Forecasting and Comparative Application of PV System Electricity Generation for Sprinkler Irrigation Machines Based on Multiple Models

Abstract

Currently, photovoltaic (PV) resources have been widely applied in the agricultural sector. However, due to the unreasonable configuration of multi-energy collaboration, issues such as unstable power supply and high investment costs still persist. Therefore, this study proposes a solution to reasonably determine the area and capacity of PV panels for irrigation machines, addressing the fluctuations in power generation of solar sprinkler PV systems under different regional and meteorological conditions. The aim is to more accurately predict photovoltaic power generation (PVPG) to optimize the configuration of the solar sprinkler power supply system, ensuring reliability while reducing investment costs. This paper first establishes a PVPG prediction model based on four forecasting models and conducts a comparative analysis to identify the optimal model. Next, annual, seasonal, and solar term scale models are developed and further studied in conjunction with the optimal model, using evaluation metrics to assess and compare the models. Finally, a mathematical model is established based on the optimal combination and solved to optimize the configuration of the power supply system in the irrigation machines. The results indicate that among the four PVPG prediction models, the SARIMAX model performs the best, as the R² index reached 0.948, which was 19.4% higher than the others, while the MAE index was 10% lower than the others. The solar term scale model exhibited the highest accuracy among the three time scale models, the RMSE index was 4.8% lower than the others, and the MAE index was 1.1% lower than the others. After optimizing the configuration of the power supply system for the irrigation machine using the SARIMAX model based on the solar term scale, it is verified that the model can ensure both power supply reliability and manage energy overflow effectively.

1. Introduction

In recent years, there has been a significant shift in the global energy structure, with the proportion of renewable energy continuously increasing [1]. With its abundance and zero greenhouse gas emissions, renewable energy has been widely adopted, and its usage is expected to continue growing. Currently, solar and wind energy account for 96% of global renewable energy consumption [2], becoming key drivers of global sustainable development. The rapid development of photovoltaic (PV) technology in particular provides immense application potential for agricultural regions, especially in arid areas with strong sunlight. In these regions, solar energy is not only a clean energy source but also offers a sustainable power source for agricultural production, showcasing vast application prospects in irrigation systems.

Agriculture is one of the major sectors of energy consumption, and traditional power systems are not only costly but also impose a certain environmental burden. To promote sustainable development, the application of renewable energy sources in agricultural equipment, such as solar-powered irrigation systems, is essential. The prediction of photovoltaic power generation (PVPG) is crucial for these systems, as it directly impacts the stable operation of irrigation systems and the reliability of energy supply. Accurate forecasting primarily relies on solar irradiance, ambient temperature, and time variables. These variables exhibit uncertainty and volatility during their dynamic changes [3,4,5], posing new challenges to the safe and stable operation of PV power supply systems. Therefore, accurately predicting PVPG is vital for enhancing the stability of power systems and improving supply reliability [6,7].

Before conducting any specific application analysis, it is essential to accurately determine the capacity or area size of the PV panels [8,9]. However, current practices often rely on experience or traditional estimation methods [8,10], such as configuring the area and capacity of PV panels based on approximate annual electricity consumption and general PV conversion coefficients. This approach fails to account for the varying solar resources in different regions and does not adaptively analyze or predict the PVPG capacity of the PV panels, leading to significant discrepancies between the results and actual application scenarios. In agriculture, due to differences in meteorological conditions, fixed PV panel capacities may not meet the power demands of irrigation machines, resulting in low supply reliability or excessive capacity leading to resource waste. Additionally, the high costs associated with installing PVPG monitoring equipment make the establishment of accurate PVPG prediction models highly valuable [11,12]. Therefore, it is necessary to configure appropriate PV panel capacities based on the meteorological conditions of different regions. To achieve a precise configuration of PV panel capacity for solar-powered irrigation machines, a PVPG prediction method based on local meteorological conditions is urgently needed. This study focuses on the case of a reel-type solar irrigation machine, as illustrated in Figure 1, with its PVPG system composition shown in Figure 2.

Therefore, reasonably configuring the capacity of PV panels based on the meteorological conditions of different regions is crucial for enhancing the application of PVPG systems in agricultural equipment. Especially in solar-powered irrigation machines, there is an urgent need for accurate PVPG forecasting methods. Traditional fixed-capacity configuration methods have many shortcomings and cannot dynamically adapt to changes in local meteorological conditions. To address this, this paper proposes a PVPG forecasting method based on local meteorological conditions. By analyzing forecasting models under different time scales and typical weather conditions, the study focuses on a type of reel solar irrigation machine to optimize the configuration of PV panel capacity, aiming to achieve a balance between power supply reliability and cost-effectiveness.

Currently, PVPG prediction has been widely applied in various fields [13], and many algorithms for predicting PVPG have been proposed by experts and scholars, achieving significant results [14,15,16,17,18,19,20,21,22,23,24,25,26]. These algorithms can be categorized into three main types: machine learning models, neural networks, and statistical models.

Machine learning models are widely used for PVPG prediction. For example, Li et al. [15] applied an improved hybrid multi-universe optimizer Support Vector Machine (SVM) model to PVPG prediction, demonstrating that this algorithm has stronger stability. Malvoni et al. [16] used Least Squares SVM (LS-SVM) to predict PVPG, validating the model based on historical data. The results indicated that this technique has very high performance in predictions. Compared to other techniques, SVM can reduce generalization errors using structural risk minimization [17]. However, when the sample size of the dataset is large, SVM requires excessive machine memory, causing the computation speed to become very slow. Extreme Gradient Boosting (XGBoost) and Light Gradient Boosting Machine (LightGBM) stand out due to their efficiency, flexibility, and portability, although they can sometimes suffer from overfitting. Random Forest (RF), XGBoost, and Gradient Boosting Decision Tree (GBDT) are categorized as ensemble machine learning models based on decision trees. Compared to SVM, they require fewer parameters, have shorter running times, exhibit greater robustness against overfitting, and achieve better predictive performance [18,19]. However, these models involve many hyperparameters that are difficult to tune manually, which limits their applications.

