Joint Scheduling Optimization of a Short-Term Hydrothermal Power System Based on an Elite Collaborative Search Algorithm

Abstract

The joint scheduling optimization of hydrothermal power is one of the most important optimization problems in the power system, which is a non-linear, multi-dimensional, non-convex complex optimization problem, and its difficulty in solving is increasing with the expansion of the grid-connected scale of hydropower systems in recent years. In this paper, three effective improvement strategies are proposed given the shortcomings of the standard collaborative search algorithm, which easily falls into local optimization and weakening of global search ability in later stages. Based on this, an elite collaborative search algorithm (ECSA) coupled with three improvement strategies is established. On this basis, taking the classic joint scheduling problem of a hydrothermal power system as an example, the optimization model with the goal of the least pollutant gas emission is constructed, and the system constraint treatment method is proposed. In addition, five algorithms, i.e., ECSA, CSA, PSO, GWO, and WOA are used to solve the model, respectively. Through the comparison of results, taking the median as an example, the emission of polluting gases of ESCA is reduced by about 1.8%, 13.1%, 38.2%, and 11.2%, respectively, and it can be found that ECSA has obvious advantages in the convergence speed and quality compared with the other four algorithms, and it has a strong ability for global search and jumps out of the local optimal.

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. 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.

.

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.

$$F=\min {\displaystyle \sum _{i=1}^{N_S}{\displaystyle \sum _{t=1}^T\left({\alpha }_i+{\beta }_i{P}_{sit}+{\gamma }_i{P}_{sit}^2+{\delta }_i{e}^{{\lambda }_i{P}_{sit}}\right)}}$$

(1)

where ${\alpha }_i$, ${\beta }_i$, ${\gamma }_i$, ${\delta }_i$ and ${\lambda }_i$ are polluting gas emission factors of the thermal power unit I. $P_{sit}$ 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

$${\displaystyle \sum _{i=1}^{N_S}{P}_{si}+}{\displaystyle \sum _{j=1}^{N_h}{P}_{hj}=P_{Dt}}$$

(2)

$$P_{hj,t}=c_{1,j}\cdot V_{hj,t}^2+c_{2,j}\cdot Q_{hj,t}^2+c_{3,j}\cdot V_{hj,t}^{}\cdot Q_{hj,t}^{}+c_{5,j}\cdot V_{hj,t}^{}+c_{5,j}\cdot Q_{hj,t}^{}+c_{6,j}$$

(3)

where $P_{si}$ is the output of the thermal power unit i; $P_{hj}$ is the output of the hydropower station j; $P_{Dt}$ is the total load demand of the system; $N_S$ is the number of thermal power units; $N_h$ is the number of hydropower stations; $c_{i,j}$ is the output algorithm parameter of the hydropower station; and $Q_{hj,t}^{}$ is the outflow of the power station hj in the period t;

(2) Unit output constraint

$$P_{si}^{\min }\le P_{si}^{}\le P_{si}^{\max }$$

(4)

where $P_{si}^{\min }$ is the minimum output power of the thermal power unit i, and $P_{si}^{\max }$ is the maximum output power of the thermal power unit i.

(3) Unit climbing constraints

$$\left\{\begin{array}{l}{P}_{si,t}-P_{si,t-1}\le U{R}_i\\ P_{si,t-1}-P_{si,t}\le D{R}_i\end{array}\right.$$

(5)

where $U{R}_i$ and $D{R}_i$ are the rate limits of the upward and downward climbing of unit i, respectively, and $P_{si,t}$ is the output for the thermal power unit i at the time t.

