## 2 Consideration of This Study  
The OPP problem delivers an adequate number of PMUs for observing a power grid and defines the health of a power grid [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12]. PMUs are placed at power network nodes, i.e., substations to which transmission lines, generators, and loads are joined [34; 35]. Due to lack of a communication facility at some substations [36; 39; 65], the PMU's cost placement is further increased and it is forbidden to be placed at substations with a lack of communication facilities [1; 34; 35; 36]. Hence, it is still significant to optimize/minimize the number of PMUs placed within the power grid transmission system [36; 37; 38; 41; 42; 51]. The optimization output gives an appropriate number of PMUs that is sufficient to monitor the power system [37; 38; 41; 51]. In practice, PMU manufacturers yield a few types of PMU, with distinct functions and varying channel capacities [56; 59; 60; 61; 62; 63; 64; 65; 66; 67]. As a consequence, specific algorithms are studied for optimally placing PMUs with a restricted number of channels at power nodes [60]. We consider an extra constraint function to the allocation problem related to the number of channels to keep complete network observability [56; 56; 61; 63; 64; 65; 66; 67]. We develop PMU placement models with multiple channels as well as a single-line placement model to detect PMU sites related to branch assignments. The optimization model is also interpreted as a minimum edge cover, otherwise known as a node set covering problem [15]. Single-line PMUs are also known as a unit channel capacity [61]. A single-line PMU is able to measure the voltage phasor of the related bus, and the phasor current of only one of the branch incidents to the bus where the PMU is placed [60; 61]. So, in the monitoring of the phasor's voltages of grid nodes, PMU measurements are significant in observing the transmission grid lines [61]. Thus, each PMU enables it to make better line capability observation with PMU measurements [56; 58; 60; 61; 64]. As a result, the installation of single-line PMUs could be a standard practice interpreted as a minimum edge covering problem [28; 29; 30; 31; 61]. So, the node set covering problem interprets the above strategy in graph theory [28; 29; 30; 31]. In practice, it can be used primarily in industries such as utility companies to construct a modern grid with the capability of optimal \"branch\" placement of PMUs [61]. Furthermore, the manufacturers supply PMUs with a varying number of channels and PMU cost ranges analogous to channel restriction. Some works study this combinatorial problem considering a restricted number of channels, which is a more realistic OPP scenario [56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69]. The principal constitution of this study is threefold:  
1. To achieve the proper PMU number by optimizing a linear or a quadratic cost function under a topological constraint function considering the limitation of the PMU channel limit capacity. 2. To accomplish this task, branch-and-bound algorithms [37], sequential quadratic programming [42], interior-point methods [42], generalized pattern search algorithms [41; 70; 71], genetic algorithms [51], and binary swarm optimization [51] are utilized to achieve optimality. 3. We focus on finding the proper PMU number with a branch assignment set solution with a fixed number of channels. Our aim is to reconsider the ILP-based model under a limited PMU number [56; 57; 58] in the YALMIP environment [37; 38]. GA and BSPO are able to handle the ILP model's constraints, and finally an equivalent nonlinear model is given to find optimality. We tested the algorithmic models on the classical IEEE power systems. BBA gives the solution under a strict optimality warranty followed by evolutionary algorithms' solutions with the same quality and quantity [37; 38; 41; 42; 51]. ZI buses are the buses without generation or load consumption [32; 33; 34; 35; 39; 40].