A failure cascading model for power systemes is proposed,in which energy dissipation and capability expansion are considered.Simulations are performed on both regular grids and scale-free networks.Self-organized critical phenomenon is observed in both cases.It means that energy dissipation and capability expansion are important factors in self-organized critical status.In addition,it shows that the maximum size of cascading failures in scale-free networks is much larger than that in regular networks.

We propose a one-dimensional sandpile model with avalanche probability.The simulation is performed with the cellular automata method.As the local slope closes to the critical slope sands avalanche with a probability.The avalanche size and the average local slope are studied at different avalanche probabilities. The self-organized critical (SOC) behavior is studied at critical exponent α=1.50±0.02 as the avalanche probability is between 0.05 and 0.98.The average local slope descends with the increase of collapse probability.A sharp transition occurs between the trival behavior and SOC behavior in this model.The SOC phenomenon in one-demensional rice-pile experiment is well explained.