## 6 Observability of the graph
### Constrained Nonlinear Programming Model
Thereby, we succeed in determining the optimal number of PMUs with the appropriate number of channels without sacrificing the system observability. ## 9 Final Remarks and Future Study

In this study, the OPP problem is reconsidered with the target of considering that the node voltage phasor and all current phasors running the connectivity lines incident to that node can be obtainable. To be accurate in the declaration of this study, the minimization model is rewritten where a channel limit capacity is considered. Each PMU may vary in terms of the number of channels to observe the voltage phasor at the node in which it is installed, as well as a number of branches observed through the channel capacity restriction. The innovation of this paper lies in using mathematical and heuristic algorithms that optimize the OPP-based model on a restricted number of channels. Utilizing these algorithms, a global solution is reached by mathematical and heuristic algorithms. An algorithmic scheme is adopted utilizing the ideas of the mathematical and heuristic algorithms, building up a procedure and a global problem solving. The new algorithmic placement model results in theoretical and calculation improvements in developing the OPP placement based on a restricted number of channels. Optimum points are derived, minimizing further the entire cost installation. As we prove, the optimal number of channels, to achieve complete observability with a reduced cost, is four. Furthermore, we present solutions with higher or highest observability, but the quantities highly depend on the number of channels [56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67]. This study presents an unambiguous combinatorial methodology, splitting into integer linear and nonlinear domains to solve the PMU allocation model, considering the effect of the restricted channel capacity per monitoring device. We study mathematical models for the varying-channel OPP based-formulation to derive appropriate and accurate solutions. We focus on a binary-integer model and an equivalent nonlinear programming model that produces globally optimal solutions. We illustrate that these optimization models are easy to implement and solvable by commercial or open-course solvers [72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85]. BBA runs the binary-integer program, whereas SQP, IPMs, and GPSA optimize the mathematical programming models to find the desired outcome in minimal runtime. Additionally, our numerical results call into question the utilization of the heuristic algorithms' solution for the problem solving for the OPP model based on a restricted number of channels. BPSO and GA were suggested for the classical OPP problem, with an unlimited number of channels being solved in [25; 26; 39; 47; 48; 50; 51; 52]. GA is known to execute the BIP and generate a global solution within a reasonable runtime [25; 26; 51]. Comparatively, BPSO offers simpler implementation than GA and demonstrates comparable results in terms of both quality and quantity, as evidenced in [51]. Following the simulation run, PMU site positions were determined along with corresponding branch assignments, allowing for the derivation of the number of channels, PMU assignments, and monitored channels within a fixed channel capacity. This assessment incorporates considerations such as restricted PMU channel capacity, voltage measurements, power flows across connectivity lines, injections at power grid nodes, and higher voltage nodes [5]. The forthcoming PMU placement is anticipated to furnish all requisite information for the state estimation model, ensuring reliable monitoring. In such contexts, leveraging graphics processing units (GPUs) and deep neural networks in state estimation applications will prove indispensable, indicative of the growing adoption of GPU architecture in smart grid applications [3; 5].