With the recent advancements in power electronics for wind turbines (WTs) and increasing penetration of wind energy, wind power plants (WPP) have become necessary contributors of reactive power support for the bulk power system. Balancing reactive power support with individual WT operating requirements in a cost-effective manner is a challenge for WPP designers. In this paper, we present a methodology to optimize the WPP reactive power capability as seen from the point of common coupling (PCC), accounting for steady-state operating capabilities of the WPP equipment. Thus, the proposed methodology determines the configuration of the tap-changing transformers within the WPP that maximizes the amount of reactive power the WPP can either consume or inject to the network, considering uncertain levels of wind power generation and voltage magnitudes at the PCC. The optimized reactive power capability (ORPC) problem is initially formulated as a mixed-integer nonlinear programming (MINLP) model. Then, a set of efficient linearization techniques are used to obtain a mixed-integer linear programming (MILP) model that can be solved via off-the-shelf mathematical programming solvers. Results demonstrate that the proposed MILP model is a scalable, flexible and accurate method to maximize the reactive power capability of WPP.