Research on Peak Load Prediction of Distribution Network Lines Based on Prophet-LSTM Model

Abstract

The increasing demand for precise load forecasting for distribution networks has become a crucial requirement due to the continual surge in power consumption. Accurate forecasting of peak loads for distribution networks is paramount to ensure that power grids operate smoothly and to optimize their configuration. Many load forecasting methods do not meet the requirements for accurate data and trend fitting. To address these issues, this paper presents a novel forecasting model called Prophet-LSTM, which combines the strengths of the Prophet model’s high trend fitting and LSTM model’s high prediction accuracy, resulting in improved accuracy and effectiveness of peak load forecasting. The proposed algorithm models the distribution network peak load using the Prophet-LSTM algorithm. The researchers then analyzed the experimental data and model of the algorithm to evaluate its effectiveness. We found that the Prophet-LSTM algorithm outperformed the Prophet and LSTM models individually in peak load prediction. We evaluate the proposed model against commonly used forecasting models using MAE (mean absolute error) and RMSE (root mean square error) as evaluation metrics. The results indicate that the proposed model has better forecasting accuracy and stability. As a result, it can predict the peak load of distribution networks more accurately. In conclusion, this study offers a valuable contribution to load forecasting for distribution networks. The proposed Prophet-LSTM algorithm provides a more precise and stable prediction, making it a promising approach for future applications in distribution network load forecasting.

1. Introduction

With the development of power systems, distribution network load forecasting has received more and more extensive attention as one of the important research areas of power grid management and dispatching [1]. The rapidly growing demand for electricity in urban areas poses a significant challenge to their power grids. The resulting increase in peak loads makes it increasingly difficult to maintain a stable grid configuration and operation [2,3]. Due to the lack of peak load monitoring distribution networks, overloads are a frequent concern in relatively small capacity urban distribution lines, which can lead to power outages, line losses, and component burnouts [4]. Accurate power load forecasting can therefore provide decision makers with important information to maintain grid line stability and achieve optimal power dispatch [5]. Currently, power load forecasting is usually performed with historical data as the main component to forecast future load data. Load forecasting is commonly divided into short-term (daily and weekly), medium-term (monthly), and long-term (annually) categories depending on the forecast horizon [6]. This research paper aims to forecast peak distribution network line currents over a two-month period, which can be classified as a medium-term load prediction. However, given the duration of the prediction, the short-term load forecasting method can also be used. Medium-term power load forecasting plays a critical role in power system planning, operation, and management. Medium-term load forecasting can help power system planning departments determine power demand and resource allocation, guide decision makers to make effective decisions in energy construction, power generation planning, renewable energy utilization, and grid upgrading, and can provide a reference basis for power generation dispatch and grid operation [7,8]. The research scope of power load forecasting is defined by a tree diagram (Figure 1).

Load forecasting is usually performed using traditional forecasting models and models with artificial intelligence. Traditional forecasting models include autoregressive (AR), mean shift (MA), and ARIMA models, which are simple to use and fast to compute. However, it can only analyze the time-dependent relationships between the data, and, in addition, the power load data have the characteristics of nonlinearity and periodicity, which cannot deal with these problems well, and the prediction accuracy and effect using the above methods are not good. Artificial intelligence methods including SVM, evolutionary algorithms, RNN, LSTM, etc., are also widely used for load prediction, but there are convergence and complexity problems. This paper will be discussed specifically in the related work section.

This paper explores the prediction of peak loads on distribution network lines by combining the Prophet and LSTM algorithms. The model is designed to handle various influencing factors, such as daily, weekly, annual, and seasonal cycles, and holidays, and can correct prediction errors through time series analysis to eliminate periodic and trend errors, thus improving accuracy and stability. Through training and testing using experimental data, the Prophet-LSTM algorithm’s effectiveness and accuracy in peak load prediction on distribution network lines are verified.

