in case off 10 , f 12 and f 13 with ihPSODE and nPSO produce equal/marginally better results than all compared algorithms but lesser in case of f 4 ; f 5 , f 6 and f 7 with ihPSODE and nDE. (ii) for fixed-dimensional multimodal functions (f 8 $ f 13 Þ: in this case presented ihPSODE, nDE and nPSO yield equal/marginally results than all compared algorithms. Moreover, presented algorithms (especially ihP-SODE) produce less S td for most of the unimodal, multimodal and fixed-dimension benchmark functions which terms their stability. This analysis shows that presented algorithms performs better in optimizing Bbf s by providing far more competitive results than the other wellknown algorithms.

Case-2
The produced result by proposed algorithms on 23 Bbfs is compared with traditional algorithms (HHO (Heidari et al. 2019) & EO (Faramarzi et al. 2019)), PSO variants (HEPSO (Mahmoodabadi et al. 2014) & RPSOLF (Yan et al. 2017)), DE variants (JADE (Zhang and Sanderson 2009) & SHADE (Tanabe and Fukunaga 2013)) and hybrid variants (FAPSO (Xia et al. 2018), & PSOSCALF (Chegini et al. 2018)). The factors (parameters) of proposed and all of the above-mentioned compared algorithms listed in Table 4. Further, in terms of average value (A vg ), standard deviation (S td ) and ranking (R ank ) of the objective function values, the comparative results shown in Table 5 over 30 independent runs. It should have been noted that the average objective function values of the suggested algorithms (ihPSODE, nPSO and nDE) are better or similar to the equated standard algorithms, PSO alternatives, DE alternatives, and hybrid variants, as shown in Table 5. The presented algorithms produce fewer S td for maximum of the Bbfs which terms their strength. Also, all methods are ranked individually (as best by1, next performer by 2, etc.) grounded on average result values in Table 5. From Table 5, it is decided that ihPSODE, nDE & nPSO ranked as 1 st , 2 nd & 4 th successively. As well, overall and average rank of    An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12645 presented versus others methods are declared in Table 5. Then, it can be say that the proposed algorithms outperform others by rankings. Furthermore, the supremacy of the proposed algorithms over other algorithms is statistically analysed by Wilcoxon signed-rank (WSR) test and one-tailed t-test at 5% significance level. The details of these tests can be found in . WSR and t-test results on Bbf s are reported in Table 6. In most of consequence, it can be perceived that from this table presented methods has ' ? ' or '&' sign in case WSR test which indicate better or equally performances and 'a ? ' or 'a' sign in case of t -test which signify highly or significantly better than other, respectively. Moreover, p-values of the presented algorithms listed in Table 6 which is lower than others and indicating that their reliability to produce best results on the majority of runs.

Convergence, feasibility and complexity analysis
Mathematical convergence analysis, graphical convergence analysis, feasibility or efficiency analysis and complexity analysis of the presented algorithms are discussed in this section.

Mathematical convergence analysis
3.2.1.1 nPSO (novel particle swarm optimization) As per the source on (van den Bergh 2006) for the convergence analysis of nPSO particle's trajectory discussed as follows. From Eq. (11) and (12) x i;j t þ 1 ð Þ written as- By Eq. (7) v i;j t ð Þ can be written as- Then equation (14) can be rewritten as- where w Ã ¼ n t x Ã i:j t ð Þ ¼ pbest t i;j and x j t ð Þ ¼ gbest j Table 3 (continued) Then the homogeneous matrix of equation (16) can be written as- Thus, the Eigen polynomial of the matrix (17) can be obtained by the following equation- The explicit form of the recurrence relation (17) is then given as- where k 1 , k 2 and k 3 are constants determined by the initial conditions of the system. From Eq. (19) and (15) v i;j t ð Þ can be written as- where So nPSO is convergent if and only if max kbk; kck ð Þ 1, i.e. if nPSO is convergent, then the particle's velocity decreases to zero continuously or holds just as its initial value during the whole searching process.

nDE (novel differential evolution)
Criteria-if X t ð Þ; t ¼ 1; 2; . . ., be a population sequence generated by nDE to solve the optimization problem. Then, nDE converges to the global optimum, if       Table 5 (continued) Proof Let t th k target population Xðt k Þ of nDE, 9 at least one vector (individual)x, which relates to the trial vector y by a crossover scheme, s.t. p y 2 S Ã d È É ! uðt k Þ [ 0 and the series P 1 k¼1 uðt k Þ diverges then nDE converges to the optimal solution set S Ã d . where t k ; ðk ¼ 1; 2; . . .Þ signifies any subsequence of natural number set, p y 2 S Ã d È É signifies the probability that y belongs to the optimal solution set S Ã d , and uðt k Þ is a positive small value which may change as t k .
In nDE, each target vector resembles to a trial vector by its crossover scheme. The probability that 8 vector of the t th k trial population Yðt k Þ 6 2 S Ã d can be defined as- So, the probability where all vectors of every trial population in previous ðt k À 1Þ iterations 6 2 S Ã d given as follows.

