Research on Economic Operation of Cascade Small Hydropower Stations Within Plants Based on Refined Efficiency Models

Abstract

In order to enhance the overall power generation efficiency of cascade hydropower, it is essential to conduct modelling optimization of its in-plant operation. However, existing studies have devoted minimal attention to the detailed modelling of turbine operating performance curves within the in-plant economic operation model. This represents a significant challenge to the practical application of the optimization results. This study presents a refined model of a hydraulic turbine operating performance curve, which was established by combining a particle swarm optimization (PSO) algorithm and a backpropagation (BP) neural network. The model was developed using a cascade small hydropower group as an illustrative example. On this basis, an in-plant economic operation model of a cascade small hydropower group was established, which is based on the principle of ’setting electricity by water’ and has the goal of maximizing power generation. The model was optimized using a genetic algorithm, which was employed to optimize the output of the units. In order to ascertain the efficacy of the methodology proposed in this study, typical daily operational scenarios of a cascade small hydropower group were selected for comparison. The results demonstrate that, in comparison with the actual operational strategy, the proposed model and method enhance the total output by 3.38%, 2.11%, and 3.56%, respectively, across the three typical scenarios. This method enhances the efficiency of power generation within the cascade small hydropower group and demonstrates substantial engineering application value.

1. Introduction

Driven by the “dual carbon goal”, clean energy construction is accelerating [1,2,3]. As the power system expands, wind and solar energy is integrated on a larger scale, and electricity market reforms deepen, the scale, operations, and coordination of hydropower systems undergo significant changes [4,5,6]. This has led to a pattern of parallel operation of multiple power sources and complementary multi-energy collaboration. These changes have posed challenges to hydropower production, expanding the operating range of units, increasing the frequency of unit starts and stops, intensifying vibration and wear of mechanical components in control systems, and raising the cost of electricity generation. Small hydropower, particularly cascade small hydropower stations, is an important component of hydropower. The scale of their development and construction will continue expanding well into the future [7]. Increasing the utilization of small hydropower can yield significant benefits. For instance, A.M. Rodríguez-Pérez et al. [8]. demonstrated the substantial impact of generating electricity by recovering pressure from irrigation systems, achieving remarkable results. Optimizing the operation of plant units in cascade small hydropower stations holds practical value in improving hydropower energy efficiency, promoting green and sustainable development of new power systems, and achieving “dual carbon goals”. The optimization of the economic operation within hydropower stations depends significantly on the hydraulic turbine operating performance curve, as it provides performance information of the units under different operating conditions. The accuracy and reliability of fitting the hydraulic turbine operating performance curve significantly affect the final outcome of the optimization. However, during actual operation, the turbine is affected by various factors, and the relationships between different parameters are extremely complex, making it currently impossible to precisely describe them with a single complete mathematical formula [9,10]. The backpropagation (BP) neural network has strong nonlinear mapping capabilities [11], which enables it to effectively capture the complex relationships between various parameters of the hydraulic turbine. By training the BP neural network, a more realistic hydraulic turbine operating performance curve model can be established. Junyi Li et al. [12] employed BP neural networks to model the nonlinear behavior of hydraulic turbines, creating a neural network representation for torque and flow characteristics, which improved the accuracy and reliability of real-time simulations.

Integrating other algorithms with BP neural networks enhances their fitting accuracy, leading to more precise representations of hydraulic turbines’ nonlinear characteristics. Baonan Liu et al. [13] proposed a method for processing pump-turbine comprehensive characteristic curves based on an improved BP neural network and logarithmic curve projection, making the processed curves consistent with the measured transient process trends and extreme values on site.

The aforementioned studies focus on modeling the characteristic curves of hydraulic turbine models, while there is less research on modeling the operating performance curves of prototype turbines.

