Using Machine Learning Models to Forecast the Conversion Coefficient between Electricity Consumption and Water Pumped for Irrigation Wells in Baicheng City, China

Abstract

Forecasting the electricity-to-water conversion coefficient (EWCC) can help manage and plan irrigation water in arid and semiarid areas. However, the EWCC is influenced by several factors, making it difficult to develop an analytical model for validation or prediction. Therefore, this study selected 206 typical irrigation wells in Baicheng City to conduct EWCC tests in a field investigation to gather information regarding the results and related influencing factors. Subsequently, machine learning models (multiple linear regression model, support vector model, and backpropagation neural network) were trained, validated, and tested, and their precisions were evaluated and compared. The backpropagation neural network model was the most accurate, followed by the support vector and multiple linear regression models. The backpropagation neural network model results were consistent with those of the field survey, and this model was thus used to forecast the EWCC for all the townships in Baicheng City. The forecasting models revealed that most towns had an EWCC from 3 to 7 m³/kW·h, with an EWCC greater than 7 observed in the Tao’er River Fan and Yueliangpao District. The BP models developed in this study proved to be dependable and applicable for forecasting the EWCC in this area.

1. Introduction

Groundwater irrigation has emerged as a key resource for ensuring the supply of water to farmers [1,2]. Agriculture accounts for approximately 70% of the world’s water consumption and is the foremost sector in terms of both the economic and social aspects of water [3,4]. With growing recognition of the significance of water resources, the challenge of gathering and documenting groundwater to manage agricultural water is becoming increasingly evident [5,6]. Because of the extensive size and widespread distribution of irrigation wells in agriculture, it is challenging to use water meter measurements from one well to another for both measurement and management [7]. Hence, the resolution of water metering in agriculture is of paramount importance for the prudent use and efficient administration of water resources.

Currently, three methods are generally used for gauging the amount of water drawn from irrigated wells in farming: water meter measurement, local water quota, and conversion by the electricity consumption of irrigated wells [8,9,10,11,12]. The first method is the most accurate if meters are installed in each well. However, the large number of irrigated wells used for agriculture leads to difficulties, including steep financial costs and the intricate task of maintaining metering systems at every juncture [13]. In contrast, the second method depends on the local water quota and magnitude of different irrigated plants to estimate water consumption. Typically, the water quota is determined based on the climate and plant type. However, this method only assigns a singular value to a single crop in a specific region and neglects the impact of varying hydrogeological conditions at diverse sites on water consumption. Thus, the statistical results of this method are only approximate. The third method initially confirmed the electricity-to-water conversion coefficient (EWCC) via a water pumping experiment. Subsequently, the volume of groundwater used for irrigation can be altered by multiplying the coefficient by the respective electricity consumption. Thus, this method is feasible and accurate for ascertaining the volume of water extracted from irrigation wells [14,15].

The EWCC is influenced by various factors, including the hydrogeological conditions (such as the depth of groundwater and the water yield property), pump power, rated pump capacity, head of the water pump, irrigation well diameter, age of the well, and stability of the power supply [16]. Consequently, the factors that influence the coefficients are highly complex. Further investigation is required to clarify the quantitative correlation between indicators and factors in specific areas.

Machine learning, a branch of artificial intelligence, can be used to train models using existing data, which can then be used to solve specific problems and extract new information from big data [17,18]. Furthermore, the development of convenient programming languages and mature algorithms has facilitated the applicability of machine learning. With recent developments in computer technology, machine learning has been used in groundwater research for water table prediction [19] and groundwater assessment and monitoring [20]. Accordingly, machine learning methods have become effective tools for obtaining predictive results from potential information in groundwater and environmental research. As tests on the coefficient have been performed extensively and data have been accumulated, the application of machine learning to the prediction of the coefficient has become possible. Through machine learning methods, existing data can be used to summarize and form a functional relationship between the coefficient and influencing factors. However, machine learning has not yet been applied to forecast the conversion coefficient between electricity consumed and water pumped.

