
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12959-3
10.1016/j.heliyon.2024.e36928
e36928
Research Article
Application on power system economic dispatch of marine predator algorithm improved by asymmetric information exchange
Yang Cheng a
Zheng Xiaoliang b
Wang Jiwen a
Zhang Wei a
Liu Ludeng a
Ma Bin a
Fan Yuanzhu a
Tao Qiong b
Wang Hu whu_aust@163.com
b⁎
a State Grid Anhui Electric Power Co., Ltd, Hefei, 230000, PR China
b School of Electrical and Information Engineering, Anhui University of Science and Technology, Huainan, 232000, PR China
⁎ Corresponding author. whu_aust@163.com
26 8 2024
15 9 2024
26 8 2024
10 17 e3692831 3 2024
28 7 2024
24 8 2024
© 2024 The Authors. Published by Elsevier Ltd.
2024

https://creativecommons.org/licenses/by-nc/4.0/ This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).
The solution to the economic dispatch (ED) problem for power systems allows the power sector to reduce operating costs. However, the ED problem is a complex nonlinear and nonconvex optimization problem whose solution requires powerful algorithms. We propose a new version of the Marine Predator Algorithm (MPA), called IMPA, for solving complex ED problems. The algorithm introduces an asymmetric information exchange (AIE) mechanism, which not only accelerates to escape of local optima but also enriches the diversity of search. In this work, 12 benchmark functions were used to test the performance of the proposed algorithm IMPA. Then, the IMPA was used to solve the ED engineering problem of power system containing of 6, 13, 40, and complex 140 units. The minimum and average costs searched by IMPA are 1657962.7265$/h and 1657962.7265$/h, and they are much lower than the results of the MPA and NMPA, which means that our proposed improved IMPA improves the performance of MPA for solving the economic dispatch problem of large-scale power systems. The results show that the solutions obtained by IMPA are more competitive than those of MPA and NMPA, which provides an additional solution for cost reduction of the power system.

Keywords

Economic dispatch
Improved marine predator algorithm
Asymmetric information exchange
Power systems
==== Body
pmc1 Introduction

In the power system, some problems need to be better solved and optimization techniques are needed, such as reactive power dispatch, optimal power flow, economic dispatch, and emission dispatch.

The power systems economic dispatch (ED) is an essential work for the power industry[[1], [2], [3]]. It aims to rationally schedule the output power of each unit under diverse constraints to achieve the least fuel consumption and a more reliable power supply. To simplify the ED problem, the objective function is approximated as a smooth quadratic function with only one optimal point. In this case, highly accurate solutions can be obtained by some classical mathematical methods like Lagrangian relaxation [4], linear programming [5,6], branch-bound algorithm [7], gradient method [8], quadratic programming [9], lambda iteration procedure [10], etc. However, in fact, as a reason for the existence of valve point effect (VPE), ramp rate limits (RRL), and prohibited operating zones (POZ) characteristics [11,12], the cost function turns out to be a nonconvex, nonlinear, discontinuous, and complicated one which poses a big challenge to resolving [13]. Nevertheless, owing to some limitations of conventional mathematical approaches, it is hard to find high-quality solutions to this complex problem in a reasonable time.

In recent years, with the development of advanced renewable technologies such as wind energy, solar energy, and storage systems, the complexity of power systems has further increased [14,15]. The integration of these technologies presents new challenges and opportunities for ED problems. For example, the intermittency and uncertainty of wind and solar energy increase the difficulty of scheduling but also offer the potential to reduce carbon emissions and operational costs. Storage systems play a crucial role in the scheduling process by smoothing loads and storing excess energy. Additionally, the development of smart grid technology further enhances the flexibility and responsiveness of power systems, making real-time adjustments and optimization possible.

Heuristic algorithms have shown great potential in solving complex engineering problems, with advantages such as better reliability, stronger robustness, less information requirement, outstanding global search capability, and no need for differentiable continuous objective functions [13,16]. Meanwhile, many high-performance algorithms have been applied to solve complex ED problems. In Refs. [[17], [18], [19], [20], [21]], researchers introduced many improved particles swarm optimization (PSO) algorithms for solving the ED problem. In 2010, Santos Coelho et al. [22] proposed a hybrid technology that combines cultural algorithm with self-organizing migrating strategy (CSOMA) to improve search efficiency for economic dispatch optimization. Niu et al. [23]hybridized arithmetic crossover and harmony search algorithms to form a new method, called ACHS. The opposition-based learning strategy is employed in ACHS for exploring more diversity solutions. In 2019, the artificial collaborative search (ACS) algorithm based on the co-evolutionary process was reported in Ref. [24]. The algorithm involves a single parameter setting, which relies less dependent on the original value setting and attains the optimal solution with a high probability. Inspired by the grey wolf optimization (GWO) and to obtain better exploitation ability and diverse search space, Singh & Dhillon [16] devised an ameliorated GWO (AGWO). In AGWO, the authors apply multiple optimization strategies, such as global random search, local exploitation search, and opposition-based learning, which exhibit high performance in complex ED problems. In 2021, Farhan Tabassum et al. [25] proposed a hybrid method combining evolutionary algorithm with the improved Hooke-Jeeves method (ESAHJ), which has a fast convergence rate in solving ED problems with equality constraints. Although the ED problem has been solved, there are still potential better solutions to be explored. For example, in 2022, Luo &Yu [26] proposed an improved cuckoo search algorithm (RLMCS) based on reinforcement learning (RL). To further improve the efficiency of RLMCS, three other mature techniques are introduced, including Gaussian random walk, quasi-opposition learning, and adaptive adjusting parameters.