Currently, the utilization efficiency of water energy in small hydropower is generally low. For the hydropower generator itself, improving power generation efficiency is technically challenging. Optimizing the operation strategy of the hydraulic turbines and achieving the optimal output allocation between units, thereby ensuring that the units operate stably in the efficient zone for a long period, are effective ways to improve power generation efficiency. The commonly used load allocation methods for units in power stations include the basic allocation method, the optimal point load method, and proportional distribution. Although these methods are simple to operate and easy to implement, they do not guarantee optimal power generation efficiency. In terms of theoretical research, Xiaoyu Wang et al. [14] proposed a multi-colony ant optimization algorithm integrated with dynamic economic distribution, effectively improving the economic performance of operations in hydropower plants. Jiao Zheng et al. [15] proposed a finite adaptive genetic algorithm to address the internal economic operation of hydropower stations, achieving the dual goals of maintaining population diversity and enhancing the convergence speed of the algorithm. Zhe Yang et al. [16] proposed a discrete hybrid frog-leaping algorithm for the traditional hybrid frog-leaping algorithm and validated the effectiveness of this method in solving the short-term economical operation of hydropower stations. Hongxue Zhang et al. [17] considered the integrated ecological water demand in the optimization scheduling model of hydropower stations and established an ecological operation model. By optimizing through a multi-objective ecological operation model, they achieved coordinated development of the economy and ecology. Tatiana Myateg et al. [18] proposed a new method based on a water inflow probability model, considering environmental and economic indicators, and verified the economic feasibility of hydropower plants that are more economical and environmentally friendly. QiaoFeng Tan et al. [19] established an optimization scheduling model for cascade hydropower stations by fitting the potential energy utility function with a quadratic function. The above research has made improvements and innovations in the strategies for the economic operation of hydropower stations from both the algorithm and model perspectives. However, there has been limited research on the fine-tuned modeling of the operational characteristics of prototype units in the economic operation of hydropower stations, which limits the practical application of the theory.

To address the above issues, this study takes cascade small hydropower stations in China as an example and combines the PSO algorithm with the BP neural network to establish a refined model of the hydraulic turbine’s operating performance curve. Based on this, following the principle of “water determines electricity”, with the objective of maximizing power generation, an inner-plant economic operation model for the cascade small hydropower stations was established. The output of the units was optimized using a genetic algorithm. Finally, the proposed method’s effectiveness and accuracy were validated using field measurement data from the hydropower station’s monitoring system across three typical scenarios.

2. Refined Model of the Hydraulic Turbine’s Operating Performance Curve

The refined model of the hydraulic turbine’s operating performance curve forms the foundation for studying the nonlinear dynamics of hydropower units and their optimal control strategies. It can also provide an accurate model that meets engineering requirements for real-time simulation of the units under normal or special operating conditions. To address the economic operation of cascade small hydropower stations and achieve more accurate fitting of the hydraulic turbine’s operating performance curve, it is necessary to conduct refined modeling of the hydraulic turbine’s operating performance curve.

For the fitting of the operating performance curve of hydraulic turbines, there are generally several methods: interpolation, function fitting, and intelligence algorithm fitting. Due to the powerful nonlinear fitting capability of neural networks, intelligence algorithms typically use neural networks or improved neural networks for fitting. The characteristics of different fitting methods are shown in

Table 1. Characteristics of different fitting methods.

Methods	Characteristics
Interpolation	Simple structure, few model parameters, high accuracy but poor generalization, high-degree interpolation prone to overfitting.
Function fitting	Simple structure, few model parameters, but poor generalization and accuracy.
BP neural network	Moderate complexity and number of model parameters, with average generalization capability and accuracy.
Fuzzy logic models	Complex structure, numerous model parameters, moderate generalization and accuracy; model quality depends on the design of fuzzy rules.
ANN-based (Artificial Neural Network)	Complex structure, numerous model parameters, moderate generalization capability, but high demands on data volume and computational resources.
PSO-BP neural network	Moderate complexity and number of model parameters, with good generalization capability and high accuracy.

.

This section employs the PSO algorithm to enhance the initial parameters of the BP neural network, addressing the issue of traditional BP neural networks easily falling into local optima due to improper weight and threshold initialization, and performs refined modeling of the prototype hydraulic turbine’s operating performance.

2.1. Basic Principles of BP Neural Network

The BP neural network, a well-established multilayer feedforward network, is trained using the error backpropagation algorithm to model complex nonlinear problems. The basic structure includes an input layer, hidden layer, and output layer [20], as illustrated in Figure 1.

The layers (input, hidden, and output) are interconnected through weighted summation and activation functions. The relationship between the input layer and hidden layer can be expressed using Equations (1) and (2).

$$h_j=\sum _{i=1}^n{x}_i{\omega }_{ij}^{21}+b_j^{21}$$

(1)

$$o{h}_j=f\left(h_j\right)$$

(2)

where n is the number of input nodes, j is the nodes in the hidden layer, $x_i$ is the nodes in the input layer, ${\omega }_{ij}^{21}$ is the weight from the input layer to the hidden layer, $b_j^{21}$ is the threshold from the input layer to the hidden layer, $o{h}_j$ is the output of each hidden layer node, and the function f is the activation function of the hidden layer.

