Energy systems (e.g. ventilation fans, refrigerators, and electrical vehicle chargers) often have binary or discrete states due to hardware limitations and efficiency characteristics. Typically, such systems have additional programmatic constraints, such as minimum dwell times to prevent short cycling. As a result, non-convex techniques, like dynamic programming, are generally required for optimization. Recognizing developments in the field of distributed convex optimization and the potential for energy systems to participate in ancillary power system services, it is advantageous to develop convex techniques for the approximate optimization of energy systems. In this manuscript, we develop the alternative control trajectory representation — a novel approach for representing the control of a non-convex discrete system as a convex program. The resulting convex program provides a solution that can be interpreted stochastically for implementation.

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