Mathematical Theory of Multi-Objective Evolutionary Algorithms with Convergence Guarantees for Complex Optimisation in Nigerian Energy Systems

This dissertation develops original mathematical theory underpinning the convergence properties of multi-objective evolutionary algorithms, establishing rigorous convergence guarantees for a new class of decomposition-based algorithms and applying the resulting framework to complex multi-objective optimisation problems in the planning and operation of Nigerian energy systems. Multi-objective evolutionary algorithms have achieved widespread engineering application, yet their theoretical foundations, particularly rigorous convergence analysis in the mathematical analysis sense, have remained substantially underdeveloped, creating a critical gap between algorithmic practice and theoretical understanding. The dissertation introduces a measure-theoretic framework for analysing the population dynamics of evolutionary algorithms operating on infinite-dimensional phenotype spaces, proving novel ergodic convergence theorems for self-adaptive operators under assumptions on the fitness landscape topology. A new decomposition-based multi-objective algorithm with provably convergent scalarisation sequence is proposed, and convergence to the true Pareto front in the Hausdorff metric sense is established under regularity conditions on the objective function mappings that are verified for the target energy system applications. The theoretical framework is applied to the joint optimisation of generation expansion planning, transmission network reinforcement, and renewable energy siting for the Nigerian electricity grid over a 20-year planning horizon, using a three-objective model minimising total cost, carbon emissions, and energy access deficit simultaneously. Pareto front approximations are computed and validated against established multi-objective benchmarks. Keywords: multi-objective optimisation, evolutionary algorithms, convergence theory, energy systems planning, Pareto optimality