This study aimed to calculate water consumption by analyzing agricultural electricity usage and its conversion coefficient. This study conducted field tests in Baicheng City, Jilin Province, during the irrigation period to achieve this goal. The conversion coefficients of the typical irrigation wells were confirmed. Subsequently, the correlation between the coefficients and the influencing factors was analyzed. Finally, three typical machine learning methods (multiple linear regression (MLR), support vector machine (SVM), and backpropagation neural network (BP)) were applied to investigate the intricate connection between the conversion coefficient and influential elements. These results can inform the statistical analysis of agricultural irrigation water consumption.

2. Materials and Methods

2.1. Study Area

Baicheng City is located in Western Jilin Province (Figure 1) and has a total area of 25,600 km². The city spans from 121°38′6″ to 124°23′56″ E and from 44°13′57″ to 46°18′15.8″ N. It has a temperate continental monsoon zone with an average annual rainfall of approximately 400 mm, distributed irregularly throughout the year. Approximately 70% of rainfall events occur between July and September. The mean evaporation is 1340 mm. The Nenjiang, Tao’er, and Huolin Rivers flow through the study area. Groundwater is the primary water source in Baicheng, and agriculture is responsible for 90.34% of the overall groundwater consumption. In 2022, 108,559 motor-pumped wells were in operation to provide irrigation for a total area of 6271.6 km². Most cultivated land is pipe-irrigated. Currently, the irrigation water usage statistics in Baicheng adhere to the quota method, meaning that water consumption depends on crop type. It is not possible to determine the actual amount of water used according to specific situations in different areas. Thus, this study analyzed agricultural electricity usage and its conversion coefficient to obtain more precise water consumption information.

Owing to the limited availability of surface water sources, rice and corn depend heavily on groundwater for primary irrigation in the study area. Most agricultural areas are irrigated by mechanical wells that extract groundwater from unconfined and confined water; however, some use surface water. Each well irrigates 4.7 ha of land on average. However, pumped water is not measured because most agricultural irrigation wells are electromechanical and metering systems incur considerable expense in terms of investment and maintenance costs.

This study chose 206 typical agricultural irrigation wells to measure the actual pumped groundwater and corresponding electricity consumption. The differences in hydrogeological conditions and their representativeness must be considered among the selection criteria for typical agricultural irrigation wells, along with the diverse target aquifers across various hydrogeological units in different districts and areas of a single well volume, to guarantee the representativeness of the selected wells. Furthermore, to reduce measurement inaccuracies, the selected agricultural irrigation wells must have sufficient irrigation area and duration in a single pumping test. Each well has its own power-consumption-measuring apparatus. To install a water quantity monitoring system and real-time transmission devices, the diameter of the pipeline must exceed 20 cm, and the outlet pipe must be made of iron and exceed 1 m in length. A minimum of three conventional agricultural irrigation wells that met these criteria in every township were studied to the greatest extent feasible. Each township is equipped with standard wells to ensure the uniform distribution of the test sites [21]. A graphical representation of the geographical spread of agricultural irrigation wells in Baicheng City is shown in Figure 2.

2.2. Field Tests for EWCC

The conversion of surrogate measures of electrical energy consumption into pumping volumes is crucial for the indirect monitoring of groundwater extraction. Initially, field tests were conducted to confirm the EWCC values in different areas and analyze these factors.

The EWCC for a single well can be expressed as the ratio of the amount of water pumped to the amount of electricity consumed by agricultural irrigation wells during the same period. The coefficient was calculated using the following equation:

$${\mu }_1=\frac{Qw}{E},$$

(1)

where ${\mu }_1$ is the EWCC, $Qw$ is the pumping volume of agricultural irrigation wells at a certain time, and $E$ is the power consumption of the well during water pumping.

The EWCC multiplies the power usage of a conventional agricultural irrigation well pump, ultimately dictating the overall water consumption for agricultural irrigation. To ascertain EWCC values, water metering devices can be installed at appropriate sites, along with field measurements of water consumption.

