## 4 Materials and Methods  
Despite the fact the PMU's cost was recently reduced, it cannot be workable to place a PMU at every power grid node in a large-size transmission grid [7; 8; 9; 10; 11]. Hence, it has an important value to attain complete observability and maximum redundancy with an optimal number of PMUs placed at buses within the power grid [33; 37; 41; 43; 44; 45; 47; 49; 51]. Many research works have considered the ZI's bus presence to further reduce the entire PMU number needed to attain power grid observability [32; 33; 35; 36; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 52; 53; 54]. A PMU-PDC monitoring system is designed in [55]. Considering the unknown tap ratios [68; 69], having all ZI nodes behave in a similar manner can provide guidance for schemes where the topological observability may not secure numerical observability being fulfilled [35; 40; 46]. Our aim is observability analysis based on a restricted number of channels per PMU device. Hence, a minimum number of such devices are optimally placed around the power transmission grid. The whole power network must be fully monitored using an optimal PMU number [37; 38; 41; 51; 56; 57; 58]. A PMU device placed at a network bus supplies the system operator with direct measurements, whereby [PERSON]'s voltage law and [PERSON]'s law are utilized to compute the pseudo measurements that are essential to maintain the system observability [31]. Optimal PMU placement models are suggested where the PMU number is found when a linear or quadratic cost function is optimized subject to several topological or numerical observability constraints in [37; 38; 41; 42; 51]. The decision vector ranges in either a binary or a continuous domain [37; 38; 41; 51]. At first, the OPP formulation was suggested in an \(0/1\) integer linear program [31; 32; 33] and a powerful ILP solver was used to obtain an optimal solution [35; 36; 37; 38; 41; 51]. Then, \(0/1\) semi-definite and nonlinear programs have been proposed to find an optimal solution from an alternative optimization point of view [37; 40]. All optimization models are robust in the implementation, with excellent outcomes ensuring the topological or the numerical approach in the state estimation tool [30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50; 51]. The high cost of PMUs related to installation limitations and commissioning procurement in substations is still a practical restriction over the wide-area power grid [36; 37; 38; 41; 51]. The PMU cost is further increased by keeping the same ratio with the number of channels [36; 36; 58; 61; 63; 64; 65; 66; 67]. In the bibliography, most articles have supposed an unlimited number of measurement channels for the PMU allocation problem solving [4; 7; 8; 9; 10; 11]. The restricted number of channels within a PMU monitoring device is an extra constraint involved in the optimization model related to finding the best PMUs to ensure the complete observability scenario [36; 56; 58]. This realistic assumption about PMU limit capacity can have an effect on the PMU arrangement in the power grid with or without contingency scenarios [56; 58]. Certainly,if the amount of PMU channel capacity is less than the number of connectivity lines to a power grid node, a combinatorial number of possible branch phasor current measurements can be found in this realistic scenario [56; 57; 58]. In such a realistic scenario, PMU monitoring devices are constructed with a restricted number of channels. A single PMU can be lost depending on the amount of current channel capacity embedded in a PMU device [53; 54]. Hence, the measurement redundancy task is very significant for the effective utilization of SE [2; 3; 4; 6]. We consider mathematical and heuristic models, with the problem solving being presented using BBA, nonlinear algorithms, derivative-free optimizers, and a BPSO.