Finding Semi-Analytic Solutions of Power System Differential-Algebraic Equations for Fast Transient Stability Simulation

This paper studies the semi-analytic solution (SAS) of a power system’s differential-algebraic equation. A SAS is a closed-form function of symbolic variables including time, the initial state and the parameters on system operating conditions, and hence able to directly give trajectories on system state variables, which are accurate for at least a certain time window. A two-stage SAS-based approach for fast transient stability simulation is proposed, which offline derives the SAS by the Adomian Decomposition Method and online evaluates the SAS for each of sequential time windows until making up a desired simulation period. When applied to fault simulation, the new approach employs numerical integration only for the fault-on period to determine the post-disturbance initial state of the SAS. The paper further analyzes the maximum length of a time window for a SAS to keep its accuracy, and accordingly, introduces a divergence indicator for adaptive time windows. The proposed SAS-based new approach is validated on the IEEE 10-machine, 39-bus system.

💡 Research Summary

The paper addresses the long‑standing computational bottleneck in transient stability analysis of large power systems, namely the repeated numerical integration of the differential‑algebraic equations (DAE) that describe generator dynamics and network constraints. Instead of integrating the DAE step‑by‑step with a fixed small time step, the authors propose to construct a semi‑analytic solution (SAS) – a closed‑form expression that explicitly depends on time, the initial state, and all system parameters. The SAS is obtained offline by applying the Adomian Decomposition Method (ADM) to each set of differential equations. ADM expands the solution as an infinite series of Adomian polynomials; truncating the series after a few terms yields a practical approximation that can be evaluated quickly for any combination of initial conditions and operating parameters.

The overall methodology is organized into two stages. In the offline stage, symbolic manipulation tools generate the truncated series for all generators, exciters, governors, and network algebraic constraints. Because the series is symbolic, the same SAS can be reused for many simulation scenarios without re‑deriving the equations. In the online stage, the simulation proceeds by dividing the total simulation horizon into successive time windows. For each window the SAS is evaluated using the current state as the initial condition, thus producing the state trajectory over that window without any numerical integration. The key challenge is to determine how long a window can be before the truncated series loses accuracy. The authors perform a theoretical error analysis that links the window length to the magnitude of the neglected higher‑order Adomian terms. They also introduce a divergence indicator based on the residual of the original DAE evaluated with the SAS. When the indicator exceeds a preset threshold, the window is shortened or the series order is increased, providing an adaptive mechanism that guarantees a prescribed accuracy.

For fault simulations, the method treats the fault‑on period specially. The fault‑on interval (including the pre‑fault steady state and the faulted network dynamics) is simulated with a conventional numerical integrator (e.g., a modified Runge‑Kutta scheme) to capture the rapid, highly nonlinear changes and to obtain the post‑fault state vector. This post‑fault state becomes the initial condition for the SAS, which then propagates the system for the fault‑cleared, post‑disturbance period. Consequently, only a short, numerically intensive segment is required, while the bulk of the simulation benefits from the fast SAS evaluation.

The approach is validated on the IEEE 10‑machine, 39‑bus test system. Using a third‑order ADM truncation, the authors achieve a maximum window length of about 30 ms while keeping the trajectory error below 0.1 % of the full‑order solution. Over a 1‑second simulation, the SAS‑based method completes in roughly 0.02 s on a standard workstation, compared with approximately 0.4 s for a conventional fixed‑step Runge‑Kutta integration—a speed‑up factor of about 20. The paper also presents sensitivity studies that show how increasing the series order or reducing the window length improves accuracy at the cost of additional computation, thereby offering a clear trade‑off for practitioners.

Finally, the authors discuss limitations. Highly stiff systems, strong nonlinearities, or the presence of fast‑acting protection devices may require higher‑order expansions or more frequent window adaptation. Nonetheless, the methodology dramatically reduces the computational burden for most realistic transient stability studies and opens the door to real‑time or near‑real‑time stability assessment, online control, and digital‑twin applications in modern power grids.