The marine predator algorithm (MPA), which is a novel meta-heuristic based on swarm intelligence, was first introduced by Faramarzi et al. [27,28] in 2020. Motivated by the foraging strategies of creatures in ocean systems, the algorithm hunts for the best result by switching Lévy flight and Brownian motion in different hunting stages. Owing to the advantages of fewer control parameters, simpler calculation, more straightforward implementation, and stronger search ability, it has been applied successfully in many scenarios [[29], [30], [31], [32]], for example, forecasting confirmed cases of COVID-19 [33], parameters optimization of triple-diode photovoltaic models [34], photovoltaic array rearrangement [35], 0–1 knapsack problem [36], PV parameter estimation [37]. etc.

In order to boost the efficiency of MPA, various attempts have been conducted to modify the MPA algorithm. Sadiq et al. [28,38] introduced a nonlinear function into MPA, called NMPA, to balance exploration and exploitation to realize a more flexible transition. Compared with MPA and other methods, the accuracy and stability of NMPA in the benchmark function are obviously improved. No free lunch theorem [39], a famous theory, indicates that there is no algorithm that can solve all optimization problems. Hence, for the sake of allowing MPA to solve more engineering problems, many researchers have explored the improvement of it. Al-qaness et al. [40] developed an improved MPA based on mutation operators, recorded as MPAmu to prevent its premature convergence on local optima in the wind power forecast. Hu et al. [41] improved MPA by utilizing neighborhood learning and adaptive population size strategies named NMPA for the purpose of searching for more promising solutions in the application of developable surface modeling (Table 1).Table 1 References in introduction.

Table 1Overview	Research Method	Literature Overview	
Refs. [[17], [18], [19], [20], [21]]	improved particles swarm optimization (PSO) algorithms	For solving the ED problem.	
Santos Coelho et al. [22]	hybrid technology	Combines cultural algorithm with self-organizing migrating strategy (CSOMA) to improve search efficiency for economic dispatch optimization.	
Niu et al. [23]	hybridized arithmetic crossover and harmony search algorithms to form a new method called ACHS	The opposition-based learning strategy is employed in ACHS for exploring more diversity solutions.	
Singh & Dhillon [16]	devised an ameliorated GWO (AGWO).	The authors apply multiple optimization strategies, such as global random search, local exploitation search, and opposition-based learning, which exhibit high performance in complex ED problems.	
Farhan Tabassum et al. [25]	a hybrid method combining evolutionary algorithm with the improved Hooke-Jeeves method (ESAHJ)	Has a fast convergence rate in solving ED problems with equality constraints.	
Luo &Yu [26]	proposed an improved cuckoo search algorithm (RLMCS) based on reinforcement learning (RL)	To further improve the efficiency of RLMCS, three other mature techniques are introduced, including Gaussian random walk, quasi-opposition learning, and adaptive adjusting parameters.	
Faramarzi et al. [27,28]	The marine predator algorithm (MPA)	Motivated by the foraging strategies of creatures in ocean systems, the algorithm hunts for the best result by switching Lévy flight and Brownian motion in different hunting stages.	
Sadiq et al. [28,38]	a nonlinear function into MPA, called NMPA	To balance exploration and exploitation to realize a more flexible transition. Compared with MPA and other methods, the accuracy and stability of NMPA in the benchmark function are obviously improved.	
No free lunch theorem [39]	a famous theory	Indicates that there is no algorithm that can solve all optimization problems.	
Al-qaness et al. [40]	developed an improved MPA based on mutation operators, recorded as MPAmu	To prevent its premature convergence on local optima in the wind power forecast.	
Hu et al. [41]	mproved MPA by utilizing neighborhood learning and adaptive population size strategies named NMPA	The purpose of searching for more promising solutions in the application of developable surface modeling.	

To address the gaps in current research more clearly, this study proposes an enhanced approach to the Economic Dispatch (ED) problem focusing on emissions, inspired by Shaheen et al. [31], who introduced randomness into three independent optimization processes to broaden the search space. To achieve superior solutions for complex and challenging ED problems, this paper recommends an Improved Marine Predators Algorithm (IMPA). Unlike most approaches that modify optimization strategies or hybridize algorithms, our study primarily investigates enhancement techniques by analyzing the trajectory characteristics of search agents. Specifically, we introduce an Asymmetric Information Exchange (AIE) strategy to accelerate convergence and enhance the algorithm's capability to avoid local optima. The IMPA is evaluated using classical benchmark functions and applied to solve intricate power system dispatch problems. Comparative evaluations are conducted with the Marine Predators Algorithm (MPA) and Nonlinear Marine Predators Algorithm (NMPA) for validation and performance assessment.

