1. Introduction
In recent years, many large hydropower bases have been connected to the grid for power generation, and with the construction and operation of pumped storage power stations, the joint scheduling of hydrothermal power has become one of the current research hotspots of power grid systems [
1,
2]. In addition, to meet the constraints of hydropower systems, joint scheduling of hydrothermal power needs to further consider the constraints of thermal power units and the overall system constraints, which is a multi-stage, high-dimensional, non-linear complex optimization solution problem [
3]. Its purpose is to give full play to the pollution-free characteristics of hydropower energy, improve the utilization efficiency of hydropower energy, increase the consumption capacity of uncertain new energy sources such as wind and solar in the power grid system [
4], and maintain the stable operation of the power grid system under large fluctuating loads.
In the joint scheduling problem of hydrothermal power, the refined management requirements and multiple complex constraints greatly increase the difficulty of solving the model, and it is difficult for traditional mathematical methods to efficiently solve it [
5,
6]. Intelligent computing methods have the characteristics of strong adaptability and high flexibility, and many researchers have used such methods to optimize the solution of joint scheduling problems of hydrothermal power [
7]. For example, Wang and Tian (2021) [
8] studied the generation cost of thermal power plants in the short-term hydro-electric control system and adopted a hierarchical structure to combine two new swarm intelligence algorithms, improved gray Wolf optimization and Harris Eagle optimization, to obtain the advantages of easy implementation, strong convergence ability, and good global search ability. The simulation results show that this algorithm is better than the traditional intelligent algorithm. Zhang and Lin (2017) [
9] proposed a parallel differential evolution algorithm based on a small population to solve the STHS problem with power flow constraints. The numerical results of two famous test systems show that the parallel differential evolution algorithm can generate the optimal solution to the STHS problem. Basu (2004). Ref. [
10] proposed an interactive fuzzy satisfaction method based on evolutionary programming techniques for short-term multi-objective joint scheduling of hydrothermal power, and the results showed that the best compromise solution obtained was closer to the expectations of decision makers. The advantages and disadvantages of some methods can be obtained from
Table 1.
On the whole, albeit the conventional algorithm has a strong background in physics or mathematics, as the problem gets more complex, its application has additionally been restricted by numerous limitations, and the application impact has progressively weakened. As an arising improvement mode, the group intelligent optimization method has been applied to the joint scheduling problem of hydrothermal power. Be that as it may, there are still imperfections such as inadequate arrangement effectiveness, a wide assortment of strategies, and temperamental outcomes. Thusly, it is fundamental and earnest to develop an improved technique that can be effectively applied to the arrangement of the hydrothermal joint scheduling model.
Considering this, in light of the arising collaborative search algorithm, taking into account the deficiencies—the individual easily falls into the local optimum, and the global search capacity is diminished—of the collaborative search algorithm in complex optimization issues, such as high-dimensional, nonlinear, multi-variable, etc., three-optimization techniques that can upgrade, the optimization proficiency of the algorithm is proposed in three parts, i.e., parameter randomization, elite reinforcement learning, and elite-assisted learning. Additionally, another new elite collaborative search algorithm is established. In the light of the basic structure of the hydrothermal power system and focusing on environmental benefits, a joint dispatching model of hydrothermal power is established, and its goal is to minimize pollutant gas emissions. The classic hydrothermal joint scheduling system is addressed and examined by this algorithm to check its solving effectiveness in this problem.
2. Joint Scheduling Model of Short-Term Hydrothermal Power System
2.1. Objective Function
Hydropower is a renewable energy and will not produce polluting gases in the operation process. By burning coal for power generation, thermal power units will emit a large number of polluting gases. Gas emissions are related to unit output and its output parameters. Therefore, this paper takes the minimum pollutant gas emissions as the objective function, describing it as the addition of a quadratic function and an exponential function [
12], and the objective function is as follows.
where
,
,
,
and
are polluting gas emission factors of the thermal power unit
I. is the output of the thermal power unit,
i.
e the natural base.
2.2. Constraints
The model includes several thermal power units and a cascade hydropower station system, and its constraints can be divided into two aspects, the thermal power constraints, and the hydropower constraints.
2.2.1. Thermal Power Constraints
(1) Load balancing constraints
where
is the output of the thermal power unit
i;
is the output of the hydropower station
j;
is the total load demand of the system;
is the number of thermal power units;
is the number of hydropower stations;
is the output algorithm parameter of the hydropower station; and
is the outflow of the power station
hj in the period
t;
(2) Unit output constraint
where
is the minimum output power of the thermal power unit
i, and
is the maximum output power of the thermal power unit
i.
(3) Unit climbing constraints
where
and
are the rate limits of the upward and downward climbing of unit
i, respectively, and
is the output for the thermal power unit
i at the time
t. 2.2.2. Hydropower Constraints
(1) Water balance constraint
where
is the initial storage capacity of the power station
n in the
t-period;
,
and
are the inflow, outflow, and interval flow of the power station
n in the period
t;
and
are the abandoned water flow and power generation flow of the power station
n in the
t period, respectively;
NU is the number of upstream hydropower stations with direct hydro-hydraulic links to the
nth hydropower station.