The relationship between the hidden layer and the output layer can be expressed by Equations (3) and (4).

$$y_k=\sum _{j=1}^{m}o{h}_j{\omega }_{jk}^{32}+b_k^{32}$$

(3)

$$o{y}_k=g\left(y_k\right)$$

(4)

where k is the number of output layer nodes, ${\omega }_{jk}^{32}$ is the weight from the hidden layer to the output layer, $b_k^{32}$ is the threshold from the hidden layer to the output layer, m is the number of hidden layer nodes, $o{y}_k$ is the output of each output layer node, and the function g is the activation function of the output layer.

The mean squared error function is used as the error calculation function for the BP neural network to evaluate the predictive accuracy of the model, as shown in Equation (5). The smaller the value, the more accurate the model’s prediction.

$$MES={\displaystyle \frac{1}{s}}\sum _{o=1}^s{\left(y_o-\widehat{y_o}\right)}^2$$

(5)

where $y_o$ is the true value of the o-th sample, $\widehat{y_o}$ is the predicted value of the o-th sample, and s is the number of samples.

2.2. Refined Model of the Hydraulic Turbine’s Operating Performance Curve Based on PSO-BP Neural Network

The PSO algorithm is a global optimization algorithm based on simulating the foraging behavior of bird flocks. By sharing information among individuals, each particle can explore the optimal solution within the search space. The algorithm is computationally simple and converges quickly, making it widely used in various optimization problems [21].

This study integrates the PSO algorithm with the traditional BP neural network, thereby enhancing the complexity of the algorithm. Specifically, the PSO algorithm is employed to conduct a global search for the initial parameters of the BP neural network. By leveraging the global search capability of the PSO algorithm, multiple potential network structures are optimized prior to fitting the hydraulic turbine’s operating performance curve. Subsequently, a network with superior performance is selected for the fitting process. Although this method requires more time than the traditional BP neural network approach, the use of intelligent algorithms for large-scale screening significantly reduces the time cost associated with manual selection, resulting in higher overall time efficiency. The specific process is shown in Figure 2.

• Step 1: The hydraulic turbine operating performance curve is read using chart-reading software to obtain a series of training data for the neural network. The input layer variable of the neural network is defined as the turbine output P and the head H. The output layer variable of the neural network is defined as the turbine efficiency η. The number of hidden layer nodes is also defined.
• Step 2: Set the algorithm parameters. Include population size N, inertia weight ω, maximum number of iterations T, and learning factors ${\mathrm{c}}_1$ and ${\mathrm{c}}_2$. Determine the BP neural network’s activation function and the number of hidden layer nodes.
• Step 3: Initialize the population. The position of each particle represents the BP neural network’s weights and thresholds. Randomly initialize the position and velocity of the particles, and set the current iteration count to 1.
• Step 4: Fitness evaluation. The current weight and threshold of each particle are fed into the BP neural network, and the mean square error (MSE) after training the BP neural network for each particle is calculated as the fitness value, as shown in Equation (5).
• Step 5: The historical best is recorded. Record the global best solution in the current particle swarm, and its position is set as the global best position.
• Step 6: Judge whether the convergence condition is met. Check if any of the following conditions are satisfied: the current global best meets the preset accuracy requirements or the maximum number of iterations has been reached. If the conditions are met, end the process and output the global best solution, representing the optimal initial parameters of the BP neural network. If not, proceed to the next step.
• Step 7: Particle position update. According to the particle position update rules, update the velocity and position of each particle, as shown in Equations (6) and (7). Increment the iteration count by 1; then, return to Step 4.

$$v_i\left(t+1\right)=\omega v_i\left(t\right)+c_1{r}_1\left(P_{best,i}-x_i\left(t\right)\right)+c_2{r}_2\left(G_{best}-x_i\left(t\right)\right)$$

(6)

$$x_i\left(t+1\right)=x_i\left(t\right)+v_i\left(t+1\right)$$

(7)

where $v_i\left(t\right)$ is the velocity of particle i at the t-th generation, $x_i\left(t\right)$ is the position of particle i at the t-th generation, $r_1$ and $r_2$ are random numbers generated within the range of [0, 1], $P_{best,i}$ is the best position of particle i, and $G_{best}$ is the best position of the entire population.

