Model Reduction for Multi-Direct-Drive Wind Turbine Systems via Enhanced Multi-Point Linearization and Balanced Truncation
Zhou Leming1, Zhen Yongzan1, Cao Tianzhi2, Gao Benfeng1
1. Hebei Key Laboratory of Distributed Energy Storage and Micro-Grid North China Electric Power University Baoding 071003 China 2. State Grid Jibei Electric Power Co. Ltd Electric Power Research Institute Beijing 100045 China
Abstract：Accurate reduced-order modeling of large-scale Type-4 direct-drive wind farm systems is essential for small-signal stability analysis and controller design. However, existing aggregation-based simplification approaches become inadequate under heterogeneous wind speed conditions, often resulting in modal drift and mismatches in frequency response. To address these limitations, this study develops a unified model order reduction (MOR) framework based on balanced truncation theory. The proposed method preserves the essential dynamic characteristics of multi-machine wind farm systems without relying on symmetry assumptions or turbine aggregation, improving adaptability to diverse operating conditions. The nonlinear model of a multi-machine direct-drive wind farm is first linearized around a steady-state operating point to derive a small-signal state-space representation. A balanced coordinate transformation is performed by jointly diagonalizing the controllability and observability Gramians, distributing controllability and observability uniformly across the transformed states. The Hankel singular values are calculated to quantify each state’s contribution to energy transfer between inputs and outputs. States with negligible energy influence are systematically truncated, yielding a reduced-order model that preserves the dominant dynamics of the original system. Unlike conventional techniques, this method does not require structural partitioning, participation factor analysis, or prior identification of dominant oscillatory modes, thereby enhancing automation and generality. Leveraging Lie group symmetry analysis, the study proves that under identical wind speeds, the reduced-order model degenerates into the traditional aggregated representation, while in non-uniform scenarios, it retains machine-level dynamics absent in conventional aggregation. The effectiveness of the proposed approach is validated through simulations under five representative wind speed scenarios, covering both symmetric and asymmetric conditions. Time-domain comparisons show that the reduced-order model closely matches the transient responses of the full-order system, with root-mean-square errors below 1%. Frequency-domain analyses confirm that subsynchronous oscillation (SSO) modes are accurately preserved, with oscillation frequency deviations under 5%. The reduced-order model also captures the sensitivity of damping ratios to variations in controller parameters, ensuring accurate reproduction of control effects on small-signal stability. Across all scenarios, the method achieves a 57.14% reduction in model size while maintaining high consistency with the full-order system. A comparative study against traditional aggregation-based models highlights several advantages of the proposed framework. First, the balanced truncation-based approach preserves dominant energy modes and avoids modal distortion under non-uniform wind speed conditions. Second, the method scales efficiently to wind farms with arbitrary numbers of turbines and layout configurations, making it suitable for large-scale systems. Third, the framework ensures robust dynamic retention across diverse operating conditions, maintaining accuracy in both global and local oscillatory behaviors. Finally, integrating Lie group symmetry theory provides a theoretical foundation for unifying model aggregation and reduction, offering a systematic perspective on simplifying multi-machine wind farm models. In summary, this study proposes a balanced truncation-based MOR methodology that provides compact, accurate, and computationally efficient reduced-order models for multi-machine Type-4 direct-drive wind farm systems. The resulting models are suitable for small-signal stability assessments, subsynchronous oscillation analysis, and control system design. By bridging the gap between aggregation techniques and model order reduction through symmetry analysis, this framework offers a generalized and automated approach for simplified modeling under both symmetric and asymmetric wind conditions.
周乐明, 甄永赞, 曹天植, 高本锋. 基于平衡截断理论的多直驱风机系统小扰动降阶技术研究[J]. 电工技术学报, 0, (): 251268-.
Zhou Leming, Zhen Yongzan, Cao Tianzhi, Gao Benfeng. Model Reduction for Multi-Direct-Drive Wind Turbine Systems via Enhanced Multi-Point Linearization and Balanced Truncation. Transactions of China Electrotechnical Society, 0, (): 251268-.
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