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• Consideration of the optimal phasor measurement unit placement problem without radial buses. • Phasor measurement unit is pre-assigned at each bus connected to radial bus. • A two-phase branch-and-bound algorithm is used to determine the strategic locations for placement of phasor measurement unit. • The redundancy measurement is performed for ranking the obtained multiples solutions. This paper proposes a mixed integer nonlinear programming (MINLP) formulation with binary-valued variables for the optimal installation of the phasor measurement units (PMUs) in a power network. The MINLP framework has been solved by a two-phase branch-and-bound algorithm (BBA) for the optimal PMU placement (OPP) problem. The present work deals with the solution of the OPP problem without considering the radial buses including in the optimal solution. PMU is pre-assigned to each bus connected to a radial bus such that the radial buses are excluded from the list of potential PMU locations. The programming technique is used to determine the optimum number of PMUs and their locations to make the interconnected power network completely observable. The proposed BBA may provide multiple solutions at the lowest objective value, each time the optimizer routine restarts. Therefore, the obtained optimal solutions are being ranked regarding the measurement redundancy index. The two-phase branch-and-bound algorithm is applied to IEEE standard test systems as well as to New England 39- bus and a northern regional power grid 246- bus system. The proposed algorithm has been compared with the existing programming techniques in the recent literature. The simulation results obtained by the proposed BBA indicate the effectiveness and the efficiency to detect the desired target. [ABSTRACT FROM AUTHOR]