RESEARCH ON LARGE-DISTURBANCE ORDER REDUCTION OF DIRECT-DRIVE WIND FARM NONLINEAR SYSTEM BASED ON IMPROVED MULTI-POINT LINEARIZATION

Zhou Leming; Zhen Yongzan; Cao Tianzhi; Xie Huan; Liang Beihua; Lu Wenqing

doi:10.19912/j.0254-0096.tynxb.2025-0665

PDF(8808 KB)

Acta Energiae Solaris Sinica ›› 2026, Vol. 47 ›› Issue (9) : 690-704. DOI: 10.19912/j.0254-0096.tynxb.2025-0665

RESEARCH ON LARGE-DISTURBANCE ORDER REDUCTION OF DIRECT-DRIVE WIND FARM NONLINEAR SYSTEM BASED ON IMPROVED MULTI-POINT LINEARIZATION

Zhou Leming¹, Zhen Yongzan¹, Cao Tianzhi², Xie Huan², Liang Beihua², Lu Wenqing²

Author information +

History +

Abstract

In large-disturbance scenarios, linearized model order reduction methods based on a single equilibrium point cannot achieve satisfactory approximation performance. To address this problem, a nonlinear model order reduction method for direct-drive wind farm systems with high accuracy, structure preservation, and system observability is proposed. First, local linearized models at multiple operating points are established to retain the nonlinear information of the original system. Second, to avoid error accumulation and matrix ill-conditioning in numerical computation, the state variables are classified into strongly and weakly correlated variables according to generalized participation factors. Then, based on singular perturbation theory and the nonlinear dynamic equations, the dynamic processes of the weakly correlated variables are neglected, while those of the strongly correlated state variables and observable state variables are retained, thereby obtaining a reduced-order model that preserves both system observability and nonlinear structure. Subsequently, through non-intrusive modeling, particle swarm optimization is employed to improve the low-order nonlinear reduced-order model, enabling the improved reduced-order model to satisfy the accuracy requirements while preserving the nonlinear structure. Finally, simulations are conducted under three large-disturbance scenarios. The results show that the reduced-order system model obtained using the proposed method exhibits dynamic response characteristics highly similar to those of the full-order system model, with a model-order reduction rate of 50% and response errors within 1%. The proposed method can significantly reduce the complexity of control strategy optimization and numerical computation, providing a more feasible and efficient approach for the design and optimization of regional automatic power control systems.

Key words

wind farm / nonlinear systems / particle swarm optimization（PSO） / model order reduction / singular perturbation theory / generalized participation factors

Cite this article

EndNote

Ris (Procite)

Bibtex

Download Citations

Zhou Leming, Zhen Yongzan, Cao Tianzhi, Xie Huan, Liang Beihua, Lu Wenqing. RESEARCH ON LARGE-DISTURBANCE ORDER REDUCTION OF DIRECT-DRIVE WIND FARM NONLINEAR SYSTEM BASED ON IMPROVED MULTI-POINT LINEARIZATION[J]. Acta Energiae Solaris Sinica. 2026, 47(9): 690-704 https://doi.org/10.19912/j.0254-0096.tynxb.2025-0665

