回声状态网络因其具有短期记忆能力和训练过程简单的特点，在预测控制领域得到了广泛的应用。然而，传统回声状态网络受到自身结构的限制，在应对复杂非线性系统时，建模精度和可靠性不足。针对这一问题，提出一种基于误差补偿的回声状态网络结构优化方法。该方法旨在提升回声状态网络在非线性系统中的建模能力，并将其应用于风力发电功率预测。本文的主要内容如下：

（1）针对传统回声状态网络在训练过程中忽略误差自相关性，导致非线性系统建模精度低的问题，提出基于迁移学习的误差补偿回声状态网络。该网络模型以处理误差的自相关性为目的，通过理论分析回声状态网络建模过程中误差产生的原因，及误差自相关性对建模精度的影响，进而使用迁移学习机制更新输出权值，实时调整预测误差，以此提高建模的精度与有效性。通过Mackey-Glass、太阳黑子和瓦斯浓度数据集进行实验，结果表明该模型能够根据误差值进行实时补偿，有效降低自相关误差对建模精度的影响，在复杂非线性系统中表现出较高的预测精度与稳定性。与其他网络模型相比，该模型的预测精度提高了约17%。

（2）针对传统回声状态网络易受系统误差带来的不确定性，导致非线性系统建模可靠性不足的问题，提出基于贝叶斯线性回归的误差分布补偿回声状态网络。该网络模型以量化预测的不确定性为目的，通过训练回声状态网络并得到预测误差，进而建立基于贝叶斯线性回归的误差分布补偿模型，采用马尔科夫链蒙特卡洛方法估计误差分布，将其用于预测的不确定性分析，以此提高预测的可靠性。此外，该模型能够利用误差分布，捕捉数据的异常情况并进行补偿。基于基准和真实数据集的实验结果表明，该模型能够根据误差分布进行有效补偿，并在95%的置信区间内尽可能覆盖更多真实值，从而显著提升预测的可靠性。

（3）针对风电功率数据具有较强的不确定性，难以实现高精度且可靠的预测，构建基于误差补偿的风力发电功率预测模型。结合数据的相关性分析与异常处理结果，发现风速与风电功率之间存在显著相关性。在此基础上，使用风电功率预测模型对风电场实际数据建模，该模型能够提供准确且可靠的风电功率预测结果，为提升能源管理应用的效能提供了理论支撑与决策依据。

Echo state networks have been widely used in predictive control due to their short-term memory capability and simple training process. However, the traditional echo state network is limited by its structure, and its modeling accuracy and reliability are insufficient when coping with complex nonlinear systems. To address this problem, a structural optimization method for an echo state network based on error compensation is proposed. The method aims to improve the modeling capability of echo state networks in nonlinear systems and apply it to wind power prediction. The main content of this paper is as follows:

(1) Aiming at the problem that the traditional echo state network ignores the error autocorrelation during the training process, which leads to the low modeling accuracy of nonlinear systems, a transfer learning-based echo state network is proposed. The network model aims to deal with the autocorrelation of errors, through theoretical analysis of the causes of errors in the modeling process of echo state network, and the impact of error autocorrelation on modeling accuracy, and then use the transfer learning method to update the output weights and adjust the prediction errors in real-time, to improve the accuracy and effectiveness of modeling. Experiments are conducted with Mackey-Glass, sunspot, and gas concentration datasets, and the results show that the model can compensate in real-time according to prediction error, effectively reduce the impact of autocorrelation error on modeling accuracy, and exhibit high prediction accuracy and stability in complex nonlinear systems. Compared with other network models, the prediction accuracy of the model is improved by 17%.

(2) Aiming at the problem that the traditional echo state network is vulnerable to the uncertainty caused by the system error, which leads to the lack of reliability in modeling nonlinear systems, a Bayesian linear regression-based echo state network is proposed. The network model aims to quantify the uncertainty of prediction by training the echo state network and obtaining the prediction error, and then establishing an error distribution compensation model based on Bayesian linear regression, estimating the error distribution by using the Markov chain Monte Carlo method, and using it for the prediction uncertainty analysis, to improve the reliability of prediction. In addition, the model can capture anomalies in the data and compensate for them using the error distribution. Experimental results based on benchmark and real datasets show that the model can compensate effectively based on the error distribution and cover as many real values as possible within a 95% confidence interval, thus significantly improving the reliability of the predictions.