Neural networks play an important role in solving classification and regression problems. Jiao et al. [20] proposed a bidirectional long-short-term memory (BiLSTM) short-term PVPG prediction model that considers data decomposition and evolutionary predation strategies, which can better mine the intrinsic features of the model and improve power prediction accuracy. Meanwhile, Du et al. [21] introduced a hybrid deep learning framework, TG-A-CNN-LSTM, for PVPG prediction, capable of exploring spatial correlations among multiple regions and temporal information based on weighted time series variables. They tested the performance of data-sparse prediction models, demonstrating the robustness and generality of the prediction model. Liu et al. [22] introduced a hybrid model that integrates the attention mechanism with convolutional neural networks (CNN) and BiLSTM, further evaluating the contribution of each model component to performance through ablation experiments. The results indicated that the integration of CNN and BiLSTM significantly improved prediction accuracy, and the introduction of the attention mechanism notably enhanced the model’s prediction stability. However, neural networks consume substantial computational resources, making the calculations more complex and requiring longer computation times.

Statistical models are typically used to explore the mathematical relationships between one or more random variables and other variables. The most widely used statistical models currently include autoregressive (AR) models, moving average (MA) models, dynamic moving average (ARMA) models, and their more complex versions such as autoregressive integrated moving average (ARIMA), autoregressive moving average with exogenous variables (ARMAX), seasonal autoregressive integrated moving average (SARIMA), and Seasonal Auto-Regressive Integrated Moving Average with eXogenous factors (SARIMAX) [23,24,25]. Li et al. [24] innovatively considered climatic conditions and proposed an ARMAX model based on the ARIMA model to allow exogenous inputs for predicting power output. Zhou et al. [25] proposed a PVPG prediction method based on the SARIMAX-SVR model, which improved prediction accuracy. Compared to SVM, these statistical models require fewer parameters and run for a shorter time, exhibiting greater robustness against overfitting and achieving better prediction performance [24,25,26]. Therefore, considering the characteristics of meteorological data in this study and the practical needs of irrigation machines, four common models—SVM, XGBoost, RF, and SARIMAX—are selected to predict PVPG based on historical meteorological data for a specific location, allowing for comparative optimization.

However, current research on PVPG primarily focuses on large PVPG areas and equipment such as grid-connected power plants. Many studies have opted for optimization scheduling on a daily or intraday basis, or have conducted comparative studies using annual, daily, or hourly time scales [14,15,16,17,18,19,20,21,22,23,24,25,26]. There is relatively little research on PV power prediction in the agricultural sector, especially regarding agricultural equipment like irrigation machines, and most time scales lack significant regularity concerning agricultural climate variations. Therefore, this study proposes establishing a PV power prediction model based on solar terms as a time scale to predict the PVPG for solar irrigation machines. The twenty-four solar terms are weather change patterns summarized by the ancient Chinese laborers from long-term production practices. They can scientifically describe the relationship between astronomical climate and agricultural production. Initially, these terms were developed by ancient peoples in accordance with agricultural seasons through celestial observations, with the Big Dipper’s rotation defining a year. The current system divides the sun’s annual motion along the ecliptic into 24 equal parts, each corresponding to 15 degrees, with each part representing a solar term [27]. For farmers, agricultural production is closely linked to the rhythms of nature, and solar terms form a knowledge system that recognizes seasonal, climatic, and phenological changes throughout the year. They cleverly combine astronomy, agriculture, phenology, and folklore, giving rise to a wealth of cultural practices related to seasonal changes. The transition of solar terms is closely related to weather changes and is recognized as part of UNESCO’s Intangible Cultural Heritage of Humanity. It holds a significant position in traditional Chinese agricultural society [28]. Therefore, this study will further explore the PVPG of solar irrigation machines based on the solar terms time scale.

PVPG is a crucial basis for the capacity configuration of solar irrigation machine power supply systems, highlighting the need for accurate predictions. This study first compares and selects the most suitable prediction models—SVM, XGBoost, RF, and SARIMAX—based on evaluation metrics, ultimately employing the SARIMAX algorithm for PV power prediction. The research will establish PVPG prediction models using SARIMAX based on three time scales: one year, four seasons, and twenty-four solar terms, followed by an assessment and comparative analysis of the model’s prediction accuracy and stability. Finally, the estimated results from the SARIMAX model can optimize the configuration of PV power supply systems for irrigation machines in specific regions, enhancing the stability of solar irrigation systems and reducing energy overflow and investment costs.

The innovation of this research lies in the introduction of multiple forecasting models to compare their performance in photovoltaic power generation forecasting, and further investigate the accuracy of power generation predictions at different time scales, such as annual, seasonal, and solar term scales, based on the optimal model (SARIMAX). In particular, incorporating China’s traditional twenty-four solar terms as a time scale in photovoltaic power generation forecasting allows for a more detailed capture of the periodic changes in meteorological conditions. The solar term scale divides annual meteorological data into 24 solar terms, generating a forecasting model for each term. The seasonal scale divides the data into four seasons, each producing a forecasting model. The annual scale models the overall meteorological data for the entire year.

This study aims to optimize the configuration of photovoltaic power generation system capacity through the comparison and optimization of these models, thereby enhancing the power supply stability of irrigation equipment while effectively reducing energy waste and investment costs. This method will provide a reference for the capacity configuration of photovoltaic irrigation systems and other agricultural equipment in different regions, promoting the widespread application of photovoltaic technology in agriculture. Through this background and research analysis, this paper not only provides a theoretical basis for the application of photovoltaic power generation systems in the agricultural sector but also offers technical support for the sustainable development of agricultural equipment.

