# Probabilistic Quantification of Distribution System Resilience for an Extreme Event

Authors: Puspendu Ghosh; Mala De
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

The occurrence of natural disasters such as cyclones, earthquakes, and floods has increased worldwide, and their effect is most intense in localized regions. These events expose weaknesses in the power system infrastructures and show how well-prepared the system is to operate its services in a resilient way. During and after an extreme event, the outage and service disruption happens due to the inability of the affected part of the grid to cope with disruptions, which leads to insufficient resilience of the system. It is becoming more critical to enhance the resilience of electrical networks to extreme weather occurrences through suitable hardening processes and smart operational techniques. A reliable prevention approach necessitates a quantitative resilience metric that can estimate the effects of the future extreme events on distribution systems and assess the possible benefits of various planning strategies. This study aims to address the issue of lack of resilience metrics and proposes probabilistic metrics for evaluating the performance of distribution systems resilience in case of extreme weather events. Specifically, this study introduces active and passive resilience concepts to provide insights into system response. The proposed resilience metrics for various weather scenarios are quantified for the IEEE 33-bus system. The simulation framework also analyzes the effects of different operational and structural resilience improvement approaches on the proposed resilience metrics.


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## Body

## 1. Introduction

The first step in assuring distribution system resilience is identifying possible vulnerabilities of the network. Usually, this comprises developing resilience indicators to evaluate the network’s existing resilience to identified vulnerabilities, recommending resilience enhancements, and assessing the resilience enhancement approach. Network-specific investigations and sophisticated analytical techniques are needed to ensure resilience. The catastrophic effects of past disaster events, such as windstorm, flooding, earthquake, and cyberattack [1–4], to mention a few, have motivated researchers to work on this area. The Texas outage [5] in 2021 demonstrated how an extreme weather event could interrupt a whole metropolis, highlighting significant inadequacies in reliability-based management. However, the absence of a strategical approach to assure that the next investment leads to increased resilience following a catastrophe is inaccessible. A proper combination of preevent investment, well-planned network hardening, and automation can minimize disruptions induced by such severe incidents. It can also assure that the time spent recovering after an incident is reduced.The radial nature of distribution system causes minor redundancies and necessitates the implementation of resilience to critical loads or segments of the distribution network in case of severe incidents. Typically, resilience is done by the way of design, with infrastructure built more durable and robust to resist the effect of catastrophic occurrences. These severe incidents may occur in either the physical or cyber sectors, requiring various strategies and procedures to improve a network’s resilience in that sector. In previous work, the present author classified distribution system resilience enhancement strategies into planning-based and operational-based methods [6]. Planning-based approaches evaluate the existing state of system resilience, analyze hardening techniques, and enhance network resilience features in the face of potential catastrophic occurrences; however, operational-based strategies make utilization of already existing assets to minimize the impacts of severe incidents occurrences on the network’s performance [7, 8]. Operational-based techniques offer several quick resolutions and actions to keep the electric network operating at a minimum acceptable level while avoiding any additional severe operational constraints that might result in a cascade outage [9].Several metrics have been utilized in the past, with techniques differing depending on how resilience was defined [11–13]. In [14], the author assesses resilience in terms of the surface area within the curves of conceptual resilience. Although various operational response curves of a network have been addressed [15, 16], the well-known “resilience triangle” [10] shown in Figure 1 was simulated in a number of studies [17–19]. The resilience triangle is a measurement of a system’s loss of functionality following a major incident and the time it takes for the system to recover to its previous state. The hypotenuse of the triangle may have several shapes, such as rectangular, triangular, or exponential, based on the performance of the restoration activities in the zone. However, this method may adequately obtain resilience restoration after an incident t∈t0,t1, but it cannot identify other extremely crucial resilience aspects that conventional electrical networks face. Also, it does not clearly define the consequences of severe incidents. Hence, the metric cannot be used to evaluate system efficiency in the face of potential future severe occurrences of such extreme events. Several authors proposed resilience metrics by evaluating the reduced effects of network failures or deviations in performance [20, 21]. However, they do not clearly represent the system response during or following a severe incident. An approach to quantifying and analyzing the distribution system resilience focused on consumer benefits is provided in [22]. The method is implemented based on the reliability concept of network service availability but does not precisely describe the effects of severe occurrences.Figure 1
The resilience triangle [10].Even though numerous research describes and assesses power system resilience, no commonly acknowledged explicit resilience metric exists. The current metrics have a few limitations highlighted in Table1.Table 1
Limitation in current metrics and/or quantifications of power system resilience.
Limitation 1. Impacts of upcoming extreme weather events1. Does not give any insight into the probable effects of an upcoming extreme event.2. Does not accurately assess the effects of extreme events on network performance loss and the expected energy not provided during the event.3. The resilience characteristics of the network change as the event progresses. The network’s resilience equates to distinct network features for every stage of an event. Not including these features based on the advancement of events leads to an under-appreciated resilience assessment that may result in network failure.Limitation 2. Dimensions of critical resilience1. The impact of operational actions aided by smart technology on a network’s resilience during extreme events has not been thoroughly examined.2. Do not consider how quickly resilience degrades when an electric network is affected by an event or how lengthy the network remains in after event degradation phases before the recovery process begins and when it has been restored.3. Be unable to provide a complete view of a network’s resilience at all stages of an event.Furthermore, these metrics make assessment of the impact of multiple adaptation and resilience improvement techniques in each stage unclear. This must also be considered because multiple stages of resilience may be accomplished from active and passive resilience viewpoints and different actions will have different impacts on both. Relevant metrics capable of effectively capturing all of these complexities are likewise unavailable. Active resilience initiates strategies and procedures for achieving resilience during and after the extreme event, and passive resilience is the existence of characteristics that provide resilience to the distribution network by boosting or hardening critical segments of the network’s response to events.In view of the above mentioned, the present paper proposes a framework for evaluating the resilience of distribution networks based on two probabilistic resilience metrics: active and passive resilience. Active resilience is achieved using tie-line switches and distributed generation (DGs), and passive resilience is enhanced through distribution line hardening. We developed a Monte Carlo Simulation (MCS) approach to model the probabilistic damage scenarios and assessed the distribution resilience metric for various weather intensities to calculate these resilience metrics. The proposed metrics have also been widely utilized to support investment decisions to address the effects of low-probability high-impact event occurrences. Similar concerns extend to monitoring the effects of extreme weather incidents, making these metrics applicable for quantifying potential impacts and comparing the potential advantages of alternative planning strategies/investments.The specific contributions of this paper are as follows:(1)
Performance-based resilience metrics for distribution systems quantify active and passive resilience when extreme weather events influence networks. The proposed metrics demonstrate the possible consequences of a future extreme event, assess the predicted system performance loss during an extreme event, and quantify the impact of different resilience enhancement approaches.(2)
The influence of extreme weather on distribution network performance and the hazards introduced by such disasters on network resilience is evaluated using an approach based on MCS studies.(3)
The proposed approach is developed to evaluate the consequences of various infrastructure and operational solutions that a utility can use to improve network resilience.(4)
The resilience of distribution networks is calculated by means of the proposed metrics by modeling different levels of degradation and the consequences of potential restoration strategies.The proposed resilience metrics are defined in the next section. The complete process of resilience evaluation for the distribution system under extreme event is presented in Section3. The simulation results in case of IEEE 33 bus test system are presented in Section 4. Finally, conclusions are drawn.