(3) Aiming at the uncertainty of wind power data, which makes it difficult to achieve high-precision and reliable prediction, a wind power prediction model based on error compensation is constructed. Combining the correlation analysis of the data with the results of anomaly processing, it is found that there is a significant correlation between wind speed and wind power. On this basis, the wind power prediction model is used to model the actual data of wind farms, and the model can provide accurate and reliable wind power prediction results, which provide theoretical support and a decision-making basis for improving the effectiveness of energy management applications.

[1]谢卓芳, 张猛. 工业互联网规模化应用提速[N]. 湖南日报, 2024-10-30(001).

[2]张楠. 工业互联网高质量发展和规模化应用提速[N]. 中国工业报, 2024-02-27(003).

[3]汤健, 崔璨麟, 夏恒, 等. 面向复杂工业过程的虚拟样本生成综述[J]. 自动化学报, 2024, 50(04): 688-718.

[4]周平, 张丽, 李温鹏, 等. 集成自编码与PCA的高炉多元铁水质量随机权神经网络建模[J]. 自动化学报, 2018, 44(10): 1799-1811.

[5]Lukoševičius M, Jaeger H. Reservoir computing approaches to recurrent neural network training[J]. Computer Science Review, 2009, 3(3): 127-149.

[6]Jaeger H, Hass H. Harnessing nonlinearity: predicting chaotic systems and saving energy in wireless communication[J]. Science, 2004, 304(5667): 78-80.

[7]Jason W Bohland, Ali A Minai. Efficient associative memory using small-world architecture[J]. Neurocomputing, 2001, 38: 489-496.

[8]伦淑娴, 林健, 姚显双. 基于小世界回声状态网的时间序列预测[J]. 自动化学报, 2015, 41(09): 1669-1679.

[9]孙晓川, 李莹琦. 小世界递归小波神经网络研究[J]. 南京邮电大学学报(自然科学版), 2017, 37(04): 97-102.

[10]Xie M, Wang Q, Yu S. Time series prediction of ESN based on chebyshev mapping and strongly connected topology[J]. Neural Processing Letters, 2024, 56(1): 30.

[11]Gauthier D J, Bollt E G, Griffith A, et al. Next generation reservoir computing[J]. Nature Communication, 2021, 12(1): 1-8.

[12]Rodan A, Tiňo P. Simple Deterministically constructed cycle reservoirs with regular jumps[J]. Neural Computation, 2012, 24(7): 1822-1852.

[13]薄迎春, 张欣, 刘宝, 等. 稀疏连接的异步池计算网络[J]. 中国科学:信息科学, 2021, 51(05): 764-778.

[14]薄迎春, 张欣, 刘宝. 时间分割的池计算网络及其动力学[J]. 控制理论与应用, 2019, 36(08): 1315-1321.

[15]Bo Y, Wang P, Zhang X. An asynchronously deep reservoir computing for predicting chaotic time series[J]. Applied Soft Computing, 2020, 95: 106530.

[16]Bo Y, Wang J. A feature-recombinant asynchronous deep reservoir computing for modeling time series data[J]. Applied Soft Computing, 2024, 151: 111167.

[17]Li Z, Liu Y, Tanaka G. Multi-reservoir echo state networks with hodrick-prescott filter for nonlinear time-series prediction[J]. Applied Soft Computing, 2023, 135: 110021.

[18]Yao X, Wang H, Huang Z. Broad fractional-order echo state network with slime mould algorithm for multivariate time series prediction[J]. Applied Soft Computing, 2024, 163: 111900.

[19]Sun X, Li Y, Gui G, et al. Echo-state restricted boltzmann machines: A perspective on information compensation[J]. IEEE Access, 2019, 7: 16281-16290.

[20]郭伟, 姚欢, 张昭昭, 等. 基于强化学习的储层神经元筛选优化方法[J]. 控制与决策, 2024, 39(09): 2876-2884.

[21]Li T, Guo Z, Li Q. Deep echo state network with projection-encoding for multi-step time series prediction[J]. Neurocomputing, 2025, 617: 128939.