The remainder of this paper is organized as follows: In Section 2, the SVM, XGBoost, RF, and SARIMAX models are introduced, along with the description of the three time scales and the criteria for dataset partitioning. The mathematical modeling methods for capacity configuration of the irrigation system using PVPG are also presented. Section 3 provides a comparative analysis of the accuracy and stability of the four PV power prediction methods, describing the model construction process based on the optimal model for predictions across different time scales and analyzing the results. And at the same time, the proposed optimal method is applied to optimize the capacity configuration for solar irrigation machines in specific regions. Section 4 contains conclusions and future work.

2. Materials and Methods

2.1. Methodology

2.1.1. SARIMAX

Time series analysis requires at least one differencing to achieve stationarity, after which the AR and MA terms are combined to form the ARIMA model. The ARIMAX model extends the ARIMA model by introducing exogenous variables to assist in modeling the time series data. This model can predict the future values of variables based on their historical values while considering the impact of external factors on those historical values. The ARIMA notation is (p, d, q), where p, d, and q represent the orders of the non-seasonal autoregressive (AR), differencing (I), and moving average (MA) components, respectively. In contrast to the ARIMAX model, the SARIMAX model accounts for seasonal factors and is denoted as (p, d, q, P, D, Q, s), where P, D, and Q represent the orders of the seasonal AR, I, and MA components, respectively. Through various parameter combinations, it encompasses ARIMA, ARMA, AR, and MA models, making it suitable for time series data that exhibit significant periodic and seasonal characteristics. Due to its robustness, high predictive accuracy, fast training speed, and simplicity of implementation, it is widely used, as expressed in Equation (1) [24,25],

$$y_t^{\prime }=c+\sum _{i=1}^p{\varphi }_i{y^{\prime }}_{t-i}+\sum _{j=1}^q{\theta }_j{{\epsilon }^{\prime }}_{t-j}+\sum _{i=1}^P{\mathsf{\Phi }}_i{y^{\prime }}_{t-is}+\sum _{j=1}^q{\mathsf{\Theta }}_j{{\epsilon }^{\prime }}_{t-j}+\sum _{l=0}^k{\beta }_l{x}_{l,t}+{\epsilon }_t$$

(1)

where $y_t^{\prime }$ represents the predicted variable, c is the constant term, ${\phi }_i$ and ${\theta }_j$ are the autoregressive and moving average coefficients, ${y^{\prime }}_{t-i}$ and ${{\epsilon }^{\prime }}_{t-j}$ are the historical values and lagged error variables, k is the number of exogenous variables, ${\beta }_l$ is the coefficient for the exogenous variables, $x_{l,t}$ is the observed value of the exogenous variables, ${\epsilon }_t$ is the error term, ${\Phi }_i$ and ${\Theta }_j$ are the seasonal autoregressive and moving average coefficients, and s is the length of the seasonal period.

2.1.2. SVM

SVM can be used to solve the problems of classification and regression by mapping low-dimensional nonlinear problems to high-dimensional spaces, improving data fitting and prediction accuracy [15,16,17,29].

Given a training sample set D = {(x_i, y_i)}, 1 ≤ i ≤ N, the SVM is used to obtain a regression model that makes f(x) close to y as much as possible, where f(x) is the regression function represented in Equation (2),

f(x_i) = ω^T φ(x_i) + b

(2)

where ω is the weight vector, b is the bias, and φ(x) is the mapping function that maps the input vector to a higher-dimensional feature space when the sample set is not classified linearly.

Suppose the error between f(x) and y is ϵ, and the loss can be calculated when the absolute error between f(x) and y is more than ϵ. Then, the regression problem can be transformed into Equation (3),

$$\min \left(\mathsf{\omega },\mathrm{b}\right){\displaystyle \frac{1}{2}}{\left|\right|\mathsf{\omega }\left|\right|}^2+C{\sum }_{i=1}^n{l}_{\mathsf{ϵ}}\left(f\right(x_i), y_i)$$

(3)

where l_ϵ is the ϵ-insensitive loss function and C is a predefined positive trade-off parameter between model simplicity and generalization ability.

Solve the regression function of the support vector machine,

$$f\left(x\right)={\sum }_{i=1}^n({\alpha }_i-{\alpha }_i^{*})K(x_i,x_j)+b$$

(4)

where K(xi,x) is a kernel function,${\alpha }_i$, ${\alpha }_i^{*}$ is a Lagrange multiplier.

2.1.3. RF

RF is a machine learning algorithm formed by combining the random subspace method and bagging ensemble learning theory [30]. A random tree is generated through bootstrap sampling of the training data and random sampling of subsets of variables. It is less prone to overfitting since each tree is trained on a unique bootstrap subsample of the original dataset [31].

Assuming that the number of training samples selected from the original sample set through multiple bootstrap sampling is T, multiple random samples are obtained to construct T decision trees, thereby forming the random forest. The marginal function defined by the random forest algorithm is shown in Equation (5),

$$mg\left(x,y\right)=a{v}_{t}I\left(f_t\left(x,\theta \right)=y_i\right)-maxa{v}_{t}I{f}_t\left(x,\theta \right)=q$$

(5)

where av_t is the averaging function; I is the indicator function; f_t(x,θ) is the ground solar radiation estimation model of a single decision tree; y_i is the training value of the i-th instance; θ is the independent distributed random variable; q is the category of the input variables.

2.1.4. XGBoost

XGBoost is developed based on gradient-boosting decision trees and is widely used in various fields of machine learning [32]. The XGBoost model can be considered as an accumulation model consisting of K decision trees. The XGBoost algorithm can be viewed as an additive model composed of T decision trees, as given in Equation (6),

$$Y_i={\sum }_{t=1}^T{f}_t\left(x_i\right)$$

(6)

where Y_i is the estimated value of solar radiation.

