  This paper studies approximate policy iteration (API) methods which use
least-squares Bellman error minimization for policy evaluation. We address
several of its enhancements, namely, Bellman error minimization using
instrumental variables, least-squares projected Bellman error minimization, and
projected Bellman error minimization using instrumental variables. We prove
that for a general discrete-time stochastic control problem, Bellman error
minimization using instrumental variables is equivalent to both variants of
projected Bellman error minimization. An alternative to these API methods is
direct policy search based on knowledge gradient. The practical performance of
these three approximate dynamic programming methods are then investigated in
the context of an application in energy storage, integrated with an
intermittent wind energy supply to fully serve a stochastic time-varying
electricity demand. We create a library of test problems using real-world data
and apply value iteration to find their optimal policies. These benchmarks are
then used to compare the developed policies. Our analysis indicates that API
with instrumental variables Bellman error minimization prominently outperforms
API with least-squares Bellman error minimization. However, these approaches
underperform our direct policy search implementation.