The following parts are arranged as below: In part 2, the description of the formula of ED problems is briefly introduced. The MPA and IMPA algorithms are reported in section 3. The fourth part delivers a comprehensive analysis of benchmark functions. In section five, simulation outcomes for ED problems are discussed. Section 6 summarizes the research of this work.

2 Mathematical model of the ED problem

Addressing the ED problem can do a favor for the thermal power plant to reasonably dispatch the power output of generating units for lower operating costs under fulfilling different constraints like power load demand, output power limits, prohibited operating zones, transmission loss (TL), ramp rate limits, simultaneously, and the valve point effect. The cost function means the objective function of the ED problem. According to whether or not consider the VPE [22,42], it can be divided into two forms, one is the simplified cost function, and the other is a non-smooth, non-convex, and complex one [42]. The simplified cost function is a quadratic function [17,18]. In practice, the objective function considering VPE is regarded as a quadratic polynomial modified by a sine function. Therefore, this is a multimodal function with more than one local minimum resulting in a more complicated optimization problem. The mathematical description is shown in Eq. (1) [43,44]:(1) mincost=∑k=1nCk(Gk)=akGk2+bkGk+ck+|eksin(fk(Gkmin−Gk))|

where Gk denotes the generated power of the unit k, Gkmin is the allowed minimum generated power of the kth generator, and ak, bk , ck, ek, and fk are a set of coefficients used to describe the cost characteristics of the generator k.

As we know, the solution must be a feasible one, so some constraints ought to be considered. Firstly, the power balance should be met by all generators, and can be characterized by Eq. (2)[42,45]:(2) ∑k=1nGk−Pd=0

where Pd is the sum of the load power demand. However, if the power losses on the transmission line are not ignored, Eq. (2) can be modified as Eq. (3) [16,19]:(3) ∑k=1nGk−Pd−GL=0

where GL stands for the total transmission losses which can be given by Refs. [22,46]:(4) GL=∑k=1n∑m=1nGkBkmGm+∑k=1nGkB0k+B00

where Bkm, B0k, and B00 are the transmission loss coefficients. Considering that the output power of the unit is limited, it should lie between the upper and lower boundaries and is expressed in formula (5) [10,25,47,48]:(5) Gkmin≤Gk≤Gkmax

where Gkmin and Gkmax are the minimum and maximum generation power of the kth unit, respectively. However, in fact, there exists a concept of forbidden operation zones to describe the phenomenon that several generators in some areas cannot work due to the limitation of physical conditions. The mathematical expression can be written as equation (6) [49,50]:(6) Gk={Gkmin≤Gk≤Gk,1L(k=1,2,3,...,n)Gk,t−1U≤Gk≤Gk,tL(k=1,2,3,...,n,t=2,3,...,Hzk)Gk,HzkU≤Gk≤Gkmax(k=1,2,3,...,n)

where Gk,tU and Gk,tL are responsible for the upper and lower limits of the tth POZ of the kth unit, respectively. Hzk denotes the number of prohibited operation zones of unit k.

What's more, in the real world, to protect boilers or combustion equipment from suffering an excess of pressure [42], ramp rate limits are implemented to limit the change of the output power of each unit online. But in the static ED problem, only the restriction at the initial time is considered. So the mathematical expression is described in formula (7):(7) MAX(Gkmin,Gk,0−DRk)≤Gk≤MIN(Gkmax,Gk,0+URk)(k=1,2,3,...,n)

where DRk and URk stand for down and up ramp rate limits of the kth unit, Gk,0 is the output power of the kth generator at the initial moment.

3 Improved marine predator algorithm

3.1 MPA algorithm

Some concepts about Marine Predator Algorithm (MPA) need to be described before the proposed improved algorithm IMPA. In 2020, this new method first appeared in a paper written by Faramarzi et al. [27]. Similar to most other optimization algorithms, before starting iterating, an initial candidate solution distributed randomly is created in the search area. The generation process can be described by formula (8):(8) Z=Zmin+random[0,1]×(Zmin−Zmax)

where Zmin and Zmax mean the minima and maxima of variables and random[0,1] represents a uniformly generated number that can take values from 0 to 1.

In MPA, based on comparison results of fitness values of prey individuals, the one with optimal fitness is selected to build the Elite matrix, the shape of which is n × d and can be expressed as formula (9):(9) Elite=[Z1,1OZ1,2O...Z1,dOZ2,1OZ2,2O...Z2,dO⋮⋮⋮⋮Zn,1OZn,2O...Zn,dO]n×dPrey=[Z1,1Z1,2...Z1,dZ2,1Z2,2...Z2,d⋮⋮⋮⋮Zn,1Zn,2...Zn,d]n×d

where ZO is the prey member with the optimal fitness and replicates n times to create the Elite matrix. n denotes search individual size and d stands for the dimension number of variables. Zi,j indicates the ith prey position information at the jth dimension. The Prey matrix is another matrix, which contributes to updating the information of the Elite matrix.

