This article considers joint active power control of wind turbines and battery storage to follow a plant-level power reference signal. The joint control dynamically curtails the energy from a subset of the wind turbines and stores or withdraws energy from the battery to meet the power reference setpoint while accounting for wind plant aerodynamic interactions, such as wake losses. As a use case, we study the performance of the controller in maintaining a constant power output over hourly periods. A wind plant operating in this way would rely much less on other grid resources to meet its contractual agreements, thereby improving grid reliability, especially in grids with high penetration of wind and solar generation. We compare the operation of the wind plant under joint active power control to standard power-maximizing control with battery support. We present an analysis of the performance of the control system architecture. To study the impact of the battery size on performance, we simulate a 50-MW wind plant supported by batteries ranging from 8 to 64 MWh. We then evaluate the over and undergeneration penalties incurred by the plant during the simulation period.