The main contributions of this paper are as follows:

• In this paper, a Prophet-LSTM-based load forecasting method is proposed, learning data trends based on the Prophet model to improve the data trend fit, while using the high prediction accuracy of LSTM model for prediction, and further improving the prediction results through the BP network to improve the prediction accuracy and effectiveness of the model.
• In Prophet model training, we optimize the parameters of the Prophet model based on PSO (particle swarm optimization algorithm), which can better capture the data mutation points.
• A complete set of experiments was designed to compare the forecasting method proposed in this paper with common forecasting methods, and to demonstrate that the method proposed in this paper can achieve better forecasting results compared with other methods.

2. Related Work

This paper categorizes load forecasting methods into three main categories: traditional forecasting models [9], machine learning [10], and deep learning [11]. Each category is presented in detail in the following sub-summaries.

2.1. Traditional Forecasting Models

Traditional forecasting methods are primarily based on historical time series data, utilizing simple models that have a relatively quick calculation speed. For example, the method based on regression analysis takes load data and influencing factors as independent and dependent variables, and load forecasting is performed by establishing regression equations. Among the traditional methods, time series methods are commonly used in load forecasting, including autoregressive (AR) [12,13], moving average (MA) [14], autoregressive integrated moving average (ARIMA) [15,16], seasonal autoregressive integrated moving average (SARIMA) [17] model, and autoregressive integrated moving average model with external inputs (ARIMAX) [18] model. These methods all rely on past data to forecast, with simple structures, without considering the correlation between data, and only treating historical and predicted data as simple mathematical relationships, with low forecasting accuracy. They are also unable to better handle nonsmooth, nonlinear data. Compared with the above methods, the Prophet model can perform nonlinear data processing and capture data trends, especially in power load time series forecasting.

2.2. Machine Learning

With the rapid development of machine learning, machine learning methods have been widely applied to image processing, and certain achievements have been made in power load forecasting. Common machine learning methods include support vector machines (SVMs), random forests (RFs), and evolutionary algorithms (EAs). There is a lot of work on SVMs for load forecasting that provides ideas for the accuracy of the forecast [19,20,21]. Pang et al. [22] used gray relational analysis and support vector machines to study short-term electrical load forecasting and proved the validity of the method. Liu et al. [23] presented a method for electric load forecasting using Elman neural network and particle swarm optimization algorithm, significantly improving load forecasting accuracy. Chen et al. [24] proposed a new SVR model based on demand response, and the experimental results showed that the proposed model has some generality. Barman et al. [25] considered climatic conditions in their experiments and performed load forecasting based on an SVR model and locust optimization algorithm, and the results showed a major breakthrough in forecasting accuracy. Random forests, which integrate multiple decision trees internally and, thus, constitute a classifier, have the advantages of fewer parameters, high generalization ability, and high forecasting accuracy [26], and are also used by some researchers in load forecasting. For example, Wu et al. [27] improved the RF model and proposed a segmentation algorithm to improve the forecasting accuracy and robustness. Although the above machine learning methods are effective in processing nonlinear data, they have weak feature extraction capability and low accuracy in forecasting highly random data.

2.3. Deep Learning

In recent years, deep learning has been gaining more and more weight in load forecasting. Deep learning neural networks have more complex structural models, stronger learning ability, generalization ability, etc. [28,29,30]. In the literature [31], a long short-term memory (LSTM) network with more advantages than recursive neural network (RNN) is used for power load forecasting, which overcomes the problems such as the gradient explosion of RNN and improves the accuracy of forecasting. In addition, Motepe et al. [32] proposed a new hybrid artificial intelligence and deep learning system for forecasting the load on the distribution network in South Africa. Li et al. [33] developed a combined forecasting model that uses LSTM and XGBoost methods to improve the accuracy of electric load forecasting. Pham et al. [34] developed a hybrid forecasting method integrating singular spectrum analysis (SSA) and deep neural network techniques. Similarly, Ciechulski et al. [35] used artificial neural networks and SSA techniques to investigate efficient and accurate methods for 1 h and 24 h power load pattern forecasting. Deep-learning-based load forecasting methods show good promise in improving grid intelligence [36].