References

[1] McKenna R, Lilliestam J, Heinrichs H U, et al. System impacts of wind energy developments: key research challenges and opportunities[J]. Joule, 2025, 9(1): 101799.
[2] 叶林, 张慈杭, 汤涌, 等. 多时空尺度协调的风电集群有功分层预测控制方法[J]. 中国电机工程学报, 2018, 38(13): 3767-3780, 4018.
Ye L, Zhang C H, Tang Y, et al.Active power stratification predictive control approach for wind power cluster with multiple temporal and spatial scales coordination[J]. Proceedings of the CSEE, 2018, 38(13): 3767-3780, 4018.
[3] 胡家兵, 袁小明, 程时杰. 电力电子并网装备多尺度切换控制与电力电子化电力系统多尺度暂态问题[J]. 中国电机工程学报, 2019, 39(18): 5457-5467, 5594.
Hu J B, Yuan X M, Cheng S J.Multi-time scale transients in power-electronized power systems considering multi-time scale switching control schemes of power electronics apparatus[J]. Proceedings of the CSEE, 2019, 39(18): 5457-5467, 5594.
[4] 王渝红, 杜婷, 廖建权, 等. 保持主导振荡模态的双馈风电场动态聚合等值方法[J]. 太阳能学报, 2024, 45(6): 518-529.
Wang Y H, Du T, Liao J Q, et al.Dynamic aggregation equivalence method for DFIG-based wind farm with maintained dominant oscillation mode[J]. Acta Energiae Solaris Sinica, 2024, 45(6): 518-529.
[5] 梁惠溉, 杨俊华, 林汇金, 等. 基于矩匹配模型降阶的直驱式波浪发电系统功率优化[J]. 太阳能学报, 2024, 45(5): 191-197.
Liang H G, Yang J H, Lin H J, et al.Power optimization of direct-drive wave power system based on moment matching model reduction[J]. Acta Energiae Solaris Sinica, 2024, 45(5): 191-197.
[6] 王晓东, 付腾, 王晓驰, 等. 基于模型降阶的双馈风电机组虚拟同步机控制参数优化[J]. 太阳能学报, 2023, 44(11): 210-216.
Wang X D, Fu T, Wang X C, et al.Control parameter optimization of virtual synchronizer for doubly-fed wind turbines based on model reduction[J]. Acta Energiae Solaris Sinica, 2023, 44(11): 210-216.
[7] 韩平平, 夏雨, 丁明, 等. 基于PCA和CA-ST方法的风电场等值建模研究[J]. 太阳能学报, 2020, 41(11): 267-277.
Han P P, Xia Y, Ding M, et al.Equivalent modeling of wind farm based on PCA and CA-ST methods[J]. Acta Energiae Solaris Sinica, 2020, 41(11): 267-277.
[8] Rafiq D, Farooq J, Bazaz M A.Synergistic use of intrusive and non-intrusive model order reduction techniques for dynamical power grids[J]. International Journal of Electrical Power & Energy Systems, 2022, 138: 107908.
[9] 李永达. 大干扰下风电并网系统动态降阶方法及适用性研究[D]. 北京: 华北电力大学, 2023.
Li Y D.Research on dynamic order reduction method and applicability for wind power grid connected system under large disturbance[D]. Beijing: North China Electric Power University, 2023.
[10] 孟潇潇, 周念成, 王强钢. 逆变型分布式电源模型的多时间尺度降阶分析及稳定一致性证明[J]. 中国电机工程学报, 2018, 38(13): 3813-3825, 4022.
Meng X X, Zhou N C, Wang Q G.Multi-time scale model order reduction and stability consistency of IIDG system[J]. Proceedings of the CSEE, 2018, 38(13): 3813-3825, 4022.
[11] 李永刚, 严风, 周一辰. 基于降阶模型的光伏发电系统振荡模式分析[J]. 电测与仪表, 2020, 57(13): 61-68.
Li Y G, Yan F, Zhou Y C.Analysis of oscillation modes based on reduced model of photovoltaic power generation system[J]. Electrical Measurement & Instrumentation, 2020, 57(13): 61-68.
[12] 谈赢杰, 黄莹, 袁智勇, 等. 基于奇异摄动理论的微电网降阶建模方法与对比研究[J]. 电网技术, 2020, 44(5): 1914-1921.
Tan Y J, Huang Y, Yuan Z Y, et al.Modeling method and comparison of order reduction for micro-grid based on singular perturbation theory[J]. Power System Technology, 2020, 44(5): 1914-1921.
[13] 李正, 郝全睿, 尹晓东, 等. 模块化多电平换流器的降阶小信号模型研究[J]. 中国电机工程学报, 2018, 38(12): 3638-3650, 24.
Li Z, Hao Q R, Yin X D, et al.Research on reduced-order small-signal model of modular multilevel converter[J]. Proceedings of the CSEE, 2018, 38(12): 3638-3650, 24.
[14] Montier L, Henneron T, Clenet S, et al.Model order reduction applied to a linear finite element model of a squirrel cage induction machine based on POD approach[J]. IEEE Transactions on Magnetics, 2021, 57(6): 1-4.
[15] 高本锋, 符章棋, 王刚, 等. 适用于次同步振荡分析的直驱风电场平衡降阶方法[J]. 电力工程技术, 2023, 42(3): 112-120.
