传统的负荷预测方法使用统计学 [1,3,4]，这在以前的研究中已经出现过。然而，需要大量精确的历史数据，这增加了准确预测的挑战。基于人工神经网络的方法因其强大的非线性逼近和自学习能力而成为数据驱动方法中最受欢迎的方法。不同类型的神经网络，例如反向传播 （BP） [5]、径向基函数 （RBF） [6] 和极限学习机 （ELM）[7,8][7,8]已被提出并应用于短期负荷预测。此外，在 [8] 中，添加了正则化项和多个 ELM 的组合，以减少传统 ELM 在光伏功率预测中的随机性。然而，低收敛速度始终是神经网络大规模应用的障碍。 传统负荷预测方法采用统计学手段[1,3,4]，这在早期研究中已有体现。但该方法需要大量精确的历史数据，这给预测精度带来了挑战。在数据驱动方法中，基于人工神经网络的方案因其出色的非线性拟合和自主学习能力而最为常用。目前已提出反向传播(BP)[5]、径向基函数(RBF)[6]和极限学习机(ELM)等多种神经网络模型，并成功应用于短期负荷预测。文献[8]还通过在光伏功率预测中引入正则化项和组合多个 ELM 模型，有效降低了传统 ELM 的随机性。但收敛速度较慢的问题，始终制约着神经网络的大规模应用。
Based on the traditional LSTM model, the bidirectional LSTM neural network will fully consider the front and back correlation of the load data in time series and improve the model performance for the sequence classification problem especially. During the training process, the input data sequence of the forward layer is the training data, and the backward layer is the reverse copy of the input data sequence. The results of bidirectional structure prediction are determined by the previous input and the latter input, which increases the dependence between the training data to avoid the forgetting of the order information. 在传统 LSTM 模型的基础上，双向 LSTM 神经网络能够充分学习负载数据在时间序列中的前后关联特性，尤其提升了模型处理序列分类问题的性能。训练时，前向层接收原始数据序列作为输入，而后向层则处理该序列的反向副本。这种双向结构的预测结果综合了历史与未来输入信息，通过增强数据间的时序依赖关系，有效避免了顺序特征的丢失。
Figure 3 shows that the forward layer calculates the forward direction from 1 to tt, and it saves the output of the forward hidden layer at each moment. The backward layer calculates the reverse time series and saves the output of the backward hidden layer at each moment. Finally, the output of the bidirectional LSTM neural network is calculated by combining the corresponding output results of the forward layer and backward layer at each time point. The bidirectional LSTM neural network can be written as: 如图 3 所示，前向层按时间顺序（从 1 到 tt ）进行计算，并逐时刻保存前向隐藏层的输出值；而后向层则逆向处理时间序列，同样保存各时刻的后向隐藏层输出。最终，通过将每个时间步的前向层与后向层对应输出结果进行整合，即可得到双向 LSTM 神经网络的输出。该网络结构可表述为：
所提出的改进 LSTM 神经网络的基本概念包括通过重建训练样本数据来获得电力负载的统计分析。多层堆叠双向 LSTM 网络经过训练，可以执行未来 24 h 的电力负荷预测。所提模型的预测过程可以分为以下步骤，如图 5 所示。
Figure 5. The framework of the proposed method for load forecasting. 图 5. 本文提出的负荷预测方法框架。
Step 1: Data preparation. Historical data of the power load profiles are collected and pre-processed to remove any outlier or incorrect data before the training process. However, the original data are not standard enough to use directly. Normalization is a common method to normalize original data structures in system modeling, and the original data become dimensionless after normalization, which can increase the convergence speed of the neural network. After normalization, the value of the original data is between the range of [0,1][0,1]. There are many normalization methods such as min-max scaling, Z-score standardization method, and decimal scaling. In this paper, a linear normalization method based on min-max scaling is used, which can be written as follows: 第一步：数据准备。在模型训练前，需先收集电力负荷历史数据并进行预处理，剔除异常或错误数据。但原始数据通常不符合标准，无法直接使用。归一化是系统建模中常用的数据标准化方法，通过消除量纲影响，能显著提升神经网络收敛速度。归一化处理后，数据值将落在 [0,1][0,1] 区间内。常用归一化方法包括最小-最大缩放法、Z-score 标准化法和十进制缩放法等。本研究采用基于最小-最大缩放原理的线性归一化方法，其计算公式如下：
x_(max)x_{\max } and x_(min)x_{\min } are the maximum and minimum values of the sample data of power load, xx is the original value of the sample data, and x^(**)x^{*} is the normalized value of the original data. " x_(max)x_{\max } 和 x_(min)x_{\min } 分别表示电力负荷样本数据的最大值与最小值， xx 为样本原始数据值， x^(**)x^{*} 则是经过归一化处理后的数值。"