[22]Ma Q, Shen L, Cottrell G W. DeePr-ESN: A deep projection-encoding echo-state network[J]. Information Sciences, 2020, 511: 152-171.

[23]Shi J, Teh J, Alharbi B, et al. Load forecasting for regional integrated energy system based on two-phase decomposition and mixture prediction model[J]. Energy, 2024, 297: 131236.

[24]Xu Y, Yao T, Yang G. An EMD-SVM model with error compensation for short-term wind speed forecasting[J]. International Journal of Information Technology and Management, 2019, 18(2-3): 171-181.

[25]Zhao X, Mu D, Gao Z, et al. Stochastic stability of the improved maximum correntropy kalman filter against non-Gaussian noises[J]. International Journal of Control, Automation and Systems, 2024, 22: 731-743.

[26]Huang J, Li Y, Shardt Y A W, et al. Error-driven chained multiple-subnetwork echo state network for time-Series prediction[J]. IEEE Sensors Journal, 2022, 20(20): 19533-19542.

[27]Han Y, Jing Y, Li K, et al. Multi-step network traffic prediction using echo state network with a selective error compensation strategy[J]. Transactions of the Institute of Measurement and Control, 2022, 44(8): 1656-1668.

[28]Yan Z, Wang J. Robust model predictive control of nonlinear systems with unmodeled dynamics and bounded uncertainties based on neural networks[J]. IEEE Transactions on Neural Networks and Learning Systems, 2014, 25(3): 457-469.

[29]杨淑凡, 王永威, 谢闻捷. 计及误差补偿的两阶段短期电力负荷预测方法[J]. 电工材料, 2024, (01): 71-76.

[30]张昭昭, 朱应钦, 余文. 具有双储层结构的动态误差补偿回声状态网络[J]. 控制理论与应用, 2024, 41(03): 385-395.

[31]陈豪钰, 李振华, 张绍哲, 等. 基于MHA-CNN-SLSTM和误差补偿的短期互感器误差预测[J]. 电力系统保护与控制, 2024, 52(24): 74-84.

[32]王萍, 张吉昂, 程泽. 基于最小二乘支持向量机误差补偿模型的锂离子电池健康状态估计方法[J]. 电网技术, 2022, 46(02): 613-623.

[33]郎坤, 张明媛, 袁永博. 基于迭代误差补偿的核极端学习机模型在短期电力负荷预测中的应用[J]. 计算机应用, 2015, 35(07): 2083-2087.

[34]耿阳, 王海龙, 张楠, 等. 基于误差补偿LSTM-GRU的综合能源系统多元负荷预测[J]. 电气工程学报, 2023, 18(04): 320-330.

[35]Qian H, Luo Y, Zhou X, et al. Ultra short-term wind power prediction beased on lightweight learning machine with error compensation[J]. International Journal of Global Energy Issues, 2024, 46(5): 463-482.

[36]邱山, 撖奥洋, 张智晟. 基于多储备池相关向量回声状态机和误差补偿的短期负荷预测研究[J]. 电力系统及其自动化学报, 2023, 35(03): 53-58.

[37]朱希, 林俊德, 施翔宇, 等. 基于VMD-SSA及误差补偿的风电功率超短期预测[J]. 福建理工大学学报, 2023, 21(06): 573-579.

[38]宫婷, 车建峰, 王勃, 等. 考虑误差概率分布及波动特性的短期风电功率预测修正方法[J]. 高电压技术, 2025, 51(01): 379-389.

[39]王健, 宋颖, 吴涛. 基于LSTM网络与误差补偿的预测模型[J]. 计算机技术与发展, 2023, 33(03): 133-138.

[40]宋瑞蓉, 王斌君, 仝鑫, 等. 基于改进果蝇的混合小波神经网络交通流预测[J]. 科学技术与工程, 2021, 21(15): 6394-6401.

[41]Adnan M, Dai H, Kisi O, et al. Modelling biochemical oxygen demand using improved neuro-fuzzy approach by marine predators algorithm[J]. Environmental Science and Pollution Research, 2023, 30(41): 94312-94333.

[42]Wu Z, Rincon D, Christofides P D. Process structure-based recurrent neural network modeling for model predictive control of nonlinear processes[J]. Journal of Process Control, 2020, 89: 74-84.