In the regression process, the objective function of the model is,

$$Obj={\sum }_{i=1}^{n}L(y_i,Y_i)+{\sum }_{t=1}^T\mathsf{\Omega }{f}_t$$

(7)

where Ω is the regularization term.

For the regularization term of each decision tree, this study uses vector mapping to improve the decision trees. Therefore, Ω can be determined by Equation (8).

$$\mathsf{\Omega }\left(\mathrm{f}\right)=\mathsf{\gamma }\mathrm{J}+{\displaystyle \frac{1}{2}}\lambda {\sum }_{j=1}^J{\omega }_j^2$$

(8)

where ω is the vector of scores in the leaves; λ is the regularization parameter; and γ is the minimum loss required for further splitting the leaf nodes.

2.1.5. Exhaustive Search Method

This study employs a penalty function-based exhaustive search optimization algorithm. It explores all solutions within a defined range and evaluates the objective function value of each solution, ultimately finding the solution that minimizes the objective function. The penalty function is set as a judgment criterion, primarily serving to impose a certain degree of penalty on solutions that violate constraints, thereby filtering out suitable objective functions based on the constraints.

• Objective Function

The annual cost of the power supply system includes initial investment and operation and maintenance costs [33]. The objective function expression is given in Equation (9),

$$\mathrm{Min}{\mathrm{C}}_{\mathrm{t}}={\mathrm{C}}_{\mathrm{acap}}+{\mathrm{C}}_{\mathrm{are}}+{\mathrm{C}}_{\mathrm{ains}}+{\mathrm{C}}_{\mathrm{aom}}$$

(9)

where $C_t$ is the annual cost of the entire power supply system, in yuan; C_acap is the annual initial investment of the system [34], in yuan; C_are is the annual replacement cost of the system [26], in yuan; C_ains is the annual installation cost of the system, in yuan; C_aom is the annual operation and maintenance cost of the system, in yuan; CRF is the capital recovery factor; SFF is the capital debt repayment coefficient.

$${\mathrm{C}}_{\mathrm{acap}}={\mathrm{C}}_{\mathrm{RF}}({\mathrm{C}}_{\mathrm{pv}}{\mathrm{N}}_{\mathrm{pv}}+{\mathrm{C}}_{\mathrm{b}}{\mathrm{N}}_{\mathrm{b}}+{\mathrm{C}}_{\mathrm{con}})$$

(10)

$${\mathrm{C}}_{\mathrm{are}}={\mathrm{S}}_{\mathrm{FF}}{\mathrm{C}}_{\mathrm{b}}{\mathrm{N}}_{\mathrm{b}}\left[{\left({\displaystyle \frac{1+\mathrm{f}}{1+\mathrm{d}}}\right)}^5+{\left({\displaystyle \frac{1+\mathrm{f}}{1+\mathrm{d}}}\right)}^{10}+{\left({\displaystyle \frac{1+\mathrm{f}}{1+\mathrm{d}}}\right)}^{15}\right]+{\mathrm{S}}_{\mathrm{FF}}{\mathrm{C}}_{\mathrm{con}}{\left({\displaystyle \frac{1+\mathrm{f}}{1+\mathrm{d}}}\right)}^{10}$$

(11)

where C_con is the cost of the controller, in yuan. The installation cost of the solar energy storage power supply system is calculated as 10% of the initial investment, and the operation and maintenance cost is calculated as 2% of the initial investment [35].

2.

Penalty Function

$$\mathrm{penalty}1={\mathrm{penalty}}_{\mathrm{f}}\times \max \left(0, \mathsf{\delta }\mathrm{LPSP}-\mathsf{\delta }\mathrm{LPS}{\mathrm{P}}_{\max }\right)$$

(12)

$$\mathrm{penalty}2={\mathrm{penalty}}_{\mathrm{f}} \times \max (0, \mathsf{\delta }\mathrm{EXC} -{ \mathsf{\delta }\mathrm{EXC}}_{\max })$$

(13)

$$\mathrm{penalty}=\mathrm{penalty}1+\mathrm{penalty}2$$

(14)

where penaltyf is penalty coefficient, set to 1000; penalty1 is the penalty term for the load shedding rate; penalty2 is the penalty term for the energy overflow ratio; and penalty is the total penalty function.

2.2. Evaluation Metrics

This study selects the coefficient of determination (R²), root mean square error (RMSE), and mean absolute error (MAE) as evaluation metrics for PV power prediction. The formulas are as follows,

$$R^2={\displaystyle \frac{{\left[{\sum }_{i=1}^n\left(E_i-\overline{E_i}\right)\times \left(M_i-\overline{M_i}\right)\right]}^2}{{\sum }_{i=1}^n{\left(E_i-\overline{E_i}\right)}^2\times {\sum }_{i=1}^n{\left(M_i-\overline{M_i}\right)}^2}}$$

(15)

$$RMSE=\sqrt{{\displaystyle \frac{{\sum }_{i=1}^n{\left(E_i-M_i\right)}^2}{n}}}$$

(16)

$$MAE={\displaystyle \frac{{\sum }_{i=1}^n\left|E_i-M_i\right|}{n}}$$

(17)

where $E_i$, $M_i$, $\overline{E_i}$, $\overline{M_i}$ and n represent the measured power, predicted power, average measured power, average predicted power, and the sample size, respectively. The closer the R² value is to 1, the better the predictive performance; the closer RMSE and MAE are to 0, the better the model’s prediction accuracy.

