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This paper utilises concavity properties in the optimal multiplier load flow method (OMLFM) to find the most suitable low voltage solution (LVS) for the systems having multiple LVS at the maximum loading point. In the previous method, the calculation of the optimal multiplier is based on only one remaining low voltage solution at the vicinity of voltage collapse point. However, this does not provide the best convergence for multi-low voltage solutions at the maximum loading point. Therefore, in this paper, concavity properties of the cost function in OMLFM are presented as the indicator to find the most suitable optimal multiplier in order to determine the most suitable low voltage solutions at the maximum loading point. The proposed method uses polar coordinate system instead of the rectangular coordinate system, which simplifies the task further and by keeping PV type buses. The polar coordinate in this method is based on the second order load flow equation in order to reduce the calculation time. The proposed method has been validated by the results obtained from the tests on the IEEE 57, 118 and 300-bus systems for well-conditioned systems and at the maximum loading point. [ABSTRACT FROM AUTHOR]