# IoT-Driven Resilience Monitoring: Case Study of a Cyber-Physical System
## 3 Theoretical Foundation of Resilience Curve and Its KPIs in This Domain
### Resilience Curve and Selected R-KPIs
#### 3.1.1 Recovery Time  
The primary objective of resilience-enhancing metrics is to enhance the three core capacities of resilience: absorption, adaptation, and restoration. This goal is achieved through the implementation of diverse strategies aimed at strengthening the functionality and performance of critical infrastructure, such as incorporating redundancy.  
Redundancy in critical infrastructure refers to integrating duplicate or backup components and systems to ensure that essential functions can persist even in the face of disruptions. By incorporating redundant elements, the system gains the ability to absorb shocks, adapt to unexpected challenges, and facilitate quicker restoration. Redundancy serves as a safeguard against single points of failure, minimizing the impact of disruptions and significantly contributing to reducing recovery time. It plays a pivotal role in enhancing overall resilience by providing alternative pathways and resources, thereby fortifying the infrastructure's ability to bounce back swiftly and effectively from unforeseen events. Moreover, shorter recovery times can also be achieved through the implementation of measures that prioritize the swift restoration and replacement of impaired infrastructure components. This proactive approach not only accelerates recovery but also contributes to the overall enhancement of resilience, as it minimizes downtime and fortifies the system against prolonged disruptions.  
The recovery time after a disruption is a crucial index for critical infrastructure resilience, particularly in sectors like energy, due to its direct impact on overall operational resilience. In the aftermath of a disturbance within critical infrastructure, performance inevitably experiences a decline, reaching its minimum level. Following this critical decrease, there are essentially two potential outcomes. Firstly, the performance may reach zero, resulting in a complete system shutdown--a scenario that is universally deemed unacceptable in any disturbance situation for critical infrastructure.  
Conversely, the system's response to the disturbance may trigger a recovery phase, characterized by an upward trajectory in performance levels. This recovery phase is crucial as it signifies the infrastructure's ability to rebound and stabilize after the initial level in performance. The stabilized performance level, as illustrated in Figure 1, can assume one of three forms: it may be equal to the prior stable level, higher than the pre-disturbance level, or lower than the pre-disturbance level. This phase is termed the recovery phase, representing a critical aspect of the system's resilience.  
The concept of recovery time encompasses both the degrading phase, where performance declines, and the subsequent recovery phase, where performance improves and stabilizes. As illustrated in Figure 1, the recovery time spans from the occurrence of the disturbance to the point of stabilization after the recovery phase. The duration of this recovery time is indicative of the system's overall robustness and its ability to efficiently  
Figure 1: Illustration of recovery time. The red box highlights the point on the resilience curve where the recovery phases duration can be identified.  
restore and stabilize its performance following disruptions, which is vital for the sustained and reliable operation of critical infrastructure systems, especially in the energy sector.