As a variation of minimum dominating set problem, the power dominating set problem is proposed to achieve the complete observation of a power system by placing the smallest number of PMUs. Under different contingencies consisting line outages or PMU losses, the reliability of observation for each bus should be guaranteed to ensure the security and robustness of the power system. Thus, the probabilistic power dominating set (PPDS) problem is proposed in this paper. Given a pre-specified reliability level for each bus in the power system to be observed and known distribution of random events, an integer programming formulation, with consideration of zero-injection property, is presented for the PPDS problem. Additionally, the reliable connected power dominating set problem is studied to meet two requirements, including the connectivity of the PMU subgraph and the reliability of the connectivity of this subgraph. Numerical experiments based on several IEEE test cases are performed to find the best deployment of PMUs satisfying different requirements.

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Recently, phasor measurement units (PMUs) are becoming widely used to measure the electrical waves on a power grid to determine the health of the system. Because of high expense for PMUs, it is important to place minimized number of PMUs on power grids without losing the function of maintaining system observability. In practice, with a budget limitation at each time point, the PMUs are placed in a multistage framework spanning in a long-term period, and the proposed multistage PMU placement problem is to find the placement strategies. Within each stage for some time point, the PMUs should be placed to maximize the observability and the complete observability should be ensured in the planned last stage. In this paper, the multistage PMU placement problem is formulated by a mixed integer program (MIP) with consideration of the zero-injection bus property in power systems. To improve the computational efficiency, another MIP, based on the equivalent network flow model for the PMU placement problem, is proposed. Numerical experiments on several test cases are performed to compare the two MIPs.