Differentiability Conditions for Stochastic Hybrid Systems with Application to the Optimal Design of Microgrids

  • Alphonse Hakizimana
  • Joseph K. ScottEmail author


This article considers the regularity of expected value minimization problems subject to discrete-time stochastic hybrid systems. A primary motivation is the optimal design of microgrids subject to detailed operational simulations with renewable resources and discrete dispatching. For such problems, hybrid behavior can make the cost function discontinuous for any fixed realization of uncertainty, which has led to the widespread use of derivative-free optimizers with well-known limitations. In contrast, we provide sufficient conditions under which the expected value of the cost is continuously differentiable. We verify these conditions for a simple example and show promising preliminary optimization results using a stochastic gradient-descent method.


Hybrid systems Sensitivity analysis Stochastic optimization Decision rules Microgrids Unit commitment Renewable energy 

Mathematics Subject Classification

37H10 37N40 60H07 90C15 93E20 

Supplementary material

10957_2017_1096_MOESM1_ESM.pdf (290 kb)
Supplementary material 1 (pdf 290 KB)


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Chemical and Biomolecular EngineeringClemson UniversityClemsonUSA

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