(2) Outflow constraints
where
and
are the upper and lower limits of the outflow of each power station in the
t-period, respectively.
(3) Water level constraints
where
and
are the upper and lower limits of the water level of each power station in the
t period, respectively.
(4) Maximum inflow constraint of the unit
where
and
are the upper and lower limits of the power generation flow of each power station in the
t-period, respectively.
(5) Power station output constraints
where
and
are the upper and lower limits of the output of the power station
n in the
t period, respectively.
(6) Beginning and end water level constraints
where
and
are the beginning and end water level of the
nth hydropower station.
3. Elite Collaborative Search Algorithms
3.1. Introduction to Collaborative Search Algorithms
The researchers [
11,
13], enlivened by the enterprise management, established a group intelligence optimization-cooperative search algorithm (CSA). In this algorithm, problem-solving optimization is compared to the development process of an enterprise, and individuals are regarded as employees of the enterprise. By imitating the cooperation conduct of the current business, utilize three modern enterprise techniques to get great arrangements step by step, i.e., team communication, reflective learning, and internal competition.
The detailed evolutionary process is as follows:
(1) Build a team
where
I is the number of populations;
J is the problem dimension;
is the position of the
ith individual, the
jth dimension at the
k iteration;
is the upper limit of the position,
is the lower limit of the position; and
represents the number of uniformly distributed random numbers over the interval
.
(2) Team communication
where
is the position of the population at the
k + 1 iteration;
is A randomly selected individual position from an outside elite group;
and
are the stochastic average of the individual optimal solution and the global optimal solution, respectively;
is the global optimal position, and
is the individual optimal position;
M is the global optimal number of individuals,
M = 3;
ind is a random integer between [0,
M]; and
α,
β are the learning coefficient,
α = 0.1,
β = 0.15.
(3) Reflective learning
where
and
are the upper and lower limits of the j-dimensional position;
is the position of the population at the (
k + 1)th iteration.
(4) Internal competition
where
indicates the fitness value of individual g;
3.2. Improvement Strategies
The standard CSA is seen as the latest published original algorithm in the field of engineering applications and has many benefits. First, the CSA can keep the global search capability by avoiding falling into local extremums for a long time. Second, with the help of team communication and reflective learning, it can nicely balance the global and local search, which can help the population to converge to a better position quickly. Third, the CSA can refresh the place of the arrangement inside the area, holding better individuals through the inward contest. Fourth, the CSA regards the objective issue as a black box that doesn’t need evolving data, utilizes different constraints processing skills to manage complex constraint optimization problems, and has the conventional benefits of meta-heuristic algorithms.
However, the algorithm shows deficient destabilization optimization capacity in complicated optimization issues, such as high-dimensional, nonlinear, and multi-variable, and it also has a disadvantage that the individuals easily fall into local optimization and the global searching ability decreases in these issues, which shows that CSA has large space for improvement when facing complex optimization problems. There are three effective strategies—parameter randomization, elite reinforcement learning, and elite-assisted learning—that can effectively improve the searching effect of the algorithm.
(1) Randomization of cj parameter
Focusing on the problem that the
cj parameter is fixed and the perturbation optimization ability is insufficient in the original CSA algorithm, it can be found that improving the parameter value can enhance the parameter random disturbance capability and increase the scope of the algorithm optimization, as follows:
where
rand is a number that is randomly generated between (0,1).
(2) Elite reinforcement learning
In an enterprise, great leaders frequently tend to lead the group in the correct direction of advancement, offer employees adequate learning chances and learning conditions in their day-to-day work, and heighten the strength of colleagues. In individual-based swarm intelligence algorithms, a similar circumstance can also be found. The global optimal location can lead the evolution of population in the optimal direction. The individual optimal location can enhance the local search ability of the population. The random combination of the two with ordinary individuals can enhance the diversity of the algorithm, which can prevent individuals from falling into the local optimum, and enhance the ability of individuals to go beyond the inherent boundaries in a later period. Subsequently, an elite reinforcement learning strategy is proposed, which randomly selects a certain proportion of individuals, and uses the random intersection of the global optimal position and the individual optimal position to carry out elite reinforcement learning. The individual selection ratio and elite reinforcement learning formula are as follows:
where
indicates a random reordering of
R; R represents the number of individuals that perform random learning; ⎡ ⎤ represents rounding up a decimal;
r2 is a random integer, r2
[1,R]; and
a1 and
a2 are random factors,
a1 = 0.3,
a2 = 0.1.