Gao B F, Fu Z Q, Wang G, et al.Balanced reduction method of direct-drive wind farm for subsynchronous oscillation analysis[J]. Electric Power Engineering Technology, 2023, 42(3): 112-120.
[16] 高本锋, 王刚, 邵冰冰, 等. 基于主导度分析的直驱风电场奇异摄动降阶方法[J]. 中国电机工程学报, 2022, 42(7): 2449-2462.
Gao B F, Wang G, Shao B B, et al.Singular perturbation approximation method based on the dominant degree analysis for direct drive wind farm[J]. Proceedings of the CSEE, 2022, 42(7): 2449-2462.
[17] Muntwiler S, Stanojev O, Zanelli A, et al.A stiffness-oriented model order reduction method for low-inertia power systems[J]. Electric Power Systems Research, 2024, 235: 110630.
[18] Garrido J.Applications of Krylov subspace and balanced truncation model order reduction methods in power systems[D]. University of Arkansas, 2021.
[19] Zulfiqar U, Sreeram V, Du X.Frequency-limited pseudo-optimal rational Krylov algorithm for power system reduction[J]. International Journal of Electrical Power & Energy Systems, 2020, 118: 105798.
[20] 许建成, 孙建军, 钟佩军, 等. 基于平衡实现理论的变流器并网系统降阶模型[J]. 电工技术学报, 2021, 36(增刊1): 255-264.
Xu J C, Sun J J, Zhong P J, et al.Reduced-order model of grid-connected converter system based on balanced realization theory[J]. Transactions of China Electrotechnical Society, 2021, 36(S1): 255-264.
[21] Barnett J, Farhat C.Quadratic approximation manifold for mitigating the Kolmogorov barrier in nonlinear projection-based model order reduction[J]. Journal of Computational Physics, 2022, 464: 111348.
[22] Bu Y P.Krylov subspace model order reduction of linear dynamical systems with quadratic output[J]. Transactions of the Institute of Measurement and Control, 2025, 47(5): 827-838.
[23] Hernández-Ortega M A, Messina A R. Recursive linearization of Carleman-based nonlinear power system models[J]. Electric Power Systems Research, 2022, 209: 107974.
[24] Cao X G, Benner P, Pontes Duff I, et al.Model order reduction for bilinear control systems with inhomogeneous initial conditions[J]. International Journal of Control, 2021, 94(10): 2886-2895.
[25] Chen P P, Motee N, Sun Q Y. Carleman-Fourier linearization of complex dynamical systems: convergence and explicit error bounds[PP/OL]. arXiv (2024-11-18)[2026-06-08]. https://doi.org/10.48550/arXiv.2411.11598.
[26] Amini A, Zheng C, Sun Q Y, et al. Carleman linearization of nonlinear systems and its finite-section approximations[PP/OL]. V2. arXiv (2022-07-19)[ 2026-06-08]. https://doi.org/10.48550/arXiv.2207.07755.
[27] 岑仲迪. 奇异摄动对流扩散问题的有限差分法[D]. 杭州: 浙江大学, 2005.
CEN Z D.Finite difference method for singularly perturbed convection diffusion problems[D]. Hangzhou: Zhejiang University, 2005.
[28] 包小兵, 刘利斌, 毛志. 奇异摄动反应扩散方程的后验误差估计及自适应算法[J]. 应用数学和力学, 2021, 42(3): 323-330.
Bao X B, Liu L B, Mao Z.A posteriori error estimation and adaptive algorithm for singularly perturbed reaction-diffusion equations[J]. Applied Mathematics and Mechanics, 2021, 42(3): 323-330.
[29] Feng L, Chellappa S, Benner P.A posteriori error estimation for model order reduction of parametric systems[J]. Advanced Modeling and Simulation in Engineering Sciences, 2024, 11(1): 5.
[30] 王培珍, 刘鸣放. 各向异性网格下非协调元的后验误差估计[J]. 河南大学学报(自然科学版), 2015, 45(3): 253-257.
Wang P Z, Liu M F.A posteriori error estimators of anisotropic nonconforming finite element[J]. Journal of Henan University(natural science), 2015, 45(3): 253-257.
[31] 邵文婷, 郑烁宇. Sinc配点法与sinh变换求解奇异摄动过渡层问题[J]. 信阳师范大学学报(自然科学版), 2025, 38(1): 92-100.
Shao W T, Zheng S Y.Sinc collocation method and sinh transform for solving singular perturbed problems with a transition layer[J]. Journal of Xinyang Normal University (natural science edition), 2025, 38(1): 92-100.
[32] Kramer B, Peherstorfer B, Willcox K E.Learning nonlinear reduced models from data with operator inference[J]. Annual Review of Fluid Mechanics, 2024, 56: 521-548.
[33] 邵冰冰, 赵书强, 裴继坤, 等. 直驱风电场经VSC-HVDC并网的次同步振荡特性分析[J]. 电网技术, 2019, 43(9): 3344-3355.