Fig. 4 a-h Convergence of different algorithms
An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12651 From the property of the infinite product (Chen et al. 2000

ihPSODE (innovative hybrid PSODE)
In each iteration ihPSODE the new population is better than the previous one. So, for random mapping u ¼ u 2 Ãu 1 ð Þ: l Â S ! S, 9 a non-negative real valued random variable 0 K w ð Þ\1 hold following principle (where l is the non-empty abstract set and w is a its basic event).
best ; X t best Þg l, then c l 0 ð Þ ¼ 1 Therefore, the mapping formed by ihPSODE, u : l Â S ! S is a random contraction operator. Hence, as per Random Contraction Mapping theorem (Lu 1990) the random mapping u is a random contraction operator and for u w ð Þ there exists a unique random fixed point. Therefore, the convergence principle of ihPSODE can be derived, which means that ihPSODE is asymptotically convergent.

Graphical convergence analysis
The convergence curves of comparative and presented algorithms on eight (f 1 x ð Þ, f 5 x ð Þ, f 6 x ð Þ, f 7 x ð Þ, f 8 x ð Þ, f 9 x ð Þ, f 10 x ð Þ and f 11 x ð Þ) typical 30-D Bbf s are plotted and presented separately in Fig. 4a-h. From this figure, it can be seen that almost all of the considered benchmark functions, either unimodal or multimodal, would be quickly optimized by the presented algorithms (nPSO, nDE, and ihPSODE).

Feasibility and/or efficiency analysis
Likewise, an effort is made and demonstrated in Fig. 5, to catch global optimal solution entire of 690 runs (in 30 population size with 30 runs for each Bbf s ). It states that the presented methods provide the best optimal solutions. Similarly, the computational time of the presented and equated methods on each Bbf s is calculated and illustrated in Fig. 6 for Case-1 (well-known and presented algorithms) and Fig 7 via box plots for Case-2 (compared and presented algorithms). These figure shows that the presented methods take less time to attain the finest value for the entire set of Bbf s .

Fig. 4 continued
In general, it can be concluded that the performance of presented methods is superior to or at least equal to other intelligent optimization algorithms (classical, PSO, DE and Hybrid variants) on most test functions. In conclusion, presented algorithms (nPSO, nDE, and ihPSODE) can be considered as an effective and efficient method.

Complexity analysis
In this section, complexity analysis of the presented methods is specified as below.
(i) Time complexity. Presented ihPSODE has the following time complexity (according to the steps). The space complexity is the maximum volume of space which is used by presented ihPSODE algorithm. Thus, the total space complexity of proposed ihPSODE algorithm is O(max( np, np; np; np 2 4 ,np 2 Þ Â D) = O(np 2 Â D).

Presented algorithms for CEED problem
To further investigate the feasibility of presented algorithms (nPSO, nDE, and ihPSODE) in real-life problems, two large scale power engineering optimization problem (ELD and CEED) are considered here. These problem include 3 test systems (3, 6 and 40-unit test system) of ELD and 3 test systems (3, 10 and 40-unit test system) of CEED problem. The obtained best solutions are utilized to evaluate the feasibility of different algorithms.

Unit test systems
Problem Unit Test Systems (U TS ) Description ELD U TS -1 (3-unit test system)  It involves load demand of 300 MW U TS -2 (6-unit test system)  It involves 700 MW total load demands U TS -3 (40-unit test system)  It consists valve loading point effect and involves load demand of 10500 MW CEED U TS -4 (3-unit test system) (Devi and Krishna 2008) It considers emission impact and involves 400 MW and 500 MW load demand as well U TS -5 (10-unit test system)  It considers valve point effects and involves 2000 MW total load demand U TS -6 (40-unit test system)  It consists of emission functions, non-smooth fuel cost and involves 10500 MW total load demand

