Incorporating convex risk measures into multistage stochastic programming algorithms

Over the last two decades, coherent risk measures have been well studied as a principled, axiomatic way to characterize the risk of a random variable. Because of this axiomatic approach, coherent risk measures have a number of attractive features for computation, and they have been integrated into a variety of stochastic programming algorithms, including stochastic dual dynamic programming (SDDP), a common class of data-driven solution methods for multistage stochastic programs. Coherent risk measures and SDDP are tools used to manage risk while solving data-driven problems. Perhaps the most prominent example involves informing operations and deriving electricity prices in power systems with significant hydro-electric power, including the Brazilian interconnected power system. We focus on incorporating the more general class of convex risk measures into an SDDP algorithm, exemplifying our approach with the entropic risk measure. It is well-known that coherent risk measures lead to an inconsistency if agents care about their state at the end of the time horizon, but control risk in a stage-wise fashion. The entropic risk measure does not have this shortcoming. We illustrate the advantages of the entropic risk measure with two small examples from transportation and finance, and test the numerical viability of our adaptation of the SDDP decomposition scheme in a large-scale hydro-thermal scheduling problem using data from the Brazilian system.