(3) Elite-assisted learning
In the process of enterprise development, some employees whose learning and communication skills are poor often lag behind the overall level of the enterprise. What the investigation discovered is that if the elites in different divisions can share their merit, and the low-level employees can learn from good workers, it can effectively improve their working ability and effectively enhance the efficiency and cohesiveness of team members. In the group intelligence algorithm, good and bad individuals also exist in the current period. It can be found that, combining the behavior characteristics of cuckoos [
14] Levi’s random walking can play a positive feedback role in the calculation of the difference between inferior individuals and global optimal individuals. Accordingly, the elite-assisted learning strategy is proposed. Firstly, it will select 30% of the currently inferior individuals for elite-assisted learning to get the same amounts of individuals by using the global optimal position and Levi’s random walk. Secondly, in order to select individuals that have better fitness values, the two will be chosen through competitive learning methods. At last, it will select the better individuals as part of the population into the next optimization calculation. The global search ability of the algorithm and the search efficiency is further developed. The specific calculation formula of elite-assisted learning is as follows:
where
is the position of the
j-dimension of the
ith individual in the
kth iteration process;
is a constant,
= 1.5;
l1 is the step parameter,
l1 = 0.001;
are the random numbers that follow a normal distribution; and
is a gamma function.
3.3. The Flowchart of the Elite Collaborative Search Algorithm
It will enhance the algorithm’s disturbance optimization ability by ameliorating the randomness of parameters in the ESCA. It will help the individuals go beyond the inherent boundaries in the later period and expand the search range by increasing the search range of the algorithm in the elite reinforcement learning strategy. Simultaneously, it can effectively improve the algorithm’s ability of global search by introducing the Cuckoo’s Levi’s Flight Evolution Strategy in elite-assisted learning. The detailed calculation processes of the ECSA algorithm are summarized as follows:
- (1)
Set parameters, such as the number of the individual population I, the maximum number of iterations K, etc.
- (2)
Set k = 1, initialize population xk.
- (3)
Calculate the fitness value of all individuals in xk, obtain the global extreme value gbest and individual extreme value pbest, and update the current population individual position.
- (4)
According to formulas (24)–(26), randomly select some individuals that contain R individuals for elite reinforcement learning according to formula (23), then, update the current individual position.
- (5)
Calculate the fitness value of all individuals in the current population xk, and a new population L can be obtained by selecting 30% of poor individuals for elite-assisted learning based on Levi’s flight using the formula (27)–(28). Individuals with good fitness values are selected to re-enter the population xk.
- (6)
Let k = k + 1, and determine whether k meets the conditions k > K. If the condition is met, enter step (7); if the condition is not met, then go to step (3) to continue iterative optimization.
- (7)
Stop the iterative calculation, calculates the fitness value, and obtain the global optimal solution.
3.4. Test Function Verification
This section selects four test functions in the CEC 2017 series for numerical verification, as shown in
Table 2; the series includes the unimodal function (F1), multimodal function (F2), fixed function (F3), and compound function (F4). The unimodal function is suitable for testing the local optimization ability of the algorithm, the multimodal function is suitable for testing the exploration ability of the algorithm, the fixed function suitable for testing the engineering application ability of the algorithm, and the compound function is suitable for testing the optimization ability of the algorithm in a highly complex environment.
To test the performance of this method, seven new swarm intelligence algorithms are introduced for comparison. In numerical experiments, these methods are developed in MATLAB language and executed on a personal computer equipped with a 1.6 GHz CPU and 8GB RAM, and run independently 50 times. For the developed methods, the number of individuals and the number of iterations is set to 30 and 500, respectively.
Table 3 shows the average value and standard deviation of each optimization algorithm after 50 independent runs, so that ECSA can get better statistical values.
Figure 1 shows the robustness of the algorithm. Among all functions, ECSA has the smallest box oscillation amplitude, the lowest point, and the shortest upper and lower shadow lines, which indicates that ECSA has better robustness.
Figure 2 shows the convergence performance of the algorithm. Compared with other algorithms, CSA and ECSA have stronger convergence effect and convergence speed. Among the unimodal function F
1, multimodal function F
2, and mixed function F
3, CSA has a better convergence effect than ECSA in the early stage, whereas the ECSA algorithm can jump out of the local optimal area in the later stage, continue to develop and search for the optimal solution position, and finally get a better convergence effect than other algorithms. In the more complex composite function, the ECSA algorithm shows superior convergence speed and convergence effect.
4. Joint Scheduling of Short-Term Hydrothermal Power System by ESCA
The purpose of the joint scheduling of hydrothermal power systems with the goal of the least-pollutant emission is to give full play to the characteristics of hydropower, maximize the suppression of the inherent shortcomings of thermal power, and make the entire system green and economical. At the same time, it is of great significance for the optimization and adjustment of the power grid system and the stable operation. In this paper, the hydrothermal power system, including three thermal power units and four hydropower stations is, used as an example to solve and analyze the joint scheduling problem of ESCA.
4.1. Introduction to the Joint System
The time lag between hydropower stations of the system is shown in
Table 4. The basic parameters of the thermal power unit are shown in
Table 5, and the algorithm coefficient of polluting gas emissions is shown in
Table 6. Since it is short-term scheduling, the water inflow delay between cascade hydropower stations is considered, and the topology of the hydrothermal power system is shown in
Figure 3. The basic parameters of the power station and the river basin are shown in
Table 7 and
Table 8. The interval inflow and system load requirements are shown in
Table 9.