2.4. Summary

We analyzed the current common load forecasting methods and related works, and summarized the advantages and disadvantages of different forecasting methods through

Table 1. Advantages and disadvantages of load forecasting methods.

Common Methods	Advantages	Disadvantages
Traditional forecasting methods (AR, MA, ARIMA)	The model is simple and the calculation is fast.	Low forecasting accuracy; unable to handle nonsmooth and nonlinear data better; poor fitting effect.
Machine learning (SVM, RF, EA)	The models are effective in dealing with nonlinear data.	The feature extraction capability is weak and the accuracy of prediction for highly random data is low.
Deep learning (RNN, LSTM, GRU, TCN)	The ability to cope with large-scale, high-dimensional, nonlinear load data and to predict future load conditions more accurately.	Deep learning algorithms have a large computational resource footprint and poor interpretation.

. Since a single forecasting method cannot simultaneously meet the requirements of high fitting of data trends and high accuracy of predicted data, we propose the Prophet-LSTM model based on these advantages and disadvantages, which is able to fit the data trend better and improve the accuracy at the same time. There are periodic trends in power load data, and the Prophet algorithm is an effective method to fit these trends accurately compared to other methods [37,38]. Parizad et al. proposed a comprehensive two-stage STLF (short-term load forecasting) method based on the Prophet algorithm [39], which can effectively forecast data trends. The paper [40] presents a Prophet-model-based electric current load forecasting, which performs well in terms of forecasting accuracy and model fit. Although not much work has been carried out using the Prophet model for power load forecasting, we can infer from the above works that this approach is effective in dealing with modeling and data fitting. Also, we add an LSTM model and a BP network to improve the accuracy of load forecasting.

3. Research Theory and Methodology

3.1. Prophet Model

Facebook has released an open-source time series forecasting algorithm called the Prophet algorithm. This algorithm is capable of managing multidimensional time series data, supporting varying units of measurement while ensuring forecast accuracy without sacrificing computational speed. It is widely used in business and scientific research for its significant advantages in time series data forecasting and analysis. Prophet uses an additive model to describe the nonlinear trend, observing periodicity and holiday effects, which enables accurate forecast time series intuitive parameters. It is a decomposable model consisting of a trend, seasonal, and holiday model. What sets Prophet apart from other classical time series forecasting models is its periodic decomposition of the series before forecasting. This approach highlights holiday effects and trends in the data, making it particularly robust in the presence of missing data, sudden changes in trends, or outliers.

The Prophet model uses Equation (1) and a time series decomposition approach to forecast time series data, dissecting the data into four essential components [41].

$$y\left(t\right)=g\left(t\right)+s\left(t\right)+h\left(t\right)+\epsilon \left(t\right)$$

(1)

Equation (1) requires g(t), which symbolizes the modification function employed to suit the nonperiodic trend in the data. The change function can consist of various options, such as the logistic regression function and segmented linear function. For instance, the trend term built on logistic regression can be indicated by Equation (2):

$$g\left(t\right)=\frac{C}{1+e^{(-k(t-m\left)\right)}}$$

(2)

Equation (2) includes C, which denotes the carrying capacity of the model, k, representing the growth rate, and m, the data offset. As t varies, g(t) gradually approaches the upper threshold C. Additionally, the segmented linear function is another alternative for the change function, which can be defined by Equation (3):

$$g\left(t\right)=\left(k+a\left(t\right)\delta \right)·t+\left(m+a{\left(t\right)}^T\gamma \right)$$

(3)

Equation (3) uses $\delta$ as a symbol for the change in growth rate over time, and m as the bias term. The change point, represented by $c_j$, refers to the time when the growth rate k shifts. The algorithm specifies the location and number of change points. Furthermore, a(t) is a time series produced by a binary function, and $\gamma$ is another time series with the same length as a(t).