[43]Hochreiter S, Schmidhuber J. Long short-term memory[J]. Neural Computation, 1997, 9(8): 1735-1780.

[44]Cho K, Van Merrienboer B, Gulcehre C, et al. Learning phrase representations using RNN encoder-decoder for statistical machine translation[J/OL]. Association for Computational Linguistics, 2025-02-18.

[45]Schuster M, Paliwal K K. Bidirectional recurrent neural networks[J]. IEEE Transactions on Signal Processing, 1997, 45(11): 2673-2681.

[46]Jaeger H. The echo state approach to analysing and training recurrent neural networks[R]. Germany: German National Research Center for Information Technology GMD, 2001.

[47]Qiao J, Li S, Li W. Mutual information based weight initialization method for sigmoidal feedforward neural networks [J]. Neurocomputing, 2016, 207: 676-683.

[48]王磊. 回声状态网络优化设计及应用研究[D]. 北京: 北京工业大学, 2019.

[49]Buehner M, Young P. A tighter bound for the echo state property[J]. IEEE Transactions on Neural Networks, 2006, 17(3): 820-824.

[50]Jaeger H. Tutorial on training recurrent neural networks, covering BPTT, RTRL, EKF, and the “echo state network” approach[R]. Bremen: National Research Center for Information Technology, 2002.

[51]张立炎, 向馗, 龙容, 等. 基于ESN的非线性系统未建模动态补偿及控制[J]. 电子学报, 2016, 44(01): 60-66.

[52]Sun F K, Christopher I L, Duane S B. Adjusting for autocorrelated errors in neural networks for Time Series Regression and Forecasting[J/OL]. NeurIPS Foundation, 2025-02-18.

[53]Zhu Y, Liu Y, Zhang Z, et al. Optimized echo state network for error compensation based on transfer learning[J]. Applied Soft Computing, 2025, 174: 112935.

[54]张昭昭, 朱应钦, 乔俊飞, 等. 一种基于行为空间的回声状态网络参数优化方法[J]. 信息与控制, 2021, 50(05): 556-565.

[55]Zhang Z, Zhu Y, Wang X, et al. Optimal echo state network parameters based on behavioural spaces[J]. Neurocomputing, 2022, 503: 299-313.

[56]Ghaderi A, Shahri A A, Larsson S. A visualized hybrid intelligent model to delineate Swedish fine-grained soil layers using clay sensitivity[J]. CATENA, 2022, 214: 106289.

[57]Zhuang F, Qi Z, Duan K, et al. A Comprehensive survey on transfer learning[J]. Proceedings of the IEEE, 2021, 109(1): 43-76.

[58]Rodan A, Tino P. Minimum complexity echo state network[J]. IEEE Transactions on Neural Networks, 2011, 22(1): 131-144.

[59]Cui H, Feng C, Chai Y, et al. Effect of hybrid circle reservoir injected with wavelet-neurons on performance of echo state network[J]. Neural Networks, 2014, 57: 141-151.

[60]Guo W, Yao H, Zhu Y, et al. A self-organization reconstruction method of ESN reservoir structure based on reinforcement learning[J]. Information Sciences, 2024, 677: 120826.

[61]Kram R M A, Mostafa R R, Chen Z, et al. Water temperature prediction using improved deep learning methods through reptile search algorithm and weighted mean of vectors optimizer[J]. Journal of Marine Science Engineering, 2023, 11(2): 259.

[62]Gallicchio C, Micheli A, Pedrelli L. Deep reservoir computing: A critical experimental analysis[J]. Neurocomputing, 2017, 268: 87-99.

[63]Morales J, Yu W. Improving neural network’s performance using Bayesian inference[J]. Neurocomputing, 2021, 461: 319-326.

[64]丁薛祥. 在联合国气候变化巴库大会世界领导人气候行动峰会上的发言[N]. 人民日报, 2024-11-14(02).

[65]Liu X, Cao Z, Zhang Z. Short-term predictions of multiple wind turbine power outputs based on deep neural networks with transfer learning[J]. Energy, 2021, 217: 119356.

[66]Li X, Zhu Y. Neural networks with transfer learning and frequency decomposition for wind speed prediction with missing data[J]. Mathematics, 2024, 12(8): 1137.