Abstract：The conventional methods of order reduction model for linear system cannot predict the time domain error of dynamic behaviors of state variables between the original and reduced models, so the reduced models derived by conventional methods maybe not suitable for analyzing the system performances because of the large variation between the reduction model and origin model. To solve this problem, this paper has proposed a L2 norm-based error prediction method for error prediction which is based on the conventional singular perturbation model order reduction method. The relative error calculation formula of the dynamic characteristic of the system state variable is deduced which ignore the variation caused by fast dynamic and fixed slow dynamic reduction. Then, the multi-time scale mathematical models of an AC system powered by one generator and a model of a DC system powered by three generators are built. They are reduced and the consequent errors are predicted. The simulation results demonstrate the effectiveness and accuracy of the proposed method.
康军, 马凡, 胡健, 孙文, 熊又星. 一种多时间尺度线性系统模型降阶的误差预测方法[J]. 电工技术学报, 2017, 32(3): 56-64.
Kang Jun, Ma Fan, Hu Jian, Sun Wen, Xiong Youxing. An Error Prediction Method of Model Order Reduction for Multi-Time Scale Linear System. Transactions of China Electrotechnical Society, 2017, 32(3): 56-64.
[1] 王阳, 鲁宗相, 闵勇, 等. 基于降阶模型的多电源微电网小干扰分析[J]. 电工技术学报, 2012,27(1):1-8. Wang Yang, Lu Zongxiang, Min Yong, et al. Small-singanl analysis of micro-grid with multiple micro soures based on reduced order model in islang operation[J]. Transactions of China Electrotechnical Society, 2012,27(1):1-8. [2] 纪锋, 王公宝, 付立军, 等. 十二相同步发电机降阶等效模型研究[J]. 电力系统保护与控制, 2012, 40(2):28-33. Ji Feng, Wang Gongbao, Fu Lijun, et al. Study on equivalent reduced model of twelve-phase synchronous generator[J].Power System Protection and Control, 2012, 40(2):28-33. [3] 李奎奎, 王克文, 邱磊, 等. 大系统特定模式动态降阶方法的误差分析[J]. 电力系统保护与控制, 2011, 39(1):67-71. Li Kuikui, Wang Kewen, Qiu Lei, et al. Error analysis of the dynamic order reduction approach for specifie model in large-scale systems[J].Power System Protection and Control, 2011, 39(1):67-71. [4] 罗建, 李阎君, 王一夫, 等. 一种基于PMU降阶的系统受扰轨迹预测方法[J]. 电力系统保护与控制, 2012,40(17):41-45. Luo Jian, Li Yanjun, Wang Yifu, et al. A perturbed trajectory prediction based on the reduced order of PMU[J].Power System Protection and Control, 2012,40(17):41-45. [5] 马凡, 付立军, 叶志浩,等. 基于降阶模型的非线性系统稳定性快速评估[J]. 中国电机工程学报, 2013, 33(19):126-134. Ma Fan, Fu Lijun, Ye Zhihao, et al. Fast assessment of nonlinear system stability based on reduced model [J].Proceedings of the CSEE,2013, 33(19):126-134. [6] Sauer P W, Ahmed-Zaid S, Pai M A. Systematic inclusion of stator transient in reduced order synchronous machine models[J]. IEEE Transactions on Power Apparatus and Systems, 1984, 6(6): 1348-1354. [7] Sauer P W, Lagesse D J, Ahmed-Zaid S, et al. Reduced order modeling of interconnected multi-machine power systems using time scale decomposition[J].IEEE Transactions on Power Systems, 1987, 2(2): 310-319. [8] 邓歆, 张广明, 王德明, 等. 基于全阶模型的异步电机磁链观测收敛性分析与对策 [J]. 电工技术学报, 2015, 30(1): 61-71. Deng Xin, Zhang Guangming, Wang Deming, et al. Convergence analysis and corresponding strategy of full model based induction motor flux observation [J]. Transactions of China Electrotechnical Society, 2015, 30(1): 61-71. [9] 舒进, 都劲松, 王靖程, 等. 基于奇异摄动降阶的风电接入系统阻尼分析[J]. 电力系统保护与控制, 2014,42(19):18-25. Shu Jin, Du Jingsong, Wang Jingcheng, et al. System damping analysis after wind power integration using singular perturbation system reduction[J].Power System Protection and Control, 2014,42(19):18-25. [10]马凡, 马伟明, 付立军. 一种多时间尺度降阶原则及其在交直流电力系统中的应用[J]. 中国电机工程学报, 2009, 29(13): 41-47. Ma Fan, Ma Weiming, Fu Lijun. A multi-time scale order reduction principle and its application in AC/DC power system[J]. Proceedings of the CSEE,2009, 29(13): 41-47. [11]Sanjay L, Jerrold E M, Sonja G. A subspace approach to balanced truncation for model reduction of nonlinear control systems[J]. International Journal of Robust and Nonlinear Control, 2002, 12(6): 519-535. [12]胡明华, 胡寿松. 平衡降阶方法及其进展[J]. 南京航空学院学报, 1990, 22(4): 92-102. Hu Minghua, Hu Shousong. Balanced reduction method and its development[J]. Journal of Nanjing Aeronautical Institute, 1990, 22(4): 92-102. [13]张喆, 赵洪山, 李志为, 等. 平衡格莱姆方法在电力系统线性模型降阶中的应用[J]. 电工技术学报, 2013, 28(6):201-207. Zhang Zhe, Zhao Hongshan, Li Zhiwei, et al. Power system linear model reduction based on the balanced gramian method[J]. Transactions of China Electrotechnical Society, 2013, 28(6):201-207. [14]Tatsushi A, Yasuyuki F. Interpolation approach to hankel-norm model reduction for rational multi-input multi-output systems[J]. IEEE Transactions on Circuits and Systems, I:Fundamental Theory and Applications, 1996, 43(12): 987-995. [15]赵洪山, 宋国维, 江全元. 利用平衡理论进行电力系统模型降阶[J]. 电工技术学报, 2010,25(2):127-133. Zhao Hongshan,Song Guowei,Jiang Quanyuan. Reduction of power systems modle using balance realization method[J]. Transactions of China Electrotechnical Society, 2010,25(2):127-133. [16]Antoulas A C, Sorensen D C. Approximation of large-scale dynamical systems: an overview[J]. Applied Mathematics & Computer Science, 2001, 11(5): 1093-1121. [17]Grimme E J, Sorensen D C, Van Dooren P. Model reduction of state space systems via an implicitly restarted lanczos method[J]. Numerical Algorithms, 1996, 12(1): 1-31. [18]Antoulas A C, Sorensen D C. Projection methods for balanced model reduction[R]. Houston: Rice University, Technical Report ECE-CAAM Depts, 2001. [19]Sen S, Datta K B. Stability bounds of singularity perturbed systems[J]. IEEE Transactions on Automatic Control, 1993, 38(2): 302-304. [20]Xu X, Mathur R M, Jiang Jin, et al. Modeling of generators and their controls in power system simulations using singular perturbations[J]. IEEE Transactions on Power Systems, 1998, 13(1): 109-114. [21]王奔. 电力系统电压稳定性[M]. 北京: 电子工业出版社, 2008: 114-117. [22]杨明, 刘先忠. 矩阵论[M]. 武汉: 华中科技大学出版社, 2005: 145-146. [23]胡适耕.应用泛函分析[M].北京:科学出版社,2003: 10-11.
2017, Vol. 32 