In the Prophet model, s(t) represents the periodic term and is used to capture the periodicity of the data, typically on a weekly, monthly, or yearly basis. The periodic term s(t) can be fitted with a flexible Fourier series to model the periodic trend of the data. The Fourier series is calculated as follows:

$$s\left(t\right)=\sum _{n=1}^N\left(a_{n}cos\left(\frac{2\pi nt}{P}\right)+b_{n}sin\left(\frac{2\pi nt}{P}\right)\right)$$

(4)

In Equation (4) of Prophet, P represents the period of the time series, which is typically in years. The parameter 2n represents the number of periods used in the algorithm.

In Prophet, h(t) represents the holiday component of the time series. In real-world scenarios, major events or holidays can significantly impact load forecasting for distribution networks, and their periodicity may not be straightforward to determine. To account for this, the model includes a holiday variable h(t) that affects the time series in the period before and after these events, and is expressed as follows:

$$h\left(t\right)=\sum _{i=1}^L{K}_{i}1\left(t\in D_i\right).$$

(5)

$$Z\left(t\right)=\left[1\left(t\in D_1\right)\cdots 1\left(t\in D_L\right)\right].$$

(6)

$$h\left(t\right)=Z\left(t\right)\kappa ,\kappa \sim Normal(0,\nu ).$$

(7)

In Equations (5)–(7) of Prophet, $K_i$ represents the impact of the time period before and after a particular holiday, where i denotes the holiday and $D_i$ denotes the window period. The value of the standard deviation $\nu$ determines the magnitude of the impact of the holiday on the model. A larger value of $\nu$ indicates that the holiday has a greater impact, while a smaller value indicates a smaller impact. The error term $\epsilon \left(t\right)$ follows a normal distribution. Figure 2 illustrates the flow of the Prophet model loop.

The Prophet algorithm is primarily based on three time-varying factors: daily, weekly, and seasonal variation. First, the time series data are segmented into multiple time periods, and a separate linear regression equation is fitted within each period, capturing the trend and cyclical patterns of different periods. Next, the algorithm incorporates seasonal and holiday components into the model to account for the impact of events and important dates on the time series. Finally, the predicted values produced by the model are used to forecast future loads.

3.2. PSO Method

The particle swarm optimization (PSO) algorithm was developed by J. Kennedy and R. C. Eberhart [42]. It is an evolutionary algorithm that takes inspiration from bird foraging. PSO has been increasingly used in solving multidimensional nonlinear optimization problems owing to its benefits like easy implementation, fast convergence, and high accuracy. The PSO algorithm considers solutions to an optimization problem as “particles”, and a group of these particles is called a “particle swarm”. The PSO algorithm overcomes many of the limitations of traditional optimization algorithms and is easy to implement. It also avoids some of the problems present in traditional algorithms, such as the need to solve for the derivative of the function and the risk of becoming stuck in a local optimum. The algorithm is also widely used. During the iteration, the velocity $v_{(i+1)j}$ and position $x_{(i+1)j}$ of each particle are updated for the next time by Equation (8).

$$\begin{array}{cc}\hfill v_{(i+1)j}=& w{v}_{ij}+c_1{R}_1\left(P_{j_{\_}\mathrm{best}}-x_{ij}\right)+c_2{R}_2\left(G_{\mathrm{best}}-x_{ij}\right)\hfill \\ \hfill x_{(i+1)j}=& x_{ij}+v_{(i+1)j}\hfill \end{array}$$

(8)

In Equation (8), $j(1⩽j⩽n)$ represents the particle number; $i(1⩽i⩽m)$ represents the number of current iteration; n, m represent the total number of particles and the total number of iterations, respectively; w is the inertia weight; $c_1$, $c_2$ are the learning factors; $R_1$, $R_2$ are random numbers from 0 to 1. The particle population calculates the fitness of each particle in this iteration according to the objective function $F_{ij}$, from which the individual extreme value $P_{j_{\_}\mathrm{best}}$ and the population extreme value $G_{\mathrm{best}}$ are obtained. Finally, the operation is terminated according to the number of iterations, and the parameter corresponding to the population extrema ($G_{\mathrm{best}}$) is calculated as the optimal solution by the objective function $F_{ij}$.

