Wide-band Modeling Method Based on the Fractional Order Differential Theory
Liu Xin1, 2, Cui Xiang1, Liang Guishu2, Ma Long2
1. State Key Laboratory of Alternate Electrical Power System with Renewable Energy Source North China Electric Power University Beijing 102206 China 2. North China Electric Power University Baoding 071003 China
Abstract：According to the problems of electromagnetic, overvoltage protection and insulation coordination in the power system, communication system and railway power system, et. al., the electric equipment, transmission line and grounding system, et. al. should be modeled considering their frequency dependent characteristic. The vector fitting method is a general method to be used to approximate the network parameter by the rational function, however, this method does not fully consider the fractional order characteristic of the frequency dependent effect, which will result in a relative complicate model. Based on the fractional order differential theory, this paper presents a general wide-band modeling method. In this method, the Levy’s identification method is used to approximate the measured or calculated frequency domain network function with the irrational function, in which the powers of the Laplace operator s is non-integer and the corresponding fractional order differential equation can be obtained by the Laplace inverse transformation. Combining with the numeric solution to the fractional order differential equation, a time domain simulation can be implemented. Considering the indispensability of the passivity verification of a system for the transient simulation, the passivity verification method for the traditional state-space system is extended to the fractional order state-space system and a practical criterion is proposed in this paper.
刘欣, 崔翔, 梁贵书, 马龙. 基于分数阶微分理论的宽频建模方法[J]. 电工技术学报, 2013, 28(4): 20-27.
Liu Xin, Cui Xiang, Liang Guishu, Ma Long. Wide-band Modeling Method Based on the Fractional Order Differential Theory. Transactions of China Electrotechnical Society, 2013, 28(4): 20-27.
[1] 梁贵书, 张喜乐, 王晓晖, 等. 特快速暂态过电压下变压器绕组高频电路模型的研究[J]. 中国电机工程学报, 2006, 26(4): 144-148. [2] 吴茂林, 崔翔. 电压互感器宽频特性的建模[J]. 中国电机工程学报, 2003, 23(10): 1-5. [3] Zhang Z, Lü F, Liang G. A high-frequency circuit model of a potential transformer for the very fast transient simulation in GIS[J]. IEEE Transactions on Power Delivery, 2008, 23(4): 1995-1999. [4] 张重远, 律方成, 梁贵书. 一种基于散射参数的电压互感器二端口高频电路模型[J]. 中国电机工程学报, 2007, 27(27): 39-43. [5] Gustavsen B. Wide band modeling of power transformers[J]. IEEE Transactions on Power Delivery, 2004, 19(1): 414-422. [6] Gustavsen B, Semlyen A. Rational approximation of frequency domain responses by vector fitting[J]. IEEE Transactions on Power Delivery, 1999, 14(3): 1052- 1061. [7] Podlubny I. Fractional differential equations[M]. New York: Academic Press, 1999. [8] Kilbas, Anatoly A, Srivastava, Hari M, et al. Theory and applications of fractional differential equations [M]. Amsterdam and Boston: Elsevier Press, 2006. [9] Ortigueira M. An introduction to the fractional continuous-time linear systems: the 21st Century Systems[J]. IEEE Transactions on Circuits and Systems Magazine, 2008, 10(3): 19-26. [10] Dalir M, Bashour M. Applications of fractional calculus[J]. Applied Mathematical Sciences, 2010, 4(21): 1021-1032. [11] Elwakil A. Fractional-order circuits and systems: an emerging interdisciplinary research area[J]. IEEE Transactions on Circuits and Systems Magazine, 2010, 10(4): 40-50. [12] Brune O. Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency[J]. Journal of Math and Physics, 1931, 10: 191-236. [13] Achar R, Gunupudi P, Nakhla M, et al. Passive interconnect reduction algorithm for distributed/ measured networks[J]. IEEE Transactions on Circuits and Systems II. Analog and Digital Signal Process, 2000, 47(4): 287-301. [14] Saraswat D, Achar R, Nakhla M. Global passivity enforcement algorithm for macromodels of interconnect subnetworks characterized by Tabulated Data[J]. IEEE Transactions on Very Large Scale Integration System, 2005, 13(7): 819-832. [15] Levy E C. Complex curve fitting[J]. IRE Transactions on Automatic Control, 1959, 4: 37-44. [16] Wing, Omar. Classical circuit theory[M]. New York: Springer Press, 2008.
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