3. The In-Plant Economic Operation Model

According to the role of the hydropower station in the power system, there are two main approaches for the economic operation of the hydropower station. One approach is to take the dispatcher’s requirement for the total output of the hydropower station as a known condition, with the objective of minimizing the water consumption of the station, and then allocate the unit load based on this. This follows the “electricity determines water” principle. The other approach is based on the known conditions of the water inflow to the hydropower station and the reservoir water usage plan, with the goal of maximizing the power output of the hydropower station for load distribution, which follows the “water determines electricity” principle [22]. This section establishes the economic operation model of a cascade small hydropower station considering the refined model of the hydraulic turbine’s operating performance curve, based on the “water determines electricity” principle.

3.1. Objective Function

The economic operation objective of the cascade hydropower stations under the “water determines electricity” principle is to maximize the total output, as shown in Equation (8) [23].

$$E=\mathit{max}\sum _{i=1}^m\sum _{j=1}^{n_i}{P}_{ij}\left(Q_{ij},H_{ij}\right)$$

(8)

where E is the total power output of the cascade hydropower stations, kW; m is the number of hydropower stations; $n_i$ is the number of units in the i-th hydropower station; $Q_{ij}$ is the flow rate of the j-th unit in the i-th hydropower station, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$; $H_{ij}$ is the water head of the j-th unit at the i-th hydropower station, m; $P_{ij}$is the output of the j-th unit at the i-th hydropower station, kW.

3.2. Constraints

1.

The reservoir

The current reservoir storage is related to the storage at the previous period, the inflow between the two periods, and the total flow from downstream cascade hydropower stations. They satisfy the water balance equation, as shown in Equation (9) [24]. The current reservoir water level and reservoir storage satisfy a specific nonlinear relationship, as shown in Equation (10).

$$V_R^t=V_R^{t-1}+\left(Q_{R,in}^t-Q_{R,out}^t\right)dt$$

(9)

$$Z_R^t=f_R\left(V_R^t\right)$$

(10)

where $Q_{R,out}^t$ and $Q_{R,in}^t$ are the outflow rate and inflow rate of the cascade reservoir at time period t, respectively, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$; $V_R^t$ and $V_R^{t-1}$ are the volumes of the cascade reservoir at time period t and t − 1, respectively, ${\mathrm{m}}^3$; $Z_R^t$ is the reservoir water level in the cascade at time period t, m; $f_R$ is the relationship curve between the reservoir capacity and water level of the cascade reservoir.

The reservoir water level should be between the maximum and minimum water levels, as shown in inequality (11) [25].

$$Z_{R,min}\le Z_R^t\le Z_{R,max}$$

(11)

where $Z_{R,min}$ and $Z_{R,max}$ are the upper and lower water level limits of the cascade hydropower station reservoir, respectively, m.

2.

The cascade hydropower station units

The total flow of the units in the cascade hydropower station is related to the water used for power generation and the spilled water of each unit. They satisfy the flow balance equation, as shown in Equation (12).

$$Q_{S_i}^t=\sum _{j=1}^2{Q}_{S_i,j}^t+Q_{S_i,a}^t=Q_{R,out}^t$$

(12)

where $Q_{S_i}^t$ is the total flow rate of the i-th station at time t, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$; $Q_{S_i,j}^t$ and $Q_{S_i,a}^t$ are the power generation flow rate and the total spilled flow rate of the i-th station at the t-th time period, respectively, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$; $Q_{R,out}^t$is the outflow rate of the cascade hydropower station reservoir or the weir between the cascade hydropower station and the upstream station at time t, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$.

The head of the cascade hydropower station units is related to the plant site water level of the cascade hydropower station and the reservoir water level of the cascade hydropower station or the water level of the weir between the cascade hydropower station and the upstream station. They satisfy the balance equations, as shown in Equations (13) [23] and (14).

$$H_{S_i}^t=Z_{R_i}^t-Z_{S_i}^t-∆h_i$$

(13)

$$∆h_i=a{\left(Q_{S_i}^t\right)}^2$$

(14)

where $H_{S_i}^t$ is the actual head of the i-th station, m; $∆h_i$ is the head loss of the i-th cascade hydropower station, m; $a$ is the head loss coefficient of the water diversion system of the i-th cascade hydropower station.

The output of a single unit in the cascade hydropower station should be between the maximum output and the minimum output, as shown in in Equation (15) [23].

$$P_{ij,min}\le P_{ij}\le P_{ij,max}$$

(15)

where $P_{ij,min}$ and $P_{ij,max}$ are the upper and lower output limits of the j-th unit in the i-th station, respectively, kW.

