  Recent studies have demonstrated the potential of flexible loads in providing
frequency response services. However, uncertainty and variability in various
weather-related and end-use behavioral factors often affect the demand-side
control performance. This work addresses this problem with the design of a
demand-side control to achieve frequency response under load uncertainties. Our
approach involves modeling the load uncertainties via stochastic processes that
appear as both multiplicative and additive to the system states in closed-loop
power system dynamics. Extending the recently developed mean square exponential
stability (MSES) results for stochastic systems, we formulate multi-objective
linear matrix inequality (LMI)-based optimal control synthesis problems to not
only guarantee stochastic stability, but also promote sparsity, enhance
closed-loop transient performance, and maximize allowable uncertainties. The
fundamental trade-off between the maximum allowable (\textit{critical})
uncertainty levels and the optimal stochastic stabilizing control efforts is
established. Moreover, the sparse control synthesis problem is generalized to
the realistic power systems scenario in which only partial-state measurements
are available. Detailed numerical studies are carried out on IEEE 39-bus system
to demonstrate the closed-loop stochastic stabilizing performance of the sparse
controllers in enhancing frequency response under load uncertainties; as well
as illustrate the fundamental trade-off between the allowable uncertainties and
optimal control efforts.
