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

Abstract

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.

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Notes

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    Dell Precision T3600, 3.0 GHz Intel Xeon, 4GB RAM, Windows 7, MATLAB R2015a

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Correspondence to Joseph K. Scott.

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Communicated by Alexander Mitsos.

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Hakizimana, A., Scott, J.K. Differentiability Conditions for Stochastic Hybrid Systems with Application to the Optimal Design of Microgrids. J Optim Theory Appl 173, 658–682 (2017). https://doi.org/10.1007/s10957-017-1096-1

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Keywords

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

Mathematics Subject Classification

  • 37H10
  • 37N40
  • 60H07
  • 90C15
  • 93E20