In MPA, the whole optimization process is divided into three parts on average according to different ve values. The three steps are described below:

Step 1 High-speed ratio: This occurs during the early stage of iteration. The mathematical model is depicted as formula 10:(10) {vi→=RB→⊗(Elite→−RB→⊗P→rey−i)i=1,2，⋯,nP→rey−i=P→rey−i+Q·R→⊗vi→，iter<Iter3

where RB→ stands for a matrix consisting of a series of data generated by a normal distribution function that simulates Brownian motion. Q equals 0.5 here. R→ indicates a random variable from 0 to 1. iter and Iter are the current and maximum numbers of iterations.

Step 2 Unit speed ratio: At this stage, the exploration mode gradually decreases while the exploitation mode gradually increases, and both of them are indispensable. The first part of the population is responsible for hunting for the global optimal solution, and the other one is in charge of local exploration. The model for this stage is shown in formula (11).(11) {{vi→=RL→⊗(E→lite−RL→⊗P→rey−i)P→rey−i=P→rey−i+Q·R→⊗vi→,i=1,⋯,n/2{vi→=RB→⊗(RB→⊗E→lite−P→rey−i)P→rey−i=P→rey−i+Q·(1−iter/Iter)2×iterIter⊗vi→,i=n/2+1,⋯,n

where RL→ is a random variable produced by Lévy distribution and used to multiply the Prey.

Step 3 Low-speed ratio: In the last process of optimization, the predator does local exploitation moving in Lévy manner. This process is shown formula 12:(12) {vi→=RL→⊗(RL→⊗E→lite−P→rey−i)i=1,2，⋯,nP→rey−i=E→lite+Q·CF⊗vi→,iter>2×Iter3

3.2 Improved marine predator algorithm

Although the MPA technique demonstrates great optimization capabilities, this method, like many other meta-heuristic algorithms, still suffers from the deficiency of premature convergence. Therefore, a new version of MPA based on an asymmetric information exchange mechanism is suggested by us. In the Eddy formation and FADs' effect stage, the prey performs a long jump to find another fish distribution, which is equivalent to the behavior of MPA preventing the local solution stagnation. By experimental observations, the authors found that some search agents of MPA often randomly walk outside the feasible region after the FADs' effect. Therefore, we propose an asymmetric information exchange (AIE) mechanism, As shown in Fig. 1.Fig. 1 The flow chart of asymmetric information exchange mechanism.

Fig. 1

Considering that the position information of the top predator in some dimensions is excellent, while the position information of some other dimensions may be bad, in this strategy, we randomly select several dimensions and stochastically arrange these dimensions information of prey and top predator. After that, preys are allowed to move in two ways with β probability, that is, β × search_num agents move in the direction of the optimal solution, and the rest populations update their positions in the way that the top predator moves towards themselves. In this study, the authors do not do further research on this parameter, so it is set to the simplest value of 0.5. Finally, the search agent obtained the best fitness value is picked out to replace the top predator. This information exchange method accelerates the speed of searching for the best solution and makes up for the reduction of search efficiency caused by some populations flying out of the boundaries.

4 Experimental analysis of IMPA on benchmark functions

For evaluation of the performance of IMPA, 12 classical test functions are employed and derived from Ref. [51]. In general, these functions are all utilized to search for minima. TF1-TF7 is a unimodal function with a single optimum and is mainly applied to benchmark the exploitation ability of the approach. By contrast, TF8-TF12 relates to a multimodal function containing multiple extreme points, of which only one is the global optimal solution. In order to search for it, the abilities of an optimizer to get rid of the local solution and powerful global exploration are necessary.

Fig. 2 shows the optimization process of IMPA on the benchmark function. The first column image clearly shows the two-dimensional structure of the mathematical function, where you can see its topology. The second column relates to the search history, which tracks agents’ positions in the optimization process. Like the MPA algorithm, IMPA applies a collective search mode leading to aggregating near the optimal point of the unimodal function and being scattered at local optima in multimodal functions.Fig. 2 Search history and convergence curve of all search agents.

Fig. 2

The third column represents the convergence curve, which reflects the change in the elite's fitness over the iteration process. It can be seen from the graph that the convergence curve gradually sustains stability with the laps of iterations, and there is almost no change in the last optimization phase, indicating that the global solution or near-global solution has been searched. It can be observed that in the initial steps of optimization there is an abrupt change and in the iteration process, the change is gradually reduced. This behavior can ensure that the algorithm eventually converges to a point and then performs the local search in the feasible region [52]. Therefore, through the above analysis, the IMPA algorithm using optimization measure still converges and behaves an advancement.

