With the incorporation of higher shares of intermittent renewable energies (RES), more flexible resources are required in power systems to keep load balance. Under some extreme circumstances, the flexible demand resources (FDRs) may have the potential to dominate and obtain excess benefits, preventing other FDRs from participating in the electricity markets. Therefore, it is of great significance to identify the key FDR market power locations and implement some corresponding regulations. However, the relevant researches in power systems focused on the supply side, rather than the demand side. In this paper, a novel nodal market power analysis method is proposed to evaluate the potential influence of FDRs on electricity markets. Firstly, a multi-state model is established to present the multiple power system operation states including the random failures of system components. Then, the nodal market power assessment model is established under each specific state and new indices are proposed to evaluate the nodal market power of FDRs quantitatively. Furthermore, the key FDR nodes in demand side with stronger power in capturing excess revenue are identified. The 24-bus IEEE Reliability Test System is modified to demonstrate the feasibility of the proposed method. The numerical results of the proposed method are capable to display the existence of market power in demand side, and provide some valuable guidance for classification and operation of electricity markets.