2.3. Dataset Partitioning

2.3.1. Meteorological Data

This study uses meteorological data from a PV power station in Northwest China, covering a period of 365 days from 4 February 2019 to 3 February 2020. The Northwest region is a typical arid and semi-arid area in China, making the climate data highly representative. This study employs the AV6592 PV cell tester (China Electronics Technology Group Corporation, Beijing, China) to measure PV power, solar irradiance, and ambient temperature, with a voltage testing accuracy of 0.01 V, a current testing accuracy of 0.001 A, and a power testing range of 0.1–500 W, with a collection interval of 10 min. The CS5M32-260 monocrystalline PV panel (Gordon Electronics Co., Ltd., Taizhou, China) has a peak power of 260 W, a peak voltage of 49.71 V, a peak current of 5.25 A, and a photoelectric conversion efficiency of 16.9%. The charging and discharging are controlled by an MPPT controller (Jiangsu Hengtong General Electric Co., Ltd., Xuzhou, China, 48 V, 20 A). The PV cell tester connects to a computer via Bluetooth for display and storage.

Based on the historical meteorological data and historical PVPG measured, a correlation coefficient matrix heatmap is plotted, as shown in Figure 3. From the figure, it can be seen that the time series changes, solar radiation, ambient temperature, and component temperature have a high correlation with PVPG. Therefore, this study uses the time series as an index and solar radiation, ambient temperature, and component temperature as covariates to establish a PVPG prediction model based on SARIMAX for predicting future PVPG.

2.3.2. Establishing Models Based on Different Time Scales

This study will establish prediction models for the following three time scales:

• Annual Time Scale: A prediction model will be established using data from the entire year, with the data directly divided into an 80% training set and a 20% testing set. After completing predictions on the testing set, evaluation metrics will be used for assessment.
• Seasonal Time Scale: The original data will be divided into four parts according to the seasons: spring, summer, autumn, and winter, for estimating PVPG performance testing. Spring is defined as March to May, summer as June to August, autumn as September to November, and winter as December to February. Each season’s data will be divided into 80% training set and 20% testing set. Finally, R2, RMSE, and MAE will be used as evaluation metrics for the models, conducting regression analysis between the estimated values and the actual measured values, and comparing the prediction accuracy and stability across the four seasons. The flowchart is shown in Figure 4.
• Model Based on Solar Terms: To test the accuracy of the PVPG estimation model, the original data were divided according to the 24 solar terms for performance testing of the power generation estimates. Each solar term’s data were split into an 80% training set and a 20% testing set. Finally, R², RMSE, and MAE were used as evaluation metrics for the model, conducting regression analysis between the estimated values and the actual measured values, and comparing the prediction accuracy and fitting effects across the 24 solar terms. The flowchart for the 24 solar terms and their models is shown in Figure 5.

2.4. Related Calculation Formulas

• There are 15 days from the beginning of “the Beginning of Summer” solar term to the next solar term, with an average PVPG of 0.985 kW/m². The power generation calculation formula for the PV panel is given in Equation (18),

$${\mathrm{P}}_{\mathrm{pv}}={\mathrm{A}}_{\mathrm{pv}}{\mathrm{p}}_{\mathrm{pv}}$$

(18)

where P_pv is the power generation of the PV panel, in watts (W); A_pv is the area of the PV module, in square meters (m²); p_pv is the power generation per unit area of the PV panel, in watts per square meter (W/m²).

2.

After the PVPG meets the load power demand, any excess electricity will be charged into the battery. The charging power of the battery over the t~t + Δt time period is given by Equation (19),

$${\mathrm{P}}_{\mathrm{charge}}=∆\mathrm{t}{\mathsf{\eta }}_{\mathrm{in}}\left[{\mathrm{P}}_{\mathrm{pv}}\left(\mathrm{t}\right)\pm {\mathrm{P}}_{\mathrm{l}}\left(\mathrm{t}\right)\right]$$

(19)

When the PVPG cannot meet the load power demand, the battery begins to discharge. The discharging power of the battery over the t~t + Δt time period is given by Equation (20),

$${\mathrm{P}}_{\mathrm{discharge}}={\displaystyle \frac{∆\mathrm{t}\left[{\mathrm{P}}_{\mathrm{pv}}\left(\mathrm{t}\right)\pm {\mathrm{P}}_{\mathrm{l}}\left(\mathrm{t}\right)\right]}{{\eta }_{out}}}$$

(20)

where Pcharge is the charging power of the battery, in watts (W); Pdischarge is the discharging power of the battery, in watts (W).

3.

The state of charge (SOC) of a battery cell at a certain moment t is given by Equations (21) and (22) [36]:

$$\mathrm{Charging}: {\mathrm{S}}_{\mathrm{oc}}\mathrm{t}={\mathrm{S}}_{\mathrm{oc}}(\mathrm{t}-∆\mathrm{t})+{\displaystyle \frac{\Delta {\mathrm{E}}_{\mathrm{store}}{\mathsf{\eta }}_{\mathrm{in}}}{{\mathrm{N}}_{\mathrm{b}}{\mathrm{E}}_{\mathrm{rate}}}}$$

(21)

$$\mathrm{Discharging}: {\mathrm{S}}_{\mathrm{oc}}\mathrm{t}={\mathrm{S}}_{\mathrm{oc}}\left(\mathrm{t}-∆\mathrm{t}\right)-{\displaystyle \frac{\Delta {\mathrm{E}}_{\mathrm{store}}}{{\mathrm{N}}_{\mathrm{b}}{\mathrm{E}}_{\mathrm{rate}}{\mathsf{\eta }}_{\mathrm{out}}}}$$

(22)

where S_oc(t − Δt) is the state of charge of the battery at time t − Δt, expressed as a percentage (%); Erate is the rated capacity of the battery, in ampere-hours (Ah), taken as 120 Ah; N_b is the number of battery units in the battery bank; ΔE_store is the amount of charge in the battery during the time period from t to t $+$ Δt.

During operation, the state of charge of the battery should be maintained within a specified range.

4.

