Optimal Scheduling of Thermostatically Controlled and Energy Constrained Electric Loads Under Real Time Pricing

In this work, we present an optimization framework for scheduling two distinct types of electrical loads under real-time pricing conditions: Thermostatically Controlled Loads (TCLs) and Energy Constrained Time Shiftable Loads (TSLs). While traditional Mixed Integer Linear Programming (MILP) approaches provide optimal solutions, their computational complexity makes them impractical for real-time applications. We propose equivalent convex optimization reformulations that achieve identical optimal solutions, making them suitable for real-time implementation in resource-constrained edge devices. Numerical simulations are used to demonstrate that our convex formulations achieve the same optimal costs as MILP while significantly reducing computational overhead. We extend our analysis to include multi-objective optimization, exploring the trade-off between energy cost and peak power consumption through Pareto front analysis. This work offers practical, computationally efficient and scalable solutions for real-time load scheduling in smart grid applications.