Step 2: Network training. The forward value of input at t=1t=1 and the reverse state value of input at t=T\mathrm{t}=\mathrm{T} ( T is the last sampling time of the training dataset) are unknown, which are generally set to a fixed value (0.5) in the training process. Additionally, the derivative of the forward value of input at T=t\mathrm{T}=\mathrm{t} and the original value of the reverse state of t=1t=1 are generally set to zero. It is assumed that the later information is not very important for the current information updated. The process of network training contains the following: 步骤 2：网络训练。输入数据在 t=1t=1 位置的前向值及 t=T\mathrm{t}=\mathrm{T} 位置的反向状态值（T 代表训练数据集的末次采样时间）通常未知，训练时默认设为固定值（0.5）。同时， T=t\mathrm{T}=\mathrm{t} 位置前向导数和 t=1t=1 位置原始反向状态值一般置零处理，这基于后续信息对当前参数更新影响较小的假设。网络训练具体包含以下环节：
(1) Forward transfer. With the time sequence of 1 < t<=T1<t<=T, training data are input from the cell of the bidirectional LSTM, and the predicted outputs are determined. Forward (1) 前向传递。按照 1 < t<=T1<t<=T 的时间顺序，将训练数据输入双向 LSTM 单元进行处理，并生成预测输出结果。前向
passes are only for forward states (from t=1t=1 to t=Tt=T ) and backward states (from t=Tt=T to t=1t=1 ). The output cells were transferred forward, and the nn-th layer forward predicted output is calculated. "该机制仅作用于前向状态（从 t=1t=1 到 t=Tt=T ）和后向状态（从 t=Tt=T 到 t=1t=1 ）。输出单元会向前传递，同时计算第 nn 层的前向预测输出。"
(2) Backward transfer: The derivative of the partial objective function is calculated for the forward transfer time period with 1 < t<=T1<\mathrm{t}<=\mathrm{T}. The backward LSTM cells are calculated based on the forward value of 1 < t<=T1<\mathrm{t}<=\mathrm{T} and the reverse value of 1 < t<=T1<\mathrm{t}<=\mathrm{T}. The reversed prediction output is calculated. (2) 反向传递：部分目标函数的导数在正向传递阶段通过 1 < t<=T1<\mathrm{t}<=\mathrm{T} 计算得出。反向 LSTM 单元则根据 1 < t<=T1<\mathrm{t}<=\mathrm{T} 的正向输出值和 1 < t<=T1<\mathrm{t}<=\mathrm{T} 的反向输出值进行计算，最终生成反向预测结果。
(3) Weight matrix updating. Based on the loss function of the neural network during the training process, the weight matrix is calculated and updated. (3) 权重矩阵更新。根据神经网络训练过程中的损失函数，对权重矩阵进行计算和更新。
(4) Result output. Based on the bidirectional calculation, the parameters of the prediction model of LSTM neural network are estimated. (4) 结果输出。通过双向计算，对 LSTM 神经网络预测模型的参数进行了估算。
3.3. Evaluation Index 3.3. 评价指标
In this paper, the mean absolute error (MAE), root mean square error (RMSE), and mean absolute percentage error (MAPE) are used to evaluate the error of prediction results. MAPE, RMSE, and MAE are common indicators to evaluate the accuracy of the proposed model based on the measurement value and estimated value. The definition of the indicators is shown in Equations (16)-(18). MAE is the estimated indictor, which is used as the measurement value. RMSE is used to evaluate the deviation between the observed value and the true value; it is sensitive to outliers. MAPE is used to evaluate the relative errors between the average observed value and the true value on the test. MAE can reflect the error distribution during the time series, while MAPE normalizes the error at different points and reduces the effect of the absolute errors of the outliers. 本研究采用平均绝对误差（MAE）、均方根误差（RMSE）和平均绝对百分比误差（MAPE）三项指标评估预测结果的误差。其中，MAPE、RMSE 和 MAE 是基于实测值与预测值衡量模型精度的常用指标，具体定义见公式(16)-(18)。MAE 作为估计指标反映测量值水平；RMSE 衡量观测值与真实值的偏离程度，对异常值较为敏感；MAPE 则用于测试集中评估观测均值与真实值的相对误差。MAE 能体现时间序列上的误差分布特征，而 MAPE 通过误差归一化处理，削弱了异常点绝对误差对整体评估的影响。
{:[MAPE=(1)/(n)sum_(i=1)^(n)(|x_(i)- hat(x)_(i)|)/(x_(i))xx100%],[RMSE=sqrt((1)/(n)sum_(i=1)^(n)|x_(i)- hat(x)_(i)|^(2))],[MAE=(1)/(n)sum_(i=1)^(n)|x_(i)- hat(x)_(i)|]:}\begin{gathered}