4.2. Coding Strategy and Constraint Processing
4.2.1. Coding Strategy
The model is composed of two different systems, so the decision variables of the two systems are respectively selected for optimization. The hydropower system selects the outflow
Q as the decision variable, the thermal power system selects the unit output
Ps as the decision variable, and the initialization expression is as follows:
4.2.2. Treatment of Thermal Power Constraints
(1) Judge whether there is a period for violating the climbing rate limit of the unit one by one for the individual population and if so, it will be treated according to the following formula:
(2) Judge whether the individual population is violating the output constraint of the unit one by one, and if so, forcibly pull the individual value back to the boundary.
(3) Adaptive constraint adjustment for the load balance constraint. For individuals who do not meet the load balance, the limit float value is set according to the degree of relaxation of each particle value in the feasible domain. That is the difference between the upper and lower limits of the particle distance variable, and its mathematical description is as follows:
where
and
are the upward and downward slack amounts of unit
i at time
t, respectively.
If
, the infeasible solution vector
violates the load balancing constraint to a degree of
. According to the limit floating value of each variable in the vector, determine the size of the extra load that the variable can bear, as shown in the following formula:
The values of the feasible solution variables after adaptive adjustment are
Conversely, if
, calculate the degree of violation of its constraints by
, and calculate the size of the additional load it can bear as:
The values of the feasible solution variables after adaptive adjustment are:
(4) In order to further ensure that the individual fully satisfies the constraints. Set the penalty function method at the end.
4.2.3. Treatment of Hydropower Constraints
Compared with pure hydropower systems and pure thermal power systems, the constraints of the hydrothermal joint system will be more complex, which involves the joint scheduling of hydrothermal power. Following the principle of first difficulty and then easy, the constraint treatment first meets the constraint conditions of the hydropower system. Then calculates the hydropower output, takes the remaining load of the system as the total load, and carries out the constraint treatment of the thermal power system. Among them, the constraints of hydropower systems is treated according to the “two-stage method” [
15,
16], and the detailed treatment steps are as follows:
- (1)
Set the relevant algorithm parameters, i = 1, t = 1, t1 = 1, Num1 = 5, e1 = 0.5, e2 = 0.001.
- (2)
Calculate the absolute difference between the end storage capacity and the constrained end storage capacity, and record it as
. If
<
e1 or
t1 >
Num1, then the first stage is completed, and go directly to step 5. If not, enter step 3.
- (3)
Adjust the outflow value of each period according to the principle of equalization of water quantity. Check whether the equalized outflow meets the constraint requirements, and if not, pull back to the boundary.
- (4)
After calculating the equalized storage capacity, t1 = t1 + 1, if t1 > Num1, the first stage is completed and go directly to step 5; otherwise, enter step 2 again.
- (5)
Set t = 1 and calculate the updated . Check that the water volume is equally divided according to formula (38), and whether the outflow after the equalization is meets the constraint requirements. If not, it is pulled back to the boundary.
- (6)
Calculate the updated . If < e2 or t > T, the second stage is completed and go to step 7; otherwise, t = t + 1, so return to step 5.
- (7)
For i = i + 1, if i = Nh, the entire hydropower system constraint treatment is completed; otherwise, return to step 2.
- (8)
Calculate the total output of the hydropower system to find the remaining load demand of the system. Take it as the total load demand of the thermal power system, and carry out the constraint treatment of the thermal power system.
- (9)
In order to ensure that the solution is completely feasible, the penalty function method is added. At this point, the constraint processing of the entire system is complete.
The constraint treatment flowchart of hydrothermal system is shown in
Figure 4.
4.3. Results and Analysis
To verify the realizability and efficiency of the ECSA method and the constraint treatment method in the joint scheduling problem of a hydrothermal system, the system with three thermal power units and four hydropower stations is independently simulated, and the mean, median, best value, worst value, and standard deviation are calculated, respectively. The detailed statistical results are listed in
Table 6. Among them, the whole scheduling period
T = 24 h, and the stage of scheduling is 1 h, the maximum number of iterations of each algorithm
K = 1000, the population size I = 100, and the algorithm parameters are set as follows:
ECSA: the global optimal number of individuals m = 3; learning coefficient = 0.1, = 0.15; random factor, a1 = 0.3, a2 = 0.1;
CSA: global optimal number of individuals M = 3, learning coefficient = 0.1, = 0.15;
PSO: The inertia weight ω decreases linearly from 0.9 to 0.4, and the learning factors c1 and c2 are both 2.0; GWO: the constant a0 is 2.0;
WOA: The variable A decreases linearly from 2 to 0. These methods are developed in MATLAB language and executed on a personal computer equipped with a 1.6GHz CPU and 8GB RAM, and run independently 10 times. As can be seen from
Table 10, ECSA performs better when solving this model compared with the other four algorithms. In the five statistical indicators, ECSA has absolute advantages in four indicators. Only in the standard deviation, it is slightly inferior to CSA. Taking the average value as an example, compared with CSA, PSO, GWO, and WOA, the emission of polluting gases of ESCA is reduced by about 1.88%, 11.57%, 27.22%, and 10.73%, respectively.