Shao B B, Zhao S Q, Pei J K, et al.Subsynchronous oscillation characteristic analysis of grid-connected DDWFs via VSC-HVDC system[J]. Power System Technology, 2019, 43(9): 3344-3355.
[34] 高本锋, 崔意婵, 邵冰冰, 等. 直驱风电机组全运行区域的次同步振荡特性分析[J]. 电力建设, 2020, 41(2): 85-93.
Gao B F, Cui Y C, Shao B B, et al.Sub-synchronous oscillation characteristics of direct-drive PMSG under all operation regions when wind farms connected to weak AC system[J]. Electric Power Construction, 2020, 41(2): 85-93.
[35] 孙圣欣, 汤晨煜, 解大, 等. 并网型直流微电网的非线性降阶建模及其估计吸引域的优化计算[J]. 中国电机工程学报, 2024, 44(2): 517-535.
Sun S X, Tang C Y, Xie D, et al.Nonlinear modeling of grid-connected DC microgrid and optimization of estimated region of attraction[J]. Proceedings of the CSEE, 2024, 44(2): 517-535.
[36] Wang L F, Peng Y L, Luo J.Model reduction method for spacecraft electrical system based on singular perturbation theory[J]. Electronics, 2024, 13(21): 4291.
[37] 衡一嘉. 基于奇异摄动理论的电力系统电力电子接口降阶建模方法研究[D]. 成都: 电子科技大学, 2021.
Heng Y J.Research on reduced order modeling method of power electronic interface in power system based on singular perturbation theory[D]. Chengdu: University of Electronic Science and Technology of China, 2021.
[38] Lu K, Zhang K Y, Zhang H P, et al.A review of model order reduction methods for large-scale structure systems[J]. Shock and Vibration, 2021, 2021(1): 6631180.
[39] Al-Iedani I, Gajic Z.Order reduction of a wind turbine energy system via the methods of system balancing and singular perturbations[J]. International Journal of Electrical Power & Energy Systems, 2020, 117: 105642.
[40] 戴诗正. 奇异摄动理论[J]. 系统工程与电子技术, 1988, 10(2): 1-12.
Dai S Z.Singular perturbation theory[J]. Systems Engineering and Electronics, 1988, 10(2): 1-12.
[41] 刘晓, 董春发, 瞿寒冰, 等. 低压直流系统高频振荡问题的降阶建模及控制参数设计[J]. 中国电力, 2024, 57(2): 82-93.
Liu X, Dong C F, Qu H B, et al.Reduced-order modeling and control parameter design of high-frequency oscillation in low-voltage DC system[J]. Electric Power, 2024, 57(2): 82-93.
[42] Yang S S, Tian Y, He C, et al.A gradient-guided evolutionary approach to training deep neural networks[J]. IEEE Transactions on Neural Networks and Learning Systems, 2022, 33(9): 4861-4875.
[43] Eberhart R C, Shi Y H. Particle swarm optimization: developments, applications and resources[C]//Proceedings of the2001 Congress on Evolutionary Computation (IEEE Cat. No.01TH8546). Seoul, Korea, 2001: 81-86 vol. 1.
[44] Li X Z, Mao K Z, Lin F F, et al.Particle swarm optimization with state-based adaptive velocity limit strategy[J]. Neurocomputing, 2021, 447: 64-79.
[45] Huang H, Qiu J N, Riedl K.On the global convergence of particle swarm optimization methods[J]. Applied Mathematics & Optimization, 2023, 88(2): 30.
[46] Pedersen M E H. Good parameters for particle swarm optimization: HL1001[R]. Copenhagen, Denmark: Hvass Laboratories, 2010.
[47] Engelbrecht A.Particle swarm optimization[C]//Proceedings of the Companion Publication of the 2014 Annual Conference on Genetic and Evolutionary Computation. 2014: 381-406.
[48] van den Bergh F, Engelbrecht A P. A study of particle swarm optimization particle trajectories[J]. Information Sciences, 2006, 176(8): 937-971.
[49] Yin S Y, Jin M, Lu H X, et al.Reinforcement-learning-based parameter adaptation method for particle swarm optimization[J]. Complex & Intelligent Systems, 2023, 9(5): 5585-5609.
[50] Kitsios V, Cordier L, O’Kane T J. Proper orthogonal decomposition reduced-order model of the global oceans[J]. Theoretical and Computational Fluid Dynamics, 2024, 38(5): 707-727.
[51] House A M, Kaminsky A, Wang Y, et al. A POD-based input-output reduced-order modeling framework for acceleration of coupled-solver workflows[C]//AIAA SCITECH2024 Forum. Orlando,FL, 2024: AIAA2024-0254.