ODE Transformations of Nonlinear DAE Power Systems
Abstract: Dynamic power system models are instrumental in real-time stability, monitoring, and control. Such models are traditionally posed as systems of nonlinear differential algebraic equations (DAEs): the dynamical part models generator transients and the algebraic one captures network power flow. While the literature on control and monitoring for ordinary differential equation (ODE) models of power systems is indeed rich, that on DAE systems is \textit{not}. DAE system theory is less understood in the context of power system dynamics. To that end, this letter presents two new mathematical transformations for nonlinear DAE models that yield nonlinear ODE models whilst retaining the complete nonlinear DAE structure and algebraic variables. Such transformations make (more accurate) power system DAE models more amenable to a host of control and state estimation algorithms designed for ODE dynamical systems. We showcase that the proposed models are effective, simple, and computationally scalable.
- L. Aolaritei, D. Lee, T. L. Vu, and K. Turitsyn, “A Robustness Measure of Transient Stability under Operational Constraints in Power Systems,” IEEE Control Systems Letters, vol. 2, no. 4, pp. 803–808, 2018.
- W. Cui and B. Zhang, “Equilibrium-Independent Stability Analysis for Distribution Systems with Lossy Transmission Lines,” IEEE Control Systems Letters, vol. 6, pp. 3349–3354, 2022.
- F. Milano, “Semi-Implicit Formulation of Differential-Algebraic Equations for Transient Stability Analysis,” IEEE Transactions on Power Systems, vol. 31, no. 6, pp. 4534–4543, 2016.
- ——, “On Current and Power Injection Models for Angle and Voltage Stability Analysis of Power Systems,” IEEE Transactions on Power Systems, vol. 31, no. 3, pp. 2503–2504, 2016.
- J. Y. Astic, M. Jerosolimski, and A. Bihain, “The mixed adams - BDF variable step size algorithm to simulate transient and long term phenomena in power systems,” IEEE Transactions on Power Systems, vol. 9, no. 2, pp. 929–935, 1994.
- R. K. Brayton, F. G. Gustavson, and G. D. Hachtel, “Differential-Algebraic Systems Using,” vol. 60, no. 1, pp. 98–108, 1972.
- C. W. Gear, “Simultaneous Numerical Solution of Differential-Algebraic Equations,” IEEE Transactions on Circuit Theory, vol. 18, no. 1, pp. 89–95, 1971.
- F. A. Potra, M. Anitescu, B. Gavrea, and J. Trinkle, “A linearly implicit trapezoidal method for integrating stiff multibody dynamics with contact, joints, and friction,” International Journal for Numerical Methods in Engineering, vol. 66, no. 7, pp. 1079–1124, 2006.
- J. D. Lara, R. Henriquez-Auba, D. Ramasubramanian, S. Dhople, D. S. Callaway, and S. Sanders, “Revisiting Power Systems Time-domain Simulation Methods and Models,” arXiv, pp. 1–11, 2023.
- T. Groß, S. Trenn, and A. Wirsen, “Solvability and stability of a power system DAE model,” Systems and Control Letters, vol. 97, pp. 12–17, 2016.
- S. A. Nugroho, A. Taha, N. Gatsis, and J. Zhao, “Observers for Differential Algebraic Equation Models of Power Networks: Jointly Estimating Dynamic and Algebraic States,” IEEE Transactions on Control of Network Systems, vol. 5870, no. c, 2022.
- J. Jaiswal and N. K. Tomar, “Existence Conditions for ODE Functional Observer Design of Descriptor Systems,” IEEE Control Systems Letters, vol. 6, no. 2, pp. 355–360, 2022.
- P. Kunkel and V. Mehrmann, “Optimal control for unstructured nonlinear differential-algebraic equations of arbitrary index,” Mathematics of Control, Signals, and Systems, vol. 20, no. 3, pp. 227–269, 2008.
- M. Althoff and B. H. Krogh, “Reachability analysis of nonlinear differential-algebraic systems,” IEEE Transactions on Automatic Control, vol. 59, no. 2, pp. 371–383, 2014.
- Y. Liu and K. Sun, “Solving Power System Differential Algebraic Equations Using Differential Transformation,” IEEE Transactions on Power Systems, vol. 35, no. 3, pp. 2289–2299, may 2020.
- M. Nadeem and A. F. Taha, “Robust Dynamic State Estimation of Multi-Machine Power Networks with Solar Farms and Dynamics Loads,” Proceedings of the IEEE Conference on Decision and Control, vol. 2022-Decem, no. Cdc, pp. 7174–7179, 2022.
- S. L. Campbell and C. W. Gear, “The index of general nonlinear DAEs,” Numerische Mathematik, vol. 72, no. 2, pp. 173–196, 1995.
- Y. Chen and S. Trenn, “The differentiation index of nonlinear differential‐algebraic equations versus the relative degree of nonlinear control systems,” Pamm, vol. 20, no. 1, pp. 2020–2022, 2021.
- J. H. Chow, J. R. Winkelman, M. A. Pai, and P. W. Sauer, “Singular perturbation analysis of large-scale power systems,” International Journal of Electrical Power and Energy Systems, vol. 12, no. 2, pp. 117–126, 1990.
- N. H. McClamroch, “On the Connection Between Nonlinear Differential-Algebraic Equations and Singularly Perturbed Control Systems in Nonstandard Form,” IEEE Transactions on Automatic Control, vol. 39, no. 5, pp. 1079–1084, 1994.
- F. Shen, P. Ju, M. Shahidehpour, Z. Li, C. Wang, and X. Shi, “Singular Perturbation for the Dynamic Modeling of Integrated Energy Systems,” IEEE Transactions on Power Systems, vol. 35, no. 3, pp. 1718–1728, 2020.
- S. P. Venkateshan and P. Swaminathan, “Chapter 8 - Numerical Differentiation,” in Computational Methods in Engineering, S. P. Venkateshan and P. Swaminathan, Eds. Boston: Academic Press, 2014, pp. 295–315.
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