When both charging and discharging cannot meet the load power demand, the system experiences a deficit. The calculation formula for the deficit is given in Equation (23) [36]:

$${\mathrm{Q}}_{\mathrm{LPS}}\left(\mathrm{t}\right)=\left\{{\mathrm{P}}_{\mathrm{l}}\left(\mathrm{t}\right)-\left[{\mathrm{P}}_{\mathrm{pv}}\left(\mathrm{t}\right)+{\mathrm{P}}_{\mathrm{discharge}}\left(\mathrm{t}\right){\mathsf{\eta }}_{\mathrm{out}}\right]\right\}∆\mathrm{t}$$

(23)

where Q_LPS is the deficit energy of the system during the time period Δt, (Wh).

5.

The calculation formula for the load-shedding rate is [37]:

$${\mathsf{\delta }}_{\mathrm{LPSP}}={\displaystyle \frac{{\sum }_{\mathrm{t}=1}^{\mathrm{T}}\left\{{\mathrm{P}}_{\mathrm{l}}\left(\mathrm{t}\right)-[{\mathrm{P}}_{\mathrm{pv}}\left(\mathrm{t}\right)+{\mathrm{P}}_{\mathrm{discharge}}\left(\mathrm{t}\right){\mathsf{\eta }}_{\mathrm{out}}]\right\}∆\mathrm{t}}{{\sum }_{\mathrm{t}=1}^{\mathrm{T}}[{\mathrm{P}}_{\mathrm{l}}\left(\mathrm{t}\right)]∆\mathrm{t}}}$$

(24)

The calculation formula for the energy overflow ratio is:

$${\mathsf{\delta }}_{\mathrm{EXC}}={\displaystyle \frac{{\sum }_{\mathrm{t}=1}^{\mathrm{T}}\left\{{\mathrm{P}}_{\mathrm{pv}}\left(\mathrm{t}\right)-[\frac{{\mathrm{P}}_{\mathrm{charge}}\left(\mathrm{t}\right)}{{\mathsf{\eta }}_{\mathrm{in}}}+{\mathrm{P}}_{\mathrm{l}}\left(\mathrm{t}\right)]\right\}∆\mathrm{t}}{{\sum }_{\mathrm{t}=1}^{\mathrm{T}}[{\mathrm{P}}_{\mathrm{pv}}\left(\mathrm{t}\right)]∆\mathrm{t}}}$$

(25)

where ${\mathsf{\delta }}_{\mathrm{LPSP}}$ and ${\delta }_{EXC}$ have value ranges of [0, 1]. The values of ${\mathsf{\delta }}_{\mathrm{LPSP}}$ and ${\delta }_{EXC}$ closer to 0 indicate a higher power supply reliability and less energy waste.

6.

The total power of the PV-driven irrigation system is calculated based on the maximum operating conditions of the unit [36], which mainly includes overcoming the resistance of the spray head and hose, overcoming the rolling resistance, and calculating the torque and speed of the traction device, taken as 1.26 kW.

2.5. Building the Model in Python

The overall process for predicting PVPG is shown in Figure 6. The specific steps to establish the model using Python 3.0 are as follows:

• Read the data and preprocess it.
• Set the time as the index and convert the data into a time series format.
• Plot the time series graph of the original data to observe the trends and periodicity in the data.
• Select exogenous factors (X) based on the correlation heatmap to determine the key meteorological factors affecting PV power. This study selects total radiation, diffuse radiation, and solar panel temperature as the three exogenous factors.
• Divide the original data into training and testing sets.
• Check the stationarity of the data and apply differencing to non-stationary data to eliminate trends.
• Use the auto ARIMA method to automatically find the optimal parameters.
• Establish the SARIMA model and fit the SARIMAX model using the optimal parameters.
• Perform model diagnostics, such as white noise testing.
• Use the fitted SARIMAX model to make predictions on the training set and evaluate the accuracy of the prediction results.
• Plot scatter plots and waveform graphs for the predicted results and actual values for visualization.

3. Results and Discussion

3.1. Comparison of Four Prediction Models

Based on the comparative analysis of

Table 1. Comparison of evaluation metrics for four prediction models.

Model	R2	RMSE	MAE
SVM	0.791	7.700	5.369
XGBoost	0.787	6.675	4.826
RF	0.789	6.658	4.910
SARIMAX	0.944	7.034	4.413

, it can be concluded that the SARIMAX model performs the best among the four PVPG prediction models with the R² index reaching 0.948, which was higher than the other 19.4%, and the MAE index was lower than the other 10%. The SARIMAX model can automatically search for the optimal prediction parameters, effectively addressing the issue of excessive hyperparameters found in models like XGBoost, such as learning rate, maximum depth of decision trees, and minimum child weight. Additionally, the proposed SARIMAX model enhances the accuracy of data fitting and prediction performance. It can analyze and learn from relevant meteorological data more accurately, making it better suited for predicting PVPG in solar-powered irrigation machines. Therefore, this study will use the SARIMAX model for subsequent comparative research and applications. The fitting evaluation graph for the SARIMAX model is shown in Figure 7.

Figure 7 presents four diagnostic plots for the residual analysis in the SARIMAX model. The top left plot shows the variation of model residuals over time, allowing for the examination of whether the residuals have a zero mean and constant variance. The residuals are standardized to have a mean of 0 and a standard deviation of 1. In this plot, the residuals fluctuate around 0 without any apparent trend or seasonality, indicating that they are random and consistent with the model assumptions. The top right corner displays the distribution of the residuals, which helps to check whether the residuals follow a normal distribution. In this plot, the residual distribution is close to a normal distribution, generally satisfying the assumption. The bottom left Q-Q plot has the theoretical normal distribution quantiles on the horizontal axis and the sample data quantiles on the vertical axis. If the data are normally distributed, the points should lie along the red 45-degree reference line, which further verifies whether the residuals are normally distributed. Most points in the plot are closely distributed around the reference line, suggesting that the normality assumption for the residuals is reasonable. The bottom right ACF plot displays the autocorrelation of the residuals. The blue bars indicate the autocorrelation coefficients at different lag orders, and the gray shaded area represents the significance boundaries. If the autocorrelation coefficients fall within the significance range, it indicates that the autocorrelation at the corresponding lag order is not significant. Most autocorrelation coefficients in the plot fall within the significance range, indicating that there is no significant autocorrelation among the residuals, meaning the residuals are independent. This suggests that the residuals of the model essentially meet the independence assumption.

