Generalized Smooth Functions for Modeling Steady-State Response of Controls in Transmission and Distribution

The promise of renewables and the consequent fluctuations in the power grid necessitate a robust simulation framework to capture the steady-state behavior of new controls. However, standard non-differentiable models of control mechanisms produce divergence and/or numerical oscillations in large power flow simulations. In this paper, we describe a methodology that introduces two generalized class C1 smooth basis functions to model the steady-state of various controls in power flow for transmission and three-phase distribution as well as optimi-zation settings. These models are accompanied by homotopy methods and limiting techniques in the simulation engine that ensure scalable and robust convergence. We map standard power flow controls, typically modeled as non-differentiable functions in commercial tools, to the proposed class C1 basis functions and demonstrate the benefits for robustness in comparison to commercial tools on test cases as large as the US Eastern Interconnection system. We further extend the approach to model non-standard devices such as STATCOMs and the steady-state effects of inverter-based generators in three-phase distribution and optimization problems.