Hydropower Plant Available Energy Forecasting Using Artificial Neural Network and Particle Swarm Optimization

Abstract

Accurate forecasting of the available energy portion that corresponds to the reservoir inflow of the month(s) ahead provides important decision support for hydropower plants in energy production planning for revenue maximization, as well as for environmental impact prevention and flood control upstream and downstream of a basin. Therefore, a reliable forecasting tool or model is deemed necessary and crucial. Considering the fluctuation and nonlinearity of data which significantly influence the forecasting results, this study develops an effective hybrid model by integrating an Artificial Neural Network (ANN) and Particle Swarm Optimization (PSO) called “PSO-ANN” model based on the hydrological and meteorological data pre-processed by cross-correlation function (CCF), autocorrelation function (AFC), and normalization techniques for predicting the available energy portion corresponding to the reservoir inflow mentioned above for a case study hydropower plant in Laos, namely, the Theun-Hinboun hydropower plant (THHP). The model was evaluated by using correlation coefficient (r), relative error (RE), root mean square error (RMSE), and Taylor diagram plots in comparison with popular single-algorithm approaches such as ANN, and NARX models. The results demonstrated the superiority of the proposed PSO-ANN approach over the other two models, in addition to being comparable to those proposed by previous studies.

1. Introduction

At present, renewable energy (RE) plays a crucial role in clean energy supply in which one of the significant sources of RE is hydropower [1]. In a hydropower plant, accurate forecasting of available energy or reservoir inflow is necessary for ensuring optimal operation of the reservoir as sufficient water needs to be maintained in the reservoir in the dry season, while unplanned or unreasonable spillage needs to be avoided in the wet season. The optimal reservoir operation is directly associated with generation planning that supports revenue maximization, as well as environmental impact prevention and flood control upstream and downstream of a basin [2]. Due to real hydrological and meteorological data having the characteristics of fluctuation and nonlinearity resulting from the extensive influence of human activity and natural factors [3], several studies have given more attention to the methodology of high precision forecasting, of which two fundamental techniques for forecasting, physically based and data-driven models, are widely implemented [4]. The results provided by the physically based models are more explainable compared to those of data-driven models [5]. The existing physically based models, including, but not limited to, CalSim, WEAP, and HEC ResSim, are regarded as powerful tools for forecasting as they are based on the principles of mass and hydrological conservation, and the laws of physics [6]; however, these physical models have some limitations resulting from their requirements for a huge effort in terms of hydrological and climate data, and they are time-consuming in the learning process [7]. Therefore, the data-driven approaches are alternative solutions as these approaches are highly adaptable and able to be constantly optimized and updated hence, promising accurate forecasting [8].

In the past decades, data-driven methods with a single algorithm have gained a lot of attention as a result of their simplicity, such as the Adaptive Neuro-Fuzzy Inference System (ANFIS) model utilized for monthly dam inflow prediction, in which the model with three input variables, i.e., monthly rainfall, dam inflow and rainfall forecast of next month, provided a higher accuracy than the other models [9]; Recurrent Neural Network (RNN) based on radar rainfall data and reservoir inflow data for multi-step-ahead reservoir inflow forecasting outperformed the Back Propagation Neural Network (FFBPNN) [10]; the multi-layer perceptron (MLP) artificial neural network model for one-month-ahead inflow forecasting for the Ubonratana reservoir situated in Thailand revealed a better performance of Type F (with inflow known and assumed to be the forecast) model over the other models [11]; Support Vector Machine (SVM) models that considered different monthly time lags for predicting inflow to Zayandehroud dam reservoir in Iran displayed more accurate results than those of the ANN models [12]; a Bayesian Networks (BNs) model for annual and monthly inflow forecasting for the Zayandehrud dam reservoir in Iran showed more reliability over the other limited research conducted in the study area [13]; the Nonlinear Auto-Regressive with exogenous input (NARX) model, and Nonlinear Auto-Regressive (NAR) model provided good alternative approaches for inflow forecasting, such as the forecast for a case study reservoir in Iran (Sefidruod dam reservoir), which depicted the superiority of the NAR model over the NARX [14]; hydropower generation prediction using SVM, ANN and ARIMA for three Gorges Dam situated in China demonstrated the superiority of the ANN model over the other models with the correlation coefficient (r) between 0.9011 and 0.8883 [15]. After reviewing the literature, some limitations of the single technique were detected, such as the overfitting of ANN, parameter uncertainty of SVM, local minimization drawback of PSO, cognitive uncertainties of Fuzzy logic, and lack of memory in genetic programming [16] especially when employed for the fluctuation and nonstationary data of different hydrological and climatic conditions, which influenced the forecasting accuracy; hence, many researchers developed hybrid models, instead of single-algorithm models, with the aim of improving the accuracy in forecasting [8].