Computational Steps of ihPSODE for CEED problem
The steps of ihPSODE for solving CEED problem are given as below: Fig. 7 Average elapsed time of compared and presented algorithms in Case-2 1st step: Read the P D (Power Demand). 2nd step: Compute h (price penalty factor). 3rd step: t (iteration) = 1. 4th step: Generate initial population vector of real power generator (based on prohibited zone and ramp limit constraints). 5th step: Evaluate the fitness function using Eq. (5) 6th step: Sort the population (as per fitness function value). 7th step: Apply new particle swarm optimization, i.e. nPSO (in best half population). 8th step: In population generated by nPSO, apply new differential evolution, i.e. nDE. 9th step: Combine the population created by nDE and nPSO. 10th step: The evolution process is repeated till the stopping condition is not satisfied. 11th step: Print the results (generator schedule, minimized operating cost, corresponding fuel cost, and emission output). An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12655 An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12657 0.122 According to the reported cost results, the proposed nPSO, nDE, and ihPSODE algorithms have the lowest fuel cost and emission when compared to other compared algorithms for all unit test systems. Furthermore, CPU average times for each unit test system are noted in the associated tables, demonstrating that the proposed algorithms produce better solutions in less time than others. As a result, the proposed algorithms outperform and outlast other compared algorithms in terms of reducing total cost in the shortest amount of time. This indicates that the presented algorithm has higher reliability/robustness, stability and convergence when compared to other algorithms.

Results and discussion
The convergence curves of presented and other algorithms are demonstrated (in terms of total cost versus iterations) in Fig. 8a-f. This figure shows that presented algorithms (nPSO, nDE, and ihPSODE) have better convergence performance. Moreover, fuel cost variations for all test systems presented in Fig. 9a-g. It shows and confirmed effectiveness of the proposed nPSO, nDE and ihP-SODE for decreasing the fuel cost. Moreover, this figure determine that the power of the proposed methods in achieving minimum fuel cost related to others. Hence, proposed algorithms are economically capable.
On the whole, it can be stating that (from the all above result investigation) presented algorithms (nPSO, nDE and ihPSODE) are performing better and/or similar with others.

2.22
An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12659 Ultimately, among three proposed algorithms ihPSODE, i.e. hybrid algorithm of nPSO and nDE, have superior competency.

Conclusion with future perspectives
In this paper, to promote the performance of PSO and DE algorithm, a novel PSO (namely nPSO), novel DE (called nDE) and their innovative hybrid (titled ihPSODE) is presented for solving CEED (combined economic and emission dispatch) problems. Presented nPSO has a new acceleration coefficient, inertia weight and position improve equation (to alleviate the stagnation) as well as nDE has a new mutation and crossover scheme (to prevent untimely convergence). After population evaluation (in ihPSODE) best half individuals has been recognized and apply nPSO (which enhanced local and global search capacity) then nDE (which ensures bring solutions with higher quality), in each iteration process. In addition, because of suitable implementation of nPSO and nDE, particle can learn from best individuals of each problem as well as from the globally individuals in ihPSODE. Altogether, quality of 'memorizing (by nPSO)' and 'diversity maintaining (by nDE)' brands ihPSODE more robust. Likewise, related novel presented control factors of nDE and nPSO makes extra support for the ihPSODE success. The presented algorithms (nPSO, nDE and ihPSODE) have been tested over 23 unconstrained benchmark functions then applied to solve two large-scale power engineering optimization problem namely economic load dispatch (ELD) and combined economic emission dispatch (CEED) problem. These problem include 3 test systems (3, 6 and 40-unit test system) of ELD and 3 test systems (3, 10 and 40-unit test system) of CEED problem. The performance of presented methods equated with the well-known and other advanced methods.
The simulation results prove that presented algorithms have more efficiency than equated algorithms in case of unconstrained benchmark functions. Moreover, presented algorithms are successfully used to solve ELD and CEED power system engineering optimization problems. The simulation results reveal that presented algorithms can attain better solutions than other compared algorithms in case of power system engineering optimization problems. Therefore, presented algorithms are economically competent. All in all, it can be summarized that the proposed algorithms (nPSO, nDE and ihPSODE) can be seen as an effective algorithm to solve power system engineering

3.25
An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12661 optimization problems. Lastly, among three proposed algorithms ihPSODE have superior competency. Furthermore, the presented algorithms have higher time complexity than some PSO, DE and hybrid variants. The matrix operation evaluation is the primary cause of the presented algorithms' time-consuming nature. This operation is repeated for each individual on each iteration, which increases the algorithm's running time in some extent. Besides, the presented algorithms may not be appropriate for all complex optimization problems.
Some novel parameters will be designed for the presented nDE, nPSO, and ihPSODE as part of our future work for finding more precise solutions and falling time complexity. Finally, this paper is expected to devote in a fruitful analysis, i.e. complete mathematical convergence analysis of the presented algorithm which may done in the coming paper with inspecting how to advance the strength for multifaceted optimization problems. Data availability Authors declare that all data analysed or generated throughout this study are involved in this paper. Similarly, availability of data is cited in references of the paper. Also, the data of this study are available from the corresponding author upon reasonable request which support the findings. No additional data archiving is required.

Declarations
Conflict of interest There are no conflicts of interest. An innovative hybrid algorithm for solving combined economic and emission dispatch problems 12663