Piecewise linear bounding of energy conversion functions and resulting MILP-based solution methods
Résumé
Different energy sources can have very different characteristics in terms of power range and energy demand/cost function also known as efficiency function or energy conversion function. Introducing these energy sources characteristics in combinatorial optimization problems such as energy resource allocation problems or energy-consuming activity scheduling problems results into mixed-integer non-linear problems neither convex or concave. Approximations via piecewise linear functions have been proposed in the literature. Non-convex optimization models and heuristics exist to compute optimal breakpoint systems subject to the condition that the piecewise linear continuous approximator (under-and overestimator) never deviates more than a given delta-tolerance from the original continuous separable function over a given finite interval, or to minimize the area between the approximator and the function. We present an alternative solution method based on the upper and lower bounding of energy conversion expressions using discontinuous piecewise linear functions with a relative epsilon-tolerance. We prove that such approach yields a pair of mixed integer linear programs with a performance guarantee. Models and heuristics to compute the discontinuous piecewise linear functions with a relative epsilon-tolerance will also be presented. Computational results have shown the efficiency of the method in comparison to state-of-the-art methods on instances derived from the literature and on real-world instances from various energy optimization problems such as energy optimization in hybrid electric vehicles.
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