3.3. LSTM Model

LSTM (long short-term memory) is a deep learning algorithm based on RNN designed to capture and learn long-term dependent information. With excellent sequence modeling capability, LSTM achieves promising results in prediction and speech recognition tasks. The primary structure of LSTM includes a storage cell that stores and propagates information explicitly at different time steps. The storage cell of LSTM leverages different cell states at different instances to remember contextual information. In contrast to RNN, which faces the problem of gradient disappearance with longer input sequences, LSTM introduces techniques such as forgetting gates, input gates, and output gates to handle long-term dependencies and resolve the issue of gradient vanishing/exploding. Its inherent ability to store and process sequence information and update itself with continuous inputs makes it an optimal choice for time series data prediction tasks. Figure 3 illustrates the addition of c, a memory unit, to the recurrent layer of a traditional RNN, forming the core idea behind the long- and short-term memory neural network. Figure 4 shows the LSTM cell structure

The first step in LSTM is to decide which information should be forgotten from the cell state by leveraging the $Sigmoid$ function. The function determines which bits of information should be discarded from the cell state, where values closer to 0 are forgotten, and those nearer to 1 are retained.

$$f_t=\sigma \left(x_t{U}^f+h_{t-1}{W}^f+b_f\right)$$

(9)

The second step in LSTM is to determine which new pieces of information should be stored in the cell state. The input gate decides whether $i_t$ should be updated, and the data is assigned a new vector with candidate values ${\tilde{C}}_t$ through the tanh function.

$$i_t=\sigma \left(x_t{U}^i+h_{t-1}{W}^i+b_i\right).$$

(10)

$${\tilde{C}}_t=tanh\left(x_t{U}^c+h_{t-1}{W}^c+b_c\right).$$

(11)

In the third step, the LSTM old cell state information is updated:

$$C_t=C_{t-1}\otimes f_t\oplus i_t\otimes C_t$$

(12)

Finally, in the output gate, the LSTM determines which information should be output. Here, $O_t$ decides which cell states should produce output, and the cell states are passed through a $tanh$ layer (with values ranging from −1 to 1) and multiplied by $O_t$ at the output gate.

$$O_t=\sigma \left(x_t{U}^o+h_{t-1}{W}^o+b_o\right).$$

(13)

$$h_t=O_t\otimes tanh\left(C_t\right).$$

(14)

4. Model Construction and Prediction

Before training the model, it is important to prepare relevant data to test the model and analyze the prediction results. In this paper, we gathered current data for a distribution network line, specifically line current values collected once every 5 min, for one year. Then, we constructed a dataset with current as the load and used the Prophet-LSTM algorithm to build a prediction model. To develop the dataset, we extracted data for daily current peaks over 368 days by preprocessing the historical load data, which included interpolating and smoothing missing points. The dataset contained 368 sample points, with each point comprising three indicators—maximum load value, minimum load value, and average load value. This allowed us to explore the highest load value of the distribution network lines—i.e., the peak load of the lines based on the actual demand. Before implementing the Prophet-LSTM model, we tested the Prophet and LSTM models separately for comparison using the dataset.

4.1. Prophet Model Construction

The Prophet model is well suited for fitting time series data, and it was used in this study to model peak line load data and perform data prediction. The Prophet algorithm requires two main parameters: “ds” column (date stamp), in YYYY-MM-DD format, and the “y” column containing the variable value. In this study, the Prophet algorithm was used to predict the peak load of distribution network lines. To construct the Prophet model and handle the experimental data characteristics, the following steps were taken: data preprocessing—raw data were screened and cleaned based on maximum load values to remove invalid sample points and outliers. Prophet model construction—the data were loaded into the Prophet model, including time and load values, and the model was built and executed. Model training and parameter optimization—the Prophet model was trained using historical data, and model parameters were adaptively optimized based on experimental results. Prediction results were then output, and the Prophet algorithm generated a graph of prediction results, showing the Prophet model’s fitting and trend curve.

