The Phasor Measurement Unit (PMU) is a modern measuring device built on the system for monitoring, controlling, and protecting power systems. Since the costs of PMU devices are very high, they must be placed in the system in optimum numbers and in a way that monitors the whole system. This study determined the locations and numbers of the optimal number of PMU devices that can monitor the whole system. Integer Linear Programming (ILP) and Binary Particle Swarm Optimization (BPSO) methods are proposed to solve the optimum PMU placement (OPP) problem. Then, the solution to the problem is carried out using Dragonfly Algorithm (DA), which is proposed as a new heuristic method. Solution methods were applied to the IEEE 14-Bus Test System and Türkiye 400 kV Interconnected Power System, and the results were compared. In addition, the results of the proposed methods were compared with the results of different studies in the literature. Thanks to the ILP, BPSO, and DA methods proposed in this study, it has been determined that power systems can be observed with fewer PMU devices. The DA method offers a great cost advantage as it is the method that provides a solution with 5 fewer PMU devices for the 400 kV Interconnected Power System in Türkiye.