As can be seen from
Table 11, first of all, in terms of calculation time, the average running time of ECSA in a water-fire system is 26 s, which is very short compared with the working time of a power grid system scheduling problem, so there is no need to worry. Secondly, these methods have little difference in internal storage and CPU ratio. Finally, the optimization effect obtained by ECSA is the best, which shows that ECSA is a relatively optimal method.
Figure 5 shows the iterative convergence process of the five algorithms. It can be seen that ECSA has a general effect in the early stage of the search, and the convergence speed and effect rank third among the five algorithms. However, at the beginning of the mid-term, ECSA can quickly find the relative optimal position, speed up the convergence process, and eventually become the one with the highest convergence accuracy.
Figure 6 shows the box diagram of the statistical results. Combined with the local enlargement in
Figure 6 and
Table 10, it can be seen that the ECSA box amplitude is smaller, and the upper and lower shadow lines are lower. This indicates that the stability and convergence of ECSA in the hydrothermal joint scheduling problem are better than the other four algorithms.
Table 12 shows the total output process of system.
Figure 7 shows the iterative process of the hydropower station output. It can be found that the total output of hydropower stations in each period meets the load requirements of the system.
Table 13 shows the change process of the outflow and storage capacity of the hydropower station in detail.
Figure 8 shows the scheduling process of storage capacity. It can be seen that the reservoir scheduling process fully meets the constraints that the system needs to meet, which proves that the constraint treatment method is feasible. In summary, ECSA and the proposed constraint treatment method can be efficiently applied to the hydrothermal joint scheduling system, which has greatly improved the solution stability and solution quality compared with other algorithms. In addition, it provides a new scientific method for solving this important hydrothermal joint scheduling model.
5. Conclusions
Aiming at the shortcomings of the standard collaborative search algorithm, three efficient improvement strategies were proposed: parameter randomization, elite reinforcement learning, and elite-assisted learning. Furthermore, an elite collaborative search algorithm coupled with three improvement strategies was established. This algorithm can enhance the global search ability, jump out of the local optimal trap, and has a better convergence effect and robustness. In addition, with the aim of minimizing pollutant gas emissions, this paper constructed a joint optimization scheduling model for short-term hydrothermal power systems, proposed a new constraint handling method, effectively solved the feasible solutions to find the characteristics of complex systems, used five kinds of swarm intelligent algorithms, and simulated and tested hydrothermal systems that contain three thermal power units and four reservoirs. Through 10 independent simulation experiments, it was found that ECSA has an absolute advantage in statistical indicators compared with the remaining four algorithms, and the emission of polluting gases was reduced by about 1.88%, 11.57%, 27.22%, and 10.73%, respectively.
In general, ECSA can obtain a relatively optimal solution by solving the hydrothermal joint scheduling optimization model, and the obtained scheduling scheme can meet the requirements of various complex constraints well, which verified the feasibility and efficiency of ECSA and the constraint treatment method in the hydrothermal joint system. Therefore, this paper provided a new scientific method for the optimal and stable operation of the power grid system. For the further application of the algorithm, the algorithm needs to verify its superiority on more models, and the calculation time also needs to be improved.
Author Contributions
Investigation, Writing—original draft, J.D.; Writing—review & editing and Methodology, Z.J. All authors have read and agreed to the published version of the manuscript.
Funding
This study was financially supported by the Natural Science Foundation of China (52179016), Natural Science Foundation of Hubei Province (2021CFB597).
Institutional Review Board Statement
Not applicable.
Informed Consent Statement
Not applicable.
Data Availability Statement
If there is a need for research, readers can obtain relevant data through the mailbox of the corresponding author.
Acknowledgments
The authors are grateful to the anonymous reviewers for their comments and valuable suggestions.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1.
Box diagram of test statistical results.
Figure 1.
Box diagram of test statistical results.
Figure 2.
Test function convergence curve.
Figure 2.
Test function convergence curve.
Figure 3.
System structure diagram.
Figure 3.
System structure diagram.
Figure 4.
Flowchart of constraint processing of hydrothermal systems.
Figure 4.
Flowchart of constraint processing of hydrothermal systems.
Figure 5.
Iterative convergence diagram of different algorithms.
Figure 5.
Iterative convergence diagram of different algorithms.
Figure 6.
Box diagram of statistical Results.
Figure 6.
Box diagram of statistical Results.
Figure 7.
Output superposition diagram of hydropower station.
Figure 7.
Output superposition diagram of hydropower station.
Figure 8.
Optimized scheduling process of storage capacity by ECSA.
Figure 8.
Optimized scheduling process of storage capacity by ECSA.
Table 1.
Pros and cons of methods and models in the literature.
Table 1.
Pros and cons of methods and models in the literature.