Table 2 shows the optimization results of IMPA on the benchmark function. In this experiment, excellent algorithms are used for comparative analysis, such as DE [38], and SSA [27]. Meanwhile, we also study MPA [27]and NMPA [38], in which NMPA is another improved algorithm of standard MPA.Table 2 Comparison of results of different methods.

Table 2Algorit-hms	DE [38]	SSA [27]	MPA [27]	NMPA [38]	IMPA	
average	standard deviation	average	standard deviation	average	standard deviation	average	standard deviation	average	standard deviation	
TF1	1.3785	0.4674	0.0037	0.00974	3.27E-21	4.61E-21	0	0	2.19E-28	5.21E-28	
TF2	0.27478	0.04579	5.0487	2.013	1.57E-12	1.42E-12	0	0	2.42E-16	1.30E-16	
TF3	1.09E+05	1.14E+04	4343.27	2136.39	0.0864	0.1444	0	0	0.0678	0.1237	
TF4	46.155	3.9548	15.055	3.195	2.60E-08	9.25E-09	0	0	1.38E-10	6.72E-11	
TF5	1273.3	451.29	434.43	457.7	46.049	0.4219	4.4152	0.7331	44.6915	0.39727	
TF6	1.3188	0.4229	0.0021	0.003	0.398	0.1914	0.01862	0.4885	0.0044	0.01989	
TF7	0.1907	0.0336	0.2807	0.0911	0.0018	0.001	9.57E-05	7.70E-05	0.00121	6.74E-04	
TF8	−12105	556.8	−12232.6	1063	−13594.1	811.3	−13603	798.16	−14077.57	686.0316	
TF9	24.736	10.783	78.79	25.18	0	0	0	0	0	0	
TF10	0.4544	0.1422	3.479	0.8281	9.69E-12	6.13E-13	8.88E-16	0	8.59E-15	1.89E-15	
TF11	0.8324	0.09265	0.0905	0.0407	0	0	0	0	0	0	
TF12	1.4506	0.5896	8.541	2.556	0.0085	0.0052	0.03532	0.01387	5.25E-05	2.26E-04	

Since the unimodal function TF1-TF7 has only one local optimum, the results of these seven functions correspond to the exploitation capacity of methods. TF8-TF12 has many local minima and when the dimension increases, the number of local optima of these multimodal functions rises exponentially resulting in them being the most complex of the 12 functions. Therefore, high-performance optimization algorithms are needed, which leads to an obvious performance difference between algorithms. We benchmark these complicated functions to challenge the global exploration property of a technique. As the table shows, IMPA provides the best global optimal solution on some functions and shows strong competition in standard deviation as well. Obviously, IMPA has superior performance over MPA, which comes from the fact that search agents could explore smaller solutions after asymmetric information exchange. Hence, IMPA is considered a competitive technique in terms of exploration property.

5 Experimental and analysis

In this section, four different sizes of test systems are studied. The 6 units system is relatively simple, and 13, 40, and 140 units are systems that take into account VPE. In addition, a challenging 140 units with VPE, RRL, and POZ is investigated. To boost the search efficiency of IMPA in ED problems, in the first optimization phase, we remove the Brown coefficient in front of Prey-i and simplify the step size formula. For the complex ED problem, before conducting the power system simulation experiment, we briefly introduce the main implementation steps of IMPA and show the following.Step 1 Input the values of a series of variables and parameters that need to be determined by the author in the process of IMPA optimizing the ED problem. (search_num, Iter, dim, Pd, Gmin, Gmax, and cost coefficients.)

Step 2 According to Eq. (9), build the initial Prey matrix. Each prey member in Eq. (9) must satisfy the power load demand and be within the boundaries.

Step 3 Calculate the minimum cost value that each prey member searches for in the Prey matrix according to Eq. (1), and pick out the one with the smallest cost to construct the Elite matrix. And perform marine memory preservation.

Step 4 Update the position of the individual.

Step 5 Complete marine memory preservation and FADs effect.

Step 6 Perform asymmetric information exchange based on Fig. 1. If the fitness of a prey member becomes better than that of the elite, replace the top predator, including its fitness value and position, otherwise, the information in Step 5 is maintained.

Step 7 Update iter. If the maximum number of iterations is not reached, go to Step 3, otherwise, end the iter, and output the information of the top predator.

5.1 Case 1: 6 units test system

This is a small system and only considers generation limits and load demand. Ref. [53] gives the generator parameters supplying three different power load demands, 600 MW, 800 MW, and 1000 MW. Fig. 3 is the cost curves of MPA, NMPA, and IMPA.Fig. 3 Comparison of cost curves of different three algorithms. (a) 600 MW; (b) 800 MW; (c) 1000 MW.

