OPTIMAL PMU PLACEMENT PROBLEM IN OCTAHEDRAL NETWORKS

. Power utilities must track their power networks to respond to changing demand and availability conditions to ensure effective and efficient operation. As a result, several power companies employ phase measuring units (PMUs) to check their power networks continuously. Supervising an electric power system with the fewest possible measurement equipment is precisely the vertex covering graph-theoretic problem, in which a set 𝐷 is defined as a power dominating set (PDS) of a graph if it supervises every components (vertices and edges) in the system (with a couple of rules). The 𝛾 𝑝 ( 𝐺 ) is the minimal cardinality of a PDS of a graph 𝐺 . In this present study, the PDS is identified for octahedral networks.


Introduction
A power network is made up of electrical hubs and transmitting cables that connect them.Electric power companies must constantly track the condition of their systems.The magnitude of the voltage at loads and the system process at generators must both be monitored.The placement of PMUs at certain locations within the device must be controlled.Due to the increasing price of PMUs, it's critical to employ as little as possible while still being able to track the entire system.This Power system observability with minimal PMU placement is a problem that Haynes et al. [19] introduced as a theoretical graph problem and dubbed the power dominating set (PDS) problem after it was demonstrated in [3].
Let  be a connected network with vertex set  () and edge set ().A dominant subset is a proper subset  of  () if any node in  () that is not in  must have at least one adjacent node in .The least cardinality of all possible  is ().If all nodes of  () can be recursively observed by either domination or propagation, a subset  is called PDS (power dominating set).
The problem of finding PDS is NP-complete in general.Even for graph classes like chordal, split and bipartite, it remains NP-complete [19].In [1,18,19], various algorithms for computing the PDS for a specific class of graphs were described.An improved algorithms with complexity results were reported in [18].This problem is studied for block graphs [34], circular-arc graphs [25], hypercubes [8,11], grids [16], generalized Petersen graphs [4,9,21,33], permutaion graphs [32], planar graphs with small diameter [36], maximal planar graphs [15], Knodel graphs and Hanoi graphs [20], de Bruijn graphs and Kautz graphs [23], regular claw-free graphs [27], and certain chemical graphs [30].This problem is also discussed for Cartesian product of graphs in [4,22], tensor and strong product in [13], corona product and join of graphs in [35], and for some other graph products were discussed in [5].The lower bounds for this problem is discussed in [17].An upper bound for one component graph with  > 4 is presented, and few extremal graphs concerning PDS are characterized in [37].The Nordhaus-Gaddum results of this problem were reported in [5].
Straight forward generalization of PDS problem is done in [9] as the -PDS problem.It is trivial to note that when  = 1, this problem converges to the original PDS problem and when  = 0 it is a traditional domination problem.This -PDS problem is discussed for Sierpiński graphs [12], block graphs [31], regular graphs [14], certain interconnection networks [29], and weighted trees [10].In [7], the complexity of power dominating throttling is discussed.The infectious power domination is introduced in [6], along with a general bound for determining the influence of particular hypergraph operations.

Octahedral and its derived networks
The octahedral structures are introduced in [2].A platonic solid's corresponds to a polyhedral graph, which is called an octahedron graph.The unit octahedron contains 6 vertices and 12 edges.We define a twin octahedron in OH() as two octahedrons sharing exactly one common vertex.The structural graph of the unit octahedron and twin octahedron and the vertex representations are depicted in Figure 1.For more information on these graphs, it can be seen from [24,28].Different ways of connecting unit octahedron derive the varieties of octahedral structures.

Octahedral network
An -dimensional octahedral network has 27 2 + 3 vertices and 72 2 edges.It is noted that OH() has 6 2 unit octahedron and 3 2 edge-disjoint twin octahedron.Figure 2a gives an idea about the extension of OH() network and its addressing scheme is depicted in Figure 2b.

Rectangular octahedral network of Type I and Type II
In line with [26], we introduce the rectangular octahedral network in this subsection.The rectangular octahedral network of Type I denoted by ROH 1 (, ) is derived by arranging octahedron in a two-dimensional plane so that the first octahedron whose apex is facing down.Rectangular octahedral network of Type II denoted by ROH 2 (, ) is derived by presenting octahedrons in a two-dimensional plane similar to ROH 1 (, ) with the condition that the first octahedron, which is the north west corner most unit octahedron whose apex should be facing the top.Rectangular octahedral network of Type I has 1  2 (9 + 2 +  − 1) vertices and Type II has  vertices.Both types of these structures have 12 edges.Different cases of ROH 1 (, ) and their addressing schemes are portrayed in Figure 4.It is clear that ROH 1 (, ) ∼ = ROH 2 (, ) for all  and  except  even and  odd.The non-isomorphic case is depicted in Figure 5.The vertex set of Type I rectangular octahedral network } for non-isomorphic case of Type II rectangular octahedral network.

Main results
Theorem 3.1.Let  be any octahedral structure,  be the edge disjoint subgraph of  isomorphic to twin octahedron and  be the PDS of .Then  () ∩  ̸ = ∅.
Proof.Since  is an edge-disjoint subgraph of , the nodes , , ,  are the only nodes through which the rest of the graph is connected to .See Figure 6.Suppose the contrary that there exists an  such that  ∩ () = ∅.Here {, , , } are dominated.But due to the fact that   () ≥ 4, for  ∈ {, , , }, the propagation at  fails.There fore the vertices {,  ′ ,  ′′ ,  ′ ,  ′ ,  ′ ,  ′ } are not observed.This contradicts the assumption of .
Since  is odd, there are −1 2 twin octahedrons in the first and last row of the  () and one unit octahedron in each of these rows.Hence there are −2 2 twin octahedrons in ROH 2 (, ) say TO 1 , TO 2 , . . .TO −2

Conclusion
This investigation offers exhaustive work on the power domination problem of various networks.It also discusses the optimal PMU placement problem in octahedral networks.The lower bound is attained using edge-disjoint subgraph-technique, and the upper bound is obtained by exhibiting the PDS in an octahedral network.Using these techniques, the tasks mentioned above of placing optimal PMU placement problem with more accurate, optimal and reliable.The problem of placing optimal PMUs for other networks are under investigation.