A Convex Optimization Framework for Probabilistic Defense Against False Data Injection Attacks

We propose a novel, strictly convex optimization framework to defend power system state estimation against stealthy False Data Injection Attacks (FDIAs).Existing multi-level game-theoretic defenses are computationally intractable (NP-hard) for large-scale grids and overwhelmingly assume rigid binary protection.To address this issue, we model defense investment as a continuous variable that probabilistically decays attack success.By integrating an explicit attacker cardinality constraint and non-uniform economic security costs, we transform the defender-attacker interaction into a highly scalable Linear Program (LP) that mathematically guarantees global optimality.Our framework demonstrates exceptional computational efficiency validated on IEEE 14-bus, 57-bus, and 300-bus systems.Simulation results prove that strategically leveraging cost-effective assets yields up to a 60.0% improvement in the optimal security level for the same budget compared to traditional topology-only baselines and up to a 35.6% enhancement over unconstrained attacker models.Comprehensive sensitivity analyses incorporating piecewise-linear effectiveness and correlated vulnerabilities confirm the model's robust adaptability, providing system operators with an actionable, economically efficient decision-support tool.