1 Introduction

Overhead electrical power distribution systems (PDS) are very susceptible to extreme wind hazards such as those associated with hurricanes and typhoons. Wind-caused disruption of PDS is responsible for most of the power outages in the United States. Power outages can cause catastrophic consequences, including economic losses, loss of critical services (for example, water, transportation, communication, and healthcare), and disruption to daily life. For example, Hurricane Sandy in 2012 left over 8.5 million customers across 21 states without power and caused USD 70 billion in damage, and 233 fatalities (Henry and Ramirez-Marquez 2016). During Hurricane Ida in 2021, up to 1.2 million customers lost their power across eight states in the United States (Shultz et al. 2022).

Extreme wind events such as hurricanes and typhoons can cause widespread damage to power poles, lines, and trees (Kocatepe et al. 2019). The falling trees or limbs can further cause extensive disruption to overhead PDS. For example, tree failures are responsible for about 55.2% of PDS failures in the Northeast United States (Li et al. 2014). Despite severe consequences caused by falling trees, the tree-failure risk has not been systematically considered in previous studies on resilience modeling of PDS against extreme winds. Most previous resilience assessment frameworks for PDS only considered wind-induced component failures, neglecting failures caused by falling trees (Salman et al. 2015; Unnikrishnan and van de Lindt 2016; Panteli et al. 2017; Hughes et al. 2021; Mohammadi Darestani et al. 2021). Although some studies considered the tree-failure risk, there are limitations in these studies. First, reasonable fragility models of tree failures and PDS component failures caused by falling trees were lacking (Ma et al. 2018; Tari et al. 2021); Second, the resilience assessment of PDS was only focused on the disruption period without considering the recovery period (Yuan et al. 2018; Hou and Chen 2020; Lu and Zhang 2022). Therefore, an integrated resilience assessment framework that can systematically consider the tree-failure risk is needed.

To model the resilience of PDS while considering the tree-failure risk, it is necessary to develop fragility models of trees first, which provide the likelihood of tree damage for a given hazard intensity. Fragility models of trees are categorized into two groups based on the approaches used—namely, empirical and analytical fragility models. Empirical fragility models are developed using historical tree damage data, while analytical models are developed by numerically simulating the response of trees with physics-based models. For example, Canham et al. (2001) developed windthrow fragility curves of seven forest tree species using windthrow data from a windstorm. The windthrow fragility curves are functions of species-specific parameters, storm severity index, and diameter at breast height (DBH). Proulx and Greene (2001) investigated the relationship between ice thickness and northern hardwood tree damage using a dataset of tree damage from an ice storm. The tree damage was measured as the mean percentage of individual tree canopy removed. In addition, the probability of tree death, which is defined as the loss of > 75% of an individual tree’s upper crown area, is expressed as functions of ice accretion. Because data availability is always a big challenge for developing empirical tree fragility curves, the simulation-based fragility modeling approach has received increasing attention. For example, Ciftci et al. (2014) numerically developed fragility curves for the bending failure of two specific trees. In this study, dynamic response of trees during windstorms was simulated with a dynamic time history analysis. Kakareko et al. (2020) and Lu and Zhang (2022) developed fragility curves for the bending failure of urban trees of different sizes under wind loads with Monte Carlo simulation. Hou and Chen (2020) further developed windthrow fragility curves of typical urban trees with numerical simulation while considering both uprooting and bending failures. However, these analytical fragility models are limited to several specific tree sizes or size groups, because the size-related parameters—for example, diameter at breast height (DBH) and height—are not explicitly included in the fragility functions. This not only limits the application of these models when the variance of tree size in an area is high but also hinders the transferability to other researchers.