## 2. Proposed Probabilistic Resilience Metrics

The multiphase resilience assessment and the time-dependent resilience metrics developed in this work to assess the resilience of a distribution system are explained in this section. The proposed time-dependent active and passive resilience metrics are defined by the distribution system performance loss (DSPL) caused by probabilistic extreme weather events and are used to assess the impact of different event intensities on the system. We define multiple terminologies needed to quantify the proposed resilience metrics for distribution systems when exposed to extreme weather occurrences. Because both metrics compute the high loss of load caused by these events, we also need a probabilistic event model and a method to estimate its effect on the performance loss DSPL function of the system.


### 2.1. Modeling the Probabilistic Extreme Weather Event’s Impact

An extreme weather eventEt is defined by the event’s intensity and the probability of its occurrence; both factors are a function of time. Wind speed’s probability density function (PDF) with a distinct geographical area is depicted in Figure 2. The X-axis represents wind speed, which is the intensity of the event, while the Y-axis is the likelihood of the event occurrence. PDFs of regional wind speed for three separate areas, observing extreme, high, and normal wind speeds (Figure 2), show that normal wind speed occurrence has a higher chance, whereas extreme wind speed event has a lower probability.Figure 2
Regional wind profile.The event’s intensity impacts the likelihood of the network failure, resulting in DSPL. When a random extreme eventE(t) affects network performance and is modeled in terms of demand load not delivered as a nonlinear function, L(t), the DSPL, that is, P(t), is defined as follows:(1)Pt=fEtLt.A network operational curve, generally referred to as a resilience curve, is used to predict the effects of a typical catastrophic incident on network performance [23]. The load demand not delivered, L(t), as a result of event E(t), is used to compute P(t) and the area under the network operational curve that quantifies the amount of energy that was not served (MWh) in the following event. However, preventive planning strategies like sophisticated recovery procedures and distribution line hardening can help to improve DSPL, and these impacts are represented accurately by the network operational curve.


### 2.2. Multiphase Resilience Assessment Framework

Figure3 shows a generalized resilience curve representing the different states where the distribution system endures following an extreme event and their time-based advancement.Figure 3
The multiphase resilience trapezoid.The different states, as represented in Figure3, are as follows.


#### 2.2.1. Phase I

Event progress (t∈te,tpe): during phase I, resilience levels decline from R0a and R0p (preevent resilience) to Rpea (postevent active resilience) and Rpep (postevent passive resilience). Rpea and Rpep may differ based on the network and the event’s intensity. The nature of the disturbance characterizes the initial effect. For instance, the network’s resilience drops sharply when an earthquake strikes, whereas a wind-based event may gradually affect the network as it moves regionally. When an event is in process, the load service is disrupted, and the loss characteristic rises. Proactive strategies can be implemented before a severe event occurs if the location and timing of the event can be correctly predicted. This will help to reduce L(t). Furthermore, increased robustness can reduce DSPL and potentially decrease the degrading resilience slope.


#### 2.2.2. Phase II

Postevent degraded state (t∈tpe,tar and t∈tpe,tpr for active and passive resilience, resp.): during phase II, the network remains in a postevent degraded state (i.e., Rpea and Rpep) until restored at tar and tpr, respectively, for active and passive resilience. Although the duration of this phase can vary based on the resilience strategies in restoration operation, a smart operational response may begin load recovery (i.e., active resilience) earlier than passive recovery, which is the desired circumstance.Whenever a distribution system component (line or/and pole) is damaged, repair action is initiated. The time to repair damaged component is a very significant concern from the point of resilience calculation. The time taken to repair it is determined by exponential distribution. The duration in normal weather is multiplied by a predefined range of random numbers with a uniform distribution to represent the expanding repair time for component damage resulting in more intense weather conditions. The time required to restore damaged components,RDChtime, in this paper is considered to be 10 hrs under normal weather situations. For different weather scenarios, the RDCtime is calculated using(2)RDCtime=RDChtime,ifw<20m/s,h1×RDChtime,if20m/s≤w<30m/s,h2×RDChtime,if30m/s≤w<40m/s,h3×RDChtime,ifw≥40m/s,where h1∼U2,3, h2∼U3,4, and h3∼U4,5 are random numbers generated within a predefined range. More appropriate recovery simulations can be developed if details on the network utility’s restore operation under extreme weather situations are available.


#### 2.2.3. Phase III

Restorative state (t∈tar,Tar and t∈tpr,Tpr for active and passive resilience, resp.): after the steps are taken in the postevent degraded state, the network reaches the restorative phase. An appropriate recovery approach is defined, and isolated loads are progressively recovered using DGs, network reconfiguration, and other smart operations. It is a systematic process in which various switching operations are conducted to reconfigure the network.The active resilience is recovered faster than passive resilience, as shown in most case study applications in real scenarios. Customers, for example, may reconnect mostly before the damaged distribution components are fully repaired. This is another factor why it is necessary to distinguish between active and passive resilience, analyze these concepts separately, and use different metrics to highlight different aspects related to these two types of resilience.