The irrigation period of the wells followed a straightforward process, with water pumped regularly in accordance with the soil moisture and rainfall. In Baicheng City, most agricultural irrigation wells have diameters exceeding 20 cm, whereas the outlet pipes have diameters exceeding 50 mm. These conduits are composed of PVC or steel and create a solitary well that generates substantial amounts of water, with the outflow pipes filled during water pumping. Therefore, for precise and easy-to-use testing, it is advisable to employ an electromagnetic flowmeter for an in-depth assessment of the pumping volume, particularly in townships lacking the necessary conditions for an integrated electromagnetic flowmeter, and to use an ultrasonic flowmeter for identical objectives. An electromagnetic flowmeter was used to test the pumping volume in this study (Figure 3a). The measurement devices for groundwater pumping were in line with national standards with a flow rate precision of 0.1 L/min. Each flowmeter was equipped with a remote water-monitoring system to store the pumping data online. The power consumption of the test wells was simultaneously measured using electricity meters (Figure 3b).

2.3. Methodology

2.3.1. Pearson Correlation Analysis

The Pearson correlation coefficient was calculated to assess the correlation between the influencing factors and the EWCC. This can be calculated using Equation (2) [22]:

$$\rho \left(x,{\mu }_1\right)=\frac{{\sum }_{i=1}^n(x_i-\overline{x})({\mu }_i-\overline{\mu })}{\sqrt{{\sum }_{i=1}^n{(x_i-\overline{x})}^2{\sum }_{i=1}^n{({\mu }_i-\overline{\mu })}^2}},$$

(2)

where $\rho$ is the Pearson correlation coefficient, x is the individual influencing factor, and ${\mu }_1$ is the measured EWCC.

2.3.2. MLR

MLR, a mathematical method, assesses the linear or nonlinear relationship among various variables by choosing several as independent variables and one as the dependent variable to analyze their interrelations, as depicted in Equation (3) [23]:

$${\mu }_2={\beta }_0+{\beta }_1{x}_1+\dots +{\beta }_k{x}_k,$$

(3)

where ${\beta }_0$ is the regression constant, ${\beta }_1$, ……, ${\beta }_k$ are the regression coefficients, ${\mu }_2$ is the projected EWCC, and $x_1$, ……, $x_k$ are the influencing factors.

PyCharm was employed to develop an MLR model by deriving an empirical formula from field test data. The influencing factors were treated as independent variables, with the EWCC serving as the dependent variable.

2.3.3. SVM

SVM employs a linear classifier that maximizes the intervals set within the feature space to differentiate them from perceptual machines. In addition, SVM incorporates kernel tricks, rendering them essentially nonlinear classifiers. As its learning strategy, SVM uses interval maximization, which can be expressed as the task of solving convex quadratic programming analogous to the task of minimizing a regularized hinge loss function. The learning algorithm of SVM is the optimization algorithm for solving convex quadratic programming [24].

PyCharm 2022.1.4 (Community Edition) was used to construct an SVM model. We developed a classifier for a support vector and divided the field-test dataset into training, validating, and testing groups, accounting for 70%, 15%, and 15% of the data, respectively. The training group was used to train the model, the test set was used for prediction, and the validation set was used to ascertain the accuracy of the model.

2.3.4. BP

As a typical feedforward network, BP surrogates the original physics-based model by recording an extensive amount of input and output data and forming novel mathematical relationships. BPs are characterized by the backward propagation of errors and forward propagation of signals. A BP model comprises three distinct layers: an input layer, a hidden layer, and an output layer (Figure 4). The neurons of each hidden layer are linked to every neuron in the subsequent layer governed by an activation function that disseminates the signals and errors. The weights of the prior neuron layers are modified by reversing the input errors from the output layer [25].

The “nntool” toolbox in Matlab was employed to develop and train the BP. A Sigmoid function was used as the activation function, as shown in Equation (4).

$$sigmoid\left(X\right)=\frac{1}{1+e^X},$$

(4)

where X is the predicted value.

The weights were optimized using the Levenberg algorithm, which is a blend of gradient descent and Gauss–Newton techniques. A total of 70%, 15%, and 15% of the datasets were allocated for training, validation, and testing, respectively.

In this study, all three models were evaluated using mean square error (MSE) and coefficient of determination (R²) [26].

3. Results and Discussion

3.1. Field Investigation

A total of 206 agricultural irrigation wells were selected from 55 townships in Baicheng City during 2-year field surveys conducted in 2022 and 2023. The water and electricity consumption of these wells were measured through field tests to calculate the EWCC, and data on well age, well depth, and pump type were collected through inquiries conducted with local farmers, as shown in

Table 1. Corresponding data collected in the field investigation.