Fragility models of PDS component failures induced by falling trees are an indispensable part of a resilience modeling framework that can consider the tree-failure risk. According to Wang (2016), there are four typical tree-caused faults of overhead PDS: (1) pole failure; (2) wire breakage; (3) short circuit fault that occurs when a bridge is formed between wires by falling trees; and (4) short circuit fault that occurs when wires are pushed together by falling trees. To fully capture the impact of falling trees on PDS resilience, fragility models of these failure modes must be developed. However, to the best of our knowledge, only fragility curves of power poles under falling trees were developed by Yuan et al. (2018). Fragility models of the other three failure modes caused by falling trees are still lacking. In addition, most existing fragility models of poles are a function of wind speed, neglecting the impact of other parameters, such as wind attack angle and some important design parameters, such as span length and wire diameter (Shafieezadeh et al. 2014; Salman et al. 2015; Salman and Li 2016; Yuan et al. 2018). First, previously the wind was often assumed to be perpendicular to power lines, which will lead to an overestimation of the failure probability of poles when the wind is not perpendicular to power lines. Second, the design parameters of components in a real PDS usually vary, but traditional generic fragility models do not differentiate PDS components with different characteristics. Therefore, to provide a component-specific estimation of the failure probability of PDS components, parameterized fragility models, which are a function of intensity measures (for example, wind speed and wind attack angle) and design parameters, are needed. Darestani et al. (2022) developed parameterized fragility functions for wood poles subjected to combined wind, surge, and wave loads. However, parameterized fragility models for wires under winds or other loads (for example, falling trees) have not been developed.

Vegetation management plays an important role in maintaining PDS reliability (Guikema et al. 2006). Vegetation management entails trimming or removing trees that contact overhead power lines and disrupt services (Kuntz et al. 2002). There are limited studies examining the effect of vegetation management on the resilience enhancement of PDS (Ma et al. 2018; Tari et al. 2021). However, these studies are limited to unrealistic and uneconomical vegetation management strategies. In these studies, it was assumed that all trees along a line up to several hundred meters pose the same risk to PDS components and need to be trimmed or removed if this line is chosen to be hardened. Hence a refined vegetation management measure focusing on hazardous trees is needed.

To overcome the above-mentioned limitations, an integrated simulation framework is proposed to assess the resilience of PDS against extreme winds, in which the tree-failure risk can be considered. In this framework, first, parameterized fragility models are developed for two common types of tree failure: stem breakage and uprooting, which are functions of tree height and wind speed. Second, parameterized fragility models of PDS components are developed by considering four different failure modes, including two tree-induced wire failure modes overlooked in previous works. The parameterized fragility models of PDS components are expressed as functions of intensity measures and design parameters. Third, PDS component failures are estimated based on fragility models of trees and PDS components, and the geometric relationship between trees and power lines. Fourth, the performance of radial PDS is evaluated with a connectivity-based method. Fifth, the PDS restoration is modeled by considering component criticality and uncertain component repair time, and the resilience index of PDS during the wind event is calculated. Finally, crown thinning of hazardous trees along power lines is used to reduce the failure probability of trees and improve PDS resilience. The framework is demonstrated with a power distribution network in Oklahoma.

2 Integrated Simulation Framework

An integrated simulation framework for assessing the resilience of PDS subjected to extreme winds is introduced in this section. The proposed framework includes six parts: tree fragility modeling, PDS component fragility modeling, PDS component failure estimation, system performance evaluation, system restoration modeling, and resilience enhancement evaluation. The details of each part of the framework are presented as follows.

