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.
杜金鹏, 王康, 汪光森, 刘著. 基于任意代数精度的电路微分方程数值积分算法[J]. 电工技术学报, 2025, 40(7): 1995-2006.
Du Jinpeng, Wang Kang, Wang Guangsen, Liu Zhu. Numerical Integration Algorithm of Circuit Differential Equations Based on Arbitrary Algebraic Precision. Transactions of China Electrotechnical Society, 2025, 40(7): 1995-2006.
[1] Wunderlich A S, Santi E.Closed-form implicit models for efficient simulation of power electronics[J]. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2023, 11(2): 1568-1577. [2] 易新强, 王东, 刘海涛, 等. 考虑原动机调速特性的中压直流发电机组动态仿真模型[J]. 电工技术学报, 2024, 39(10): 2974-2983. Yi Xinqiang, Wang Dong, Liu Haitao, et al.Dynamic simulation model of MVDC generating set consi-dering the speed regulation characteristics of prime movers[J]. Transactions of China Electro-technical Society, 2024, 39(10): 2974-2983. [3] 虞竹珺, 赵争鸣, 施博辰, 等. 基于刚性检测的电力电子系统自适应仿真算法[J]. 电工技术学报, 2023, 38(12): 3208-3220, 3247. Yu Zhujun, Zhao Zhengming, Shi Bochen, et al.An adaptive simulation method for power electronics systems based on stiffness detection[J]. Transactions of China Electrotechnical Society, 2023, 38(12): 3208-3220, 3247. [4] Bai Hao, Huang Gang, Liu Chen, et al.A controller HIL testing approach of high switching frequency power converter via slower-than-real-time simulation[J]. IEEE Transactions on Industrial Electronics, 2024, 71(8): 8690-8702. [5] Chalangar H, Ould-Bachir T, Sheshyekani K, et al.Methods for the accurate real-time simulation of high-frequency power converters[J]. IEEE Transactions on Industrial Electronics, 2022, 69(9): 9613-9623. [6] 管乐诗, 程怡, 施震宇, 等. 一种10 MHz高频DC-DC功率变换器及其同步整流技术[J]. 电工技术学报, 2023, 38(18): 5029-5038. Guan Yueshi, Cheng Yi, Shi Zhenyu, et al.A 10 MHz high frequency DC-DC power converter and its synchronous rectification technology[J]. Transactions of China Electrotechnical Society, 2023, 38(18): 5029-5038. [7] Gray W S, Verriest E I.Linear operator differential equations with applications to linear time-varying systems[C]//2023 59th Annual Allerton Conference on Communication, Control, and Computing (Allerton), Monticello, IL, USA, 2023: 1-7. [8] Baiburin V B, Karabutov M V.Method for accelerated numerical solution of nonlinear differential equations[C]//2023 IEEE XVI International Scientific and Technical Conference Actual Problems of Electronic Instrument Engineering (APEIE), Novosibirsk, Russian Federation, 2023: 1190-1193. [9] 何绍民, 张喆, 卢倚平, 等. 基于计算前沿面的实时仿真数值积分并行构造及其数值模型解耦加速方法[J]. 电工技术学报, 2023, 38(16): 4246-4262. He Shaomin, Zhang Zhe, Lu Yiping, et al.Numerical model decoupling acceleration method with numerical integration parallelism construction based on com-putation front in real-time simulation[J]. Transactions of China Electrotechnical Society, 2023, 38(16): 4246-4262. [10] Watson N, Arrillaga J.Power systems electromagnetic transients simulation[M]. 2nd ed. Stevenage: IET, 2018. [11] Clitan I, Colosi T, Unguresan M L, et al.Prospects of algebraization of deformable exponential functions through ordinary differential equations and partial differential equations[C]//2020 IEEE International Conference on Automation, Quality and Testing, Robotics (AQTR), Cluj-Napoca, Romania, 2020: 1-5. [12] Tzounas G, Dassios I, Milano F.Small-signal stability analysis of numerical integration methods[J]. IEEE Transactions on Power Systems, 2022, 37(6): 4796-4806. [13] Bai Hao, Liu Chen, Majstorovic D, et al.Real-time simulation technology for modern power electronics [M]. Singapore: Academic Press, 2023. [14] Boldo S, Faissole F, Chapoutot A.Round-off error and exceptional behavior analysis of explicit Runge-Kutta methods[J]. IEEE Transactions on Computers, 2020, 69(12): 1745-1756. [15] Lebedka S M, Petrovskyi M V, Veprik Y N.Discrete models of