County	Number of Wells	EWCC (m3/kW·h)	Groundwater Depth (m)	Usage Life of the Well (a)	Well Depth (m)	Pump Type
Taobei	57	3.14–8.75	3.11–12.42	1–20	14–120	1, 3, 5
Taonan	48	3.64–10.48	1.52–12.13	4–23	15–75	1, 2, 3
Zhenlai	39	3.57–11.28	2.26–10.88	5–21	45–87	1, 2
Tongyu	38	3.13–10.46	1.71–8.24	3–25	25–100	1, 2
Da’an	24	3.86–11.38	3.25–8.36	6–20	83–100	3, 4

. The findings indicated that the majority of the EWCC fell between 3 and 8 m³/kW·h, with only 10 wells exceeding 8 m³/kW·h. There are five typical types of water pumps in Baicheng City (Figure 5), with the first three types making up 85%.

3.2. Correlation Analysis

The Pearson correlation analysis of the data from the field tests of the agricultural irrigation wells and the EWCC values was conducted to obtain a correlation heat map, as shown in Figure 6. The strongest correlation with the EWCC was observed for groundwater depth, followed by the pump head, pump power, and the age of the well, whereas the pump flow exhibited the lowest correlation with the EWCC. Consequently, this study chose the four most correlated initial influencing factors—groundwater depth, pump head, pump power, and well age–for the subsequent development of the model.

3.3. Model Development

After identifying the influencing factors, the EWCC of the study area was predicted using the MLR, SVM, and BP models using the empirical formula derived from the MLR model, as shown in Equation (5):

$${\mu }_2=-0.6628431x_1-0.03310846x_2+0.02102649x_3+0.0240252x_4,$$

(5)

where ${\mu }_2$ is the predictive EWCC (m³/(kW·h)), $x_1$ is the groundwater depth (m), $x_2$ is well age (a), $x_3$ is pump power (kW·h), and $x_4$ is pump head (m).

Tests were conducted using the MLR, SVM, and BP methods, and the results are shown in Figure 7. The results demonstrated a significant parallel between the predicted and observed values.

Table 2. Error and corresponding parameter statistics of the three methods.

Method	Maximum Prediction Error (%)	Minimum Prediction Error (%)	Average Prediction Error (%)	MSE	R2
MLR	65.68	0.95	16.74	1.40	0.63
SVM	42.58	0.34	8.57	0.54	0.83
BP	31.72	0.14	6.29	0.48	0.87

presents a comparison of the determination factors, mean squared bias, and error between the forecasts and actual measurements of the three methods.

The MLR model exhibited the poorest prediction accuracy, with the MSE at its peak, indicating a larger variance between the predicted and actual values. Conversely, the lower R² implied a weak correlation between the predicted and actual values. Compared with the MLR model, the SVM model exhibited superior predictive precision. A reduced MSE implied less variance between the model’s predicted and actual values, whereas an increased R² indicated a more robust correlation between the measured and predicted values. The BP model exhibited superior predictive precision, as evidenced by the minimal MSE, indicating the least variance between the model’s actual and forecasted values. R² stood out as the most accurate, denoting the most robust link between the actual and predicted values.

Field data such as groundwater depth, well age, pump power, and pump head data were analyzed and fed into the BP model to confirm the efficiency of the BP model. A curve diagram of the change between the EWCC and a single influencing factor was plotted according to the output results. The graphs illustrate the correlation between the sole influencing factors and the EWCC (Figure 8). With increasing groundwater depth, the EWCC decreased (Figure 8a). The depth of the groundwater exerted the most significant effect on the EWCC, with the EWCC decreasing by an average of 0.634 m³/kW·h per meter increase in groundwater depth. As the depth of the groundwater decreased, its impact on the EWCC decreased. When the groundwater depth exceeded 6 m, the inclination intensified. The EWCC decreased by an average of 0.014 m³/kW·h for each additional year of well age (Figure 8b). Thus, the variation in well age had a very small effect on the EWCC compared with groundwater depth. With increasing pump power and pump head, the EWCC increased (Figure 8c,d). These outcomes mirror those derived from the data collected through our field surveys.