2.1 Tree Fragility Modeling

A fragility modeling method proposed by Hou and Chen (2020) is used to develop the tree fragility model. The modeling method is briefly introduced as follows; further details can be found in Hou and Chen (2020). Allometric equations are developed based on measured data. With allometric equations, some hard to measure parameters can be predicted with an easily measured one. For the allometric equations developed in this study, parameters such as diameter at breast height (DBH), crown height, and crown diameter are expressed as functions of the tree height. Tree data including DBH, crown height, crown diameter, and height used in this study were collected by the U.S. Forest Service Pacific Southwest Research Station (McPherson et al. 2016). A linear model developed by McPherson et al. (2016) is used for the regression. A finite element tree model shown in Fig. 1 is built to compute the internal forces of the tree structure under the impact of winds. A tapered tree stem is discretized into multiple beam elements with uniform cross-sections. The cross-section in the middle of an element is used to represent the property of this element. Static wind loads are applied to both the stem and crown. The 3-s gust wind speed is used to calculate the wind forces; the wind profile is assumed to follow the power-law form. The weight of the stem and crown are also considered in the finite element model. The tree structure is fixed at the bottom. To consider the P-Delta effects due to the tree self-weight, the second-order analysis is performed by including the geometric stiffness matrix in the total stiffness matrix. Two common failure modes of trees under the impact of winds are considered: stem breakage and uprooting. Stem breakage is defined as when the maximum compressive stress in the stem exceeds the stem modulus of rupture while uprooting refers to the situation when the critical turning moment provided by the root-soil plate anchorage is exceeded by the base turning moment produced by wind. Logistic regression is used to derive parameterized fragility functions because of its advantages in providing a closed form equation and having good predictive accuracy (Balomenos et al. 2020). The logistic fragility model for estimating the probability of failure has a general form as follows:

$$ P\left( X \right) = \frac{1}{{1 + e^{ - l\left( X \right)} }}, $$
(1)

where \(X\) is the vector of input parameters (for example, geometrical data, intensity measures); \(l\left(X\right)\) is the logit function. For the fragility model of stem breakage and uprooting of trees, the input parameters are tree height \(H\) and wind speed \(U\), so the logit function takes the following form:

$$l\left(X\right)={a}_{0}+{a}_{1}H+{a}_{2}U,$$
(2)

where \({a}_{0}\) to \({a}_{2}\) are coefficients obtained from logistic regression. By including tree height in the fragility model of trees, the limitation in previous studies, that is, fragility models are not explicitly expressed as functions of tree sizes, is overcome.

Fig. 1
figure 1

Source Hou and Chen (2020)

Finite element model of a tree.

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2.2 Power Distribution System Component Fragility Modeling

Power distribution system component fragility modeling includes three steps: structural analysis of pole-wire systems, failure modes under the wind and falling-tree scenarios, and fragility model of different failure modes. First, internal forces and displacements of PDS components are calculated through structural analysis of pole-wire systems; Monte Carlo simulation is used in this step to generate random samples for structural analysis. Second, based on the results of structural analysis, failures of PDS components are determined with limit state functions of different failure modes. Third, based on the sampling data generated in step 1 and the binary state (Fail/Not Fail) of PDS components obtained in step 2, fragility functions are developed with logistic regression.

2.2.1 Structural Analysis

Traditionally, the individual pole model is widely used for fragility analysis. However, such model has two limitations: first, it cannot reflect the 3D behaviors of power poles and wires; and second, it can only model the bending failure of poles, while other failure modes related to wires cannot be captured. To overcome these limitations, a three-span pole-wire model developed with ANSYS (Fig. 2) is used for the structural analysis (Hou et al. 2023). There are three spans of wires and four poles in the model. Two guy wires are used to support two end poles. The poles and guy wires are fixed at the bottom. Poles, crossarms, and braces are modeled with the Beam 4 element. Power lines and guy wires are modeled with the Link 8 element. The initial tension in power lines controls the initial sag and shape of power lines under the self-weight condition, which must be determined before running analysis with applied loads. A recommended tension by Ausgrid (2011), which is 6% of the rated tension strength for power lines spanning from 30 to 90 m, is used in the pole-wire model considering that the mean span length in this study falls in this range.

Fig. 2
figure 2

Finite element model of the three-span power-wire system in ANSYS (cyan lines are wires; pink lines are poles)