Literature | Advantages | Disadvantages |
---|
parallel differential evolution algorithm [9] | Make better use of the advantages of computer multithreading, and have less CPU running time. | The comparison algorithm is too old to prove to be excellent among many new algorithms. |
stochastic dynamic approach [2] | In the model, the uncertainty of inflow and the influence of electricity market demand on the operation of the power stations are considered. | The innovation lies in the model, whereas the method is ordinary, so there can be a better solution. |
interactive fuzzy satisfying method [10] | It is the first time adopting an interactive fuzzy satisfactory method to solve this kind of problem, and the effect is remarkable. | It is unconvincing to convert multi-objective decision making into a single-objective choice. |
hybrid Gray Wolf–Harris hawks algorithm [8] | This method combines the advantages of the method Gray Wolf and Harris hawks and has a good optimization effect. | The method is too complicated, there are too few comparative verification methods, and the result analysis is not sufficient. |
cooperation search algorithm [11] | The CSA method can avoid falling into the local extremum trap for a long time and keep the global search ability of the algorithm. | The searchability of the algorithm is weakened in the later stage, and it may fall into the local optimal trap. |
Table 2.
Basic information on four different types of functions.
Table 2.
Basic information on four different types of functions.
Function Name | Dim | Range | fmin | Type |
---|
F1: Shifted and Rotated Bent Cigar Function | 10 | [−100, 100] | 100 | Unimodal |
F2: Shifted and Rotated Rastrigin’s Function | 10 | [−100, 100] | 500 | Multimodal |
F7: Hybrid Function 6 (N = 4) | 10 | [−100, 100] | 1600 | Fixed |
F14: Composition Function 8 (N = 6) | 10 | [−100, 100] | 2800 | Compound |
Table 3.
Test function statistics results.
Table 3.
Test function statistics results.
Function | Item | MFO | GSA | SCA | GWO | CSA | SSA | WOA | ECSA |
---|
F1 | Average | 1.60 × 108 | 4.17 × 102 | 4.11 × 109 | 2.77 × 108 | 3.42 × 103 | 4.17 × 103 | 7.09 × 107 | 2.99 × 103 |
| STD | 4.28 × 108 | 6.58 × 103 | 1.22 × 109 | 1.45 × 108 | 3.30 × 103 | 3.90 × 103 | 9.80 × 107 | 2.81 × 103 |
F2 | Average | 5.33 × 102 | 5.44 × 102 | 5.95 × 102 | 5.40 × 102 | 5.16 × 102 | 5.32 × 102 | 5.62 × 102 | 5.15 × 102 |
| STD | 1.10 × 101 | 1.08 × 101 | 1.39 × 101 | 7.19 × 100 | 7.30 × 100 | 1.36 × 101 | 2.06 × 101 | 5.48 × 100 |
F3 | Average | 1.79 × 103 | 2.10 × 103 | 2.33 × 103 | 1.77 × 103 | 1.74 × 103 | 1.80 × 103 | 1.96 × 103 | 1.73 × 103 |
| STD | 1.42 × 102 | 1.60 × 102 | 1.94 × 102 | 1.18 × 102 | 1.38 × 102 | 1.45 × 102 | 1.57 × 102 | 1.27 × 102 |
F4 | Average | 3.35 × 103 | 3.51 × 103 | 3.60 × 103 | 3.36 × 103 | 3.36 × 103 | 3.38 × 103 | 3.47 × 103 | 3.35 × 103 |
| STD | 9.31 × 101 | 4.61 × 101 | 1.20 × 102 | 9.20 × 101 | 1.19 × 102 | 1.99 × 102 | 1.90 × 102 | 1.14 × 102 |
Table 4.
Water flow lag data.
Table 4.
Water flow lag data.
Table 5.
Basic parameters of thermal power units.
Table 5.
Basic parameters of thermal power units.
Unit | UR (MW) | DR (MW) | (MW) | (MW) |
---|
1 | 40 | 40 | 20 | 175 |
2 | 60 | 60 | 40 | 300 |
3 | 115 | 115 | 50 | 500 |
Table 6.
Gas emission parameters of thermal power units.
Table 6.
Gas emission parameters of thermal power units.
Unit | | | | | |
---|
1 | 60 | −1.355 | 0.0105 | 0.4968 | 0.01925 |
2 | 45 | −0.6 | 0.008 | 0.486 | 0.01694 |
3 | 30 | −0.555 | 0.012 | 0.5035 | 0.01478 |
Table 7.
Output coefficient hydropower stations.
Table 7.
Output coefficient hydropower stations.
Plant | c1 | c2 | c3 | c4 | c5 | c6 |
---|
1 | −0.0042 | −0.42 | 0.03 | 0.9 | 10 | −50 |
2 | −0.004 | −0.3 | 0.015 | 1.14 | 9.5 | −70 |
3 | −0.0016 | −0.3 | 0.014 | 0.55 | 5.5 | −40 |
4 | −0.003 | −0.31 | 0.027 | 1.44 | 14 | −90 |
Table 8.
Power station basic parameters (unit of storage capacity: 10,000 m3, unit of inflow rate: 10,000 m3, unit of output: MW).
Table 8.