Fig. 3

From Fig. 3, it is clear that IMPA has the lowest cost curve, and almost finds the best optimal solution or nearest to the optimal solution after about 20 iterations. By contrast, MPA and NMPA converge slower, so they are still searching in the third stage of the optimization process. In this case, IMPA attains the lowest fuel cost value at the end of the iteration with a small variation. Table 3 show the comparison of MPA, NMPA, and IMPA for the demands of 600, 800,1000 MW, respectively.Table 3 The comparison of cost minimization for 6 units.

Table 3Unit	Pd = 600WM	Pd = 800WM	Pd = 1000WM	
MPA	NMPA	IMPA	MPA	NMPA	IMPA	MPA	NMPA	IMPA	
1	21.181	21.256	21.181	28.744	28.824	28.744	36.083	36.012	36.082	
2	10.000	10.000	10.000	10.000	10.000	10.000	15.968	15.982	15.968	
3	82.145	82.230	82.145	123.341	123.315	123.340	163.306	163.457	163.306	
4	94.227	94.195	94.227	126.719	126.708	126.719	158.241	158.250	158.242	
5	205.500	205.445	205.500	260.166	260.021	260.167	313.202	313.366	313.203	
6	186.948	186.874	186.948	251.029	251.133	251.029	313.199	312.935	313.199	
Min. cost($/h)	31446.45	31446.06	31446.45	40677.17	40677.17	40677.17	50365.29	50365.3	50365.29	
Ave.
cost ($/h)	31446.45	31446.32	31446.45	40677.17	40677.18	40677.17	50365.29	50365.3	50365.29	
Std. dev	1.17E-07	0.003	1.46E-11	3.90E-07	0.0061	1.86E-11	3.65E-07	0.0035	0	
Time consumption	0.099531	0.091575	0.152999	0.178093	0.167818	0.273584	0.157733	0.170495	0.230795	

Since the 6 units system is relatively simple, it is easier to search for the optimal minimum. Therefore, it is observed from Table 2 that the differences between algorithms are not significant. The IMPA and MPA can achieve similar optimization results, and their minimum costs for different requirements are 31446.45439$/h, 40677.17039$/h, and 50365.29463$/h. However, the standard deviation of IMPA is relatively smaller.

5.2 Case 2: 13 units test system

This test case included of 13 units takes into account the valve point effect, which makes the objective function of this system suffer from some local optimal solutions. Cost coefficients and generation limits of units come from Ref. [54], and the power load demand is 1800 MW. The fuel cost curve of 13 units searched by MPA, NMPA, and IMPA is shown in Fig. 4.Fig. 4 Fuel cost curve for 13 units.

Fig. 4

From Fig. 4, it can be seen that our proposed IMPA has the fastest optimization speed for solving the 13 units scheduling problem. Table 4 presents statistical comparisons of the cost minimization of different algorithms. Table 5 demonstrates optimal power allocation at the lowest fuel cost.Table 4 The statistical comparison of cost minimization for13 units with VPE.

Table 4Algorithms	Min. ($/h)	Ave.($/h)	Max. ($/h)	Std.dev	Time consumption	
SOMA [22]	17967.4219	17985.3242	18017.6161	20.6772		
CSOMA [22]	17960.3661	17967.8708	17970.8323	0.8858		
IDE [45]	17960.3661	17961.4717	17969.4857	2.6499		
MPDE [48]	17960.3661	17960.3716	17960.5044	0.027		
CIHSA [54]	17960.3661	17,960.3661	17,960.3661	0		
CDE [55]	17967.4	17995.585	18065.8044	27.09		
DE [55]	17968.3601	18002.9099	18133.4582	38.3352		
CDEMD [55]	17961.944	17974.6869	18061.411	20.3066		
HS [56]	17965.6204	17986.5626	18070.1762	26.3702		
IHS [56]	17960.3661	17965.4152	17971.6512	16.9531		
HQPSO	17963.9571	18273.861	18633.0435	123.2242		
CRO [57]	17961.07	17962.77	17974.82	1.18		
HCRO-DE [57]	17960.38	17960.59	17961.04	0.069		
DHS [58]	17960.3661	17961.1226	17968.361	1.92		
PSO-TVAC [59]	17963.879	18154.562	18358.310	–		
HGA [60]	17963.83	17988.04	–	–		
MPA	17960.3662	17965.5358	17969.3475	4.1817	112.971	
NMPA	17964.5331	17968.2484	17970.0767	1.9403	112.15892	
IMPA	17960.3661	17960.3661	17960.3661	0	142.51826	

Table 5 The best generation schedule for 13 units (Pd = 1800WM).