The actual head of the unit in the cascade hydropower station should be between the maximum head and the minimum head, as shown in Equation (16).

$$H_{S_i,min}\le H_{S_i}^t\le H_{S_i,max}$$

(16)

where $H_{S_i,min}$ and $H_{S_i,max}$ are the upper and lower limits of the head for the i-th station, m.

3.

The plant site of the cascade hydropower station

The water level at the plant site of the cascade hydropower station is related to the total flow of the station through a specific nonlinear relationship, as shown in Equation (17).

$$Z_{S_i}^t=f_{S_i}\left(Q_{S_i}^t\right)$$

(17)

where $f_{S_i}$ is the flow water level relationship curve for the i-th station; $Z_{S_i}^t$ is the plant site water level of the i-th station.

The plant site water level of the cascade hydropower station should be between the maximum and minimum water levels, as shown in in Equation (18) [25].

$$Z_{S_i,min}\le Z_{S_i}^t\le Z_{S_i,max}$$

(18)

where $Z_{S_i,min}$ and $Z_{S_i,max}$ are the upper and lower limits of the plant site water level for the cascade hydropower station, m.

4.

The weirs

There may be weirs between cascade small hydropower stations, and their inflow flow rate is related to the interval inflow between the cascade hydropower stations and the total flow rate of the upstream hydropower station. They satisfy the flow balance equation, as shown in Equation (19) [26]. The water level of the weir satisfies a nonlinear relationship, as shown in Equation (20).

$$Q_{R_{i+1},in}^t=Q_{li}^t+Q_{Si}^t$$

(19)

$$Z_{R_{i+1}}^t=f_{R_{i+1}}\left(Q_{R_{i+1},in}^t\right)$$

(20)

where $Q_{R_{i+1},in}^t$ is the inflow rate of the weir between the i-th and (i + 1)-th stations, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$; $Q_{li}^t$ is the inflow rate between the i-th and (i + 1)-th stations, ${\mathrm{m}}^3/\mathrm{ }\mathrm{s}$; $f_{R_{i+1}}$ is the flow water level relationship curve for the weir between the i-th and (i + 1)-th stations; $Z_{R_{i+1}}^t$ is the water level of the weir between the i-th and (i + 1)-th stations, m.

The water level of the weir between the power stations should be between the maximum and minimum water levels, as shown in in Equation (21) [25].

$$Z_{R_{i+1},min}\le Z_{R_{i+1}}^t\le Z_{R_{i+1},max}$$

(21)

where $Z_{R_{i+1},min}$ and $Z_{R_{i+1},max}$ are the upper and lower limits of the plant site water level of the cascade hydropower station, respectively, m.

3.3. Solution Method

This section mainly introduces the solution of the cascade hydropower station plant economic operation model with a daily scheduling period and an hourly calculation time interval. As the plant economic operation model contains nonlinear functions reflecting the operating characteristics of the hydraulic turbines, a genetic algorithm is employed for solving. The initial state of the system should first be determined. The initial reservoir capacity and active power of the units for the initial period are obtained from the actual signals of the hydropower station’s monitoring system. The initial reservoir capacity is then input into the mathematical model to obtain the reservoir water level for the initial period. After completing the calculation of the system’s initial state, the goal is to maximize the power generation of the cascade hydropower station. The unit flow and spilled flow are set as the optimization variables. The basic process of solving the plant economic operation model using the genetic algorithm is shown in Figure 3. The detailed steps are as follows.