A hybrid model of SARIMA–GEP integrated a Seasonal Autoregressive Integrated Moving Average (SARIMA) and the gene expression programming (GEP) carried out for monthly reservoir inflow forecasting provided more accurate forecasting over single-algorithm models like ANN, GEP, and SARIMA models [17]; a DEL-LSTM model combined a Decomposition-Ensemble Learning (DEL) and Long Short-Term Memory (LSTM) for predicting daily reservoir inflow outperformed the standalone ARIMA, DNN, MDFL, and LSTM [18]; a Hybrid Deep Learning Inflow Prediction–Rolling Window called “HDeepLIP-RW models” provided superiority over several single-algorithm models, namely, ANN, RNN, MLR, GRU, and LSTM [19]; a fusion-based model combined several individual artificial neural networks (ANNs) with moderate-resolution imaging spectroradiometer (MODIS) data increased the accuracy in forecasting monthly reservoir inflow [20]. As an alternative, this study develops effective hybrid models by integrating an Artificial Neural Network (ANN) and Particle Swarm Optimization (PSO) based on the hydrological and meteorological data, properly pre-processed by cross-correlation function (CCF), autocorrelation function (AFC), and normalization techniques, to forecast the available energy portion corresponding to the reservoir inflow mentioned above.

Previously, some researchers adopted a hybrid of Particle Swarm Optimization (PSO) and Artificial Neural Network (ANN)-based models known as “PSO-ANN models” in forecasting works such as electrical load prediction in which the PSO algorithm was used for solving the problem related to day-ahead load shifting and cost saving with demand response (DR) program in combination with the multilayer feed-forward model [21]; a water level prediction by employing the PSO as an optimizer to search for the optimal parameter values for the ANN training process [22]; an application of PSO in optimizing the weights and biases of the ANN to enhance the accuracy and performance of the prediction model for a gas metering system [23]; a preference prediction in Multi-Criteria Recommender System (a powerful online tool that helps overcome problems of information overload) by training the feed-forward neural networks with the Particle Swarm Optimization (PSO) algorithm and then used the neural network as an aggregation function for predicting the preferences of the users [24]; behavior prediction and structural optimization of lightweight sandwich composite heliostats by using the PSO-ANN approach [25]; crack prediction in pipeline using PSO-ANN, whereas the PSO algorithm was employed to enhance ANN training parameters (biases and weights) by minimizing the difference between actual and desired outputs and then using these parameters to generate the network [26]. Based on the aforementioned, we found that the hybrid of ANN and PSO has been effectively employed in several fields of forecasting, except in the reservoir inflow or available energy prediction area; hence, this study aims to take advantage of the PSO-ANN approach to forecast the available energy portion corresponding to the Nam Gouang reservoir inflow of the case study hydropower plant in Laos (THHP), which constitutes a novelty approach for the forecasting on this field. To reflect the real application in the commercial hydropower plants and/or the case study hydropower plant, this study converted the historical monthly reservoir inflow into the energy unit (available energy in GWh) before forecasting. By forecasting the available energy portion corresponding to reservoir inflow, the case study hydropower plant can significantly improve its decision support for energy production planning that maximizes revenue. This forecasting also aids in environmental impact prevention and flood control upstream and downstream of the basin. In essence, the hydropower plant will consider the forecasted available energy portion alongside the actual available energy portion, which corresponds to the existing water in the reservoir, to determine the amount of energy it can declare for sale in each month ahead. This decision aligns with annual supply targets and ensures sufficient available energy throughout the year. Additionally, this approach helps prevent unreasonable or unplanned reservoir spillage during the wet season, which could lead to lost generation opportunities.

This study has two main goals. The first is to develop the proposed PSO-ANN models in the MATLAB environment. The second is to evaluate the prediction performance of the proposed models using the testing data cluster (not included in the training data cluster), and statistical indicators, including correlation coefficient (r), relative error (RE), and root mean square error (RMSE) coupled with Taylor diagrams in comparison with single-algorithm models, i.e., ANN, and NARX models.

The remainder of this paper is divided into three main sections. Section 2 introduces the methodology. Section 3 describes the results and discussion. Finally, Section 4 presents the conclusions.

2. Methodology

In this section, we introduce the methodology, including the study area, data preparation, research framework, and detailed descriptions of the Artificial Neural Network (ANN), Particle Swarm Optimization (PSO), and the proposed PSO-ANN hybrid model.

2.1. Study Area

This study implemented the Energy Forecasting for the Theun-Hinboun Hydropower Plant (THHP) situated in the co-area of Khammouane and Bolikhamxay Provinces in Laos approximately 217 km to the east of Vientiane capital and owned by the Theun Hinboun Power Company (THPC). A 220 MW run-of-river project, consisting of two spillway radial gates, a head pond with a full supply level of 400 masl, and generation capacity of 1100 GWh per year, was constructed, of which 95% of electricity is purchased by the Electricity Generating Authority of Thailand (EGAT). The remaining 5% is for domestic supply to Electricité du Lao (EDL). After operating for a decade, the THPC expanded the project to a total installed capacity of 500 MW by constructing a new reservoir on the Nam Gnouang River, which is a tributary of the mainstream of the existing run-of-river project. The new Nam Gnouang Dam, hereafter referred to as “NG”, is constructed approximately 20 km upstream of the existing original Weir Dam in the form of a concrete gravity dam with a structure 480 m wide and 65 m high. The NG reservoir has a storage capacity of up to 2430 million cubic meters (MCM) at the full supply level of 455 masl as shown in

Table 1. Basic design specifications of the case study hydropower plant.

Design Feature	Specification
Original Weir Dam (run-of-river type):
Concrete gravity-free overflow Weir:	268 × 27 m (L × H)
Spillway radial gates	2
Head pond full supply level	400 masl
Two concrete-lined headrace tunnels length	5289 m and 5496 m
Installed capacity	2 × 110 MW (+220 MW)
NG Dam (storage type):
Concrete gravity dam	480 × 65 m (L × H)
Reservoir full supply level	455 masl
Max actual storage volume	2430 MCM
Spillway radial gates	5
Installed capacity	2 × 30 MW

. The NG Dam consists of five spillway gates operated only to discharge water during the wet season to avoid flood event (s).

