A self-adaptive memetic algorithm with Lyapunov damping for stochastic OPF: evidence from South Africa's grid crisis

Abstract

This study presents Lyapunov-SQP-MA (Lyapunov-guided Sequential Quadratic Programming Memetic Algorithm), a self-adaptive memetic algorithm that embeds an online stability estimator directly into the local Sequential Quadratic Programming (SQP) solver of a hybrid evolutionary optimizer. The central innovation is a closed-loop framework in which a finite-time local divergence rate ρ ^ k , computed via parallel shadow trajectory propagation, simultaneously controls the Levenberg–Marquardt damping parameter and validates the Damping–Stability Coupling Theorem in real-time. The theorem shows that the maximal Lyapunov exponent of the deterministic optimization skeleton increases (toward zero from below) with damping in the deterministic case, whereas the stochastic noise-amplification term strictly decreases with damping. Above a critical noise level, the full spectral bound decreases with damping, confirming that Lyapunov-guided damping provides a net stability benefit precisely when stochastic perturbations are large, and that the sufficient stabilizing threshold rises with noise amplitude. The online estimator operationalizes this result: its output drives a tanh-saturating feedback controller that self-adjusts regularization without manual re-tuning, and its observed behavior confirms the predicted coupling between detected instability and the need for increased damping across 35,064 hourly South African grid scenarios (2022–2025) spanning Eskom Stages 0–6. On IEEE 30-bus, 118-bus, and 300-bus benchmark systems, Lyapunov-SQP-MA achieves cost reductions of 0.20%–10.57%, 100% convergence success, and 39% variance reduction over the standard baseline. The monotone scaling of improvement with uncertainty level directly validates the Damping–Stability Coupling Theorem; ablation analysis confirms that the estimator is the decisive mechanism; and parameter sensitivity shows full robustness across a 3 × 3 controller grid without tuning. These results establish Lyapunov-guided adaptive damping as a principled mathematical framework for self-regulating optimization under stochastic perturbations, with direct applicability to large-scale energy dispatch and non-convex stochastic programming broadly.