### 2.3. Active and Passive Resilience Metrics

The active and passive resilience metrics are determined by the network operation betweent∈te,tpr as indicated in phases I-II. DSPL, Pit, is used to determine the amount of energy that was not served (MWh) following an event. It is a multifunctional conception that needs both the demand load not delivered Lit and the time it takes to shift between two phases. As a result, operational responses throughout phases I-II of the resilience curves significantly impact active resilience. To assess resilience metrics, we first compute the DSPL function, Pit, by selecting a random event from PDF (Figure 2). The active and passive resilience in Figure 3 can be evaluated using equations (3a) and (3b) and equations (4a) and (4b), respectively:(3a)Rat=∫tetprP1tdt,(3b)P1t=tar−12tpe+teL1+12tpr−tarL3,(4a)Rpt=∫tetprP2tdt,(4b)P2t=tpr−12tpe+teL2.The time needed to transition fromtar to Tar is the same as the time taken to complete the switching operations for restoration using the proactive islanded mode configuration and the other responses for active resilience. The switching operation is considered sequential, with the meantime of operating a remote-controlled switch (RCS) and manual switch to be 15 seconds and 25 minutes, respectively.


## 2.1. Modeling the Probabilistic Extreme Weather Event’s Impact

An extreme weather eventEt is defined by the event’s intensity and the probability of its occurrence; both factors are a function of time. Wind speed’s probability density function (PDF) with a distinct geographical area is depicted in Figure 2. The X-axis represents wind speed, which is the intensity of the event, while the Y-axis is the likelihood of the event occurrence. PDFs of regional wind speed for three separate areas, observing extreme, high, and normal wind speeds (Figure 2), show that normal wind speed occurrence has a higher chance, whereas extreme wind speed event has a lower probability.Figure 2
Regional wind profile.The event’s intensity impacts the likelihood of the network failure, resulting in DSPL. When a random extreme eventE(t) affects network performance and is modeled in terms of demand load not delivered as a nonlinear function, L(t), the DSPL, that is, P(t), is defined as follows:(1)Pt=fEtLt.A network operational curve, generally referred to as a resilience curve, is used to predict the effects of a typical catastrophic incident on network performance [23]. The load demand not delivered, L(t), as a result of event E(t), is used to compute P(t) and the area under the network operational curve that quantifies the amount of energy that was not served (MWh) in the following event. However, preventive planning strategies like sophisticated recovery procedures and distribution line hardening can help to improve DSPL, and these impacts are represented accurately by the network operational curve.


## 2.2. Multiphase Resilience Assessment Framework

Figure3 shows a generalized resilience curve representing the different states where the distribution system endures following an extreme event and their time-based advancement.Figure 3
The multiphase resilience trapezoid.The different states, as represented in Figure3, are as follows.


### 2.2.1. Phase I

Event progress (t∈te,tpe): during phase I, resilience levels decline from R0a and R0p (preevent resilience) to Rpea (postevent active resilience) and Rpep (postevent passive resilience). Rpea and Rpep may differ based on the network and the event’s intensity. The nature of the disturbance characterizes the initial effect. For instance, the network’s resilience drops sharply when an earthquake strikes, whereas a wind-based event may gradually affect the network as it moves regionally. When an event is in process, the load service is disrupted, and the loss characteristic rises. Proactive strategies can be implemented before a severe event occurs if the location and timing of the event can be correctly predicted. This will help to reduce L(t). Furthermore, increased robustness can reduce DSPL and potentially decrease the degrading resilience slope.


### 2.2.2. Phase II

Postevent degraded state (t∈tpe,tar and t∈tpe,tpr for active and passive resilience, resp.): during phase II, the network remains in a postevent degraded state (i.e., Rpea and Rpep) until restored at tar and tpr, respectively, for active and passive resilience. Although the duration of this phase can vary based on the resilience strategies in restoration operation, a smart operational response may begin load recovery (i.e., active resilience) earlier than passive recovery, which is the desired circumstance.Whenever a distribution system component (line or/and pole) is damaged, repair action is initiated. The time to repair damaged component is a very significant concern from the point of resilience calculation. The time taken to repair it is determined by exponential distribution. The duration in normal weather is multiplied by a predefined range of random numbers with a uniform distribution to represent the expanding repair time for component damage resulting in more intense weather conditions. The time required to restore damaged components,RDChtime, in this paper is considered to be 10 hrs under normal weather situations. For different weather scenarios, the RDCtime is calculated using(2)RDCtime=RDChtime,ifw<20m/s,h1×RDChtime,if20m/s≤w<30m/s,h2×RDChtime,if30m/s≤w<40m/s,h3×RDChtime,ifw≥40m/s,where h1∼U2,3, h2∼U3,4, and h3∼U4,5 are random numbers generated within a predefined range. More appropriate recovery simulations can be developed if details on the network utility’s restore operation under extreme weather situations are available.


### 2.2.3. Phase III

Restorative state (t∈tar,Tar and t∈tpr,Tpr for active and passive resilience, resp.): after the steps are taken in the postevent degraded state, the network reaches the restorative phase. An appropriate recovery approach is defined, and isolated loads are progressively recovered using DGs, network reconfiguration, and other smart operations. It is a systematic process in which various switching operations are conducted to reconfigure the network.The active resilience is recovered faster than passive resilience, as shown in most case study applications in real scenarios. Customers, for example, may reconnect mostly before the damaged distribution components are fully repaired. This is another factor why it is necessary to distinguish between active and passive resilience, analyze these concepts separately, and use different metrics to highlight different aspects related to these two types of resilience.


## 2.2.1. Phase I

Event progress (t∈te,tpe): during phase I, resilience levels decline from R0a and R0p (preevent resilience) to Rpea (postevent active resilience) and Rpep (postevent passive resilience). Rpea and Rpep may differ based on the network and the event’s intensity. The nature of the disturbance characterizes the initial effect. For instance, the network’s resilience drops sharply when an earthquake strikes, whereas a wind-based event may gradually affect the network as it moves regionally. When an event is in process, the load service is disrupted, and the loss characteristic rises. Proactive strategies can be implemented before a severe event occurs if the location and timing of the event can be correctly predicted. This will help to reduce L(t). Furthermore, increased robustness can reduce DSPL and potentially decrease the degrading resilience slope.