• Step 1: Set the algorithm parameters. Include population size ${\mathrm{N}}_{\mathrm{G}\mathrm{A}}$, maximum iteration count ${\mathrm{T}}_{\mathrm{G}\mathrm{A}}$, crossover probability ${\mathrm{p}}_{\mathrm{c}}$, and mutation probability ${\mathrm{p}}_{\mathrm{m}}$.
• Step 2: Determine whether the economic operation of the cascade hydropower stations in the basin has been optimized for the final time period. If optimized, output the optimal solutions for each time period; otherwise, proceed to Step 3.
• Step 3: Initialize the population. Each individual in the population represents the possible generation flow rates of all units ${\mathrm{Q}}_{\mathrm{i}\mathrm{j}}$. Record the current time period: $\mathrm{t}=0$.
• Step 4: Calculate the fitness evaluation of each individual in the initial population, representing the power output of units. The steps inside the dashed box in Figure 3 represent the process.
• Step 5: The individual with the highest fitness value is identified and recorded. This individual is then compared with the historical best individual, and updates are made based on the fitness value.
• Step 6: Check whether the maximum number of iterations has been reached. If it has, stop the calculation, output the optimal solution, and proceed to Step 11. Otherwise, proceed to Step 7.
• Step 7: Selection, crossover, and mutation operations generate offspring; the offspring are recombined with the previous generation to form the new population.
• Step 8: Calculate the fitness of the new population. Compute the fitness of each individual in the new population.
• Step 9: Update the historical best individual information: If an individual in the new population has a higher fitness than the previous best individual, update the historical best individual; otherwise, retain the historical best individual.
• Step 10: Increment the iteration count and return to Step 6.
• Step 11: Calculate the initial state of the next period for the cascade hydropower station. Use the output optimal solution, the hydrological relationship model, and the dynamic characteristic model of the cascade hydropower station to calculate the initial state for the next period. At the same time, increment the current period number by 1 and proceed to Step 3 for iteration.

4. Case Study Analysis

The object of this study is a cascade small hydropower station in Songyang County, Zhejiang Province, China, as shown in Figure 4.

The power station belongs to a single-basin small hydropower cluster, consisting of reservoirs and two cascade power stations. A water dam is set before the second-stage power station. The hydraulic structure topology of the station is shown in Figure 5, and the specific data of the power station are listed in

Table 2. Parameters of cascade hydropower stations.

Cascade Hydropower Stations	Installed Capacity (MW)	Maximum Head (m)	Minimum Head (m)	Head Loss Coefficient
Primary cascade hydropower station	2 × 8	245.58	194.27	0.1755
Secondary cascade hydropower station	2 × 2	50.2	47.3	0.02596

.

4.1. The Refined Modeling Results of the Hydraulic Turbine’s Operating Performance Curve

The hydraulic turbine’s output power P and head H serve as the input layer of the BP neural network, with the efficiency η of the turbine as the output layer. Thus, the input layer has two nodes, and the output layer has one node. The task is relatively simple. Selecting a single hidden layer can ensure performance while reducing the risk of overfitting. Through trial calculations, the hidden layer is determined to have five nodes, which can avoid overfitting. The hidden layer uses the tansig activation function, which is a hyperbolic tangent function, and the expression is $f(x)={\displaystyle \frac{2}{1+e^{-2x}}}-1$. The output layer uses the purelin activation function, which is a linear function, and the function expression is $g\left(x\right)=x$.

Fit the hydraulic turbine operating performance curves for the primary cascade hydropower station and secondary cascade hydropower station. The training data for the primary cascade hydropower station and secondary cascade hydropower station consist of 266 and 160 sets, respectively. The sample ratios for training, validation, and testing are set to 70%, 15%, and 15%, respectively. The specific algorithm parameters are shown in

Table 3. Parameters of the algorithm.

Parameters	Training Iterations	Number of Hidden Layer Nodes	Training Function	Number of Particles	Maximum Number of Iterations	Inertia Weight	Acceleration on Factors
BP	1000	5	Trainlm	—	—	—	—
PSO-BP	1000	5	Trainlm	30	100	0.6	2

.

A comparison of the fitting results of the traditional BP neural network model and the PSO-BP neural network model for the turbine operating performance curves of the cascade hydropower stations is shown below.

Figure 6 shows the error variation curves of each cascade station during the training process of the two models. For the primary cascade hydropower station, the traditional BP neural network model, after 62 iterations, reaches the best result at the 56th iteration, with the smallest error being $4.2731\times {10}^{-6}$. The PSO-BP neural network model, after 42 iterations, reaches the best result at the 36th iteration, with the smallest error being $1.8912\times {10}^{-6}$. For the secondary cascade hydropower station, the traditional BP neural network model, after 21 iterations, reaches the best result at the 15th iteration, with the smallest error being $8.2376\times {10}^{-7}$. The PSO-BP neural network model, after 136 iterations, reaches the best result at the 136th iteration, with the smallest error being $7.5306\times {10}^{-8}$. Compared to the traditional BP neural network model, the PSO-BP neural network model has smaller fitting result errors and more accurate outcomes.