Finally, the trained BP model was applied to predict the EWCC in the townships of Baicheng City (Figure 9). The EWCC typically ranged from 3 to 7 m³/kW·h. The areas with an EWCC greater than 7 m³/kW·h were distributed in the Tao’er River Fan and Yueliangpao District in the northern part of Da’an City.

In addition, the predictive outcomes from the BP model were used to generate EWCC query curves within the standard agricultural irrigation well models in Baicheng City (Figure 10).

3.4. Analysis of Influencing Factors

The use of a static conversion factor is justifiable, provided that the correlation between the energy expended and the volume of water pumped remains consistent across both the spatial and temporal dimensions. Hydrogeological conditions, pumping conditions, and variations in groundwater depth can affect the EWCC. The magnitude of the EWCC is influenced by a variety of elements, including the depth of groundwater; water yield property; power, head, age, and installation depth of the pump; age, depth, and diameter of the well; length of the diversion pipes; and types of irrigation [27,28]. The determining factors in this study were among the ranges of these elements. According to the results of the field investigation, we observed obvious differences in the EWCC in areas with different groundwater depths. The EWCC decreased significantly as groundwater depth increased. When the depth of the groundwater increased by 1 m, the EWCC decreased by 0.5 or more. An increase in well age resulted in a reduction in the EWCC; however, the impact of well age on the EWCC was minimal, with only a decade difference in age significantly altering the EWCC. Altering the well depth did not affect the EWCC. For all five pump types investigated, increasing pump power and head increased the EWCC. The increase in pump power was accompanied by an increase in the pump head; however, there was no linear relationship between the pump-rated flow rate and pump power. Consequently, establishing a link between the pump-rated flow rate and the EWCC is unfeasible. Correlation analysis of the finalized dataset from the field survey yielded similar results. Finally, the main factors affecting the EWCC in Baicheng City were groundwater depth, well age, pump power, and pump head. In addition, to validate the constructed BP model, a graph of the variation between the EWCC and a single influencing factor was plotted (Figure 8). The results were the same as those obtained from the field investigation and correlation analysis.

After analyzing multiple elements, it was determined that the depth of groundwater has a substantial effect on the EWCC. Information was collected from a pair of wells in Baicheng City by monitoring the depth of groundwater, and we mapped out the variations in this depth throughout the years (Figure 11). According to the curve, the annual variation in groundwater depth remains below 1 m. Yet, the most significant yearly fluctuation in precipitation during these 23 years amounted to 400 mm. There was no observed link between the amount of rainfall and the depth of groundwater. This suggests a minor influence of rainfall on groundwater depth alterations and a negligible effect of rainfall on the EWCC.

3.5. Comparison and Analysis of the Models

The most accurate prediction model was the BP model, followed by the SVM and MLR models, which is consistent with the results of previous studies [29,30]. This is because the MLR model can easily establish linear causality among various variable groups, simplifying the analysis; however, this overlooks the interaction effect and nonlinear causality. The MLR model can be employed when influencing factors are clear and a substantial quantity of samples have been measured. The SVM model can establish linear causal relationships between multiple sets of variables and can be introduced through the kernel function in the case of nonlinearity to map it to linear. However, even when the SVM of nonlinear data is mapped to a high-dimensional space, the calculation is still low-dimensional. An SVM can be used to make predictions when the exact influencing factors are unclear, but the quantity of data is adequate. The BP model has a powerful nonlinear mapping capability, allowing it to approximate any nonlinear continuous function. Thus, not including physical BP when investigating the EWCC may lead to unreliable predictions. Although the BP model was expected to be superior to the MLR model, their comparison was significant as if the MLR regression model was satisfactory, and there would be no requirement to construct the BP model [31].