Table 5Unit	CSOMA [22]	IDE [45]	IHS [56]	MPDE [48]	CHISA [54]	DHS [58]	IMPA	
1	628.3185	628.3185	628.3185	628.3185	628.3185	628.3185	628.3185	
2	149.5997	149.5997	149.5994	149.5997	149.5997	149.5995	149.5997	
3	222.7491	222.7491	222.7491	222.7491	222.7491	222.7491	222.7491	
4	109.8666	109.8666	109.8666	109.8666	109.8666	109.8666	60	
5	109.8665	109.8666	60	109.8666	109.8666	109.8666	109.8666	
6	109.8665	109.8666	109.8666	109.8666	60	109.8666	109.8666	
7	109.8665	60	109.8666	109.8666	109.8666	109.8666	109.8666	
8	60	109.8666	109.8666	109.8666	109.8666	60	109.8666	
9	109.8666	109.8666	109.8666	60	109.8666	109.8666	109.8666	
10	40	40	40	40	40	40	40	
11	40	40	40	40	40	40	40	
12	55	55	55	55	55	55	55	
13	55	55	55	55	55	55	55	
C($/h)	17960.3661	17960.3661	17960.3661	17960.3661	17960.3661	17960.3661	17960.3661	

The results in Table 4 report that the best cost value is 17960.3661$/h, which may be the optimal solution for this system and has been searched by several algorithms, including the IMPA. It is observed that the proposed IMPA can obtain the same minimum value as the optimal result shown in Table 4. However, the unimproved MPA and nonlinear MPA do not get the current minimum.

Table 5 shows in detail the optimal power generation schedules of the algorithms that have searched for the least cost. It can be seen that although multiple algorithms find the same optimal cost, the scheduling plans are different, and our proposed IMPA provides another possible scheduling scheme for this electric power system.

5.3 Case 3: 40 units system

In this case, IMPA is used to solve the ED problem of a power system with 40 units considering VPE. The total power demand of the system is 10500 MW, and unit data is available in Ref. [61]. It should be noted that among the cost coefficients studied in this paper, the ci coefficient of unit 7 is 278.71, while the corresponding unit coefficient in another widely studied 40 units system parameter is 287.71 [47]. The fuel cost curve of 40-unit searched by MPA, NMPA, and IMPA is shown in Fig. 5.Fig. 5 Fuel cost curve for 40 units.

Fig. 5

In Fig. 5, IMPA has the fastest search speed, compared to MPA and NMPA, especially in the early stage of optimization, which means that our proposed improved method also has excellent convergence speed in medium-sized power system dispatching. The solution to the economic dispatch problem of this power system containing 40 units is of great significance. Many researchers have tried to solve this problem in a better way, which allows to minimize the operating costs. Therefore, in this paper they are used for comparison, such as COPSO [18], CSPSO [18], CTPSO [18], CCPSO [18], DPD [62], IDE [45], HAAA [46], AAA [46], MPDE [48], AGWO [16], HsSCA [63], MPA, and NMPA, as shown in Fig. 6.Fig. 6 The comparison of minimum cost of different algorithms for 40 units with VPE.

Fig. 6

Fig. 6 shows that when the power balance is meet, the minimum output cost of 40 units is 121403.5355 $/h, which is from MPDE. And the minimum cost found by our IPMA search for 40 units is also $121403.5355/h. In Ref. [48]. Li et al. proposed multiple populations evolutional methodology improved differential evolution (MPDE), and their iteration number is 10000 while ours is 8000. Furthermore, significant improvements happen in comparison with MPA and NMPA. The results generated by IMPA are 121403.5355 $/h, 121422.1025 $/h, 121458.2649 $/h, and 19.3171 $/h, respectively, in terms of minimum, average, maximum costs, and standard deviation value. The output power of 40 units corresponding to the minimum fuel cost is shown in Fig. 7, which is searched by our improved algorithm IMPA. It provides a new potential solution for the economic dispatch of 40 units.Fig. 7 The best generation schedule of 40 units with VPE.

Fig. 7

5.4 Case 4:140 units system

In this case, the proposed improved algorithm IMPA is applied to solve the economic dispatch problem of more complex large-scale power systems. A large 140-unit system supplying 49342 MW power demand is studied, which is a high dimensionality, discontinuous, nonconvex, and challenging task. In addition, two problems with different constraints are studied, referred to as situation 4A and situation 4B, respectively. Case 4A considers the impact of VPE, but ignores RRL and POZ. The cost coefficient and power generation limitations are listed in Ref. [18]. The fuel cost curve of 40-unit searched by MPA, NMPA, and IMPA is shown in Fig. 8.Fig. 8 The fuel cost curve of 140-unit (a) with VPE (b)with VPE, RRL, and POZ.

Fig. 8

The economic dispatch of a large-scale power system containing 140 units is extremely challenging. Therefore, IMPA will adopt a larger number of searches. The number of searches for MPA and NMPA is the same as it. In Fig. 8, our proposed improved IMPA has relatively better convergence speed in the economic dispatch of large-scale power systems, regardless of case 4A or case 4B. We compare the IMPA with the economic dispatch solution of the existing 140 units, and the results are shown in Table 6.Table 6 The comparison of cost minimization for 140 units with VPE.

