  This work addresses the distributed frequency control problem in power
systems considering controllable load with a nonsmooth cost. The nonsmoothness
exists widely in power systems, such as tiered price, greatly challenging the
design of distributed optimal controllers. In this regard, we first formulate
an optimization problem that minimizes the nonsmooth regulation cost, where
both capacity limits of controllable load and tie-line flow are considered.
Then, a distributed controller is derived using the Clark generalized gradient.
We also prove the optimality of the equilibrium of the closed-loop system as
well as its asymptotic stability. Simulations carried out on the IEEE 68-bus
system verifies the effectiveness of the proposed method.