M A P E=\frac{1}{n} \sum_{i=1}^{n} \frac{\left|x_{i}-\hat{x}_{i}\right|}{x_{i}} \times 100 \% \\
R M S E=\sqrt{\frac{1}{n} \sum_{i=1}^{n}\left|x_{i}-\hat{x}_{i}\right|^{2}} \\
M A E=\frac{1}{n} \sum_{i=1}^{n}\left|x_{i}-\hat{x}_{i}\right|
\end{gathered}
nn is the number of sample data, x_(i)x_{i} is the real value, and hat(x)_(i)\hat{x}_{i} is the predicted value. "其中， nn 表示样本数据的数量， x_(i)x_{i} 为真实值， hat(x)_(i)\hat{x}_{i} 为预测值。"
4. Simulation and Experimental Analysis 4. 仿真与实验分析
In this section, we evaluate the performance of the proposed multi-layer stacked bidirectional LSTM neural network for short-term load forecasting, and the key parameters of the model are discussed as well. Moreover, the comparison between the proposed method and previous work are also assessed. All models were executed in a computer with a CPU clock speed of 3.0 GHz and 8 GB of RAM. The hidden layer of the proposed model is 100 , the hidden node is 8 , the initial value of training learning rate is 0.01 , and the number of model training iterations is 100 . 本节对提出的多层堆叠双向 LSTM 神经网络在短期负荷预测中的性能进行评估，并探讨模型关键参数设置。同时，将所提方法与已有研究进行对比分析。实验环境为 3.0 GHz CPU、8 GB 内存的计算机。模型参数设置为：隐藏层数 100 层，隐藏节点 8 个，初始学习率 0.01，训练迭代次数 100 次。
4.1. Dataset for Load Forecasting 4.1. 负荷预测数据集
The databases used in the paper were obtained from the station in the southwest of China with an AC power voltage of 35 kV . The dataset contains a 3-year power load profile with the sampling time of 15 min . The dataset is a mixed dataset that contains different types of loads such as resident load, commercial load, and industrial load. The dataset was pre-processed in order to separate the relevant data and select the predictive features in the models. Here, we separated the dataset into different types for the load forecasting based on days and season characteristics. The pre-processing of the dataset is shown in Section 3.2, and the forecasting models were trained and tested using a 1-year sample dataset where the first 80%80 \% is used for model training and the remaining 20%20 \% is used to test the performance of the proposed model. 本研究所用数据库采集自中国西南地区某 35 千伏交流变电站，包含为期 3 年、采样间隔 15 分钟的电力负荷数据。该混合数据集涵盖居民用电、商业用电及工业用电等多元负荷类型。研究中对原始数据进行了预处理，包括数据分类和预测特征筛选，具体根据日期和季节特性将负荷数据重新归类以适配预测需求。数据预处理方法详见第 3.2 节，模型训练与测试采用 1 年期的样本数据，其中前 80%80 \% 时段数据用于模型训练，剩余 20%20 \% 时段数据则用于验证模型性能。
Prediction accuracy has a significant relationship with the depth of the bidirectional LSTM neural network. The dynamic characteristics of the load data will be extracted based on the interaction of the different layers of the neural network. The internal relevance information of the load profiles will be deep learned with the different stacked layers, and the nonlinearity of the load sequence can be described in different dimensions. The parameters of the input units, forget units, and output units of the proposed model are shown in Table 1. 预测精度与双向 LSTM 神经网络的深度密切相关。通过神经网络各层间的交互作用，可有效提取负载数据的动态特征。利用不同堆叠层结构，能够深入挖掘负载曲线的内在关联信息，并从多维角度描述负载序列的非线性特性。表 1 展示了该模型输入单元、遗忘单元及输出单元的具体参数。
Table 1. The parameters of the equivalent model. 表 1. 等效模型参数列表。
The prediction accuracy of the different layers of the LSTM neural work is shown in Figure 6. It can be seen that the proposed multi-layer bidirectional LSTM neural network is an effective method and is accurate enough for the load forecast problem. Furthermore, with the increasing numbers of layers, the prediction result will be more accurate. However, when there are four layers, the prediction accuracy will increase, on the contrary. It is said that three layers is suitable for the prediction of the load sequence data in this paper. Table 2 shows the prediction errors of MAPE between the different layers of the different neural network model. 图 6 显示了 LSTM 神经网络各层的预测精度。结果表明，本文提出的多层双向 LSTM 神经网络方法有效且能准确进行负荷预测。随着网络层数增加，预测精度会相应提高。但当层数达到四层时，预测精度反而会出现下降。研究发现，三层网络结构最适合本文的负荷序列数据预测需求。表 2 对比了不同神经网络模型各层的平均绝对百分比误差(MAPE)。
