Analytic solution to swing equations in power grids with ZIP load models

OBJECTIVE: This research pioneers a novel approach to obtain a closed-form analytic solution to the nonlinear second order differential swing equation that models power system dynamics. The distinctive element of this study is the integration of a generalized load model known as a ZIP load model (constant impedance Z, constant current I, and constant power P loads). METHODS: Building on previous work where an analytic solution for the swing equation was derived in a linear system with limited load types, this study introduces two fundamental novelties: 1) the innovative examination and modeling of the ZIP load model, successfully integrating constant current loads to augment constant impedance and constant power loads; 2) the unique derivation of voltage variables in relation to rotor angles employing the holomorphic embedding (HE) method and the Padé approximation. These innovations are incorporated into the swing equations to achieve an unprecedented analytical solution, thereby enhancing system dynamics. Simulations on a model system were performed to evaluate transient stability. RESULTS: The ZIP load model is ingeniously utilized to generate a linear model. A comparison of the developed load model and analytical solution with those obtained through time-domain simulation demonstrated the remarkable precision and efficacy of the proposed model across a range of IEEE model systems. CONCLUSION: The study addresses the key challenges in power system dynamics, namely the diverse load characteristics and the time-consuming nature of time-domain simulation. Breaking new ground, this research proposes an analytical solution to the swing equation using a comprehensive ZIP model, without resorting to unphysical assumptions. The close-form solution not only assures computational efficiency but also preserves accuracy. This solution effectively estimates system dynamics following a disturbance, representing a significant advancement in the field.