Wind energy systems (WESs) based on doubly-fed induction generators (DFIGs) have enormous potential for meeting the future demands related to clean energy. Due to the low inertia and intermittency of power injection, a WES is equipped with a virtual inertial controller (VIC) to support the system during a frequency deviation event. The frequency deviation measured by a phase locked loop (PLL) installed on a point of common coupling (PCC) bus is the input signal to the VIC. However, a VIC with an improper inertial gain could deteriorate the damping of the power system, which may lead to instability. To address this issue, a mathematical formulation for calculating the synchronizing and damping torque coefficients of a WES-integrated single-machine infinite bus (SMIB) system while considering PLL and VIC dynamics is proposed in this paper. In addition, a power system stabilizer (PSS) is designed for wind energy integrated power systems to enhance electromechanical oscillation damping. A small-signal stability assessment is performed using the infinite bus connected to a synchronous generator of higher-order dynamics integrated with a VIC-equipped WES. Finally, the performance and robustness of the proposed PSS is demonstrated through time-domain simulation in SMIB and nine-bus test systems integrated with WES under several case studies.