  In power distribution systems, the growing penetration of renewable energy
resources brings new challenges to maintaining voltage safety, which is further
complicated by the limited model information of distribution systems. To
address these challenges, we develop a model-free optimal voltage control
algorithm based on projected primal-dual gradient dynamics and continuous-time
zeroth-order method (extreme seeking control). This proposed algorithm i)
operates purely based on voltage measurements and does not require any other
model information, ii) can drive the voltage magnitudes back to the acceptable
range, iii) satisfies the power capacity constraints all the time, iv)
minimizes the total operating cost, and v) is implemented in a decentralized
fashion where the privacy of controllable devices is preserved and
plug-and-play operation is enabled. We prove that the proposed algorithm is
semi-globally practically asymptotically stable and is structurally robust to
measurement noises. Lastly, the performance of the proposed algorithm is
further demonstrated via numerical simulations.