Figure 7 shows the error curves of sample points for the two models, where the red line represents the traditional BP neural network model and the blue line represents the PSO-BP neural network model. For the primary cascade hydropower station, the two boundaries of the sample point error of the traditional BP neural network model are 0.004898 and −0.009107, respectively, and the variance is $3.783\times {10}^{-6}$. The two boundaries of the PSO-BP neural network model through sample point error are 0.004534 and −0.007303, respectively, and the variance is $2.4347\times {10}^{-6}$. For the secondary cascade hydropower station, the two boundaries of the sample point error of the traditional BP neural network model are 0.002116 and −0.004154, respectively, and the variance is $1.32\times {10}^{-6}$. The two boundaries of the PSO-BP neural network model through sample point error are 0.000794 and −0.000738, respectively, and the variance is $5.90006\times {10}^{-5}$.

Table 4. Comparison of sample point errors.

Cascade Hydropower Stations	Upper Limit of Error (BP/PSO BP)	Lower limit of Error (BP/PSO BP)	Variance (BP/PSO BP)
Primary cascade hydropower station	0.004898/0.004534	−0.009107/−0.007303	$3.783\times {10}^{-6}/2.4347\times {10}^{-6}$
Secondary cascade hydropower station	0.002116/0.000794	−0.004154/−0.000738	$1.32\times {10}^{-6}/5.90006\times {10}^{-5}$

shows the data.

The fitting results of the PSO-BP neural network model and the error range of the samples are generally within the error range of the traditional BP neural network model, with fluctuations being smaller compared to the traditional BP neural network model.

Figure 8 shows the fitting results of the hydraulic turbine operating performance curves for the primary and secondary cascade hydropower units. The refined model, optimized by PSO, provides a more comprehensive coverage of the sample points, resulting in more accurate fitting.

The fitting results of the sample data through the model show that by using the PSO algorithm to initialize the thresholds and weights of the BP neural network, the initial parameters can be better distributed globally, providing a more optimized initial parameter set. This improves the training performance, resulting in higher accuracy and smaller fitting errors and enables better capture of the hydraulic turbine’s operating characteristics, leading to a more precise fitting curve.

Furthermore, comparing the research results of this study with findings from the literature can further demonstrate that the PSO-BP neural network significantly outperforms the standard BP neural network in terms of fitting accuracy. For example, Yuanqi Li et al. [27] demonstrated that, although applied in different scenarios, the PSO-BP neural network achieves superior performance over the BP neural network in photovoltaic power prediction, which aligns with the findings of this study. The global optimization capability of the particle swarm algorithm effectively mitigates the local minima issue inherent in the BP neural network.

4.2. The Optimization Results of the Internal Economic Operation Model

Three typical days from January, July, and October within a year are selected to represent the operation of the hydropower station in different seasons. Based on the known inflow, changes in reservoir storage, and interval inflows, the internal economic operation optimization of the target cascade hydropower station is carried out. The genetic algorithm parameters are shown in

Table 5. Genetic algorithm parameters.

Parameters	Population Size	Individual Length (Byte)	Maximum Number of Iterations	Crossover Probability ${\mathit{p}}_{\mathit{c}}$	Mutation Probability ${\mathit{p}}_{\mathit{m}}$
Genetic algorithm	200	60	100	0.6	0.1

.

The optimization results of the model for the selected typical days at each time period, along with the corresponding changes in reservoir storage and inflow, are shown in Figure 9, Figure 10 and Figure 11.

From the comparison of the output of the cascade hydropower stations before and after optimization in each time period shown in Figure 8, Figure 9 and Figure 10, it can be observed that the output of the primary cascade hydropower unit in the target basin is relatively stable. After optimization, the output of the station is significantly increased, enhancing the power generation efficiency. The output of the secondary cascade hydropower unit fluctuates significantly. Due to the large variation in the inflow of the cascade basin, the power generation flow of the secondary hydropower unit is influenced by the tailwater from the primary cascade hydropower unit, rainfall, and ecological runoff. During the period from 10:00 to 23:00, the station output is relatively low and stable, while from 24:00 to 09:00, the station output is relatively high and stable. After optimization, the overall output of the hydropower station has significantly increased, and the fluctuation of the output during each period has noticeably decreased. The stability has been improved, enhancing the operational reliability of the station.

On the typical day in January, the aggregate electricity production of the cascade hydropower station before optimization was 456.35 MW, which increased to 471.77 MW after optimization, representing a growth of 3.38%. On the typical day in July, the aggregate electricity production of the cascade hydropower station before optimization was 460.21 MW, which increased to 469.92 MW after optimization, representing a growth of 2.11%. On the typical day in October, the aggregate electricity production of the cascade hydropower station before optimization was 456.86 MW, which increased to 473.12 MW after optimization, representing a growth of 3.56%.

