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Simulating the random changes of power prices is a crucial task for operational and trading decisions. Currently, models stemming from econometrics and financial mathematics represent the dominating approach to the stochastic simulation of electricity prices. This work proposes a novel methodology based on frequency-domain techniques for simulating the random fluctuations of hourly electricity prices according to probabilistic and spectral properties observed in historical data. The developed nonparametric algorithm is based on the well-known spectral representation method. The method has been extended to accurately reproduce the remarkable non-Gaussian and local nonstationary features of power prices. An iterative procedure and a nonlinear memoryless transformation have been applied to simultaneously match the observed evolutionary spectral content and the marginal non-Gaussian probability density function (PDF) of the random power price fluctuations. The proposed method is general and can be applied to any power market as it does not require the postulation of a model structure and the calibration of model parameters. The method is computationally very efficient as it takes advantage of fast Fourier transform techniques. Spot prices quoted on the German EEX have been selected for extensively testing the quality of the synthetically generated prices. Results show that price samples simulated with the proposed model replicate very accurately both the distributional and time-varying spectral features of the stochastic electricity price dynamics.