## 6 Observability of the graph
### Constrained Nonlinear Programming Model
For this solution, Newton (or quasi-Newton) methods are utilized for convergence [19; 20; 21] whereas some algorithms utilize SQP (or trust-region) methods [19; 20; 21]. The Newton methods utilize a search direction strategy from solving a Newton equation of the perturbed optimality system [19; 20; 21]. To obtain a local convergence algorithm in a fast manner, the barrier parameter needs to be updated carefully [19]. The SQP and IPMs are performed through the Fmincon optimizer function included in the MATLAB optimization toolbox [41]. The optimizer solver is able to initialize the iterative process using a multi-start search procedure in order to terminate to a local solution point, which is approximately the global one with high accuracy [72]. Furthermore, IPMs can be performed by calling the IPOPT solver and reaching a local point [73]. Both algorithms end the iterative process with specific tolerances [19; 20; 21]. The criteria tolerances are adequate to guarantee convergence to a local point [21]. The objective function is estimated at each iteration taking into account specific tolerances such as the \(\ |(x_{i}-x_{i+1})|<StepTolerance\times(1+|x_{i}|)\) and the function tolerance \(|f(x_{i})-f(x_{i+1})|<FunctionTolerance\times(1+|f(x_{i})|)\)[19; 20; 21; 21; 73; 22; 73]. Furthermore, the constraint tolerance is considered an upper limit for any constraint violation at each iteration [21]. The constraint function magnitude must not exceed the constraint tolerance to avoid the violation of the feasibility [19; 20; 21; 41; 72; 73; 79]. Considering the importance of the selection option in handling the constraint function, a variety of mathematical and heuristic algorithms are adopted for solving the OPP-based model with a limited amount of channel capacity. Our aim is the following:  
* Instead of searching local solution points for the optimization models, we examine all the sufficient and guaranteed optimality conditions for finding global solutions. * Hence the testing of optimality yields the exact solutions by utilizing the BBA, SQP, IPMs, GPSA, and BPSO for simulation purposes. All mathematical algorithms evaluate a linear objective of a quadratic function through the iterations required to first ensure a feasible trial point, and then optimality, etc. [13; 14; 15; 16; 19; 20; 21; 37; 38; 41; 42; 51; 82]. The other concern is how to improve the performance of convergence towards an optimal solution using BBA, nonlinear, and heuristic algorithms [37; 38; 41; 42; 51]. Finally, the simulation running of the algorithms reveals the possible combinations of PMUs, the PMU locations, and the branch assignments [56; 57; 58; 59; 60; 61]. The IEEE-14-Bus System for Illustrating the Mathematical Models and Numerical Results Obtained by the Optimization Procedure  
In line with the power grid utility industry's concern, the number of constraints is increasing and it is strongly dependent on the quantity of channel capacity [56; 57; 58; 60; 61]. We first declare the \(0-1\) integer program and the mixed-integer nonlinear program, and finally obtain the pure nonlinear mathematical models [13; 14; 15; 23]. The \(0-1\) integer program is declared with a linear objective function to be minimized under specific inequality constraints under a topological analysis. We consider that the power grid is fully observable with an appropriate PMU number under the channel restriction capacity. The solution is given by BBA, Gas, and BPSO. Then, we transform it into a \(0-1\) nonlinear model solved by BBA. Finally, we examine the conditions under which the pure nonlinear problem is solved using SQP, IPMs, and GPSA. As a benchmark case study, we use the IEEE-14-bus system and a binary-integer linear program is firstly studied following the works of [PERSON] et al. [56; 57; 58]. The matrix B, which consists of rows representing the possible combinations, is depicted in Equation (32) for the IEEE-14-bus system (Figure 3) [52; 56; 57; 58; 59; 60; 62; 63; 64]. \[B=