## 2.2.2. Phase II

Postevent degraded state (t∈tpe,tar and t∈tpe,tpr for active and passive resilience, resp.): during phase II, the network remains in a postevent degraded state (i.e., Rpea and Rpep) until restored at tar and tpr, respectively, for active and passive resilience. Although the duration of this phase can vary based on the resilience strategies in restoration operation, a smart operational response may begin load recovery (i.e., active resilience) earlier than passive recovery, which is the desired circumstance.Whenever a distribution system component (line or/and pole) is damaged, repair action is initiated. The time to repair damaged component is a very significant concern from the point of resilience calculation. The time taken to repair it is determined by exponential distribution. The duration in normal weather is multiplied by a predefined range of random numbers with a uniform distribution to represent the expanding repair time for component damage resulting in more intense weather conditions. The time required to restore damaged components,RDChtime, in this paper is considered to be 10 hrs under normal weather situations. For different weather scenarios, the RDCtime is calculated using(2)RDCtime=RDChtime,ifw<20m/s,h1×RDChtime,if20m/s≤w<30m/s,h2×RDChtime,if30m/s≤w<40m/s,h3×RDChtime,ifw≥40m/s,where h1∼U2,3, h2∼U3,4, and h3∼U4,5 are random numbers generated within a predefined range. More appropriate recovery simulations can be developed if details on the network utility’s restore operation under extreme weather situations are available.


## 2.2.3. Phase III

Restorative state (t∈tar,Tar and t∈tpr,Tpr for active and passive resilience, resp.): after the steps are taken in the postevent degraded state, the network reaches the restorative phase. An appropriate recovery approach is defined, and isolated loads are progressively recovered using DGs, network reconfiguration, and other smart operations. It is a systematic process in which various switching operations are conducted to reconfigure the network.The active resilience is recovered faster than passive resilience, as shown in most case study applications in real scenarios. Customers, for example, may reconnect mostly before the damaged distribution components are fully repaired. This is another factor why it is necessary to distinguish between active and passive resilience, analyze these concepts separately, and use different metrics to highlight different aspects related to these two types of resilience.


## 2.3. Active and Passive Resilience Metrics

The active and passive resilience metrics are determined by the network operation betweent∈te,tpr as indicated in phases I-II. DSPL, Pit, is used to determine the amount of energy that was not served (MWh) following an event. It is a multifunctional conception that needs both the demand load not delivered Lit and the time it takes to shift between two phases. As a result, operational responses throughout phases I-II of the resilience curves significantly impact active resilience. To assess resilience metrics, we first compute the DSPL function, Pit, by selecting a random event from PDF (Figure 2). The active and passive resilience in Figure 3 can be evaluated using equations (3a) and (3b) and equations (4a) and (4b), respectively:(3a)Rat=∫tetprP1tdt,(3b)P1t=tar−12tpe+teL1+12tpr−tarL3,(4a)Rpt=∫tetprP2tdt,(4b)P2t=tpr−12tpe+teL2.The time needed to transition fromtar to Tar is the same as the time taken to complete the switching operations for restoration using the proactive islanded mode configuration and the other responses for active resilience. The switching operation is considered sequential, with the meantime of operating a remote-controlled switch (RCS) and manual switch to be 15 seconds and 25 minutes, respectively.


## 3. Fragility Modeling and Monte Carlo Simulation to Characterize Resilience Metrics

This section discusses how to compute the resilience metric by assessing the damage caused by the probabilistic event. The PDF for DSPLPt, when influenced by an extreme weather event, is required by the resilience metrics specified in the previous section. The PDF for the specific event for a particular location is used to calculate the probability and intensity of an event Et. Figure 2 presents a PDF of three different regions’ wind speeds, having normal, high, and extreme speeds. The effect of an event on the distribution system is utilized to calculate the DSPL function Pt. First, the component level is used to estimate the probability of influence of an extreme weather event, in this scenario, a wind-related occurrence. Following that, a framework is created to represent the weather event’s system effect.Fragility curves at the component level have been developed in related work to predict the effects of high wind speed occurrences on the electricity network [24, 25]. A fragility curve, in particular, depicts the likelihood of electrical network component failure as a function of risk severity, with the curve shape varying and depending on the application. Figure 4 represents the fragility curve, which analytically expresses the likelihood of failure of distribution network components as a function of wind speed, as presented by (5)Pcw=0,ifw<wcritical,Pcw,ifwcritical≤w<wcollapse,1,ifw>wcollapse,where Pcw is a component’s failure probability at any wind speed w; the probability of failure increases rapidly at critical wind speed wcritical. At wcollapse, distribution network components have a very low chance of survival. Empirical data obtained during catastrophic occurrences can be used to build the fragility curve. Damage assessments for a particular wind profile can be derived from observed damage in previous extreme weather events. Damage assessments can be combined with network infrastructure and spatial data to create an analogous fragility curve for distribution network components. The fragility curve values are chosen at random in this work for simulation purpose.Figure 4
Fragility curve [23].The failure probability of the components of distribution network for any event,Pcw, is utilized to characterize the current status of the specific components. A random sample with a uniform distribution number, r∼U0,1, is produced for each component. The current status of the component, Fcw, is defined by comparing these random numbers to the event-based failure probability, Pcw, for a particular wind speed w. Fcw is obtained using (6)Fcw=0,ifPcw<r,1,ifPcw>r,where Fcw is the distribution network component's failure function, Fcw=1 indicates component failure, and Fcw=0 means that the component will remain in a healthy state. The operating status of all distribution network components that are influenced by the extreme event is obtained.To compute the network loss function, the proposed method uses a fragility curve. The overall procedure introduced in Section3 is presented in Figure 5. The approach needs weather details and a complete network model and generates resilience metrics as outputs. Because extreme weather severity and its influence on a component are not predictable, the probabilistic implications of an event on the distribution network are evaluated using an MCS approach, commonly used in bulk-system reliability calculations [26, 27]. MCSs are ideal for assessing and developing multidimensional networks, despite their computing requirements, as evidenced by associated important research in this area [28, 29]. The methodology is generally applicable and comparatively more accurate because it does not make any assumptions about probabilistic occurrences, as is common in comparable analytical approaches to decrease the difficulty of probabilistic calculations. This paper assesses a distribution network’s resilience to future severe occurrences. The Monte Carlo approach allows for modeling such unexpected events while including the low likelihood of approaching the events, resulting in a realistic evaluation of hazards related to the extreme circumstances.Figure 5
Flowchart for computation of resilience metrics by proposed probabilistic method.The PDF of relevant weather occurrence for selected region is generated from weather information gathered by meteorological indicators. The failure probability of the components,Pcw, is calculated by plotting the geographical wind speed PDF to the fragility curves for distribution network components. The fragility curve will shift rightward as the distribution lines are hardened (and vice versa), enabling components to be more resilient to higher intensity meteorological events as shown in Figure 6. As a result, when hardened, the likelihood of failure of the associated lines is reduced.Figure 6
The basic case and robust case scenarios used wind fragility curves.MCSs are conducted to predict damage probabilities for a specific wind speed. The component failure probability,Pcw, is utilized in every trial to evaluate the operating situation of the specific component Fcw using (6). The operating status of all distribution network components is created using a similar approach, defining the damage scenario for a specific MCS. The resilience curve is built using the provided damage scenario to determine the DSPL function. The restoration issue is addressed as a mixed-integer linear program (MILP) with the goal of maximizing the load recovered using all accessible feeders and DGs with premeditated island initiation [30]. All three phases of the resilience curve are evaluated after the smart operations are executed, and resilience metrics are computed using (3) and (4). To generate statistically relevant results, several MCSs are performed. The procedure is performed for several different wind speeds. After running a sufficient number of MCSs, the average demand load loss, Lt, for each wind speed sample is calculated. The average Lt is then translated onto the PDF for the meteorological conditions to obtain a probabilistic characterization of DSPL, Pt, when subjected to a specific event. The PDF for the DSPL function and the energy not served (ENS) during events (in MWh) is obtained at the completion of MCSs for all sampling intensities of the meteorological event. The ENS (MWh) is used in this paper to assess the active and passive resilience of the distribution network subjected to a severe windstorm.


