**5. Application of the modulus of resiliency**

The power industry was selected to provide an example for applying the modulus of resiliency using empirical data. The aim of the electric subsystem is to deliver electricity to all end use customers; therefore, data regarding the number of customers out of service can be used to quantify subsystem performance. The use of customers out of service in quantifying subsystem performance was supported by a review of regulatory reliability metrics used by Public Utility Commissions. For major electric utility disruptions, DOE situation reports provide customer outage information for and are publicly available from the DOE website. One of the most prominent events to challenge utilities is hurricane, and as a result, multiple

hurricane events have data on the DOE website. Following data collection, plots can be constructed of the electric utility response in restoring customers. The inflection points were identified, and a yield point designated by reviewing disaster preparedness data from the Capital Region Study [22]. The study indicated that 73% of survey respondents had less than 10 days of food stored. Therefore, an event lasting greater than 10 days would most likely result in scarcity from food spoilage and diminished retail capabilities. With a known bifurcation and yield point, analysis can be completed.

Hurricanes Wilma and Irma presented an opportunity to compare resiliency of separate events in the same region. Following Wilma, the ability of several infrastructures to recover from severe events was reviewed in the Florida region. "[M] ore than \$141.5 million has been obligated by FEMA for 119 Hazard Mitigation Grant Program projects to build stronger, safer more resilient communities in Florida" [23]. Florida was once again subjected to a hurricane when Irma came ashore 12 years later. More than six million customers lost power as a result of Irma; compared to 4 million from Wilma. Although more than a decade apart, these two storms provide an opportunity to compare the recoveries following significant investment in resiliency. The comparison of the two resiliency indices can present an opportunity to calculate a cost per unit of resiliency and explore concepts such as diminishing returns or optimization from multi-hazard investment. Multi-hazard resiliency actions would provide an ability to address multiple HILP scenarios with a single investment. A resiliency index for each of the scenarios would be computed in order to create a composite change in resiliency for a given investment. The goal of this composite approach is to provide a means for justifying highly adaptable subsystem structures based on resiliency benefits.

The example demonstrates the process of calculating the resiliency index for a power utility scenario and comparing the response before and after the investment in resiliency. The values shown in **Table 2** were extracted from United States Energy Information Administration (EIA) data. The additional data points associated with 0.5 and 1.5 days were included due to nonlinearities in customer outages associated with Hurricanes Wilma and Irma, respectively. Similarly, day 9 for Hurricane Wilma was approximated for the purpose of this analysis. The data required to calculate the change in length was available by collecting System

Average Interruption Duration Index (SAIDI) data. SAIDI data provides a basis for the average duration a customer faces and can be compared to the protracted

Following the collection of empirical data, the total area under the curve was calculated by dividing the outage curve into time steps and summing the areas of

The study region had a SAIDI of 60 minutes and a protracted outage duration of 12,960 minutes. Therefore, the resiliency index (RI) for Hurricane Wilma is deter-

> ð Þ <sup>0</sup>*:*<sup>35</sup> <sup>2</sup> ð Þ 0*:*12�1*:*758 *=*0*:*258 ð Þ 12, 960�<sup>60</sup> *<sup>=</sup>*<sup>60</sup> � �

ð Þ <sup>0</sup>*:*<sup>64</sup> <sup>2</sup> ð Þ 0*:*11�2*:*175 *=*0*:*56 ð Þ 12, 960�<sup>57</sup> *<sup>=</sup>*<sup>57</sup> � � 1

1

A ¼ 16*:*07 (21)

A ¼ 89*:*14 (22)

2

2

0 @

12,960 minutes. Therefore, the resiliency index (RI) for Hurricane Irma is determined as shown in Eq. (22) based on EIA data [24] (**Tables 3** and **4**).

0 @

The study region had a SAIDI of 57 minutes and a protracted outage duration of

system disruption as a change in length.

mined as shown in Eq. (7).

*Hurricane Irma restoration plot.*

*Hurricane Wilma restoration plot.*

*The Modulus of Resilience for Critical Subsystems DOI: http://dx.doi.org/10.5772/intechopen.93783*

**Figure 8.**

**29**

**Figure 7.**

each time step as shown in **Figures 7** and **8**, respectively.

Resiliency index RI ð Þ¼ <sup>1</sup>

Resiliency index RI ð Þ¼ <sup>1</sup>


**Table 2.** *Outages for Hurricanes Wilma and Irma.*

*The Modulus of Resilience for Critical Subsystems DOI: http://dx.doi.org/10.5772/intechopen.93783*

**Figure 7.** *Hurricane Wilma restoration plot.*

hurricane events have data on the DOE website. Following data collection, plots can be constructed of the electric utility response in restoring customers. The inflection points were identified, and a yield point designated by reviewing disaster preparedness data from the Capital Region Study [22]. The study indicated that 73% of survey respondents had less than 10 days of food stored. Therefore, an event lasting greater than 10 days would most likely result in scarcity from food spoilage and diminished retail capabilities. With a known bifurcation and yield point, analysis

Hurricanes Wilma and Irma presented an opportunity to compare resiliency of separate events in the same region. Following Wilma, the ability of several infrastructures to recover from severe events was reviewed in the Florida region. "[M] ore than \$141.5 million has been obligated by FEMA for 119 Hazard Mitigation Grant Program projects to build stronger, safer more resilient communities in Florida" [23]. Florida was once again subjected to a hurricane when Irma came ashore 12 years later. More than six million customers lost power as a result of Irma; compared to 4 million from Wilma. Although more than a decade apart, these two storms provide an opportunity to compare the recoveries following significant investment in resiliency. The comparison of the two resiliency indices can present an opportunity to calculate a cost per unit of resiliency and explore concepts such as diminishing returns or optimization from multi-hazard investment. Multi-hazard resiliency actions would provide an ability to address multiple HILP scenarios with a single investment. A resiliency index for each of the scenarios would be computed in order to create a composite change in resiliency for a given investment. The goal of this composite approach is to provide a means for justifying highly adaptable

