In large-scale energy market models, the price-demand relationship is usually represented by a linear function. In this paper, nonlinear demand functions are fitted to electricity market bid data; in particular, exponential and polynomial (cubic) functions are estimated from EPEX day-ahead data (i.e. Central Western European market area). The corresponding game-theoretic, large-scale electricity models were successfully solved using the Extended Mathematical Programming framework after a suitable adaptation for conjectural variations. Additionally, sufficient conditions for the existence of equilibrium solutions are tested. Numerical results show that nonlinear demand curves lead to an improved modeling especially in high price (peak) load periods and to lower levels of implied market power, which can be considered to be more realistic for markets that have strong transparency measures.