Power station basic parameters (unit of storage capacity: 10,000 m3, unit of inflow rate: 10,000 m3, unit of output: MW).
Plant | Vmin | Vmax | Vbegin | Vend | Qmin | Qmax | | |
---|
1 | 80 | 150 | 100 | 120 | 5 | 15 | 0 | 500 |
2 | 60 | 120 | 80 | 70 | 6 | 15 | 0 | 500 |
3 | 100 | 240 | 170 | 170 | 10 | 30 | 0 | 500 |
4 | 70 | 160 | 120 | 140 | 6 | 20 | 0 | 500 |
Table 9.
Interval inflow and system load requirements.
Table 9.
Interval inflow and system load requirements.
Time | Plant1 | Plant2 | Plant3 | Plant4 | PD | Time | Plant1 | Plant2 | Plant3 | Plant4 | PD |
---|
1 | 10 | 8 | 8.1 | 2.8 | 750 | 13 | 11 | 8 | 4 | 0 | 1110 |
2 | 9 | 8 | 8.2 | 2.4 | 780 | 14 | 12 | 9 | 3 | 0 | 1030 |
3 | 8 | 9 | 4 | 1.6 | 700 | 15 | 11 | 9 | 3 | 0 | 1010 |
4 | 7 | 9 | 2 | 0 | 650 | 16 | 10 | 8 | 2 | 0 | 1060 |
5 | 6 | 8 | 3 | 0 | 670 | 17 | 9 | 7 | 2 | 0 | 1050 |
6 | 7 | 7 | 4 | 0 | 800 | 18 | 8 | 6 | 2 | 0 | 1120 |
7 | 8 | 6 | 3 | 0 | 950 | 19 | 7 | 7 | 1 | 0 | 1070 |
8 | 9 | 7 | 2 | 0 | 1010 | 20 | 6 | 8 | 1 | 0 | 1050 |
9 | 10 | 8 | 1 | 0 | 1090 | 21 | 7 | 9 | 2 | 0 | 910 |
10 | 11 | 9 | 1 | 0 | 1080 | 22 | 8 | 9 | 2 | 0 | 860 |
11 | 12 | 9 | 1 | 0 | 1100 | 23 | 9 | 8 | 1 | 0 | 850 |
12 | 10 | 8 | 2 | 0 | 1150 | 24 | 10 | 8 | 0 | 0 | 800 |
Table 10.
Statistical table of pollutant gas emission structure of five algorithms (unit: kg).
Table 10.
Statistical table of pollutant gas emission structure of five algorithms (unit: kg).
Method | Average Value | Median | Optimal Value | Worst Value | Standard Deviation |
---|
ECSA | 16,169.18 | 16,180.19 | 16,051.55 | 16,313.26 | 88.21 |
CSA | 16,478.39 | 16,475.76 | 16,329.90 | 16,615.14 | 77.86 |
PSO | 18,284.97 | 18,300.63 | 17,995.63 | 18,622.83 | 163.87 |
GWO | 22,216.65 | 22,353.61 | 19,963.20 | 26,070.94 | 2037.74 |
WOA | 18,112.36 | 17,995.92 | 17,683.99 | 18,513.42 | 294.32 |
Table 11.
Running time comparison.
Table 11.
Running time comparison.
Method | Average CPU Time (s) | CPU Ratio | Internal Storage (M) |
---|
ECSA | 26.45 | 21.1% | 1002.5 |
CSA | 21.75 | 22.3% | 990.9 |
PSO | 13.22 | 22.3% | 1012.2 |
GWO | 12.72 | 19.7 | 980.2 |
WOA | 10.32 | 19.9 | 980.0 |
Table 12.
Table of output process of hydropower and thermal power units (unit: MW).
Table 12.
Table of output process of hydropower and thermal power units (unit: MW).
