## 3 Contribution of This Study  
PMU brings a revolution to the manner in which power networks are observed and operated. Most of the works assume that these monitoring devices have an unrestricted amount of channel capacity to measure as many phasors' voltage and currents running the lines as are required. In real conditions, manufacturers produce PMUs with different characteristics.  
One characteristic is the channel limit capacity, which allows the PMU's availability onto the market with a varying cost [52]. Therefore, an optimal investment can be made based on the selection of appropriate PMU types utilizing efficient algorithms.  
This decision leads to a decrease in the cost of hardware needed for the development of the WAMS system based on synchronized measurements [52].  
In this study, we consider that optimal PMU numbers are installed around the power grid having a restricted number of measurement channels. Although there exist some works in this domain [56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67], we present a revised optimization model and a number of algorithms related to its solution. The solution of this model will keep the network completely observable relying only on synchronized measurements [37].  
This study takes into account the realistic scenario where the channel capacity is varied to measure phasors' voltage and current running in the connectivity lines and produces global solutions with strict optimality conditions [13; 14; 15; 16; 17; 18; 19; 20; 21].  
The optimization study does not consider the existence of ZI buses at this time. This approach does not take away any truth of models presented in [40; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67] when ZI, active injections, and flow measurements are taken into account for further consideration.  
Our aim is to give a comprehensive output of our algorithmic approach to the PMU allocation problem solving based on BBA considering a restricted number of channel capacities. Furthermore, we examine how the entire observability is preserved with less measurement redundancy in the state estimation [1; 2; 3; 4; 6].  
To extend the mentioned target, we illustrate a number of algorithmic models to enhance our approach. In this direction, we adopt nonlinear, derivative-free, and heuristic algorithms to find alternative PMU set solutions with the branch assignments covered to keep the system functional and observable. Our primary outputs are:We create an algorithmic connection between the OPP based-model with a restricted channel capacity and the optimal solutions produced by the suggested mathematical and heuristic algorithms.
2. The branch-and-bound algorithm is used in a comparison study with nonlinear and heuristic algorithms.
3. We present a state-of-the-art solution produced by the \(0/1\) integer program solving with a zero absolute gap calculated with zero percentage gap tolerances; this means that this solution is a global one returned by the solver.
4. We extend the ILP model into a heuristic search of the feasible set to show that evolutionary algorithms can efficiently perform the optimization following the BBA's solution.
5. Finally, a novel nonlinear program is created and solved by either continuous algorithms or a BBA embedded in the SCIP optimizer performed in MATLAB.