During normal operation, the NG Dam releases water through the NG Powerhouse, which can generate power up to 60 MW for domestic supply to the EDL from the NG Powerhouse. After the NG reservoir, the water flows into the Theun River and the existing head pond above the original Weir Dam. The water finally flows through the new intake and tunnel system to the expanded Theun-Hinboun Powerhouse located approximately 240 m below in the Na Hin town of Khounkham District. At this location, a new 220 MW Francis generator was installed alongside the existing two (2) 110 MW generators. After the powerhouse, the water flows to a spillway and then a regulating pound before being released into the Nam Hai (river), which is a tributary of the Nam Hinboun (river) as illustrated in Figure 1. The new NG reservoir enables THPC to export up to 440 MW to EGAT, in which the power is transmitted to a designated delivery point at the Thakhek substation on the Lao PDR-Thailand border via a 230 kV double-circuit transmission line with a total length of approximately 86 km, while the remaining 60 MW are allocated for the domestic supply, which is purchased by the EDL.

The power purchase agreement between the THPC and EGAT is in the form of the take-or-pay principle with a contract term of 25 years commencing from the commercial operation date (COD). To ensure optimal generation planning associated with revenue maximization as well as environmental impact prevention, and flood control at the upstream and downstream of the basin, the THPC needs to take into account two key energy portions before declaring the energy amount (availability) for selling, including the actual available energy portion that corresponds to the existing water in the Nam Gnouang reservoir and head pound at the time of declaration, and the future energy (forecasted energy), which corresponds to the natural inflow (reservoir inflow) expected to flow into the reservoir and head pound in the days and/or months ahead, as illustrated in Figure 2.

2.2. Data Preparation

Initially, the historical hydrological and meteorological data including rainfall, and reservoir inflow of the Nam Gnouang reservoir of the THHP were collected from the measurement stations, and some missing data were filled by using the backpropagation technique. The total duration of the data was 11 years (1 January 2013 to 31 December 2023) as shown in Figure 3. To reflect real-world application in the commercial hydropower plants and/or the case study hydropower plant as illustrated in Figure 2, the historical reservoir inflow of the time step (t) was converted into the energy unit (available energy in GWh) before being applied to the models based on the THHP energy coefficient at 0.56117 kWh/m³. The collected data were then analyzed, normalized, and partitioned into four clusters: (i) training input cluster (rainfall and lagged inflows); (ii) training target cluster (converted available energy); (iii) testing input cluster (same parameters as the training input cluster); and (iv) testing target cluster (same parameters as the training target cluster). The duration of the training input cluster and training target clusters were the same at 9 years (approximately 80%), ranging from 1 January 2013 to 31 December 2021, and that of the testing input cluster and testing target clusters was 2 years thereafter (approximately 20%), ranging from 1 January 2022 to 31 December 2023, in compliance with the partition ratio recommended by the previous study [4]. The training input and target clusters were collectively referred to as the “training dataset”, and the testing input and target clusters were collectively referred to as the “testing dataset” in this study.

To determine the number of delays (input time lags) and input combinations for the proposed models, the correlation of every single parameter as well as between each input parameter, and the target parameter were analyzed with the autocorrelation function (AFC), and cross-correlation function (CCF). In addition to the data analysis by ACF and CCF, all the data were also normalized into numbers in [0, 1] to ensure faster learning and achievement of better results [27] by using Equation (1):

$$x{\prime }_k={\displaystyle \frac{(x_k-x_{\min })}{(x_{\max }-x_{\min })}}$$

(1)

where $x_k$ is the sampling data; $x_{\min }$ is the minimum value of the sampling data; and $x_{\max }$ is the maximum value of the sampling data.

2.3. Research Framework

This study was carried out to develop the hybrid models for the forecasting of the month-ahead available energy portion corresponding to the reservoir inflow by integrating the ANN and PSO based on the available historical hydrological and meteorological data of the Nam Guouang reservoir of the Theun-Hinboun hydropower plant, which is the case study hydropower plant situated in Laos, hereafter referred to as “THHP”. The flowchart of the proposed approach is illustrated in Figure 4. In the data preparation and preprocessing phase, the hydrological and meteorological data, including historical monthly rainfall, and inflow data (available in the case study hydropower plant), influencing the prediction results were collected; then, the cross-correlation function (CCF) and autocorrelation function (AFC) were employed to determine the number of input lags and combinations for the proposed models. For the achievement of better results and faster learning, the datasets were normalized into numbers between 0 and 1 before plugging into the proposed PSO-ANN models. With respect to the modeling process, four main steps were carried out: (1) optimizing the ANN weights and biases using the PSO algorithm, thus obtaining the optimal network; (2) training the models by the “trainlm” function, and using a training cluster; (3) testing the trained models by using a separated testing cluster (not included in the training cluster); and (4) evaluating the model by calculating the correlation coefficient (r), relative error (RE), root mean square error (RMSE) coupled with Taylor diagrams.