Taking the typical day in October as an example, different population sizes, crossover probabilities, and mutation probabilities were selected for optimization. The specific results are shown in

Table 6. Comparison of different genetic algorithm parameters.

Population Size/ Crossover Probability/ Mutation Probability	150/0.6/0.1	250/0.6/0.1	200/0.5/0.1	200/0.6/0.1	200/0.6/0.15	200/0.6/0.05
Generation increase rate	2.93%	3.16%	3.11%	3.35%	3.20%	3.29%

.

It can be seen from Table 6 that keeping the crossover probability and mutation probability unchanged, the lower and higher population sizes compared with this study are selected, and the optimization results are 2.93% and 3.16%, respectively. Keeping the population size and mutation probability unchanged, the low and high crossover probability compared with this study were selected, and the optimization results were 3.11% and 3.35%, respectively. Keeping the population size and crossover probability unchanged, the lower and higher mutation probabilities compared with this study were selected, and the optimization results were 3.20% and 3.29%, respectively. Through a comparison of optimization results, it can be seen that the parameters of the genetic algorithm selected in this study are reasonable.

For the optimization results of the three typical days, they are closely related to the typical daily water distribution, reservoir regulation capacity, abandoned water volume, and unit efficiency of the example. From the changes in reservoir capacity and inflow of cascade hydropower stations on typical days, it can be seen that compared with typical days in January, the inflow of water on typical days in October is larger and more evenly distributed, which enables the units to operate efficiently. Moreover, the changes in reservoir capacity are relatively gentle, indicating that the reservoir has stronger regulation capacity and less abandoned water. Therefore, the optimization results are the best, followed by January. On typical days in July, the water inflow is concentrated in a few periods, which may limit the optimization effect, and the reservoir capacity decreases rapidly and changes greatly, which may affect the efficiency of the units.

5. Conclusions

This study constructs a refined model for the hydraulic turbine operating performance curve and optimizes the initial thresholds and weights of the BP neural network using the PSO algorithm. By leveraging the global search capability of the particle swarm algorithm, the initial thresholds and weights of the BP neural network are more effectively distributed across the global range, providing a more advantageous initial parameter set. The results show that the BP neural network optimized by the PSO algorithm significantly improves the training performance, achieving higher accuracy and smaller fitting errors. This enables more accurate capture of the hydraulic turbine’s operating characteristics and provides high-precision fitting curves for optimizing the internal economic operation of the cascade small hydropower plant.

The operating performance curve of a hydraulic turbine reveals the relationship between power, head, and flow during its operation. In the mathematical model for the economic operation optimization of cascade small hydropower plants, the coupling relationship of these parameters is involved. Calculation results based on an actual hydropower station in a coastal province of southeastern China demonstrate that the internal economic operation model, combined with refined modeling of the unit’s operating performance, can significantly enhance the overall operational efficiency of the station.

This high-precision hydraulic turbine operating performance curve can more accurately reflect the performance changes of the unit under different operating conditions, making the optimization results more valuable for practical application. It not only improves the accuracy of the results but also provides strong support for enhancing unit operating efficiency, reducing energy consumption and operational costs and maximizing economic benefits.

This study demonstrates this economic operation model for small cascade hydropower stations. The calculation results confirm the reliability of the optimized operation, providing a scientific basis for achieving economic operation in small-scale hydropower systems.

Small hydropower stations, with their limited installed capacity, are at a disadvantage in the power market compared to medium and large hydropower stations. However, small hydropower has its unique characteristics. Its flexible operation mode and minimal impact on the grid, due to its small proportion in the power network, often exempt it from grid dispatching. Therefore, it only needs to maximize power generation based on the incoming water flow, a strategy known as “water determines electricity”. This dispatching mode differs from medium and large hydropower stations, which must follow the grid’s power generation plan and aim to minimize water consumption, a strategy referred to as “electricity determines water”. The economic operation model constructed in this study is suitable for small cascade hydropower stations. For medium and large hydropower stations, the economic operation model proposed here has certain limitations, but the refined model of the hydraulic turbine operating performance curve still holds practical application value.

Furthermore, there is still much work to be done in the future. To facilitate practical application, further research on more real-world cases is necessary to explore how to improve the internal economic operation models for small cascade hydropower stations under various conditions.