## 3 Methodology
### Description of Network Functionality and Network Contingency Quantification  
Unintended power outages are invariably unwelcome due to the associated costs arising from the duration of the disruption. Components of CI are significantly dependent on electrical power. In the event of power disruptions to these CI components, the consequent financial consequences can be significant. Moreover, extended or large-scale outages can bring in not just financial detriments but also social, cyber, and other multifaceted issues. The optimization effort of this paper primarily aims to minimize these associated losses and curtail the threat risk associated with the malfunction of CI's crucial components, as detailed in [28]. To delve deeper into the risk concerning the power failure of such pivotal components and consequently enhance the overall system's reliability, the paper clarifies these issues using the test network spotlighted in Section 3.1. To commence, it is imperative to define the precise operational dynamics of the entire system. In this regard, this paper defines a Point of Delivery (POD) that is supplied from the LV bus via the C4 cable, which in turn caters to the CI (this could include key data centers or analogous entities). The overarching correct operation of the system is characterized by the seamless electricity transition from the HV stratum right down to the CI's load junction at the LV echelon. Under this operational definition, specific components such as DT1, DT4, POS, OLT, and relevant conduits, which do not play a decisive role in the outlined function, can be avoided  
Figure 3: Considered representative topology of the control and communication network part including the most common components. in the holistic test network representation. The topology of this distributed network is visually represented in Figure 4. Given the premise that reclosers (denoted as R1-R4) do not spontaneously interrupt the power supply during malfunctions, they are omitted from the scheme. Nonetheless, the LBS, pivotal for swift backup power provisioning, remains integral and is thus incorporated within the schematic. For accurate comprehension of dependencies, it is essential to correctly interpret both individual systems and their interconnections. The comprehensive diagram that illustrates these systems, referenced in Figure 4, can be complex and potentially confusing for some. Consequently, this study uses the oriented AG modelling, as it brings clear and unambiguous representation of these interconnected systems. AGs are graph structures prevalent in both computer science and mathematics. They consist of nodes and directed edges, ensuring there are no directed cycles. Such a design makes them highly versatile and invaluable for a multitude of applications. Not only can they illustrate complex relationships between elements, but they are also instrumental in scheduling tasks based on dependencies. In the specialized domain of power electrical engineering, AGs play a crucial role in various functions:  
* **Power Flow Analysis**: Here, AGs act as a representation of the power flow in a grid. The nodes within these graphs stand for substations, while the edges denote power transmission lines. By using AGs, engineers and researchers can determine the most efficient power flow routes and detect potential bottlenecks within the grid. For a deeper dive into this application, readers can refer to [29; 30]. * **Maintenance Optimization**: Maintenance within the power grid often requires intricate scheduling to account for dependencies and constraints. AGs assist in this endeavor by helping to prioritize tasks. With the help of these graphs, it becomes easier to determine which tasks need immediate attention and ensure a systematic and efficient completion sequence. More on this can be explored in [26; 31; 32]. * **Power and Data Outage Modeling**: AGs also find their application in modeling power and data outages.