After comparison, we used the BP model with the highest accuracy to predict the EWCC of the townships in Baicheng City (Figure 9). The EWCC of most townships was less than 7 m³/kW·h. The EWCC exceeded 7 m³/kW·h in the Tao’er River Fan and Yueliangpao District, which, according to the hydrogeological map of Baicheng City (Figure 2), are extremely rich in groundwater resources. In particular, the Tao’er River Fan is the most water-rich area in Baicheng. According to the field investigation, the depth of the groundwater in the Yueliangpao District was the closest to the surface. Therefore, the larger EWCC in these two areas is consistent with the actual situation. The southern part of Tongyu County has poor groundwater resources, which was mirrored in the forecasted results. After examining and contrasting the forecasted EWCC with the actual values from the field survey, the greatest discrepancy between the forecasted and actual EWCC was 31.72%, with an average of 6.29%. The allocation of regions with varying EWCC sizes aligned with real circumstances. Consequently, we concluded that the BP model is reliable. The query curves of the EWCC under typical pump types in Baicheng City based on the reliable BP model are presented in Figure 10. The query curves revealed a consistency akin to the field investigation, showing a decrease in EWCC with increasing groundwater depth and well age and an increase in the EWCC with higher pump power and head. Therefore, we believe that the query curve aligns with the real scenario and, thus, serves as a reference for the relevant departments in Baicheng City when predicting the EWCC.

Given that the groundwater usage charges for agricultural irrigation coincide with the electricity charges, it is both practical and economical to gather water fees via the electricity fee collection system rather than establishing an additional parallel system. Data on the electrical consumption of agricultural irrigation wells can be gathered via the State Grid Corporation of China, employing the EWCC query curve (Figure 10) to acquire the EWCC for determining water consumption and recording the water bill when adequate data are available. In the case of insufficient information, water consumption is calculated, and water bills are collected according to the predicted EWCC for each township (Figure 9). By levying fees on water via electricity collection, farmers will be motivated to save water during irrigation, guaranteeing that the harvested groundwater meets only the irrigation requirements. Such charging methods will motivate farmers to use groundwater resources wisely.

Despite the BP model’s demonstrated efficacy in forecasting the EWCC in Baicheng City, certain constraints remain. Constructing the BP model necessitates the acquisition of sufficient authentic and credible data, implying the existence of an adequate number of qualified agricultural irrigation wells in the study area. Researchers are required to undertake extensive fieldwork. Moreover, the installation of electromagnetic flow meters in agricultural irrigation wells is costly. These challenges are prevalent in EWCC research.

4. Conclusions

Although smart meter-based direct water metering offers advantages in data accuracy and equity, its steep expense renders it the most expensive and difficult to implement. On the other hand, indirect metering of groundwater pumping, treating electricity consumption as an alternative, is found to be more economical, reliable, and simpler to implement. Using electricity to oversee groundwater extraction significantly reduces the labor and resources required for maintaining systems and collecting data. This approach results in savings and also facilitates the use of data to charge farmers for extracting groundwater. Because of the inadequate water metering rate of agricultural irrigation wells in Baicheng City, water consumption cannot be directly obtained. Therefore, the EWCC can be used to significantly enhance agricultural water metering efficiency, reduce labor expenditures, and provide technical assistance to advance the use of electricity to measure water. The primary findings of the EWCC in Baicheng City are as follows:

(1) A total of 206 agricultural irrigation wells were selected from 55 townships in Baicheng City. The water consumption and power consumption of these wells were measured through field tests, and the relevant data and EWCC of the 206 agricultural irrigation wells were organized and calculated (Table 1). The four main factors influencing the EWCC were groundwater depth, well age, pump power, and head.

(2) Of the three machine learning methods used to develop a prediction model for EWCC using field investigation data, the BP model was the most accurate. Thus, it can serve as a tool for forecasting the EWCC across the townships of Baicheng City. EWCC curves of typical pump types in Baicheng City were plotted using a BP model. Finally, the BP model was employed to forecast the EWCC distribution anywhere in Baicheng City.

(3) This approach can be easily adapted to various regions or nations, such as the U.S. and Mexico, where wells predominantly operate on electric power. Owing to the requirement of comprehensive data throughout the irrigation phase, it is imperative to safeguard all flow meters from acts of vandalism, interference, and theft. Conducting inspections on location and analyzing data are crucial components of field operations. There is a necessity to train farmers in the proper use and upkeep of flow meters.