Mathematical modeling of energy mix and optimization of renewable resources

Abstract

Energy, as both a direct and indirect fundamental life-supporting resource, has experienced a steady rise in domestic and industrial demand, driven by technological advancement, population growth, and economic expansion. Various sources of energy including fossil fuels, hydroelectric power, geothermal energy, wind, solar, and nuclear are available in different proportions, each with distinct cost structures and environmental impacts. The challenge of meeting these diverse needs while minimizing production and distribution costs, conserving the environment, and reducing wastage has evolved into a complex multi-objective problem. This research focuses on the mathematical modelling of the optimal energy mix and the optimization of renewable resources, with particular emphasis on individualized demand profiles. The objectives are threefold: first, to formulate a mathematical model for analysing the dynamics of energy demand, production, and distribution; second, to determine the parameter thresholds that guarantee stability and robustness of the optimal energy mix; and third, to develop a smart grid feedback model using adaptive neural networks capable of automatically maintaining the desired energy balance. The methodology entails formulating a system of differential equations to represent the energy system, expressing it in state-space form, and applying Laplace transforms to derive transfer functions. These will be analysed for sensitivity, stability, and robustness using Nyquist and Bode plot criteria. MATLAB–Simulink, equipped with neural network modules, will then be employed to simulate and implement an intelligent, adaptive feedback control system. Through these simulations, the study will integrate real-time learning and self-adjustment capabilities to align production with demand in the most efficient manner. The anticipated outcome is an automated, smart distribution system capable of dynamically meeting individualized energy requirements at the lowest possible cost, while enhancing the utilization of renewable sources and reducing reliance on non-renewable options. Ultimately, this approach aims to promote environmental sustainability through increased adoption of green energy technologies.