the electric grid elements, obtained with using the second-order formula of implicit gear method[C]//2017 IEEE First Ukraine Conference on Electrical and Computer Engineering (UKRCON), Kyiv, UKraine, 2017: 584-588. [16] Song Xinli, Tang Yong, Zhong Wuzhi, et al.New mixed integral algorithm for unified dynamic power system simulations of transient, medium-term and long-term stabilities[C]//2011 IEEE/PES Power Sys-tems Conference and Exposition, Phoenix, AZ, USA, 2011: 1-7. [17] Yushkova M, Sanchez A, de Castro A. The necessity of resetting memory in Adams-bashforth method for real-time simulation of switching converters[J]. IEEE Transactions on Power Electronics, 2021, 36(6): 6175-6178. [18] Aghahassani M, Castronuovo E D, Ledesma P, et al.Application of linear multi-step methods to a transient stability constrained optimal power flow model[C]// 2020 IEEE International Conference on Environment and Electrical Engineering and 2020 IEEE Industrial and Commercial Power Systems Europe (EEEIC/I& CPS Europe), Madrid, Spain, 2020: 1-6. [19] Zhao Zhengming, Yuan Liqiang, Bai Hua, et al.Electromagnetic transients of power electronics sys-tems[M]. Singapore: Singapore Springer, 2019. [20] Veigend P, Nečasová G, Šátek V.Solving linear and nonlinear problems using Taylor series method[J]. Open Computer Science, 2024, 14(1): 302-309. [21] Wang Bin, Xu Xin, Sun Kai.Power system transient stability analysis using high-order Taylor expansion systems[C]//2019 IEEE Texas Power and Energy Conference (TPEC), College Station, TX, USA, 2019: 1-5. [22] Trotsenko Y, Brzhezitsky V, Protsenko O, et al.Calculation of high voltage divider accuracy using Duhamel’s integral[C]//2018 IEEE 17th International Conference on Mathematical Methods in Electro-magnetic Theory (MMET), Kyiv, UKraine, 2018: 213-216. [23] Sevastianov L A, Lovetskiy K P, Kulyabov D S.Multistage collocation pseudo-spectral method for the solution of the first order linear ODE[C]//2022 Ⅷ International Conference on Information Technology and Nanotechnology (ITNT), Samara, Russian Federation, 2022: 1-6. [24] Yaghoobnia A, Kazemi M.A collocation method for the numerical solution of a class of linear stochastic integral equations based on Legendre polynomials[C]//2022 Second International Conference on Distributed Computing and High Performance Computing (DCHPC), Qom, Iran, 2022: 26-30. [25] Conte D, D’Ambrosio R, D’Arienzo M P, et al. Singly diagonally implicit multivalue collocation methods[C]// 2020 International Conference on Mathematics and Computers in Science and Engineering (MACISE), Madrid, Spain, 2020: 65-68. [26] 周斌, 汪光森, 李卫超, 等. 基于FPGA的电力电子系统电磁暂态实时仿真通用解算器[J]. 电工技术学报, 2023, 38(14): 3862-3874. Zhou Bin, Wang Guangsen, Li Weichao, et al.An FPGA-based general solver for electromagnetic transient real-time simulation of power electronic systems[J]. Transactions of China Electrotechnical Society, 2023, 38(14): 3862-3874. [27] 付浩, 李鹏, 富晓鹏, 等. 面向多FPGA实时仿真器的资源优化配置方法[J]. 电力系统自动化, 2023, 47(11): 88-100. Fu Hao, Li Peng, Fu Xiaopeng, et al.Optimal resource allocation method for real-time simulator based on multiple field programmable gate arrays[J]. Automation of Electric Power Systems, 2023, 47(11): 88-100. [28] Boldo S, Faissole F, Chapoutot A.Round-off error analysis of explicit one-step numerical integration methods[C]//2017 IEEE 24th Symposium on Computer Arithmetic (ARITH), London, UK, 2017: 82-89. [29] 王钦盛, 王灿, 潘学伟, 等. 基于FPGA的电力电子恒导纳开关模型修正算法及实时仿真架构[J]. 电力系统自动化, 2024, 48(1): 150-159. Wang Qinsheng, Wang Can, Pan Xuewei, et al.Fixed-admittance switch model correction algorithm and real-time simulation architecture of power electronics based on field programmable gate array[J]. Auto-mation of Electric Power Systems, 2024, 48(1): 150-159.
2025, Vol. 40 