## 4. Application for Test System

### 4.1. Test System

The proposed approach is numerically tested on a modified IEEE 33-bus test system under the assumption that it is operating in isolation. The single-phase diagram of the system is presented in Figure7. The load data and feeder parameters for the system are taken from reference [30]. In the preevent setup, all tie-line switches are considered to be open. MCSs are run on a PC with a 2666 MHz processor and 8 GB of RAM. For simulation, the following assumptions are made.(1)
In the base scenario, we only evaluate extreme wind profiles (Figure2) and utilize the same fragility curves for all distribution network components. It covers a smaller geographic region and observes a uniform meteorological state when considering the failure probability of distribution components.(2)
We considered no recovery action is conducted while the extreme weather event is ongoing (te to tpe) and also assume that all distribution components are active before the event.(3)
Three tie switches are used with restoration scenarios to model proactive interruption operations for active resilience. Utility-owned DGs with network formation are used to aid the restoration process through premeditated islanding for the IEEE 33-bus test system. Fifteen overhead lines are selected randomly and hardened to model proactive interruption operations for passive resilience (Figure7).(4)
The availability of resources, such as repair crews (RCs), for responding fast and efficiently to damaged components following weather events is crucial. However, we assume that only four damaged components can be repaired simultaneously due to the limitation of RCs.Figure 7
The modified IEEE 33 bus system.


