Resource Allocation with Unknown Constraints: An Extremum Seeking Control Approach and Applications to Demand Response*

This paper studies a resource allocation problem with unknown functions in the constraints. The resource allocation problem is formulated as a nonlinear optimization problem. A sufficient condition is established to guarantee a unique global optimal solution in the optimization problem, and an extremum seeking control (ESC)-based primal-dual algorithm is developed to generate the optimal solution. To implement extremum seeking, the gradients of the unknown functions are estimated by adding dither signals to the measurable inputs and outputs. We prove the semi-globally practically asymptotically (SPA) stability of the ESC-based primal-dual algorithm. The results are further applied to the demand response program with distributed heating ventilation air conditioning (HVAC) systems with unknown relationship between the temperature settings and the power consumption. Simulation results demonstrate that the ESC-based primal-dual algorithm converges to a neighborhood of the optimal solution and achieves the balance between supply and demand in electricity markets.