Homotopy-Based Re-Initialization for Switched DAEs in Power System Transient Simulation

Technical Summary: Homotopy-Based Re-Initialization for Switched DAEs in Power System Transient Simulation

Problem Statement
Power system transient simulation is formulated as the numerical solution of switched differential-algebraic equations (DAEs). During discontinuous events—such as breaker operations, faults, or controller limiting—the system mode ($\sigma$) changes, replacing one set of continuous equations with another. While a valid post-event solution often exists, the simultaneous solution of these DAEs frequently suffers from convergence loss in the nonlinear solver (typically Newton-Raphson, NR).

Standard remedies, such as reducing the integration step size or performing algebraic re-initialization (solving algebraic equations with fixed differential variables), often fail. The paper argues that this failure is not merely a numerical artifact of discretization error but a fundamental geometric inconsistency. When a mode transition occurs, the pre-event state, which lies on the pre-event solution manifold ($M_{\sigma-}$), becomes inconsistent with the post-event manifold ($M_{\sigma+}$). Because the DAE vector field is tangent to the solution manifold, no valid continuous trajectory exists to connect an inconsistent point to the new manifold. Consequently, the NR method, initialized at this inconsistent point, attempts to resolve an impulsive transient (where algebraic variables must change instantaneously), leading to poor conditioning, stagnation, or divergence.

Methodology
The paper proposes a homotopy-continuation (HC)-based globalized re-initialization scheme to restore convergence. The methodology proceeds as follows:

1. Geometric Interpretation: The authors interpret the algebraic equations $g_\sigma(x, y) = 0$ as defining constraint manifolds in the state space. A mode transition shifts the solution from manifold $M_{\sigma-}$ to $M_{\sigma+}$. The core difficulty is that the inherited pre-event point $z_s^-$ is not on $M_{\sigma+}$.
2. Homotopy Construction: Instead of attempting a direct "one-shot" solve for the post-event algebraic equations (which may lie outside the NR basin of attraction), the method constructs a continuous path $\gamma: [0, 1] \to X$ connecting the pre-event solution to the post-event solution.
3. Continuation Process: A homotopy function $H(z; \lambda) = 0$ is defined, where $\lambda \in [0, 1]$ is a continuation parameter.
   • At $\lambda = 0$, the system $H(z; 0)$ is an auxiliary problem (e.g., the pre-event algebraic system or a modified post-event system) for which the inherited point $z_s^-$ is an exact solution.
   • At $\lambda = 1$, the system $H(z; 1)$ corresponds to the target post-event algebraic equations $g_{\sigma+}(z) = 0$.
4. Iterative Solution: The parameter $\lambda$ is incremented in steps ($\Delta \lambda$). At each step, the nonlinear system is solved using the converged solution from the previous step as the initial guess. This sequence of nearby sub-problems ensures the solver remains within the basin of attraction, effectively tracing a path from the inconsistent pre-event state to a consistent post-event state.
5. Re-Initialization: Once $\lambda = 1$ is reached, the resulting point $z_s^+$ is consistent with the new mode. This point is then used to initialize the first post-event nonlinear solve for the time-stepping integration.

Key Contributions

• Geometric Framework: The paper provides a geometric explanation for convergence failure in switched DAEs, identifying the root cause as the loss of consistency between the inherited state and the post-event constraint manifold, rather than simple discretization error.
• Globalized Re-Initialization: It develops a homotopy-continuation scheme specifically for in-simulation re-initialization at mode transitions. Unlike prior homotopy applications that connect an auxiliary problem to a single target system (e.g., for cold-start initialization), this method connects two distinct manifolds ($M_{\sigma-}$ to $M_{\sigma+}$) through a family of intermediate manifolds.
• Robustness: The scheme extends Brown's single-step re-initialization by adding a globalization mechanism that guarantees a valid starting iterate regardless of the distance between the pre- and post-event manifolds.

Results
The proposed method was validated through two case studies on power system transient simulations:

1. Control Mode Switching: In a grid-forming (GFM) converter transitioning from voltage-source mode to current-limiting mode, direct NR iterations failed to converge, with residuals stagnating. The HC-based scheme successfully traced the solution path through intermediate manifolds, allowing the solver to converge to the desired post-event solution.
2. Topology Change: In a modified IEEE 39-bus system with a simulated line-to-ground fault, reducing the integration step size by an order of magnitude (from 500 $\mu$s to 50 $\mu$s) failed to restore convergence. The NR solver stagnated due to manifold inconsistency. The HC-based re-initialization computed a consistent post-event point, enabling the subsequent nonlinear solve to converge in just three iterations with a significantly reduced initial residual.

Significance and Claims
The paper claims that the primary significance of this work lies in addressing a specific class of convergence failures where standard remedies (step-size reduction and direct re-initialization) are insufficient. By interpreting the problem geometrically, the authors demonstrate that the difficulty stems from the non-existence of a continuous trajectory from an inconsistent point to the new manifold. The proposed homotopy-based scheme offers a robust mechanism to recover convergence in these scenarios, ensuring that transient simulations can proceed reliably even when discontinuous events cause the system state to jump between incompatible solution manifolds. The work is presented as a "letter" offering a specific framework and validation, without claiming universal applicability to all types of power system nonlinearities or future extensions beyond the demonstrated cases.