2.2.2. Hydropower Constraints

(1) Water balance constraint

$$\left\{\begin{array}{l}{V}_{n,t+1}=V_{i,t}+3600\times {\Delta }_t\times \left[R_{n,t}-O_{n,t}\right]\\ R_{n,t}=r_{n,t}+{\displaystyle \sum _{u=1}^{NU}{O}_{n,t}}\\ O_{n,t}=S_{n,t}+Q_{n,t}\end{array}\right.$$

(6)

where $V_{n,t}$ is the initial storage capacity of the power station n in the t-period; $R_{n,t}$, $O_{n,t}$ and $r_{n,t}$ are the inflow, outflow, and interval flow of the power station n in the period t; $S_{n,t}$ and $Q_{n,t}$ 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

$$O_{n,t}^d\le O_{n,t}\le O_{n,t}^u$$

(7)

where $O_{n,t}^u$ and $O_{n,t}^d$ are the upper and lower limits of the outflow of each power station in the t-period, respectively.

(3) Water level constraints

$$Z_{n,t}^d\le Z_{n,t}\le Z_{n,t}^u$$

(8)

where $Z_{n,t}^u$ and $Z_{n,t}^d$ 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

$$Q_{n,t}^d\le Q_{n,t}\le Q_{n,t}^u$$

(9)

where $Q_{n,t}^u$ and $Q_{n,t}^d$ 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

$$P_{n,t}^d\le P_{n,t}\le P_{n,t}^u$$

(10)

where $P_{n,t}^u$ and $P_{n,t}^d$ 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

$$\left\{\begin{array}{l}{Z}_{n,0}=Z_n^{begin}\\ Z_{n,T}=Z_n^{end}\hspace{0.33em}\hspace{0.33em}\end{array}\right.$$

(11)

where $Z_n^{begin}$ and $Z_n^{end}\hspace{0.33em}\hspace{0.33em}$ 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

$$x_{i,j}^k=\varphi (x_u,x_d),i\in [1,I],j\in [1,J],k=1$$

(12)

where I is the number of populations; J is the problem dimension; $x_{i,j}^k$ is the position of the ith individual, the jth dimension at the k iteration; $x_u$ is the upper limit of the position, $x_d$ is the lower limit of the position; and $\varphi (l,u)$ represents the number of uniformly distributed random numbers over the interval $[l,u]$.

(2) Team communication

$$u_{i,j}^{k+1}=x_{i,j}^k+A_{i,j}^k+B_{i,j}^k+C_{i,j}^k,i\in [1,I],j\in [1,J],k=\in [1,K]$$

(13)

$$A_{i,j}^k=\log (\frac{1}{\varphi (0,1)})\times (gbes{t}_{ind,j}^k-x_{i,j}^k)$$

(14)

$$B_{i,j}^k=\alpha \times \varphi (0,1)\times (\frac{1}{M}{\displaystyle \sum _{m=1}^{M}gbes{t}_{m,j}^k-x_{i,j}^k})$$

(15)

$$C_{i,j}^k=\beta \times \varphi (0,1)\times (\frac{1}{I}{\displaystyle \sum _{i=1}^{I}pbes{t}_{i,j}^k-x_{i,j}^k})$$

(16)

where $u_{i,j}^{k+1}$ is the position of the population at the k + 1 iteration; $A_{i,j}^k$ is A randomly selected individual position from an outside elite group; $B_{i,j}^k$ and $C_{i,j}^k$ are the stochastic average of the individual optimal solution and the global optimal solution, respectively; $gbest$ is the global optimal position, and $pbest$ 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

$$v_{i,j}^{k+1}=\left\{\begin{array}{l}{r}_{i,j}^{k+1} if (u_{i,j}^{k+1}\ge c_j)\\ p_{i,j}^{k+1} if (u_{i,j}^{k+1}<c_j)\end{array}\right\}$$

(17)

$$r_{i,j}^{k+1}=\left\{\begin{array}{l}\varphi (x_{u,j}^{}+x_{d,j}^{}-u_{i,j}^{k+1},c_j) if(\left|u_{i,j}^{k+1}-c_j\right|<\varphi (0,1)\times \left|x_{u,j}^{}-x_{d,j}^{}\right|)\\ \varphi (x_{d,j}^{},x_{u,j}^{}+x_{d,j}^{}-u_{i,j}^{k+1}) otherwise\end{array}\right\}$$