The example demonstrates the process of calculating the resiliency index for a power utility scenario and comparing the response before and after the investment in resiliency. The values shown in **Table 2** were extracted from United States Energy Information Administration (EIA) data. The additional data points associated with 0.5 and 1.5 days were included due to nonlinearities in customer outages associated with Hurricanes Wilma and Irma, respectively. Similarly, day 9 for Hurricane Wilma was approximated for the purpose of this analysis. The data required to calculate the change in length was available by collecting System

**Day. % Out of service (Hurricane Wilma 2005) % Out of service (Hurricane Irma 2017)**

0 0 0 0.5 34 20 1 35 40 1.5 34 64 2 31 56 3 28 40 4 21 31 5 18 20 6 12 11 7 10 7 8 9 4 9 6 1

can be completed.

**Table 2.**

**28**

*Outages for Hurricanes Wilma and Irma.*

subsystem structures based on resiliency benefits.

*Operations Management - Emerging Trend in the Digital Era*

**Figure 8.** *Hurricane Irma restoration plot.*

Average Interruption Duration Index (SAIDI) data. SAIDI data provides a basis for the average duration a customer faces and can be compared to the protracted system disruption as a change in length.

Following the collection of empirical data, the total area under the curve was calculated by dividing the outage curve into time steps and summing the areas of each time step as shown in **Figures 7** and **8**, respectively.

The study region had a SAIDI of 60 minutes and a protracted outage duration of 12,960 minutes. Therefore, the resiliency index (RI) for Hurricane Wilma is determined as shown in Eq. (7).

$$\text{Resilieuy index} \ (\text{RI}) = \frac{1}{2} \left( \frac{\text{(0.35)}^2}{\left( \frac{(0.12 \times 1.758)/0.258}{(12, 960 - 60)/60} \right)} \right) = 16.07 \tag{21}$$

The study region had a SAIDI of 57 minutes and a protracted outage duration of 12,960 minutes. Therefore, the resiliency index (RI) for Hurricane Irma is determined as shown in Eq. (22) based on EIA data [24] (**Tables 3** and **4**).

$$\text{Resilieuy index (RI)} = \frac{1}{2} \left( \frac{(0.64)^2}{\left( \frac{(0.11 \times 2.175)/0.56}{(12,960 - 57)/57} \right)} \right) = 89.14\tag{22}$$

#### *Operations Management - Emerging Trend in the Digital Era*


The determination of a change in resiliency allows for a quantitative measurement related subsystem response. The use of resiliency indices can aid in quantify-

In this chapter, a comparison to mean-based reliability was contrasted with the use of resiliency calculations for HILP events. Resiliency calculations are required, given the infrequent nature of protracted subsystem disturbances. Following a review of resiliency computations, a gap analysis was used to identify the opportunities for ensuring a resiliency calculation can capture the nonlinearities observed in empirical data. Parallels are provided between the modulus of resilience construct from materials science and an isomorphic application defined. In conclusion, an example is presented for the power utility sector demonstrating the methods of collecting the inputs and completing the computations. These inputs include defining the aim of the system and failure point, data collection, determination of bifurcation point, and the use of reliability data for calculating a change in length. The ability to calculate resiliency regardless of the subsystem or scenario can assist in the evaluation of resiliency actions already taken or planning for new investment. The ability to compute resiliency on a common base may also offer opportunities to optimize investment based on interconnectedness to the subsystems which yield the greatest improvement. A more integrated approach may lead to increased systemic resiliency as opposed to more common heuristics-based subsystem specific approaches. The proposed method more closely adheres to the ontological and conceptual frameworks associated with initial references of resiliency. Furthermore, subjective inputs are avoided increasing the replicability and repeatability of associated research. By acknowledging a yield point specific to the aim of the subsystem, results from the resiliency index better represent the outcomes of real-world subsystems. Lastly, bifurcating the event curve allows the onset characteristics of the disruptive event to normalize the resiliency performance

Further research on the distribution of events by type will be conducted to validate the anecdotal evidence regarding common cause and special cause events. This additional data will assist in the development of statistics for assessing the correlation between increasing interdependence and HILP events for critical subsystems. In order to test a wider array of empirical data sets, resiliency indexes will be calculated using both historical and future HILP event data. The results of these analyses will be used to continually evaluate the efficacy of the metrics and identify

ing the efficacy of resiliency investment.

*The Modulus of Resilience for Critical Subsystems DOI: http://dx.doi.org/10.5772/intechopen.93783*

**6. Conclusions**

metric.

**Acronyms**

**31**

opportunities for enhancements.

RI resiliency index

HILP high impact, low probability MTBF mean time between failure MTTF mean time to failure MTTR mean time to repair DOE department of energy

FEMA Federal Emergency Management Agency EIA energy information administration

SAIDI system average interruption duration index

#### **Table 3.**

*Resiliency index calculation for Hurricane Wilma.*


#### **Table 4.**

*Resiliency index calculation for Hurricane Irma.*

Change in resiliency is found by Eq. (3).

$$
\Delta RI = \frac{RI\_{final} - RI\_{initial}}{RI\_{initial}} = \frac{89.14 - 16.07}{16.07} = 4.55\tag{23}
$$

The determination of a change in resiliency allows for a quantitative measurement related subsystem response. The use of resiliency indices can aid in quantifying the efficacy of resiliency investment.
