A nonlinear phase-field model is proposed for modeling microstructure evolution during highly nonequilibrium processes. We consider electrochemical reactions at electrode-electrolyte interfaces leading to electroplating and electrode-electrolyte interface evolution. In contrast to all existing phase-field models, the rate of temporal phase-field evolution and thus the interface motion in the current model is considered nonlinear with respect to the thermodynamic driving force. It produces Butler-Volmer-type electrochemical kinetics for the dependence of interfacial velocity on the overpotential at the sharp-interface limit. At the low overpotential it recovers the conventional Allen-Cahn phase-field equation. This model is generally applicable to many other highly nonequilibrium processes where linear kinetics breaks down.