(18)

$$p_{i,j}^{k+1}=\left\{\begin{array}{l}\varphi (c_j,x_{u,j}^{}+x_{d,j}^{}-u_{i,j}^{k+1}) if(\left|u_{i,j}^{k+1}-c_j\right|<\varphi (0,1)\times \left|x_{u,j}^{}-x_{d,j}^{}\right|)\\ \varphi (x_{u,j}^{}+x_{d,j}^{}-u_{i,j}^{k+1},x_{d,j}^{}) otherwise\end{array}\right\}$$

(19)

$$c_j^{}=(x_u+x_d)\times 0.5$$

(20)

where $x_{u,j}^{}$ and $x_{d,j}^{}$ are the upper and lower limits of the j-dimensional position; $v_{i,j}^{k+1}$ is the position of the population at the (k + 1)th iteration.

(4) Internal competition

$$x_i^{k+1}=\left\{\begin{array}{l}{u}_{i,j}^{k+1} if(f(u_i^{k+1})\le f(v_i^{k+1}))\\ v_{i,j}^{k+1} if(f(u_i^{k+1})>f(v_i^{k+1}))\end{array}\right\}$$

(21)

where $f(g)$ 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 c_j parameter

Focusing on the problem that the c_j 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:

$$c_j^{}=(x_u+x_d)\times rand$$

(22)

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:

$$x_{h(i),j}^{k+1}=pbes{t}_{i,j}+rand\times (gbes{t}_j-x_{r2,j}^k)$$

(23)

$$h=randperm(R)$$

(24)

$$R=⌈p\times I⌉$$

(25)

$$p=a1\times \frac{k}{K}+a2$$

(26)

where $randperm(R)$ 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 $\in$ [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:

$$L_{{}_{i,j}}^k=x_{{}_{i,j}}^k+l1\cdot (x_{{}_{i,j}}^k-gbest)\times levy(\gamma )$$

(27)

$$levy(\gamma )=\frac{\sigma \times \upsilon }{{\left|\nu \right|}^{1/\beta }}$$

(28)

$$\sigma ={\left\{\frac{\Gamma (1+\beta )\times \sin (\frac{\pi \times \beta }{2})}{\beta \times \Gamma (\frac{\beta }{2})\times 2^{\frac{\beta -1}{2}}}\right\}}^{\frac{1}{\beta }}$$

(29)

where $L_{{}_{i,j}}^{k+1}$ is the position of the j-dimension of the ith individual in the kth iteration process; $\beta$ is a constant, $\beta$ = 1.5; l1 is the step parameter, l1 = 0.001; $\nu ,\upsilon$ are the random numbers that follow a normal distribution; and $\Gamma$ 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 x^k.

(3)

Calculate the fitness value of all individuals in x^k, 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 x^k, 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 x^k.

(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. 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

; 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. 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

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 F₁, multimodal function F_2, and mixed function F₃, 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. Water flow lag data.

Plant	1	2	3	4
Ru	0	0	2	1
td	2	3	4	0

Note: no of upstream plants; t_d: time delay to immediate downstream plant.

. The basic parameters of the thermal power unit are shown in

Table 5. Basic parameters of thermal power units.

Unit	UR (MW)	DR (MW)	${\mathit{P}}_{\mathit{s}}^{\mathit{m}\mathit{i}\mathit{n}}$(MW)	${\mathit{P}}_{\mathit{s}}^{\mathit{m}\mathit{a}\mathit{x}}$(MW)
1	40	40	20	175
2	60	60	40	300
3	115	115	50	500

, and the algorithm coefficient of polluting gas emissions is shown in

Table 6. Gas emission parameters of thermal power units.

Unit	${\mathit{\alpha }}_{\mathit{i}} \left(\mathbf{kg}\right)$	${\mathit{\beta }}_{\mathit{i}} (\mathbf{kg}/\mathbf{MWh})$	${\mathit{\gamma }}_{\mathit{i}} (\mathbf{kg}/(\mathbf{MW}{)}^2\mathbf{h})$	${\mathit{\delta }}_{\mathit{i}} (\mathbf{kg}/\mathbf{h})$	${\mathit{\lambda }}_{\mathit{i}} (1/\mathbf{MW})$
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

. 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. 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

and

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	${\mathit{P}}_{\mathit{h}}^{\mathit{m}\mathit{i}\mathit{n}}$	${\mathit{P}}_{\mathit{h}}^{\mathit{m}\mathit{a}\mathit{x}}$
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

. The interval inflow and system load requirements are shown in

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

.