We optimize the Prophet model parameters using PSO. The method for optimizing Prophet parameters using PSO is as follows: (1) Partitioning the electric load dataset into a training set and a test set: set the parameters of the Prophet model to be optimized; initialize the hyperparameters such as the total number of particles n, the maximum number of iterations m, the learning factors $c_1$ and $c_2$, and the inertia weights w in the PSO method. (2) At each iteration, each particle in the swarm is used as the parameter value to be optimized in the Prophet model: Prophet models on the training set and predictions on the test set; the root mean square error (RMSE) of the predicted values on the test set is used as the target value for each particle iteration in the particle swarm; compute the current $P_{j_{\_}\mathrm{best}}$ and $G_{\mathrm{best}}$ by RMSE. (3) After the last iteration is completed, the particle corresponding to $G_{\mathrm{best}}$ is used as the optimal solution of the Prophet model for the parameters to be optimized.The flow chart for optimizing the parameters of Prophet model based on the PSO method is shown in Figure 5.

According to the selection range of the number of populations n in the optimization algorithm $(100⩽n⩽300)$, $n=300$ is set in this paper. It is found that the number of iterations has a large impact on the experimental running time, so the number of iterations for each particle is taken as 20 and the inertia weight w is taken as 0.8; the learning factors $c_1$ and $c2$ are both set as 0.5. The PSO method parameters are shown in

Table 2. PSO method parameters.

Parameters	Description
$n=300$	Number of populations
$m=20$	Number of iterations
$w=0.8$	Inertia weights
$c_1=c_2=0.5$	Learning factors

. We adjust the trend parameters and seasonal parameters of Prophet model to minimize RMSE as the objective function, select key parameters, optimize the parameters by particle swarm optimization algorithm (PSO), set $changepoint\_range=0.37$, and adjust the model fitting flexibility $changpoint\_prior\_scale=0.67$. The parameters after PSO optimization are given in

Table 3. Prophet section parameter settings.

Parameters	Description
$growth=linear$	Set the growth model; this paper is set to linear model (linear).
$changepoint\_prior\_scale=0.67$	Flexibility of the growth trend model.
$holidays=holidays$	Set special dates and holidays.
$weekly\_seasonality=True$	Analyze weekly seasonality of data.
$n\_changepoints=25$	Set the number of potential variables.
$changepoint\_range=0.37$	The location of the change point needs to be set in a time series as long as the first $changepoint\_range$.

. The change of the objective function (RMSE) value optimized by the PSO method is shown in Figure 6, where the RMSE value becomes smaller and smaller by iterative optimization of the PSO method (the Prophet model optimized by the PSO method is still essentially the Prophet model, hereafter referred to as Prophet model).

Figure 7 illustrates the model’s predictive performance on the past data and the following two months or so. The y-axis represents the current value in amperes (A), while the x-axis indicates time. The dark blue line depicts the model predicted values (yhat), whereas the shaded blue area represents the upper and lower limits of the predicted values, and the black dots are the true data. As observed, the peak load during January to February, coinciding with the Spring Festival holiday, is lower, while during July, it is higher, which indicates the seasonal trend of the data. The model data mostly lie within the prediction confidence interval, thus fitting the current trend and allowing accurate forecasting.

The graph in Figure 8 compares the model’s predicted values with the real values. Here, “actual” represents the real value, and “predict” represents the predicted value. It can be observed that the real value exhibits a cyclical trend, which is estimated to be about two weeks by the curve. The model can better capture these apparent changes in the early stage and make timely adjustments; however, it is not capable of making good changes in the later stage due to the long prediction window.

This study utilized the Prophet algorithm to predict the maximum load of distribution network lines, with a training-to-testing data ratio of 8:2. The effectiveness and accuracy of this approach were further demonstrated by analyzing and predicting experimental data. Although the Prophet model optimized using PSO performs well in fitting the data, it suffers from the problem of low accuracy, and we add the LSTM model to solve this problem.