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The structural responses of the pole-wire system under wind and tree load are analyzed separately. For the wind scenario, wind load is applied to poles and wires. According to the National Electrical Safety Code (NESC) (IEEE 2017), the wind pressure \(q\) on poles and wires can be calculated with Eq. 3:

$$q=0.5{\rho }_{a}{k}_{z}G{C}_{f}{{U}_{p}}^{2},$$
(3)

where \({\rho }_{a}\) = air density; \({k}_{z}\) = velocity pressure exposure coefficient; \(G\) = gust response factor; \({C}_{f}\) = force coefficient; \({U}_{p}\) = projected 3-s gust wind speed. The projected 3-s gust wind speed \({U}_{p}\) is equal to 3-s gust wind speed \(U\) for poles. Because wires are only subject to the wind component in the perpendicular direction, the projected 3-s gust wind speed \({U}_{p}\) for wires is equal to \(Usin(\alpha )\), where \(\alpha \) is the wind attack angle. For the falling-tree scenario, the load of the falling tree is modeled as a point load on the middle of the phase wire, which represents the most unfavorable scenario.

The response of the power-wire system is determined through a two-step nonlinear static analysis. First, the initial shape of the power-wire system under self-weight is determined through the shape-finding analysis (Kim et al. 2002). Second, external wind or tree loads are applied to the deformed structure; the solutions are obtained with the geometric nonlinear analysis. Force and displacement results of members in the middle span are used for the fragility analysis considering that two side spans mainly serve as boundary conditions through balancing the load transmitted from the middle span.

2.2.2 Failure Modes

During extreme wind events, PDS components can be damaged by either winds directly or falling trees indirectly. In the wind scenario, common failure modes are bending failure of poles and wire breakage. In the falling-tree scenario, wire breakage and short circuit are prevalent failure modes. Pole failure occurs when the maximum stress due to external loads exceeds the fiber strength of poles. Wire breakage occurs when the breaking strength of wires is exceeded by the maximum stress caused by external loads. A short circuit occurs when the phase wire is close enough to the neutral wire under the load of falling trees. This type of short circuit only applies to a 1-phase line system. For a 3-phase line system, the short circuit is assumed to occur once a tree falls on it. This is because wires in a 3-phase line system are in an approximately horizontal plane and a bridge between different wires is easily formed in this case. Limit state functions for pole failure, wire breakage, and short circuit can be expressed with Eqs. 4, 5, 6, respectively:

$$ g_{p} = \sigma_{{{\text{max}},p}} - \sigma_{f} , $$
(4)
$$ g_{c} = \sigma_{{{\text{max}},c}} - \sigma_{b} , $$
(5)
$$ g_{s} = d_{pn} - d_{0} , $$
(6)

where \({g}_{p}\), \({g}_{c}\), and \({g}_{s}\) are the limit state functions for pole failure, wire breakage, and short circuit, respectively; \({\sigma }_{\mathrm{max},p}\) = maximum stress in the pole; \({\sigma }_{f}\) = fiber strength of the pole; \({\sigma }_{\mathrm{max},c}\) = maximum stress in the wire; \({\sigma }_{b}\) = breaking strength of the wire; \({d}_{pn}\) = distance between the phase wire and neutral wire under the tree load; \({d}_{0}\) = distance threshold that, if exceeded, can cause a short circuit, which is assumed to follow a normal distribution \({d}_{0}\sim N\left(0, 0.1 \mathrm{m}\right)\) considering the uncertainties caused by tree branches.

2.2.3 Fragility Model

Logistic regression is used to develop fragility functions for predicting PDS component failures. Similar to the fragility model for trees, the logistic fragility model for PDS components has a form shown in Eq. 1. The input parameters for the PDS component failures under the wind scenario, namely, pole failure and wire breakage, are wind speed \(U\) and wind attack angle \(\alpha \). Therefore, the logit function for estimating wind-induced pole failure and wire breakage can be expressed as:

$$l\left(X\right)={b}_{0}+{b}_{1}U+{b}_{2}Usin\left(\alpha \right),$$
(7)

where \({b}_{0}\) to \({b}_{2}\) are coefficients obtained from logistic regression. In contrast, the logit function for wire breakage and short circuit under the falling-tree scenario is a function of wire diameter \({d}_{c}\), span length \(SP\), tree load \({F}_{t}\), which has the following form:

$$ l\left( X \right) = c_{0} + c_{1} { }d_{c} + c_{2} SP + c_{3} F_{t} , $$
(8)

where \({c}_{0}\) to \({c}_{3}\) are coefficients obtained from logistic regression.

