Electricity prices in liberalized electricity markets exhibit extreme price spikes,not seen in other commodity markets. Existing models for electricity prices, such as mean-reverting jump diffusions or regime-switching models, are only partially successful in modeling price spikes. In this paper we develop a new approach to electricity price modeling: a class of Lévy diffusion models driven by potential functions, which allow for a continuum of state-dependent reversion rates. The stochastic fluctuations in the price are modeled by either a compound Poisson process with a new form of jump-size distribution, or by a Lévy process with an α - stable distribution. Both approaches correctly model skewness and excess kurtosis (ie, heavy tails) of the distribution of price returns. We develop model estimation procedures and apply them to the data from electricity markets. We demonstrate that the potential Lévy diffusion with an α-stable distribution is particularly successful in capturing the first four moments of the returns distribution. The models generate realistic price paths and are widely applicable in applications such as risk management and scenario simulations.