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 P_s as the decision variable, and the initialization expression is as follows:

$$X=\left[\begin{array}{l}{Q}_{1,1} ,Q_{1,2} \cdots \cdots Q_{1,T} \\ ⋮\\ Q_{4,1} ,Q_{n,2} \cdots \cdots Q_{n,T}\\ P_{s1,1} ,P_{s1,2} \cdots \cdots P_{s1,T}\\ ⋮\\ P_{s3,1} ,P_{s3,2} \cdots \cdots P_{s3,T}\end{array}\right]$$

(30)

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:

$$\left\{\begin{array}{l}{P}_{si,t}=U{R}_i+P_{si,t-1} if P_{si,t}-P_{si,t-1}\le U{R}_i\\ P_{si,t-1}=D{R}_i+P_{si,t} if P_{si,t-1}-P_{si,t}\le D{R}_i\end{array}\right.$$

(31)

(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:

$$\left\{\begin{array}{l}\stackrel{-}{P_{si,t}}=P_{si}^{\max }-P_{si,t}\\ \underset{-}{P_{si,t}}=P_{si,t}-P_{si}^{\min }\end{array}\right.$$

(32)

where $\stackrel{-}{P_{si,t}}$ and $\underset{-}{P_{si,t}}$ are the upward and downward slack amounts of unit i at time t, respectively.

If ${\displaystyle \sum _{i=1}^{N_S}{P}_{si}}>P_{Dt}$, the infeasible solution vector $P_s$ violates the load balancing constraint to a degree of ${\displaystyle \sum _{i=1}^{N_S}{P}_{si}}-P_{Dt}$. 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:

$$\Delta P_{si,t}=\frac{{\displaystyle \sum _{i=1}^{N_S}{P}_{si}}-P_{Dt}}{{\displaystyle \sum _{i=1}^{N_S}\underset{-}{P_{si,t}}}}\times \underset{-}{P_{si,t}}$$

(33)

The values of the feasible solution variables after adaptive adjustment are

$$P_{si,t}=P_{si,t}-\Delta P_{si,t}$$

(34)

Conversely, if ${\displaystyle \sum _{i=1}^{N_S}{P}_{si}}<P_{Dt}$, calculate the degree of violation of its constraints by $P_{Dt}-{\displaystyle \sum _{i=1}^{N_S}{P}_{si}}$, and calculate the size of the additional load it can bear as:

$$\Delta P_{si,t}=\frac{P_{Dt}-{\displaystyle \sum _{i=1}^{N_S}{P}_{si}}}{{\displaystyle \sum _{i=1}^{N_S}\stackrel{-}{P_{si,t}}}}\times \stackrel{-}{P_{si,t}}$$

(35)

The values of the feasible solution variables after adaptive adjustment are:

$$P_{si,t}=P_{si,t}+\Delta P_{si,t}$$

(36)

(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 $\Delta$. If $\Delta$ < e1 or t1 > Num1, then the first stage is completed, and go directly to step 5. If not, enter step 3.

$$\Delta =V_{i,T}-V_{end}$$

(37)

(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.

$$\begin{array}{l}{Q}_{i,t}=Q_{i,t}+eV\\ eV=\frac{\Delta }{N_h}\end{array}$$

(38)

(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 $\Delta$. 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 $\Delta$. If $\Delta$ < 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 = N_h, 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. 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

, 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. 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

, 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. 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

Note: Ph1 indicates the output of hydropower unit 1: Ps1 indicates the output of thermal power unit 1.

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. 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

Note: Q1 represents the outbound flow of reservoir 1: V1 stands for reservoir 1 capacity.

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.