Figure 6. The load forecasting of different layers of LSTM. 图 6. LSTM 各网络层的负荷预测结果。
Table 2. Error of different stacked layers of bidirectional LSTM. 表 2. 双向 LSTM 各堆叠层的误差对比。
In order to show the high performance of the multi-layer stacked bidirectional LSTM neural network in short-term load forecasting, different methods that contain a BP neural network, ELM, traditional LSTM, and multi-layer stacked bidirectional LSTM model are discussed in this paper. The prediction results all the methods tested in this paper followed the same trend with the real load power shown in Figure 7. It can be seen that the multilayer stacked bidirectional LSTM neural network will be more competitive, and the error comparison of those methods is shown in Table 3, where the MAPE, RMSE, and MAE index are calculated and compared for one day over 24 h . From Table 3, the average MAPE of the proposed method prediction model is 0.4137%0.4137 \%; however, the average MAPE values of the BP, LSTM, and ELM models are 1.485%,1.030%1.485 \%, 1.030 \%, and 0.77%0.77 \%, respectively. The average RMSE of the proposed method prediction model is 0.706 , and those of the BP, LSTM, and ELM models are 2.95, 1.921, and 1.369, respectively. 为验证多层堆叠双向 LSTM 神经网络在短期负荷预测中的优越性能，本文对比分析了 BP 神经网络、极限学习机(ELM)、传统 LSTM 及多层堆叠双向 LSTM 等模型的预测效果。如图 7 所示，所有测试模型的预测结果均与实际负荷功率变化趋势相符。实验结果表明，多层堆叠双向 LSTM 神经网络具有显著优势。表 3 详细列报了各模型在 24 小时周期内的平均绝对百分比误差(MAPE)、均方根误差(RMSE)和平均绝对误差(MAE)指标对比数据。其中，本文提出的预测模型平均 MAPE 为 0.4137%0.4137 \% ，明显优于 BP 神经网络( 1.485%,1.030%1.485 \%, 1.030 \% )、LSTM( 0.77%0.77 \% )和 ELM 模型。在 RMSE 指标方面，新模型取得 0.706 的优异表现，相较之下 BP 模型(2.95)、LSTM 模型(1.921)和 ELM 模型(1.369)的误差水平显著偏高。
Figure 7. Load power of different forecasting methods. 图 7. 不同负荷预测方法的功率对比。
Table 3. Error comparison of different prediction models. 表 3. 各类预测模型的误差对比。
Prediction Model 预测模型
BP
LSTM
ELM
Proposed Method 建议方案
MAPE (%) 平均绝对百分比误差(%)
1.485
1.03
0.77
0.405
RMSE
2.95
1.921
1.369
0.706
MAE
33.564
23.236
17.07
9.341
Prediction Model BP LSTM ELM Proposed Method
MAPE (%) 1.485 1.03 0.77 0.405
RMSE 2.95 1.921 1.369 0.706
MAE 33.564 23.236 17.07 9.341| Prediction Model | BP | LSTM | ELM | Proposed Method |
| :---: | :---: | :---: | :---: | :---: |
| MAPE (%) | 1.485 | 1.03 | 0.77 | 0.405 |
| RMSE | 2.95 | 1.921 | 1.369 | 0.706 |
| MAE | 33.564 | 23.236 | 17.07 | 9.341 |
计算了 ELM 、 LSTM 、 BP 和表 4 中提出的方法之间的不同时间间隔误差。2 h间隔预测结果在不同评价指标下存在波动。然而，所提出的方法的总评价指标在一天内是最低的，并且计算了定量分析预测误差，如图 8 所示。由此可见，基于多层双向LSTM预测模型所提出的方法更能把握预测样本信息，具有更具竞争力的预测性能。多层堆叠双向 LSTM 神经网络模型可以保留载荷序列的原始特性，并通过在无监督训练中加入误差来减少数据误差，从而增强预测模型的鲁棒性。
Figure 10. The load forecasting for a weekend with a 0.5 h sample training dataset. 图 10. 基于 0.5 小时样本训练数据的周末负荷预测结果。
Figure 11. The load forecasting for a weekend with a 1 h sample training dataset. 图 11. 基于 1 小时样本训练数据集的周末负荷预测结果。
5. Conclusions 5. 结论
