AC Dynamics-aware Trajectory Optimization with Binary Enforcement for Adaptive UFLS Design

TL;DR

This work addresses the need for AC dynamics–feasible adaptive UFLS under high DER backfeed by formulating UFLS as a trajectory optimization problem that includes full nonlinear network dynamics. It introduces a homotopy-driven approach that relaxes binary switching into continuous surrogates and solves a sequence of NLPs, guiding the solution toward near-integer feasibility while preserving AC feasibility. The framework supports multi-stage UFLS with sequential shedding actions and yields practical relay settings through post-processing. Demonstrations on multiple synthetic networks up to 1648 nodes show scalable computation, near-perfect binary feasibility, and effective arrest of frequency decline under worst-case disturbances, highlighting potential for real-world deployment in DER-rich grids.

Abstract

The high penetration of distributed energy resources, resulting in backfeed of power at the transmission and distribution interface, is causing conventional underfrequency load shedding (UFLS) schemes to become nonconforming. Adaptive schemes that update UFLS relay settings recursively in time offer a solution, but existing adaptive techniques that obtain UFLS relay settings with linearized or reduced-order model formulations fail to capture AC nonlinear network behavior. In practice, this will result in relays unable to restore system frequency during adverse disturbances. We formulate an adaptive UFLS problem as a trajectory optimization and include the full AC nonlinear network dynamics to ensure AC feasibility and time-coordinated control actions. We include binary decisions to model relay switching action and time-delayed multi-stage load-shedding. However, this formulation results in an intractable MINLP problem. To enforce model tractability, we relax these binary variables into continuous surrogates and reformulate the MINLP as a sequence of NLPs. We solve the NLPs with a homotopy-driven method that enforces near-integer-feasible solutions. We evaluate the framework on multiple synthetic transmission systems and demonstrate that it scales efficiently to networks exceeding 1500+ nodes with over 170k+ continuous and 73k+ binary decision variables, while successfully recovering binary-feasible solutions that arrest the frequency decline during worst-case disturbance.

Figures (7)

• Figure 1: Illustration of our adaptive UFLS design trajectory optimization framework applied to the WSCC 9-bus system (A) Trapezoidal direct collocation discretizes the nonlinear AC network dynamics with binary decision variables ($z$). (B) The reformulation converts the MINLP $(\mathbf{P}_1)$ into a sequence of NLPs $(\mathbf{P}_3^1, \mathbf{P}_3^2, \ldots, \mathbf{P}_3^\text{n})$ solved through successive homotopy steps (C) A 25% generator-loss disturbance at $t=1$s produces binary switch trajectories (moving from near 0.5 to binary {0, 1}) across the homotopy steps, with the final step yielding near-integer feasibility and a frequency response within acceptable bounds.
• Figure 2: Validation of the AC dynamic model against PSS/E using the savnw test case, with a 25% generator outage introduced at 1 second.
• Figure 3: (a) Contour plot for original problem $\mathbf{P}_3$ objective with quadratic penalty (b) Perturbed convex objective after introducing the barrier term with the value of $\mu > \gamma$ (c) Solution trajectory with the decreasing values of the homotopy parameter $\mu$. As $\mu \to 0$, the function's minimum is driven to a vertex of the original feasible region, yielding an integer feasible solution.
• Figure 4: Frequency responses at all buses of the savnw system, showing that optimal load shedding satisfies the frequency bound with $\underline{\Delta \omega}$ set to 59.0 Hz.
• Figure 5: Near Integer feasible solution showing stage activations for loads at Bus 153 and Bus 205 from the savnw test case.

Theorems & Definitions (2)

• Remark 1
• Remark 2