### 4.2. Simulation Results and Discussion

Initially, sufficient MCSs are run to determine the number of trials necessary to converge the network failure probability for extreme wind profile (Figure2) and DSPL function for a specific damage scenario. As shown in Figures 8–10, it can be seen that 1000 trials are sufficient for convergence in all scenarios. The procedures for conducting the simulation are described in detail here. First, we take a wind speed sample from the PDF for the extreme wind profile and calculate the failure probability for the distribution network component as shown in Figure 8. The MCSs converge after 1000 trials, and the failure probability for the distribution network component is about 0.23, 0.152, and 0.303 for extreme wind profile events, extreme wind profile events with 20% more robust, and extreme wind profile events with 20% less robust system, respectively.Figure 8
Monte Carlo Simulation convergence for case I: (a) extreme wind profile event, (b) extreme wind profile event with 20% more robust system, and (c) extreme wind profile event with 20% less robust system.
(a)(b)(c)Figure 9
ENS for multiple scenarios and DSPL convergence of Monte Carlo Simulations for case I: (a) extreme wind profile event, (b) extreme wind profile event with 20% more robust system, and (c) extreme wind profile event with 20% less robust system.
(a)(b)(c)Figure 10
ENS for multiple scenarios and DSPL convergence of Monte Carlo Simulations for case II: (a) 20 m/s wind speed, (b) 30 m/s wind speed, and (c) 40 m/s wind speed.
(a)(b)(c)For the sampled wind data, we run a total of 1000 MCSs. The DSPL is calculated for each trial by analyzing the different phases of the resilience curve, as described in Section2, and the DSPL functions computed for the 1000 MCSs are averaged. Multiple wind data are sampled from the PDF for the extreme wind profile, and the process is continued. Finally, calculate the ENS during phases I and II after obtaining the average DSPL function. This aids in evaluating the system’s performance to different extreme weather event intensities. The event is assumed to last 8 hours, with hourly time resolution, the severe windstorm strikes the system 24 hours after the simulation begins, and the simulation framework determines the duration of phases II and III.Various operational and infrastructure factors can influence the resilience performance of a distribution network subjected to extreme weather events. Different case studies have been methodically built in this context to analyze the influence of these characteristics on the test system’s resilience, as measured by the proposed active and passive resilience metric framework, and test its appropriateness. Specifically, the following illustrative cases are considered:Case I: the resilience trapezoid is simulated using Section4.1 data and 20% more and less robust distribution network component to the windstorm by modifying the fragility curves (Figure 6).Case II: three wind speed profiles are tested on a modified 33-bus test system (Figure7), with maximum wind speeds of 20 m/s, 30 m/s, and 40 m/s, respectively.Figures9 and 10 show the numerical values for the different phases of the resilience curve under study, that is, the DSPL function (KW), Pt, based on the computed numeric value of phase I and the ENS (KWh) during phases I and II, for cases scenarios I and II, respectively. For case I, making distribution lines more robust decreases the likelihood of line failure during a windstorm, reducing DSPL in phase I and reducing the maximum values of ENS (Figure 9(b)) for multiple scenarios and vice versa (Figure 9(c)). After 1000 case scenarios, the simulation result shows that the average DSPL values for extreme wind profile events, extreme wind profile events with 20% more robust, and extreme wind profile events with 20% less robust system are about 1121 kW, 918 kW, and 1411 kW, respectively.As expected, for case II, higher windstorm intensity (i.e., greater wind speed) increases the likelihood of lines failure, increasing the average DSPL function value (approximately 907 kW, 1983 kW, and 2354 kW for 20 m/s, 30 m/s, and 40 m/s wind speed event, resp.) and the ENS (Figure10). Also, it has been observed that making the smart operation by enhancing performance and proactive islanding-based restoration strategies (i.e., DGs and tie-line switches) enables a large amount of load to be restored during phase II for active resilience, as shown in Figure 11. It is worth noting that a maximum wind speed of 40 m/s event restores less load than a maximum wind speed of 30 m/s events (Figure 11(a)), although the event’s maximum wind speed of 40 m/s results in more DSPL function (Figures 10(b) and 10(c)). This is because, based on real-time data from local stations and weather prediction data, the highest sustained wind speed is expected to be about 45 m/s [31], and for the highest wind speed of 40 m/s, maximum nonhardened lines (Figure 7) are damaged and unable to restore a large amount of load.Figure 11
Load restoration by active resilience for (a) case I (extreme wind profile event, extreme wind profile event with 20% more robust system, and extreme wind profile event with 20% less robust system) and (b) case II (20 m/s, 30 m/s, and 40 m/s wind speed).
(a)(b)Passive resilience is calculated in this paper when no restoration options (tie-line switches and DGs) are applied. Tables 1 and 2 represent the active and passive resilience during phases I and II for different scenarios in cases I and II, respectively. It is observed that the predicted resilience loss (MWh) is higher for passive resilience and smaller for active resilience for each scenario in both cases, and a more robust network and less intensity wind speed event has less resilience loss. This demonstrates that the proposed method can accurately estimate the resilience of a particular geographic area to meteorological occurrences.Figures12 and 13 show the active distribution lines throughout all phases of the extreme event, with and without tie-line switches and DGs, for the different scenarios in cases I and II, respectively. These curves closely resemble the resilience trapezoid of Figure 3 (Figure 12(a) and 13(a) for passive resilience and Figure 12(b) and 13(b) for passive resilience), enabling adequate modeling and characterization; that is, the three phases of event progression, postevent degraded state, and restoration state can all be differentiated. The duration of phase I is predefined and equivalent to the duration of the severe weather event (in this study, 8 hours); however, the duration of phases II and III is dependent on the electricity utility’s responses (i.e., the tie-line switches, DGs, and RCs case assumed in this study). Phase II begins when the extreme event ends (tpe) and finishes when active and passive resilience starts to recover (tar and tpr, resp.), for example, after a demand or a damaged component has been recovered. Furthermore, phase III begins at tar and tpr and finishes at Tar and Tpr, that is, when active and passive resilience have recovered completely. It can be observed that active resilience recovers a large amount of load much faster than passive resilience. This demonstrates the significance of differentiating between the concepts of active and passive resilience and the use of different resilience factors to quantify both. It should be noted that when wind speeds increase, the resilience degradation slopes and the level of phase I of the resilience trapezoid increase (Figure 13). Furthermore, it is clear that the repair RCs’ limitations have the most significant impact on the restoration capabilities of the active and passive resilience restoration slopes. Additionally, the more the RCs there is, the more the distribution component that can be repaired at one time is and therefore the higher the restoration slopes of both resilience metrics are.Figure 12
Active distribution lines during the extreme event (phases I, II, and III) for different scenarios in case I: (a) without tie-line switches and DGs and (b) with tie-line switches and DGs.
(a)(b)Figure 13
Active distribution lines during the extreme event (phases I, II, and III) for different scenarios in case II: (a) without tie-line switches and DGs and (b) with tie-line switches and DGs.
(a)(b)The values of the active and passive resilience, as defined by equation (3) and equation (4), are presented in Tables 2 and 3 during phase I, that is, extreme event progress state, and phase II, that is, postevent degrade state for both cases. From Table 2, we see that both the active and passive resilience values improve (lower numerical value indicates improved resilience) when the system is made 20% more robust than the base case by hardening. The resilience values degrade when the robustness of the system is decreased by 20%; that is, the lines are weakened.Table 2
Active and passive resilience during phases I and II for different scenarios in case I.
EventActive resilienceRat (MWh)Passive resilienceRpt (MWh)Base20.7343.4820% more robust15.3335.5220% less robust29.6857.16Table 3
Active and passive resilience during phases I and II for different scenarios in case II.
EventActive resilienceRat (MWh)Passive resilienceRpt (MWh)20 m/s4.5811.3730 m/s51.1675.7140 m/s89.42117.16Similarly, active and passive resilience values degrade (numerical value increases) when the wind speed increases, as shown in Table3. These results are as per expectation for a distribution system under extreme wind condition.During load restoration of active resilience in phase II (Figure3), it is realized that changing the location of the DG causes a new value for the resilience metric. This is due to the fact that when the DG location is changed, a new DG-supported island is generated to pick loads during the recovery process following an event. Hence, optimizing the placement of DG is critical to the overall resilience of the network. The primary purpose of this paper is to present probabilistic distribution network metrics and illustrate their application in evaluating network resilience during extreme weather events and under various resource allocations to increase resilience. However, we do not optimize DG position or capacity in this paper; instead, we show how to use the metric to quantify operational resilience for any network configuration by randomly placing and scaling the DGs.The resources allocated to resilience improvement can either mitigate the effects of a catastrophic event or accelerate the network’s recovery process. Minimizing predicted resilience loss necessitates an optimal allocation of operational strategies that reduce both network losses and restoration time at the same time. However, we do not evaluate the related costs of investment options in this paper, and it is widely acknowledged that hardening procedures are much more costly than smart operational investments, although smart operational procedures are not sufficient to keep the lights on in the case of a major disaster [19]. Moreover, as shown in Figure 14, the appropriate network response varies between smart operations and network hardening options depending on the intensity of the event. Due to the unpredictable nature of weather events and their impacts, having an appropriate planning strategy that includes both smart operations and hardening can give the best-suited road-map for building resilience.Figure 14
Expected resilience loss for different wind speeds for IEEE 33-bus test scenario.The proposed active and passive resilience metrics can be used to allocate resources appropriately for distinct planning processes in order to improve network resilience. Notably, the proposed resilience metric will be utilized to prepare resources to increase network performance when natural disasters occur. This necessitates a method for quantifying the reductions in hazards related to the severe occurrences under a particular resource allocation. These features are precisely modeled and quantified in the proposed resilience metrics, making them useful for resilience aid operations.