In summary, Figure 7 demonstrates that the residual characteristics of the model conform to the basic assumptions of time series models, indicating that the model is valid and can be used for prediction and analysis.

3.2. Constructing Models for Three Time Scales

• The evaluation metric results for the annual time scale model and the scatter plot of the estimated PVPG values versus the actual measured values during the testing phase are shown in Figure 8.
• The evaluation metric results for the seasonal time scale model and the scatter plot of the estimated PVPG values versus the actual measured values during the testing phase are shown in Figure 9. From the figure, it can be observed that in the four seasons, the estimation errors between the PVPG values and the actual measured values are smaller in spring and winter, indicating better prediction performance.
• The evaluation metrics for the solar terms time scale model and the scatter plot of the estimated PVPG values versus the actual measured values during the testing phase are shown in Figure 10. From the figure, it can be observed that among the 24 solar terms, the PVPG evaluation metrics RMSE and MAE are the lowest during the period from the start of “The Beginning of Summer” solar term to the end of the next solar term. The scatter points are closer to the 1:1 line, indicating the smallest error between the estimated and actual values, with the highest accuracy. Additionally, during the period from the start of “The Beginning of Winter” solar term to the end of the next solar term, the model achieves the highest R² value for PVPG, indicating the best fitting performance. Overall, except for certain solar term periods where data quality was affected by overcast and rainy days and the uniform dataset partitioning, the fitting performance and accuracy during other solar terms are better compared to the annual and seasonal scales. This also addresses the issue of poor prediction performance during the summer and autumn seasons on the seasonal scale.

3.3. Comparative Analysis of Three Time Scale Predictions Based on SARIMAX

Using evaluation metrics, the three models were assessed, and the results are shown in

Table 2. Comparison of evaluation metrics for the three scales.

Time Scale	R2	RMSE	MAE
Annual Scale	0.9442	0.0703	0.0441
Seasonal Scale	0.9448	0.0688	0.0443
Solar Terms Scale	0.9478	0.0655	0.0436

. By examining the scatter plots and waveforms of the three time scale models, along with the data in the table, it can be concluded that the model based on SARIMAX using solar terms as the time scale outperforms the seasonal and annual scale models in terms of accuracy and fitting performance. The RMSE for the solar term scale is 6.8% lower than that for the annual scale, while the MAE is 1.1% lower. Additionally, the RMSE for the solar term scale is 4.7% lower than that for the seasonal scale, and the MAE is 1.6% lower. This is because the climate variation between the two solar terms is relatively more stable and similar. The model can reasonably configure the power supply system capacity for the irrigation machine based on different solar terms; however, it may be significantly affected during periods of extreme weather conditions within certain solar terms. Although the estimation accuracy of the annual and seasonal models is lower compared to the solar term and seasonal models, they can be used to estimate photovoltaic power generation across various regions worldwide, providing broader applicability.

3.4. Analysis of Estimation Accuracy of the Three Time Scale Models Under Typical Weather Conditions

To analyze the estimation accuracy of models under three different time scales, this study compares the results of PVPG estimates under typical sunny and overcast days, considering only the time periods when PVPG occurs. The typical sunny day selected is 18 May 2019 (with a maximum temperature of 32.34 °C and a minimum temperature of 13.41 °C), while the typical overcast day selected is 19 May 2019 (with a maximum temperature of 22.55 °C and a minimum temperature of 11.71 °C). The temperature and radiation variation curves for these typical weather conditions are shown in Figure 11. Figure 12, Figure 13 and Figure 14 present the comparison of estimated PVPG values against actual measured values for each algorithm under typical weather conditions. The horizontal axis represents the number of measurements of PVPG taken in one day, while the vertical axis represents the normalized value of PVPG. Figure 12a, Figure 13a and Figure 14a corresponds to a typical sunny day, and Figure 12b, Figure 13b and Figure 14b corresponds to a typical overcast day.

Table 3. Comparison of the three scales based on typical weather.

	Sunny Day			Overcast Day
Time Scale	R2	RMSE	MAE	R2	RMSE	MAE
Annual Scale	0.9614	0.0692	0.0424	0.9482	0.0710	0.0475
Seasonal Scale	0.9721	0.0675	0.0494	0.9364	0.0781	0.0528
Solar Terms Scale	0.9789	0.0578	0.0310	0.9397	0.0726	0.0407

shows the evaluation indexes based on the three time-scale models under typical weather.

Since irrigation is not required on rainy days, the solar reel irrigation system operates only under two conditions: sunny and cloudy weather. Therefore, this study randomly selected two days (one sunny and one overcast) to measure meteorological data and PVPG data to verify the prediction accuracy of the proposed models. Data from other weather conditions showed similarities, making the use of just these two weather types sufficient.

From the figures, it can be observed that significant fluctuations in solar radiation affect the model’s fitting performance, which in turn impacts the accuracy of PVPG estimates. Among the three models, those using solar terms as the time scale provide the most accurate predictions. Overall, the SARIMAX-based models across all three scales can meet prediction requirements, allowing for selection based on actual application scenarios and reasonable configuration of power supply system capacity.

