Dynamic Equivalencing Method for Power Systems Based on Koopman Theory
Zhang Zi′ao1, Li Yansong1, Ren Bixing2, Liu Jun1, Li Qiang2
1. School of Electrical and Electronic Engineering North China Electric Power University Beijing 102206 China; 2. State Grid Jiangsu Electric Power Company Ltd. Research Institute Nanjing 211103 China
Abstract：There is a lack of dynamic equivalent modeling methods for power systems that can be applied to both large and small disturbances. To enhance the adaptability of the equivalence method to uncertainties and time-varying operating conditions in power systems, a Koopman dynamic equivalence method for power systems under different levels of disturbance is proposed, which unifies the dynamic equivalence methods for power systems under different operating conditions. First, establish the state space equation of the external system, perform high-dimensional linearization mapping based on the Koopman operator, and establish its global linearized equivalent model in high-dimensional space. Through this mathematical mapping, the model is linearized without losing key information, so the model is applicable to all operating conditions of the power system. Then, based on the data-driven approach, the parameters of the equivalent model are identified by collecting equivalent boundary data. The dynamic mode decomposition (DMD) and least squares are used to identify the state coefficient matrix and output coefficient matrix of the equivalent model, respectively. When the measurement data contains noise, the model parameters are adjusted using least squares dynamic mode decomposition (LS-DMD) and total least squares (TLS) to enhance the model's noise resistance. This process does not require explicit system detailed structure and operating parameters, thus reducing the workload of detailed modeling. The low-dimensional identification DMD matrix of the external system's state coefficient matrix shares the same state-space characteristics with the original matrix. Therefore, the numerical solution for the state of the original model can be obtained through the spectral decomposition of the DMD matrix. Then, the Koopman modes are extracted by spectral decomposition of the system state coefficient matrix. The mode with eigenvalues closer to the imaginary axis has a greater impact on the system and will dominate the dynamic characteristics of the system. The rapidly decaying modes have a relatively small impact on the dynamic characteristics of the system. By retaining the analysis of modes that have a greater impact on the system and filtering out rapidly decaying modes, it is possible to achieve reduced order of the equivalent model and significantly reduce the computational load. Finally, the proposed method is tested on examples of external systems experiencing transient (large disturbances) and steady-state (small disturbances), respectively. The results show that the proposed equivalent model has basically consistent power angle swing curves and equivalent boundary tie line power output curves with the original system, regardless of whether the system is under large or small disturbances. Quantitative analysis was conducted under large disturbance cases, and the fitting accuracy of the equivalent model and the original model for the transmission of active and reactive power on the five tie lines was 99.35%, 95.47%, 98.21%, 99.49%, 95.51%, 99.34%, 99.32%, 95.73%, 98.30%, and 96.72%, respectively, with an average model accuracy of 97.74%. The following conclusions are obtained through the verification of the examples: (1) The proposed model, based on the high-dimensional linear Koopman operator, fully preserves the nonlinear dynamic characteristics of the original system. So the model has no restrictions on operating points and is suitable for dynamic equivalence of power systems under varying degrees of disturbance, enhancing the robustness and universality of power system equivalence models. (2) The parameters of the equivalent model are identified based on the measured data of the system boundary, so the proposed method does not require explicit detailed structural and operational parameters of the system, reducing the workload associated with detailed modeling of the internal structure of the system. (3) The effectiveness of the proposed equivalent model was verified through examples. The results show that the proposed model has basically the same power angle swing curve and power output curve as the original system under large and small disturbances, with a model accuracy of 97.74%. The model can replace the original system for dynamic stability analysis under different levels of disturbance.
张子傲, 李岩松, 任必兴, 刘君, 李强. 电力系统Koopman动态等值方法[J]. 电工技术学报, 2026, 41(1): 111-126.
Zhang Zi′ao, Li Yansong, Ren Bixing, Liu Jun, Li Qiang. Dynamic Equivalencing Method for Power Systems Based on Koopman Theory. Transactions of China Electrotechnical Society, 2026, 41(1): 111-126.
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2026, Vol. 41 