2.4. Artificial Neural Networks (ANNs)

Artificial neural network models were first introduced by McCulloch and Pitts in 1943 [28]. The ANNs are computer software or hardware models inspired by the structure and behavior of neurons in the human nervous system [29]. The ANN model is a part of the artificial intelligence modeling techniques, it can learn from the training data and find the data pattern to estimate the results [30]. Due to the fluctuation and nonlinearity behavior of the hydrological and meteorological time-series data, the ANN is widely used to develop forecasting models that produce results more accurately than statistical models. In general, the ANNs consist of a three-layer architecture including the input layer, hidden layer, and output layer, which is sufficient to solve a complex nonlinear problem like time-series forecasting. Figure 5 illustrates an ANN architecture with three layers.

Where $x_1,x_2,x_3,\dots ,x_n$ are the input signals; $w_{\mathrm{1,1}},w_{\mathrm{1,2}},w_{\mathrm{1,3}},\dots ,w_{n,n}$ are the respective weights for the connection between the input layer and hidden layer of the network; $w_{1,O},w_{2,O},w_{3,O},\dots ,w_{n,O}$ are the respective weights for the connection between the hidden layer and output layer of the network; $b_1,b_2,b_3,\dots ,b_n$ are the biases of the hidden nodes; $b_O$ is the bias of the output node; ${\phi }_1(\cdot ),{\phi }_2(\cdot ),{\phi }_3(\cdot ),\dots ,{\phi }_n(\cdot )$ are the activation functions of the hidden nodes; ${\phi }_O(\cdot )$ is the activation function of the output node; and $y_O$ is the output signal of the network. Mathematically, the ANN can be represented as follows:

$$u_1=(w_{\mathrm{1,1}}{x}_1+w_{\mathrm{2,1}}{x}_2+w_{\mathrm{3,1}}{x}_3+\dots +w_{n,1}{x}_n)$$

(2)

$$u_2=(w_{\mathrm{1,2}}{x}_1+w_{\mathrm{2,2}}{x}_2+w_{\mathrm{3,2}}{x}_3+\dots +w_{n,2}{x}_n)$$

(3)

$$u_3=(w_{\mathrm{1,3}}{x}_1+w_{\mathrm{2,3}}{x}_2+w_{\mathrm{3,3}}{x}_3+\dots +w_{n,3}{x}_n)$$

(4)

$$u_n=(w_{1,n}{x}_1+w_{2,n}{x}_2+w_{3,n}{x}_3+\dots +w_{n,n}{x}_n)$$

(5)

$$h_1={\phi }_1(u_1+b_1)$$

(6)

$$h_2={\phi }_2(u_2+b_2)$$

(7)

$$h_3={\phi }_3(u_3+b_3)$$

(8)

$$h_4={\phi }_4(u_4+b_4)$$

(9)

$$h_n={\phi }_n(u_n+b_n)$$

(10)

$$u_O=(w_{1,O}{h}_1+w_{2,O}{h}_2+w_{3,O}{h}_3+\dots +w_{n,O}{h}_n)$$

(11)

$$y_O={\phi }_O(u_O+b_O)$$

(12)

where $u_1,u_2,u_3,\dots ,u_n$ are the linear combiners of the hidden nodes; $h_1,h_2,h_3,\dots ,h_n$ are the output signals of the hidden nodes; $u_O$ is the linear combiner of the output node; and $y_O$ is the output signal of the network. The optimal choice in selecting the activation functions can depend on the specific problem and/or dataset. However, this study selected the tangent sigmoid activation function based on its superior advantages in terms of balanced range that can be advantageous for PSO as particles do not become confined to a specific positive region, allowing for a more balanced exploration during optimization, and its gradient consistency, as it generally has smoother gradients compared to other activation functions like ReLU. This smoother gradient can be beneficial for PSO, as the particles rely on these gradients to update their positions and potentially lead to faster convergence during optimization.

2.5. Particle Swarm Optimization (PSO)

Particle Swarm Optimization, widely known as “PSO”, was introduced by Kennedy and Eberhart in 1995. It is a computational swarm-based search method inspired by the flocking behavior of birds in search of food. With its strong optimization ability and ease of implementation, the PSO has become a popular option for solving various optimization problems [31,32]. Like other evolutionary algorithms, the PSO algorithm begins with an initial random population of candidate solutions called “particles”. These particles iteratively explore the search space, adjusting their positions based on their own flying experience and the best solution discovered by the entire swarm. This process is guided by a few key parameters to lead the swarm towards the optimal solution with minimal computational cost [33]. Here, a swarm $P\left(t\right)$ of $N$ particles initialized the $t=0$, a random position $x_i\left(t\right)$ was assigned to each particle $P_i\left(t\right)$, and the cost of each particle evaluated. During subsequent iterations, each particle performance was assessed at its current location to determine and update the personal best $\left(P_{best}\right)$ and global best $\left(G_{best}\right)$ experiences. $P_{best}$ refers to the best personal position that a particular particle has encountered, and $G_{best}$ refers to the best position discovered by the entire swarm. The flight velocity and position of each particle were updated according to the following equations.

$$v_t\left(t\right)=w{v}_i(t-1)+{\rho }_1(P_{best}-x_i(t-1\left)\right)$$

$$+{\rho }_2(G_{best}-x_i(t-1\left)\right)$$

(13)

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

(14)

Regarding the random variables, ${\rho }_1=r_1{C}_1$ and ${\rho }_2=r_2{C}_2$, whereas $r_1,r_2$∼$U$ [0, 1], $C_1$ and $C_2$ represent positive acceleration constants, and $\omega$ represents an inertia weight. The second and last portions of Equation (13) are referred to as the cognitive component and social component, respectively. Therefore, $C_1$ and $C_2$ are the cognitive and social acceleration constants, respectively.

