Accurate physical modeling and parameter extraction for the nonlinear current-voltage (Ⅰ-Ⅴ) characteristics of photovoltaic (PV) cells and modules are essential prerequisites for the design calculation, performance analysis, and optimal control of PV generation systems. In contrast to the traditional implicit single-diode models, this paper first derives the explicit single-diode models of PV cells and modules using the Lambert Wfunction, and then proposes a simple and efficient parameter extraction method on the basis of restarting the bound constrained Nelder-Mead simplex method (rbcNM). For comparing and analyzing the accuracy of implicit and explicit single-diode models, experimental data of the two typical PV cells and modules are tested and verified. Simulation results indicate that the proposed rbcNM method can rapidly and accurately extract the optimal parameters of implicit and explicit single-diode models, the simulation data produced by the extracted parameters of rbcNM method are in very good agreement with the experimental data in all cases. Comparison results show that the accuracy of rbcNM method is quite promising and outperforms the existing methods reported in the literature. Furthermore, the accuracy of explicit single-diode models is significantly higher than that of implicit single-diode models, and thus fit the Ⅰ-Ⅴ characteristic curves better.