Accurate short-term load forecasting is a huge challenge due to the complexity of the electrical load composition in modern power systems. In this paper, based on the traditional LSTM neural network, a multi-layer stacked type short-term load forecasting method is proposed. Reverse computing combined with forward computing is designed to solve the unidirectionality of the memory process during the training period. The output gate can collaborate the implied information in the historical load series. Furthermore, a multi-layer stacked deep learning style for the neural network is proposed to perceive a low-level features form of power load and form a more abstract high-level representation of load characteristics. At last, a load forecasting frame based on the multi-layer bidirectional LSTM neural network is proposed that contains neural network model construction, historical load profile training, and load forecasting. In the experiments, the real operational load data of a substation are tested, and the performance of the proposed method is tested and evaluated. The results show that the proposed multi-layer stacked bidirectional LSTM neural network method has high performance and is more accurate than the others. The proposed method can retain the original information as much as possible and has a strong memory function to extract the relevant information from historical load sequences. 现代电力系统的负荷构成复杂，使得短期负荷精准预测面临巨大挑战。本研究在传统 LSTM 神经网络基础上，创新性地提出了一种多层堆叠式短期负荷预测方法。通过设计反向计算与正向计算相结合的训练机制，有效解决了传统方法在训练过程中记忆单元单向传递的局限性。输出门的设计能够有效整合历史负荷序列中的隐含特征信息。此外，该方法采用多层堆叠的深度学习架构，既能捕捉电力负荷的底层特征，又能构建更高层次的负荷特征抽象表示。最终建立的多层双向 LSTM 神经网络预测框架，完整涵盖了网络模型构建、历史负荷数据训练和负荷预测三个关键环节。基于某变电站实际运行数据的实验表明，该多层堆叠双向 LSTM 神经网络方法预测性能优异，准确度显著优于其他方法，不仅能最大限度保留原始数据信息，还具备强大的时序记忆能力，可有效提取历史负荷序列中的关键特征信息。
However, with the increase in the sequence length of the problem, the efficiency of the proposed method will reduce because the capacity of the memory units is limited. There are four fully connected layers in each cell in the LSTM neural network; it needs a lot of computing time in a deep stacked LSTM neural network. Future works will focus on the industrial application of the proposed method with a more complex dataset. (1) We built an online load forecasting system. The application of load forecasting is employed for the dispatch of the power system, which is working all the time. Hence, an online and rolling load forecasting system using the historical load data is the basis of this work. (2) We corrected the load forecasting results. If the load forecast results deviate greatly, the forecast points are corrected based on the data before and after time points. 然而，随着问题序列长度的增加，由于内存单元容量有限，所提方法的效率会有所下降。LSTM 神经网络中每个单元包含四个全连接层，在深度堆叠的 LSTM 网络中需要耗费大量计算时间。后续研究将重点探索该方法在更复杂工业数据集中的应用。(1) 我们开发了一套在线负荷预测系统。该系统持续运行，为电力系统调度提供负荷预测服务。因此，基于历史负荷数据构建在线滚动预测系统是本项工作的基础。(2) 我们对预测结果进行了修正。当负荷预测出现较大偏差时，会根据前后时间节点的数据对预测点进行校准。
Author Contributions: Conceptualization, C.C. and Y.T.; methodology, Y.T., T.Z.; validation, C.C. and Y.T.; writing-review and editing, C.C. and Y.T. and Z.D. All authors have read and agreed to the published version of the manuscript. 作者贡献：研究构思由 C.C.和 Y.T.共同完成；研究方法由 Y.T.和 T.Z.设计；验证工作由 C.C.和 Y.T.进行；文稿撰写、审阅及修改由 C.C.、Y.T.和 Z.D.共同完成。全体作者均已审阅并同意论文的最终发表版本。
Funding: This research was funded by “National Natural Science Foundation of China, grant number 51607057”, “The Fundamental Research Funds for the Central Universities, grant number 2020B22514” and “The open funding of Jiangsu Key Laboratory of Power Transmission & Distribution Equipment Technology, grant number 2021JSSPD07”. "基金项目：本研究获得国家自然科学基金（项目编号 51607057）、中央高校基本科研业务费专项资金（项目编号 2020B22514）以及江苏省输配电装备技术重点实验室开放基金（项目编号 2021JSSPD07）的资助。"
Institutional Review Board Statement: Not applicable. 机构审查委员会声明：无需适用。
Informed Consent Statement: Not applicable. 知情同意声明：无需提供。
Conflicts of Interest: The authors declare no conflict of interest. 利益冲突声明：作者宣称不存在利益冲突。
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