A larger PSO population explores a wider search space, potentially finding better solutions; however, it can be computationally expensive. In contrast, a smaller population leads to faster computation but might get stuck in local optima. Higher $C_1$ emphasizes individual learning, potentially leading to faster convergence towards individual bests but risking local optima; meanwhile, higher $C_2$ emphasizes swarm learning, promoting the exploration of the search space but potentially slowing down convergence. Thus, adjusting $C_1$ and $C_2$ helps strike a balance between individual and swarm learning. With respect to the inertia weight ($\omega$), a higher weight promotes exploration in the early stages, allowing particles to move further and explore a wider search space while the lower weight encourages exploitation in later stages, focusing particles closer to promising areas. Gradually reducing the weight over iterations helps with this transition. To ensure the PSO algorithm’s stability, previous research [34] suggested $C_1+C_2\le 4$. Considering the aforementioned, and based on the real forecasting results, this study finally configured $C_1$ and $C_2$ at 1.5 and 2, respectively. For the inertia weight $\omega$, it was updated in each iteration according to Equation (15).

$$\omega ={\omega }_{damp}\times \omega$$

(15)

where ${\omega }_{damp}$ is the inertia weight damping ratio finally set to 0.99 in this study.

2.6. Proposed PSO-ANN Hybrid Models

ANNs are good for learning input–output relationships or patterns. However, they could face local minima and network paralysis problems due to rough weights assigned by learning algorithms such as backpropagation [35]. To overcome these drawbacks, we propose the PSO algorithm to optimize the weights and biases for the ANNs so that the results produced by the ANNs are more accurate as the prediction errors will be minimized. The descriptions of the ANN and PSO algorithms were clearly explained in Section 2.4 and Section 2.5, respectively. To enhance the prediction performance, this study also employed the autocorrelation function (ACF), cross-correlation function (CCF), and data normalization techniques during the data pre-processing phase, as described in Section 2.2. For a better understanding, the proposed PSO-ANN approach can also be represented and described by the flowchart in Figure 6:

In Figure 6, the output vector of the ANN network is compared with the target vector, and their difference, called “error”, is calculated in the form of a normalized mean squared error (NMSE). If the average change in the global best (Gbest), corresponding to the NMSE, is greater than the options of the function tolerance (1 × 10⁻¹⁵), the weights and biases of the network will be adjusted or optimized by using the PSO algorithm. The iteration continues until the average change is smaller than the options of the function tolerance. The proposed PSO-ANN models for predicting the available energy portion corresponding to the reservoir inflow are summarized in the

Table 2. Summary of the various combination input designs.

Model Scenario	Input Combinations	Output
SC1	R(t), I(t − 1)
SC2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)
SC3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)
SC4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)
SC5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)

Notes: R(t) = Rainfall of month (t) [mm]. R(t − 1) = Rainfall of month (t − 1) [mm]. R(t − 2)= Rainfall of month (t − 2) [mm]. I(t − 1) = Inflow of month (t − 1) [MCM]. I(t − 2) = Inflow of month (t − 2) [MCM]. I(t − 3) = Inflow of month (t − 3) [MCM]. E(t) = Energy of month (t) [GWh].

below:

Based on the ACF and CCF analyses, the rainfall of the timestep (t) and the inflow of the timestep (t − 1) have a good correlation with their own lagged values as well as with the target values, i.e., with the converted available energy of the timestep (t) up to three timesteps. Hence, this study performed the simulation in several scenarios by alternating various input combinations, as summarized in Table 2, to seek the best prediction model or scenario of the proposed approach to compare with the ANN and NARX approaches as mentioned above, whereas the number of output variable for the models was determined to be equal to 1, corresponding to the forecasting objective, and the number of hidden nodes or neurons was determined by using the trial-and-error method.

2.7. Prediction Error

To evaluate the performance of the proposed PSO-ANN prediction models, the three measurement or statistical indicators, including root mean square error (RMSE), relative error (RE), and correlation coefficient (r), were calculated and analyzed using Equations (16)–(18) coupled with the Taylor diagrams.

$$r={\displaystyle \frac{{\displaystyle \sum _{i=1}^N}(E_{mi}-{\overline{E}}_m)\times (E_{ci}-{\overline{E}}_c)}{\sqrt{\left[{\displaystyle \sum _{i=1}^N}(E_{mi}-{\overline{E}}_m{)}^2\times {\displaystyle \sum _{i=1}^N}(E_{ci}-{\overline{E}}_c{)}^2\right]}}}$$

(16)

$$RMSE={\left({\displaystyle \frac{{\displaystyle \sum _{i=1}^N}(E_{mi}-E_{ci}{)}^2}{N}}\right)}^{0.5}$$

(17)

$$RE={\displaystyle \frac{\left|E_{ci}-E_{mi}\right|}{E_{ci}}}\times 100\%$$

(18)

where $E_{mi}$ represents the target or observed available energy at the time $i$; ${\overline{E}}_m$ represents the average value of the target or observed available energy at the time $i$; $E_{ci}$ represents the predicted available energy at the time $i$; ${\overline{E}}_{ci}$ represents the average value of predicted available energy at the time $i$; and $N$ represents the number of data points.

