基于任意代数精度的电路微分方程数值积分算法

doi:10.19595/j.cnki.1000-6753.tces.240572

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Abstract The circuit differential equation is the foundation of electrical engineering, spanning various fields such as circuit principle analysis, topology design, and power system engineering applications. However, with the increasing scale of the power system and the widespread application of high-frequency semiconductor switches, solving circuit differential equations quickly and accurately has become increasingly challenging. Numerical integration is one of the classic methods for solving circuit differential equations, but traditional numerical integration methods are fixed in form, with non-adjustable integration accuracy, and the relationship between accuracy, computation effort, and integration is not well-defined. Moreover, traditional indicators such as stability, accuracy, and calculation effort only allow for qualitative comparisons of numerical integration, making it difficult to accurately evaluate integration accuracy and computation efficiency. To address these problems, an explicit or implicit numerical integration algorithm based on arbitrary algebraic precision is proposed. Meanwhile, the computational effort per unit time is defined to supplement the traditional integral evaluation indicators.
Firstly, the construction method of explicit and implicit numerical integration is given. By independently selecting the differentiation items and their orders to construct a new type of integration, the structure of the integration is simplified, thereby enhancing the efficiency of the integration calculation. Secondly, a method for determining integral parameters based on arbitrary algebraic precision is proposed to improve integration accuracy, and the accuracy and stability of the proposed algorithm are proved. Finally, computational effort per unit time is defined to clarify the quantitative relationship between accuracy, computational effort, and numerical integration. And the patterns of variation with respect to the scale of circuit equations and factors like accuracy are investigated.
Using classic RLC circuits and a two-level converter as experimental objects, a real-time simulation hardware platform is built to verify the effectiveness of the proposed integration algorithm and evaluation indicator. The simulation results indicate that compared to low-order integrations, high-order integrations have an absolute advantage in accuracy. Increasing the integral order by one reduces the maximum absolute error by an order of magnitude. When the integral order is the same, compared to traditional high-order integration, the accuracy of the proposed integration algorithm can be improved by 33% to 72%, the computational effort can be reduced by 9% to 42%, and computational effort per unit time can be reduced by 16% to 61%. Furthermore, the proposed integration algorithm has higher stability and stronger ability to suppress numerical oscillations compared to traditional numerical integration.
The following conclusions can be drawn from the simulation analysis: (1) Compared to traditional numerical integration, the proposed integration structure is simpler and more computationally efficient. The parameter determination method based on algebraic precision makes the integration more accurate, more stable, and better at suppressing numerical oscillations. (2) The computational effort per unit time can simultaneously characterize the quantitative relationship between accuracy, calculation effort, and numerical integration, allowing for accurate assessment of integration performance. For the proposed numerical integration algorithm and evaluation indicator, the main aspects of application prospects and future research directions are as follows: (1) The proposed integration algorithm can be used in power system transient response analysis, device modeling and electromagnetic transient simulation. (2) The computational effort per unit time can be used to determine the discrete method in switch model and to select the integration algorithm in real-time simulation. (3) The structure and accuracy of variable-step numerical integration need to be further studied.

Key words： Circuit differential equation numerical integration algebraic precision calculation effort

Received: 11 April 2024

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TM133

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	Du Jinpeng
	Wang Kang
	Wang Guangsen
	Liu Zhu

Cite this article:

Du Jinpeng,Wang Kang,Wang Guangsen等. Numerical Integration Algorithm of Circuit Differential Equations Based on Arbitrary Algebraic Precision[J]. Transactions of China Electrotechnical Society, 2025, 40(7): 1995-2006.

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https://dgjsxb.ces-transaction.com/EN/10.19595/j.cnki.1000-6753.tces.240572 OR https://dgjsxb.ces-transaction.com/EN/Y2025/V40/I7/1995

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