4.2. LSTM Model Construction

The collected point-to-point load data were first analyzed and processed into time series data that conformed to the input format required by the LSTM model. Subsequently, the Keras framework was employed to build the LSTM model, modifying the model structure and adjusting the hyperparameters to enhance the prediction accuracy of the model.

The prediction of load data by the LSTM model is not only influenced by historical data but also by neighboring data. The window size (w) affects the model input, whereas the level factor (h) affects the model output (indicating the distance for predictions). Figure 9 displays the load dataset (dataset), training results (train), and test results (test). Considering the periodic pattern of the data and the length of the time series, the window size was 7 and the level factor was 1, and the update gradient was based on Adam optimizer. Finally, a load prediction experiment was conducted using the trained model, with training and test data at a ratio of 8:2. By calculation, we find that the accuracy of the load prediction data is higher in the training and test sets. We combine the advantages of the Prophet model and the LSTM model to build the Prophet-LSTM model.

4.3. Prophet-LSTM Combined Model

In this paper, we propose an optimal combined prediction model based on the Prophet model and LSTM neural network for time series prediction. The predicted values of these two models are used as input to a BP neural network to reduce errors and increase accuracy gradually. The processing flow is shown in Figure 10. We divide the electric load dataset into training sets, and determine the Prophet model parameters and LSTM model weights through training set training. The prediction is performed by the trained Prophet model and LSTM model, and the prediction results of the two models are fed into the BP neural network as feature values to obtain more accurate prediction results.

As shown in Figure 11 and Figure 12, we compared the prediction results of six models with the true values. We observed that the predicted values of the Prophet-LSTM model were closer to the true values compared to the other models. This indicates that the Prophet-LSTM model had higher prediction accuracy and its predicted values were closer to the true curve.

In order to test the accuracy of the model’s predictions, this paper uses root mean square error (RMSE) and mean absolute error (MAE).

$$MAE\left(y,y^{\ast }\right)=\frac{1}{n}\left(\sum _{i=1}^n\left|y-y^{\ast }\right|\right)$$

(15)

$$RMSE\left(y,y^{\ast }\right)=\sqrt{\frac{1}{n}{\left(\sum _{i=1}^n\left(y-y^{\ast }\right)\right)}^2}$$

(16)

Based on these two metrics, the metrics for Prophet, LSTM, GRU [43], TCN [44], Transformer [45], and Prophet-LSTM were computed and are presented in

Table 4. Comparison of metrics of different models.

Model	MAE	RMSE
$Prophet$	15.74	19.26
$LSTM$	12.32	15.55
$GRU$	11.67	15.45
$TCN$	12.63	17.57
Transformer	12.73	18.44
Prophet-LSTM	8.569	11.68

.

The statistical indicators in Table 4 of this paper show that the proposed Prophet-LSTM model outperforms the Prophet, LSTM, GRU, TCN, and Transformer alone. These metrics show that the Prophet-LSTM model has the smallest mean absolute error (MAE) and root mean square error (RMSE) values, which further supports the conclusion that the proposed model is more accurate in predicting the target variable.

5. Conclusions

This paper proposes a novel hybrid prediction model, namely, Prophet-LSTM, for forecasting the peak load of distribution network lines.The performance of the proposed model is verified by power load data and the test results show that the Prophet-LSTM model has a lower MAE value (8.569) and RMSE value (11.68) compared to the single Prophet, LSTM, GRU, TCN, and Transformer models. This model has significant practical value for load prediction, which can improve the operational efficiency and economic efficiency of the power system, reduce the cost of electricity consumption and load volatility, and guarantee the stability and reliability of power supply. Although there are many load forecasting methods, the existing analysis mainly focuses on the time series data themselves with a single predictive factor. Therefore, future research in this area should focus on adding more influencing factors, such as calendar, weather information, etc., and optimizing the models to obtain more accurate prediction results.