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ODE Transformations of Nonlinear DAE Power Systems

Mohamad H. Kazma, Ahmad F. Taha

TL;DR

Power-system dynamics are traditionally modeled as nonlinear DAEs coupling generator dynamics and network constraints. The paper presents two transformations to convert the nonlinear DAE model into nonlinear ODE-structured representations that preserve the algebraic variables: (i) an IFT-based exact reformulation and (ii) an approximate mu-augmented DAE. These methods enable leveraging the extensive ODE-based control and state-estimation theory for DAE-based power systems, without losing information in the algebraic constraints. Validation on WSCC 9-bus and ACTIVSg200-bus networks under load disturbances shows accurate transient trajectories with distinct trade-offs: the IFT-based method incurs higher computational cost due to Hessian-like terms, while the Approx-DAE offers tunable stiffness via $\mu$ and good fidelity for small $\mu$.

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.

ODE Transformations of Nonlinear DAE Power Systems

TL;DR

Power-system dynamics are traditionally modeled as nonlinear DAEs coupling generator dynamics and network constraints. The paper presents two transformations to convert the nonlinear DAE model into nonlinear ODE-structured representations that preserve the algebraic variables: (i) an IFT-based exact reformulation and (ii) an approximate mu-augmented DAE. These methods enable leveraging the extensive ODE-based control and state-estimation theory for DAE-based power systems, without losing information in the algebraic constraints. Validation on WSCC 9-bus and ACTIVSg200-bus networks under load disturbances shows accurate transient trajectories with distinct trade-offs: the IFT-based method incurs higher computational cost due to Hessian-like terms, while the Approx-DAE offers tunable stiffness via and good fidelity for small .

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.
Paper Structure (7 sections, 1 theorem, 11 equations, 5 figures, 2 tables)

This paper contains 7 sections, 1 theorem, 11 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Proposed Nonlinear DAE Transformations
  3. IFT-Based Nonlinear DAE Model: $\mathrm{ODE}$-$\mathrm{DAE}$
  4. Approximate Nonlinear DAE Model: $\mathrm{Approx}$-$\mathrm{DAE}$
  5. Time-domain Modeling for Transient Stability Analysis
  6. Case Studies
  7. Summary and Future Work

Key Result

Proposition 1

Consider the nonlinear descriptor system eq:semi_NDAE_rep and that Assumption assump:index holds true for any initial condition and control input, then the descriptor system can be restructured into a nonlinear ODE system by applying the IFT Krantz2013 to resolve the algebraic constraints into ODEs.

Figures (5)

  • Figure 1: Transient differential and algebraic state trajectories under load disturbance: (a) $\mathrm{case}$-$\mathrm{9}$ ($\alpha_{L} = 2\%$) and (b) $\mathrm{case}$-$\mathrm{200}$ ($\alpha_{L} = 15\%$).
  • Figure 2: RMSE on dynamic and algebraic state trajectories for the $\mathrm{Approx}$-$\mathrm{DAE}$ model while changing $\mu$-value.
  • Figure 3: Error norm on discrete-time domain state trajectories---under load disturbance---for the proposed model simulated under different discretization methods: (a) $\mathrm{case}$-$\mathrm{9}$ ($\alpha_{L} = 2\%$) and (b) $\mathrm{case}$-$\mathrm{200}$ ($\alpha_{L} = 15\%$).
  • Figure 4: RMSE on transient dynamic and algebraic states under varying load disturbances.
  • Figure 5: Computational time for the different time-domain methods and power system models.

Theorems & Definitions (6)

  • Definition 1
  • Definition 2
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2