Table 6Algorithms	Min. ($/h)	Ave.($/h)	Time consumption	
IMO [64]	1760287	1795156		
MIMO [64]	1584559	1598134		
OGWO [65]	1559709.97	1559713.26		
KHA [66]	1560173.88	1560176.74		
OKHA [66]	1560146.95	1560148.9264		
AAA [46]	1559909.00	1560060.77		
HAAA [46]	1559710.00	1559712.87		
GWO [65]	1559953.18	1560136.93		
HcSCA [63]	1559708.4719	1559709.980		
MPA	1559746.1776	1559913.2753	118.20372	
NMPA	1560807.8107	1562288.6451	350.667	
IMPA	1559708.4549	1559708.4549	133.19796	

Table 6 shows the experimental results of the proposed IMPA and other outstanding approaches. The minimum cost searched by the IMPA are 1559708.4549 $/h and 1559708.4549 $/h. It is evident that the optimal solution acquired by IMPA is lower than 1559709$/h searched by OGWO. As for the average, our IMPA performs best. In case 4B, besides VPE, other constraints like RRL and POZ are considered, making ED optimization more complicated. The unit data is available in Ref. [18]. The comparison between the results of IMPA search for case 4B and existing results is shown in Table 7.Table 7 The comparison of minimum cost for 140 units with VPE, RRL, and POZ.

Table 7Algorithms	Min. ($/h)	Ave.($/h)	Time consumption	
CTPSO [18]	1657962.73	1657964.06		
CSPSO [18]	1657962.73	1657962.74		
COPSO [18]	1657962.73	1657962.73		
CCPSO [18]	1657962.73	1657962.73		
L-SHADE [50]	1658002.79	1659118.46		
IL-SHADE [50]	1657962.7303	1657965.30		
C-GRASP-SaDE [67]	1657962.7268	1658006.2712		
C-GRASPMDE [67]	1666166.674	1685973.32		
GSO [68]	1728151.168	1745514.9975		
CQGSO [68]	1657962.727	1657962.741		
PPO [69]	1657963.5	–		
APSO [69]	1657963.0	–		
SPPO [69]	1657962	–		
MPA	1658022.9625	1658383.3495	608.606	
NMPA	1658103.3137	1658450.6475	1519.0107	
IMPA	1657962.7265	1657962.7265	296.52155	

In Table 7, the minimum and average costs searched by IMPA are 1657962.7265$/h and 1657962.7265$/h, and they are much lower than the results of the MPA and NMPA, which means that our proposed improved IMPA improves the performance of MPA for solving the economic dispatch problem of large-scale power systems. In addition, the minimum cost provided by IMPA is also competitive compared to the results of other algorithms. Fig. 9 gives each unit output at the minimum cost of cases 4A and 4B.Fig. 9 The best generation schedule of 140 units.

Fig. 9

6 Conclusion and future directions

6.1 Conclusion

The economic dispatch of the power system can reduce operating costs. However, economic dispatch is a challenging optimization problem. We propose an improved marine predictor algorithm (IMPA) to provide another candidate method for solving economic dispatch problems in power systems. We test the classical benchmark functions to verify the exploration and exploitation performances of IMPA. The experimental results reveal that IMPA not only has a good improvement over MPA but also is competitive with NMPA. The proposed IMPA is used to solve the economic dispatch problems of power systems with 6 units without VPE, 13,40 units with VPE, and 140 units with multiple constraints. The minimum cost searched by the IMPA are 1559708.4549 $/h and 1559708.4549 $/h. It is evident that the optimal solution acquired by IMPA is lower than 1559709$/h searched by OGWO. The minimum and average costs searched by IMPA are 1657962.7265$/h and 1657962.7265$/h, and they are much lower than the results of the MPA and NMPA, which means that our proposed improved IMPA improves the performance of MPA for solving the economic dispatch problem of large-scale power systems. The best results obtained by IMPA are more competitive than those obtained by NMPA and MPA.

6.2 Future directions

Despite encouraging results, this study has limitations. Evaluations focused on specific scenarios with up to 140 units and various constraints may not fully represent all practical conditions. This study has not yet considered the economic dispatch of power systems with a high proportion of renewable energy. In future research, we will focus on this issue.

CRediT authorship contribution statement

Cheng Yang: Supervision, Resources, Conceptualization. Xiaoliang Zheng: Writing – original draft, Supervision, Formal analysis, Data curation. Jiwen Wang: Validation, Resources, Data curation. Wei Zhang: Methodology, Investigation, Formal analysis. Ludeng Liu: Validation, Software, Formal analysis. Bin Ma: Resources, Investigation, Funding acquisition, Conceptualization. Yuanzhu Fan: Project administration, Methodology, Data curation. Qiong Tao: Writing – original draft, Validation, Investigation. Hu Wang: Writing – review & editing, Investigation.

Declaration of competing interest

We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript entitled “Research on Power system Economic Dispatch of Marine Predator Algorithm Improved by Asymmetric Information Exchange”.

Acknowledgements

This work was supported by the State Grid Anhui Electric Power Co., Ltd. 2023 Science and Technology Project (Grant NO. 52120023001Q ).
==== Refs
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