## 4.1. Test System

The proposed approach is numerically tested on a modified IEEE 33-bus test system under the assumption that it is operating in isolation. The single-phase diagram of the system is presented in Figure7. The load data and feeder parameters for the system are taken from reference [30]. In the preevent setup, all tie-line switches are considered to be open. MCSs are run on a PC with a 2666 MHz processor and 8 GB of RAM. For simulation, the following assumptions are made.(1)
In the base scenario, we only evaluate extreme wind profiles (Figure2) and utilize the same fragility curves for all distribution network components. It covers a smaller geographic region and observes a uniform meteorological state when considering the failure probability of distribution components.(2)
We considered no recovery action is conducted while the extreme weather event is ongoing (te to tpe) and also assume that all distribution components are active before the event.(3)
Three tie switches are used with restoration scenarios to model proactive interruption operations for active resilience. Utility-owned DGs with network formation are used to aid the restoration process through premeditated islanding for the IEEE 33-bus test system. Fifteen overhead lines are selected randomly and hardened to model proactive interruption operations for passive resilience (Figure7).(4)
The availability of resources, such as repair crews (RCs), for responding fast and efficiently to damaged components following weather events is crucial. However, we assume that only four damaged components can be repaired simultaneously due to the limitation of RCs.Figure 7
The modified IEEE 33 bus system.