The root mean square error (RMSE) and relative error (RE) were used to evaluate the performance of the models, whilst the correlation coefficient (r) was used to assess the relationship between the target and predicted values as well as to measure the strength in terms of the relationship between the two variables [15].

3. Results and Discussion

A hybrid model has been developed in this study to predict the available energy portion corresponding to the reservoir incoming flow of the Nam Gouang reservoir of the THHP based on the combination of ANN and PSO algorithms described in Section 2.6 together with ACF, CCF, and data normalization techniques. The historical hydrological and meteorological data of the case study hydropower plant were initially collected ranging from 2013 to 2023 and then some missing data were filled with the backpropagation technique before pre-processing by proper analysis and normalization. The normalized data were finally partitioned into training and testing clusters at a ratio of approximately 80:20, as described in Section 2.2. In addition to the proposed model, the same datasets were also applied for the ANN and NARX models for comparison purposes.

3.1. Hyperparameter Determination

Input combinations for the feedforward neural network of the proposed approach were determined based on the correlation analysis results, in which five scenarios of input combinations were selected and tested, as summarized in Table 2. The number of hidden neurons for the neural network was determined by the trial-and-error method, and, finally, 30 were selected, which is the number that provided the best prediction results. Lastly, the number of output variables was determined to be equal to 1, corresponding to the forecasting objective. With respect to the PSO algorithm, all the hyperparameters were determined by the trial-and-error method, and, finally, a $C_1$ and $C_2$ of 1.5 and 2, respectively, ${\omega }_{damp}$ (inertia weight damping ratio) of 0.99, and a population (swarm) size of 200 were obtained.

3.2. Results of the Proposed Approach

Table 3. Results by statistical indicators of the PSO-ANN models 2022.

Model Scenarios	Model Input Combinations	Model Output	Different Models	Model Structures	r	RMSE	RE
PA1	R(t), I(t − 1)	E(t)	PSO-ANN	(2,30,1)	0.951	38.168	8.261
PA2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(4,30,1)	0.968	33.261	18.459
PA3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(5,30,1)	0.973	22.994	1.038
PA4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(5,30,1)	0.942	32.924	7.847
PA5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(6,30,1)	0.930	34.354	3.662

and

Table 4. Results by statistical indicators of the PSO-ANN models 2023.

Model Scenarios	Model Input Combinations	Model Output	Different Models	Model Structures	r	RMSE	RE
PA1	R(t), I(t − 1)	E(t)	PSO-ANN	(2,30,1)	0.905	44.925	15.059
PA2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(4,30,1)	0.965	30.668	5.832
PA3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(5,30,1)	0.966	24.846	2.853
PA4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(5,30,1)	0.930	36.757	7.928
PA5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(6,30,1)	0.956	27.934	3.001

reveal that the proposed PSO-ANN model of the third scenario (PA3) with structure (5,30,1) has given the best prediction results with the statistical indicators r = 0.973, RMSE = 22.994 and RE = 1.038% for the single year of 2022, and r = 0.966, RMSE = 24.846 and RE = 2.853% for 2023, which were superior over those of the other scenarios, i.e., the RMSE and RE of the proposed scenario, for both years, which were less than those of other scenarios, indicating more accuracy in the forecasting. On the other hand, the correlation coefficient (r) of the proposed scenario was closer to 1 than that of the other scenarios (both years), demonstrating a stronger relationship between the predicted and the target values. Hence, the third scenario (PA3) was selected to be the best structure for the proposed approach and for later comparison with the other prediction approaches in this study.

3.3. Training Results

The proposed approach started by adjusting or optimizing the weights and biases of the feedforward neural networks, whereas the particle swarm optimization (PSO) algorithm was employed based on the training dataset. The weights and biases were then updated a slightly more with the default function of the network namely, “Trainlm” (a network training function that updates weight and bias values according to Levenberg–Marquardt optimization) during the training process. Several models with different structures were trained and tested. The best model was then selected based on the results of the statistical indicators; the proposed PSO-ANN model of the third scenario (PA3) with structure (5,30,1) gave the best prediction results.

A cost function applied in the model is the Normalized Mean Square Error (NMSE). The cost function returns an output value, called the “Cost”, which is a numerical value representing the deviation, or degree of error, between the model prediction and the target data; the lower the cost, the lower the deviation (error); hence, the optimal prediction model would have a cost close to 0. Figure 7 shows the convergence of the best cost in PSO versus the updating iteration for the selected scenario (PA3), in which the best cost value started from 0.61479 in the first iteration and gradually declined to 0.10206 in iteration 842, regarded as the last iteration of the PSO running process.

The error histogram represents the histogram of the errors between the target values and the predicted values after training a feedforward neural network. As these error values indicate how predicted values are different from the target values, these can be negative or positive. The graph of the error histogram in Figure 8 shows the errors between the proposed model output and target values during the training process, in which the vertical line represents the error at the zero point. In this graph, most of the errors are close to zero points, which demonstrates minor errors in all data clusters, i.e., the training, testing, and validation clusters.

The training regression graphs in Figure 9 representing the correlation between the model output and target values of each data cluster during the process of training (training, validation, testing, and all data clusters) show that all the calculated regression coefficients (R) have directions close to 1, i.e., 0.94182, 0.95648, 0.96596 and 0.94786 for training, validation, test, and all data clusters, respectively, which means that the output values of the proposed model (PA3) have a very good relationship with the target values.