3.5. Application

To further apply the SARIMAX-based PVPG prediction model in practical agricultural production, this section will take the solar reel irrigation machine JP300-75 as an example. Based on the data obtained from the prediction model, the unit area PVPG capacity in the region will be calculated to reasonably configure the capacity of the solar energy storage power supply system. This power supply system primarily includes solar power generation and batteries. Since the solar terms time scale model demonstrated more advantages in the comparative analysis, this section will use the most accurate prediction results from the “The Beginning of Summer” solar term for configuration optimization. At the same time, the annual scale model and seasonal scale model will also be optimized for configuration, allowing for comparison with the solar term scale model, and selecting summer meteorological data corresponding to the “Beginning of Summer” solar term as the seasonal scale data.

The battery selected is the Hebei Wind Sailing Battery Co., Ltd. 190H52 valve-regulated sealed lead-acid battery with a rated voltage of 12 V and a capacity of 120 Ah. The PV module selected is the CS5M32-260 monocrystalline PV module from Jinyuan Electronics, with a photoelectric conversion efficiency of 16.9%. The input parameters for the optimization model are shown in

Table 4. Input parameters of the optimization model for simulation.

Parameter	Numeric	Parameter	Numeric
PV panel cost	USD 141.58	Charging efficiency	90%
The cost of the battery	USD 84.95	Discharge efficiency	85%
Maximum energy spillover ratio	0%	Debt repayment coefficient of funds	0.037
Inflation rate	3.5%	Fund recovery factor	0.067
Years of operation	20a	Controller cost	USD 226.52
Maximum load power deficit rate	0%	Interest rate	3.1%
The lower limit of the allowable state of charge	20%	The upper limit of the allowable state of charge	80%
Initial state of charge	60%	Motor work efficiency	80%

, and the flowchart for the optimization model calculation is illustrated in Figure 15.

Based on the results from the aforementioned model, The optimal configuration for the solar energy storage irrigation machine during the “Beginning of Summer” solar term is determined to be: 6 m² of solar panels and 2 batteries of 120 Ah each. Based on the objective function, the minimum cost for this period (from the “Beginning of Summer” solar term to the end of the next solar term, a duration of 14 days) is calculated to be approximately USD 5.04. Based on the seasonal scale, the optimal configuration for the summer solar energy storage irrigation machine is determined to be 7 m² of solar panels and one battery of 120 Ah, and the minimum annual cost is calculated to approximately USD 146.5, which amounts to approximately USD 6.11 for each solar term over 14 days. Based on the annual scale, the optimal configuration for the solar energy storage irrigation machine for the entire year is determined to be: 10 m² of solar panels and two batteries of 120 Ah each, and the minimum season cost is calculated to be USD 40.5, and amounts to approximately USD 6.75 for each solar term over 14 days. Based on experience, to meet the power supply demand, it is necessary to install enough photovoltaic panels, specifically 12 m² of PV panels and two batteries. The cost, after accounting for this period, is approximately USD 8.46. Comparing the results of the four configurations, it can be seen that using the solar term scale, which has the highest accuracy, allows for a lower cost while ensuring power supply reliability and effective energy overflow management, for the same quarter, the cost obtained from the model established based on the solar term scale is 17.5% lower than that from the seasonal scale, 25.3% lower than that from the annual scale, and 40% lower than that from traditional methods.

Additionally, the application of photovoltaic power supply in reel-type irrigation machines helps reduce fuel consumption and carbon emissions. This configuration method can also be applied to various other solar energy equipment, particularly electromechanical devices in the agricultural sector. The SARIMAX model for photovoltaic power prediction based on solar terms can be widely used in various climate-affected fields, effectively predicting photovoltaic power generation in different regions during specific solar terms based on historical meteorological data. Similarly, seasonal and annual scale models can be utilized to predict the power generation of other solar energy equipment in different regions.

4. Conclusions and Future Work

4.1. Conclusions

This study has significant practical implications for improving the reliability of PV power systems, accurately predicting solar power generation in specific regions, and rationally configuring the capacity of multi-energy-driven power supply systems for solar reel irrigation machines. In this study, we compared four models: XGBoost, RF, SVM, and SARIMAX, ultimately selecting the most accurate SARIMAX model for further research, and using the automatic optimization method to search for the optimal hyperparameters of the SARIMAX model to achieve effective prediction of PVPG.

We established multiple models at different time scales to predict solar power generation in specific regions. This study proposed a solar term-based time scale for modeling, we highlighted the advantages over annual and seasonal scale prediction models, improving prediction accuracy and stability while offering greater flexibility.

Through simulation analysis, we reached the following conclusions: Among the four models—XGBoost, RF, SVM, and SARIMAX, the SARIMAX model performed the best [24,25,29,30,31,32]. In this paper, the SARIMAX model is used to evaluate the prediction accuracy and stability of PVPG prediction at three time scales, and the results show that the model proposed in this study for predicting PVPG based on the solar term time scale, the effect is higher than that of the annual scale and seasonal scale models. It has also demonstrated better performance compared to the commonly used short-term predictions on a daily basis and long-term predictions based on annual units [14,15,16,17,18,19,20,21,22,23,24,25,36,37]. Additionally, testing under typical weather conditions yielded the same results.

Finally, based on the prediction results, we optimized the capacity configuration of the PV irrigation machine’s power supply system. Compared to previous studies that focused heavily on PVPG prediction in the industrial sector, this research reasonably applies it in the agricultural irrigation field [14,15,16,17,18,19,20,21,22,23,24,25]. A mathematical model was established using a penalty function-based exhaustive search method, enhancing the rationality of various configurations, improving supply reliability, and reducing costs and energy overflow. The research results provide a theoretical reference for accurate PVPG predictions while laying a foundation for the practical application of PV-driven reel irrigation machines.

4.2. Future Work

However, this work has some limitations. For instance, using the twenty-four solar terms as a time scale to establish the model is only applicable to PVPG prediction studies in China, while other regions can only utilize annual or seasonal scales. In future research, we will conduct further studies in this area to enhance the model’s universality. Additionally, the use of data collected from a typical region for prediction research led to an insufficient sample size. Future research will aim to increase the sample size.