## 4.2. Simulation Results and Discussion

Initially, sufficient MCSs are run to determine the number of trials necessary to converge the network failure probability for extreme wind profile (Figure2) and DSPL function for a specific damage scenario. As shown in Figures 8–10, it can be seen that 1000 trials are sufficient for convergence in all scenarios. The procedures for conducting the simulation are described in detail here. First, we take a wind speed sample from the PDF for the extreme wind profile and calculate the failure probability for the distribution network component as shown in Figure 8. The MCSs converge after 1000 trials, and the failure probability for the distribution network component is about 0.23, 0.152, and 0.303 for extreme wind profile events, extreme wind profile events with 20% more robust, and extreme wind profile events with 20% less robust system, respectively.Figure 8
Monte Carlo Simulation convergence for case I: (a) extreme wind profile event, (b) extreme wind profile event with 20% more robust system, and (c) extreme wind profile event with 20% less robust system.
(a)(b)(c)Figure 9
ENS for multiple scenarios and DSPL convergence of Monte Carlo Simulations for case I: (a) extreme wind profile event, (b) extreme wind profile event with 20% more robust system, and (c) extreme wind profile event with 20% less robust system.
(a)(b)(c)Figure 10
ENS for multiple scenarios and DSPL convergence of Monte Carlo Simulations for case II: (a) 20 m/s wind speed, (b) 30 m/s wind speed, and (c) 40 m/s wind speed.
(a)(b)(c)For the sampled wind data, we run a total of 1000 MCSs. The DSPL is calculated for each trial by analyzing the different phases of the resilience curve, as described in Section2, and the DSPL functions computed for the 1000 MCSs are averaged. Multiple wind data are sampled from the PDF for the extreme wind profile, and the process is continued. Finally, calculate the ENS during phases I and II after obtaining the average DSPL function. This aids in evaluating the system’s performance to different extreme weather event intensities. The event is assumed to last 8 hours, with hourly time resolution, the severe windstorm strikes the system 24 hours after the simulation begins, and the simulation framework determines the duration of phases II and III.Various operational and infrastructure factors can influence the resilience performance of a distribution network subjected to extreme weather events. Different case studies have been methodically built in this context to analyze the influence of these characteristics on the test system’s resilience, as measured by the proposed active and passive resilience metric framework, and test its appropriateness. Specifically, the following illustrative cases are considered:Case I: the resilience trapezoid is simulated using Section4.1 data and 20% more and less robust distribution network component to the windstorm by modifying the fragility curves (Figure 6).Case II: three wind speed profiles are tested on a modified 33-bus test system (Figure7), with maximum wind speeds of 20 m/s, 30 m/s, and 40 m/s, respectively.Figures9 and 10 show the numerical values for the different phases of the resilience curve under study, that is, the DSPL function (KW), Pt, based on the computed numeric value of phase I and the ENS (KWh) during phases I and II, for cases scenarios I and II, respectively. For case I, making distribution lines more robust decreases the likelihood of line failure during a windstorm, reducing DSPL in phase I and reducing the maximum values of ENS (Figure 9(b)) for multiple scenarios and vice versa (Figure 9(c)). After 1000 case scenarios, the simulation result shows that the average DSPL values for extreme wind profile events, extreme wind profile events with 20% more robust, and extreme wind profile events with 20% less robust system are about 1121 kW, 918 kW, and 1411 kW, respectively.As expected, for case II, higher windstorm intensity (i.e., greater wind speed) increases the likelihood of lines failure, increasing the average DSPL function value (approximately 907 kW, 1983 kW, and 2354 kW for 20 m/s, 30 m/s, and 40 m/s wind speed event, resp.) and the ENS (Figure10). Also, it has been observed that making the smart operation by enhancing performance and proactive islanding-based restoration strategies (i.e., DGs and tie-line switches) enables a large amount of load to be restored during phase II for active resilience, as shown in Figure 11. It is worth noting that a maximum wind speed of 40 m/s event restores less load than a maximum wind speed of 30 m/s events (Figure 11(a)), although the event’s maximum wind speed of 40 m/s results in more DSPL function (Figures 10(b) and 10(c)). This is because, based on real-time data from local stations and weather prediction data, the highest sustained wind speed is expected to be about 45 m/s [31], and for the highest wind speed of 40 m/s, maximum nonhardened lines (Figure 7) are damaged and unable to restore a large amount of load.Figure 11
Load restoration by active resilience for (a) case I (extreme wind profile event, extreme wind profile event with 20% more robust system, and extreme wind profile event with 20% less robust system) and (b) case II (20 m/s, 30 m/s, and 40 m/s wind speed).
(a)(b)Passive resilience is calculated in this paper when no restoration options (tie-line switches and DGs) are applied. Tables 1 and 2 represent the active and passive resilience during phases I and II for different scenarios in cases I and II, respectively. It is observed that the predicted resilience loss (MWh) is higher for passive resilience and smaller for active resilience for each scenario in both cases, and a more robust network and less intensity wind speed event has less resilience loss. This demonstrates that the proposed method can accurately estimate the resilience of a particular geographic area to meteorological occurrences.Figures12 and 13 show the active distribution lines throughout all phases of the extreme event, with and without tie-line switches and DGs, for the different scenarios in cases I and II, respectively. These curves closely resemble the resilience trapezoid of Figure 3 (Figure 12(a) and 13(a) for passive resilience and Figure 12(b) and 13(b) for passive resilience), enabling adequate modeling and characterization; that is, the three phases of event progression, postevent degraded state, and restoration state can all be differentiated. The duration of phase I is predefined and equivalent to the duration of the severe weather event (in this study, 8 hours); however, the duration of phases II and III is dependent on the electricity utility’s responses (i.e., the tie-line switches, DGs, and RCs case assumed in this study). Phase II begins when the extreme event ends (tpe) and finishes when active and passive resilience starts to recover (tar and tpr, resp.), for example, after a demand or a damaged component has been recovered. Furthermore, phase III begins at tar and tpr and finishes at Tar and Tpr, that is, when active and passive resilience have recovered completely. It can be observed that active resilience recovers a large amount of load much faster than passive resilience. This demonstrates the significance of differentiating between the concepts of active and passive resilience and the use of different resilience factors to quantify both. It should be noted that when wind speeds increase, the resilience degradation slopes and the level of phase I of the resilience trapezoid increase (Figure 13). Furthermore, it is clear that the repair RCs’ limitations have the most significant impact on the restoration capabilities of the active and passive resilience restoration slopes. Additionally, the more the RCs there is, the more the distribution component that can be repaired at one time is and therefore the higher the restoration slopes of both resilience metrics are.Figure 12
Active distribution lines during the extreme event (phases I, II, and III) for different scenarios in case I: (a) without tie-line switches and DGs and (b) with tie-line switches and DGs.
(a)(b)Figure 13
Active distribution lines during the extreme event (phases I, II, and III) for different scenarios in case II: (a) without tie-line switches and DGs and (b) with tie-line switches and DGs.
(a)(b)The values of the active and passive resilience, as defined by equation (3) and equation (4), are presented in Tables 2 and 3 during phase I, that is, extreme event progress state, and phase II, that is, postevent degrade state for both cases. From Table 2, we see that both the active and passive resilience values improve (lower numerical value indicates improved resilience) when the system is made 20% more robust than the base case by hardening. The resilience values degrade when the robustness of the system is decreased by 20%; that is, the lines are weakened.Table 2
Active and passive resilience during phases I and II for different scenarios in case I.
EventActive resilienceRat (MWh)Passive resilienceRpt (MWh)Base20.7343.4820% more robust15.3335.5220% less robust29.6857.16Table 3
Active and passive resilience during phases I and II for different scenarios in case II.
EventActive resilienceRat (MWh)Passive resilienceRpt (MWh)20 m/s4.5811.3730 m/s51.1675.7140 m/s89.42117.16Similarly, active and passive resilience values degrade (numerical value increases) when the wind speed increases, as shown in Table3. These results are as per expectation for a distribution system under extreme wind condition.During load restoration of active resilience in phase II (Figure3), it is realized that changing the location of the DG causes a new value for the resilience metric. This is due to the fact that when the DG location is changed, a new DG-supported island is generated to pick loads during the recovery process following an event. Hence, optimizing the placement of DG is critical to the overall resilience of the network. The primary purpose of this paper is to present probabilistic distribution network metrics and illustrate their application in evaluating network resilience during extreme weather events and under various resource allocations to increase resilience. However, we do not optimize DG position or capacity in this paper; instead, we show how to use the metric to quantify operational resilience for any network configuration by randomly placing and scaling the DGs.The resources allocated to resilience improvement can either mitigate the effects of a catastrophic event or accelerate the network’s recovery process. Minimizing predicted resilience loss necessitates an optimal allocation of operational strategies that reduce both network losses and restoration time at the same time. However, we do not evaluate the related costs of investment options in this paper, and it is widely acknowledged that hardening procedures are much more costly than smart operational investments, although smart operational procedures are not sufficient to keep the lights on in the case of a major disaster [19]. Moreover, as shown in Figure 14, the appropriate network response varies between smart operations and network hardening options depending on the intensity of the event. Due to the unpredictable nature of weather events and their impacts, having an appropriate planning strategy that includes both smart operations and hardening can give the best-suited road-map for building resilience.Figure 14
Expected resilience loss for different wind speeds for IEEE 33-bus test scenario.The proposed active and passive resilience metrics can be used to allocate resources appropriately for distinct planning processes in order to improve network resilience. Notably, the proposed resilience metric will be utilized to prepare resources to increase network performance when natural disasters occur. This necessitates a method for quantifying the reductions in hazards related to the severe occurrences under a particular resource allocation. These features are precisely modeled and quantified in the proposed resilience metrics, making them useful for resilience aid operations.


## 5. Conclusion

This paper quantifies the resilience of distribution systems using probabilistic metrics and a comprehensive simulation approach. The estimated resilience loss resulting from extreme weather events is represented using two system performance-based metrics: active and passive resilience. These metrics effectively evaluate the performance resilience of the electricity distribution system since they quantify the worst consequences of low-probability, higher intensity events such as extreme weather events. We also layout a method for calculating the impact of proactive operational approaches and network hardening options on enhancing network resilience. The proposed resilience metrics appear suitable for comparing potential resilience enhancement approaches based on simulation results. The proposed methodology demonstrates that load loss caused by extreme events can be minimized, and the power supply can be maintained to the maximum level possible. This approach could be utilized as a platform for further study into various resilience-enhancing methods and a reference for a more resilient distribution system. Finally, it can be concluded that a combination of smart operation and hardened networks may provide higher resilience while reducing the investment costs for potential resilience enhancement actions. This work in progress aims to demonstrate how the proposed resilience metrics can be utilized in a comprehensive cost/benefit assessment approach to help statistical decision-making for resilience and economic purposes. Future research will include cosimulation of distribution and transmission systems and how to integrate resilience-enhancing schemes with the costs associated with achieving the most cost-effective progress, which will aid in understanding the impact of a wide range of factors that can influence a power system’s overall resilience performance.



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*Source: 3838695-2022-06-01.xml*