Figure 10 depicts the prediction results of the third scenario (PA3) of the proposed PSO-ANN approach compared to the target values with a total duration of two years, i.e., from 2022 to 2023. We observed that the prediction values, represented by the blue curve, agree quite well with the target values, represented by the red curve, all along the testing duration. However, there are still some deviations visualized from the graph both in the dry period and wet/peak period of the years; the significant deviation mainly resulted from the high fluctuation and nonstationary of data in the periods of extreme weather conditions.

3.4. Comparison of PSO-ANN, ANN and NARX Models

To compare the performance of the proposed PSO-ANN approach, of which the third scenario (PA3) has been proven and selected to be the best scenario, as described in Section 3.2, the same datasets, i.e., the training and testing datasets, comprising of rainfall and lagged inflow, were also applied for the ANN and NARX prediction models and then the best scenario from each approach was selected and compared among the three approaches. In addition to the datasets, the same numbers of input combinations and hidden neurons were also applied for both the ANN and the NARX model. The comparison among the best models of the three approaches was made based on the statistical indicators coupled with the Taylor diagrams.

Table 5. Results by statistical indicators of all models 2022.

Model Scenarios	Model Input Combinations	Model Output	Different Models	Model Structures	r	RMSE	RE
PA1	R(t), I(t − 1)	E(t)	PSO-ANN	(2,30,1)	0.951	38.168	8.261
PA2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(4,30,1)	0.968	33.261	18.459
PA3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(5,30,1)	0.973	22.994	1.038
PA4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(5,30,1)	0.942	32.924	7.847
PA5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(6,30,1)	0.930	34.354	3.662
A1	R(t), I(t − 1)	E(t)	ANN	(2,30,1)	0.894	43.444	18.178
A2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	ANN	(4,30,1)	0.900	55.512	41.521
A3	R(t), R(t − 1), I(t − 1), I(t − 2),I(t − 3)	E(t)	ANN	(5,30,1)	0.905	36.851	7.872
A4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	ANN	(5,30,1)	0.821	76.032	27.723
A5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2),I(t − 3)	E(t)	ANN	(6,30,1)	0.775	55.106	10.051
N1	R(t), I(t − 1)	E(t)	NARX	(2,30,1)	0.894	40.182	3.030
N2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	NARX	(4,30,1)	0.832	55.317	16.462
N3	R(t), R(t − 1), I(t − 1), I(t − 2),I(t − 3)	E(t)	NARX	(5,30,1)	0.939	30.315	3.491
N4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	NARX	(5,30,1)	0.796	51.605	7.338
N5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2),I(t − 3)	E(t)	NARX	(6,30,1)	0.844	45.577	4.250

and

Table 6. Results by statistical indicators of all models 2023.

Model Scenarios	Model Input Combinations	Model Output	Different Models	Model Structures	r	RMSE	RE
PA1	R(t), I(t − 1)	E(t)	PSO-ANN	(2,30,1)	0.905	44.925	15.059
PA2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(4,30,1)	0.965	30.668	5.832
PA3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(5,30,1)	0.966	24.846	2.853
PA4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	PSO-ANN	(5,30,1)	0.930	36.757	7.928
PA5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)	E(t)	PSO-ANN	(6,30,1)	0.956	27.934	3.001
A1	R(t), I(t − 1)	E(t)	ANN	(2,30,1)	0.909	44.268	24.058
A2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	ANN	(4,30,1)	0.873	51.048	28.574
A3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)	E(t)	ANN	(5,30,1)	0.942	37.238	3.619
A4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	ANN	(5,30,1)	0.910	55.710	25.550
A5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)	E(t)	ANN	(6,30,1)	0.821	57.602	11.466
N1	R(t), I(t − 1)	E(t)	NARX	(2,30,1)	0.905	43.566	14.521
N2	R(t), R(t − 1), I(t − 1), I(t − 2)	E(t)	NARX	(4,30,1)	0.922	39.719	5.724
N3	R(t), R(t − 1), I(t − 1), I(t − 2), I(t − 3)	E(t)	NARX	(5,30,1)	0.960	28.320	3.548
N4	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2)	E(t)	NARX	(5,30,1)	0.943	34.179	6.101
N5	R(t), R(t − 1), R(t − 2), I(t − 1), I(t − 2), I(t − 3)	E(t)	NARX	(6,30,1)	0.911	39.604	7.106