Hour | Ph1 | Ph2 | Ph3 | Ph4 | Ps1 | Ps2 | Ps3 |
---|
1 | 59.17 | 49.94 | 13.40 | 132.23 | 170.54 | 190.87 | 133.85 |
2 | 83.98 | 51.60 | 32.01 | 129.00 | 174.26 | 188.03 | 121.12 |
3 | 74.93 | 55.02 | 7.42 | 125.72 | 175.00 | 149.00 | 112.91 |
4 | 61.23 | 52.70 | 36.60 | 121.60 | 146.79 | 137.68 | 93.39 |
5 | 55.29 | 56.33 | 32.59 | 115.80 | 154.91 | 155.02 | 100.06 |
6 | 74.59 | 54.87 | 38.20 | 133.98 | 170.70 | 187.05 | 140.61 |
7 | 81.85 | 56.23 | 36.91 | 210.83 | 171.16 | 233.56 | 159.46 |
8 | 83.95 | 55.99 | 32.75 | 226.28 | 174.18 | 244.04 | 192.81 |
9 | 89.54 | 57.21 | 35.82 | 284.44 | 174.99 | 245.03 | 202.97 |
10 | 81.50 | 77.05 | 32.45 | 284.91 | 173.38 | 257.11 | 173.60 |
11 | 89.62 | 78.60 | 35.34 | 289.42 | 174.79 | 256.57 | 175.65 |
12 | 91.48 | 72.01 | 37.76 | 304.29 | 174.37 | 272.40 | 197.69 |
13 | 92.92 | 70.96 | 35.66 | 299.39 | 175.00 | 245.29 | 190.79 |
14 | 84.66 | 77.72 | 36.20 | 272.85 | 175.00 | 238.53 | 145.03 |
15 | 92.04 | 88.62 | 42.53 | 279.47 | 167.02 | 194.94 | 145.39 |
16 | 82.06 | 89.10 | 46.14 | 286.88 | 174.75 | 233.18 | 147.88 |
17 | 82.95 | 87.18 | 37.48 | 283.59 | 174.78 | 220.78 | 163.24 |
18 | 67.45 | 82.64 | 41.80 | 301.94 | 174.79 | 255.81 | 195.57 |
19 | 80.67 | 65.14 | 52.06 | 301.99 | 174.67 | 205.70 | 189.77 |
20 | 81.28 | 73.19 | 46.97 | 298.12 | 174.14 | 207.39 | 168.92 |
21 | 63.90 | 45.61 | 55.46 | 293.73 | 172.57 | 164.91 | 113.82 |
22 | 62.11 | 44.16 | 58.15 | 292.57 | 151.22 | 170.66 | 81.12 |
23 | 57.92 | 51.51 | 59.50 | 290.80 | 161.94 | 123.38 | 104.96 |
24 | 57.53 | 60.00 | 41.41 | 273.20 | 148.22 | 150.07 | 69.56 |
Table 13.
Scheduling process of hydropower station (unit: 10,000 m3).
Table 13.
Scheduling process of hydropower station (unit: 10,000 m3).
Hour | Q1 | Q2 | Q3 | Q4 | V1 | V2 | V3 | V4 |
---|
1 | 5.79 | 6.13 | 25.46 | 6.03 | 104.21 | 81.87 | 152.64 | 116.77 |
2 | 9.31 | 6.21 | 21.53 | 6.00 | 103.90 | 83.65 | 139.30 | 113.17 |
3 | 7.80 | 6.56 | 24.58 | 6.00 | 104.10 | 86.09 | 124.51 | 108.77 |
4 | 5.93 | 6.04 | 18.07 | 6.00 | 105.17 | 89.06 | 123.88 | 102.77 |
5 | 5.20 | 6.32 | 19.03 | 6.00 | 105.97 | 90.73 | 121.86 | 122.24 |
6 | 7.67 | 6.00 | 17.28 | 6.01 | 105.30 | 91.73 | 121.07 | 137.76 |
7 | 8.87 | 6.12 | 17.58 | 11.18 | 104.43 | 91.61 | 117.73 | 151.16 |
8 | 9.29 | 6.09 | 18.34 | 11.52 | 104.14 | 92.52 | 115.39 | 157.71 |
9 | 10.47 | 6.20 | 17.19 | 17.15 | 103.67 | 94.32 | 114.06 | 159.59 |
10 | 8.89 | 9.22 | 18.00 | 16.98 | 105.78 | 94.10 | 112.47 | 159.89 |
11 | 10.36 | 9.53 | 16.95 | 17.53 | 107.42 | 93.57 | 113.08 | 159.94 |
12 | 10.67 | 8.39 | 16.19 | 19.73 | 106.75 | 93.18 | 113.97 | 158.56 |
13 | 11.08 | 8.25 | 17.05 | 19.16 | 106.67 | 92.94 | 120.50 | 156.58 |
14 | 9.30 | 9.48 | 17.72 | 15.84 | 109.37 | 92.45 | 125.98 | 158.75 |
15 | 10.66 | 11.99 | 16.32 | 16.39 | 109.71 | 89.46 | 132.12 | 159.31 |
16 | 8.70 | 12.65 | 15.72 | 17.27 | 111.01 | 84.81 | 135.95 | 158.23 |
17 | 8.80 | 12.97 | 19.02 | 16.98 | 111.20 | 78.84 | 139.07 | 158.31 |
18 | 6.55 | 12.81 | 18.17 | 19.61 | 112.66 | 72.03 | 143.60 | 156.41 |
19 | 8.37 | 9.57 | 14.28 | 19.95 | 111.29 | 69.46 | 151.77 | 152.79 |
20 | 8.52 | 12.00 | 17.90 | 19.94 | 108.77 | 65.45 | 154.39 | 148.58 |
21 | 6.15 | 6.91 | 11.90 | 19.96 | 109.62 | 67.54 | 165.67 | 147.64 |
22 | 5.92 | 6.46 | 12.95 | 19.94 | 111.70 | 70.08 | 172.81 | 145.87 |
23 | 5.39 | 7.34 | 13.48 | 19.96 | 115.31 | 70.74 | 178.48 | 140.19 |
24 | 5.31 | 8.74 | 21.31 | 18.09 | 120.00 | 70.00 | 170.00 | 140.00 |
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