reveal that the third scenario of each prediction approach, with structure (5,30,1), i.e., PA3, A3, and N3 for PSO-ANN, ANN, and NARX models, respectively, showed the superior prediction results. We also observed that the third scenario (PA3) of the proposed approach with r = 0.973, RMSE = 22.994 and RE = 1.038% for the single year of 2022, and r = 0.966, RMSE = 24.846 and RE = 2.853% for 2023 outperformed those of the ANN and NARX models, of which scenario A3 of the ANN given r = 0.905, RMSE = 36.851 and RE = 7.872% for the single year of 2022, and r = 0.942, RMSE = 37.238 and RE = 3.619% for 2023, and scenario N3 of the NARX, given r = 0.939, RMSE = 30.315 and RE = 3.491% for the single year of 2022, and r = 0.960, RMSE = 28.320 and RE = 3.548% for 2023, i.e., the root mean square error (RMSE), and relative error (RE) of the proposed PSO-ANN model, for both years, were less than those of the ANN and NARX models, indicating more accuracy in the forecasting. On the other hand, the correlation coefficient (r) of the proposed model was closer to 1 than that of the other models (both years), which a stronger correlation between the predicted and target values. In terms of industry benchmarks, the proposed model also demonstrated competitive performance, as the correlation coefficient (r) and root mean square error (RMSE) were found to be comparable to those proposed by Babaei et al. (2019) for the Za-yandehroud dam reservoir [12], in which the SVM model achieved an r of 0.9622 and RMSE of 23.56193, and the ANN model attained an r of 0.95333 and RMSE of 28.5125, while the proposed PSO-ANN model exhibited an r of 0.973 and the RMSE of 22.994. However, it has been noted that dataset variations can influence the comparisons.

In addition to the clear comparison in Table 5 and Table 6, Figure 11 depicts the predicted values of the three prediction models (scenarios PA3, A3, and N3) versus the target values. We observed that all three prediction curves can track changes in the target (red curve) throughout the testing duration, whereas the curve of the proposed PSO-ANN model has a better coincidence than the others, especially during the peak periods of the years, indicating superiority in the forecasting. However, the ANN, and NARX models, in some months during the dry season, performed slightly better than the proposed model; this depicted the advantages and limitations of each approach.

The testing regression graphs with a testing duration of 2 years in Figure 12 illustrate the correlation between each model prediction and target values, of which the calculated regression coefficients (R) of the proposed PSO-ANN model at 0.95758 were superior to those of the ANN and NARX models (0.93290, and 0.90303, respectively). In general, correlation coefficients with a magnitude between 0.9 and 1.0 indicate variables that can be considered very highly correlated. Correlation coefficients with a magnitude between 0.7 and 0.9 indicate variables that can be considered highly correlated. Correlation coefficients with a magnitude between 0.5 and 0.7 indicate variables that can be considered moderately correlated. Correlation coefficients with a magnitude between 0.3 and 0.5 indicate variables that have a low correlation, and correlation coefficients with a magnitude lower than 0.3 have little if any (linear) correlation.

The Taylor diagram in Figure 13 shows that the root mean square deviation (RMSD), and correlation coefficient (r) of the proposed PSO-ANN model are closer to the target location compared to the ANN and NARX models. With respect to the standard deviation, the ANN and NARX models are depicted slightly closer to the target location; however, the standard deviation just indicates how far each value lies from the mean or expected value (a high standard deviation means that values are generally far from the mean, while a low standard deviation indicates that values are clustered close to the means); thus, the size of the standard deviation does not reflect a better or worse prediction performance. That of the proposed PSO-ANN model herein depicted was less than that of the target and of the ANN and NARX models. Based on the comparison results illustrated by the Taylor diagram in combination with the results of statistical indicators summarized in Table 5 and Table 6, the proposed model demonstrated promising performance and was comparable to those in the existing literature.

4. Conclusions

To forecast the available energy portion that corresponds to the reservoir inflow of the month(s) ahead for ensuring optimal generation planning associated with revenue maximization as well as environmental impact prevention, and flood control upstream and downstream of the basin, this study has developed an effective hybrid model by integrating an Artificial Neural Network (ANN) and Particle Swarm Optimization (PSO) based on hydrological and meteorological data pre-processed by autocorrelation function (AFC), cross-correlation function (CCF), and normalization techniques. The Nam Gnouang reservoir of THHP in Laos was selected to be the case study hydropower plant. Based on the comparison results among the proposed PSO-ANN, ANN, and NARX models, the PSO-ANN model exhibited superior accuracy, achieving an r of 0.973, RMSE of 22.994, and RE of 1.038% for the year 2022, and an r of 0.966, RMSE of 24.846, and RE of 2.853% for the year 2023. Comparisons with the existing literature, such as the work of Babaei et al. (2019), who reported an r of 0.9622 and RMSE of 23.56193 for the SVM model and an r of 0.95333 and RMSE of 28.5125 for the ANN model on the Zayandehroud dam reservoir [12], indicated that the proposed PSO-ANN model performance is competitive. However, it is important to note that direct comparisons between studies can be influenced by dataset variations, regions, and hydrological conditions. While these results are encouraging, the proposed PSO-ANN model performance should be interpreted cautiously due to this study’s scope limitation, using only two years of data from a single reservoir for model testing. Consequently, the findings offer preliminary insights rather than definitive conclusions. Further research is imperative to evaluate the model’s generalizability across diverse hydrological conditions, regions, and extended timeframes.

The potential applications of this proposed model in optimizing energy production planning for revenue maximization, mitigating environmental impacts, and supporting flood control are promising. To realize these benefits fully, addressing this study’s limitations is crucial. Expanding the dataset, incorporating additional hydrological and meteorological variables or factors, and employing advanced techniques to handle non-stationary and fluctuating data are essential for improving model robustness and reliability. Moreover, integrating data from multiple sources, such as satellite imagery and reanalysis products, can enhance the ability of the model. This study serves as a foundation for future research aimed at developing more advanced and reliable hydropower-forecasting methodologies. By addressing the limitations identified above and expanding the scope of analysis, subsequent investigations can contribute to optimal power generation planning and the sustainable management of water resources for hydropower worldwide.