2.3 Power Distribution System Component Failure Estimation

With tree fragility and PDS component fragility, the failure probability of different PDS components can be calculated. The failure probability of pole \(i\) can be computed with Eq. 9:

$$ P_{{{\text{f}},{\text{pole}},i}} = P_{{{\text{bd}},{\text{w}},i}} , $$
(9)

where \({P}_{\mathrm{bd},\mathrm{w},i}\) is the bending failure probability of pole \(i\) due to wind load. The failure probability of wire \(j\) can be calculated by Eq. 10 (Winkler et al. 2010; Ouyang and Duenas-Osorio 2014):

$$ P_{{{\text{f}},{\text{wire}},j}} = {\text{max}}\left( {P_{{{\text{br}},{\text{w}},j}} , P_{{{\text{tree}},j}} P_{{{\text{br}},{\text{t}},j}} f_{j} , P_{{{\text{tree}},j}} P_{{s{\text{t}},{\text{t}},j}} f_{j} } \right), $$
(10)

where \({P}_{\mathrm{br},\mathrm{w},j}\) is the breakage probability of wire \(j\) due to wind load; \({P}_{\mathrm{br},\mathrm{t},j}\) and \({P}_{\mathrm{st},\mathrm{t},j}\) are the breakage and short circuit probability of wire \(j\) due to falling trees, respectively; \({P}_{\mathrm{tree},j}\) is the damage probability of trees along wire \(j\), \({P}_{\mathrm{tree},j}=1-{(1-{P}_{\mathrm{tree},\mathrm{w}})}^{n}\); \({P}_{\mathrm{tree},\mathrm{w}}\) is the failure probability of a single tree due to wind load; \(n\) is the number of hazardous trees along wire \(j\); \({f}_{j}\) indicates the consequence of falling trees along wire \(j\), if a falling tree falls on it, \({f}_{j}=1\), otherwise, \({f}_{j}=0\). As shown in Fig. 3, if Eq. 11 or Eq. 12 is satisfied, a broken or uprooted tree will fall on the power line. A broken tree falls on the power line:

Fig. 3
figure 3

Source Hou and Chen (2020)

Schematic diagrams of falling-tree scenarios. a Stem breakage; b Uprooting.

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$${H}_{b}>{H}^{^{\prime}}=\sqrt{{{h}^{^{\prime}}}^{2}+{d}^{^{\prime}}}$$
(11)

An uprooted tree falls on the power line:

$$H>{H}^{^{\prime}}=\sqrt{{h}^{2}+{d}^{^{\prime}}}$$
(12)

where \(h\) is the height of the power line; \({H}_{b}\) is the length of the broken tree; \(H\) is the height of the tree; \({d}^{^{\prime}}=d/sin\left(\theta \right)\), \(d\) is the distance between the tree and the power line (m); \({h}^{^{\prime}}=h-\left(H-{H}_{b}\right)\). There are not sufficient data or suitable models that can quantify the dynamic effects of falling trees/branches, which can be influenced by many factors such as the branch shape, lateral and vertical distance between the tree and power lines, and wind speed. Currently, the dynamic effect of falling trees is not considered; only the static load is applied to the power lines. The load of a falling tree acting on the power line under the broken and uprooted tree scenarios can be calculated with Eqs. 13 and 14, respectively.

$${F}_{t}=\frac{{W}_{tb}{H}_{b}}{2{H}^{^{\prime}}},$$
(13)
$${F}_{t}=\frac{{W}_{t}H}{2{H}^{^{\prime}}},$$
(14)

where \({W}_{tb}\) is the weight of the broken tree; \({W}_{t}\) is the weight of the uprooted tree. Under scenarios when there is more than one tree along a power line, it is assumed that the \({f}_{j}\) and \({F}_{t}\) is determined by the biggest tree.

2.4 System Performance Evaluation

A connectivity-based method, as shown in Fig. 4, is used in this study to evaluate the performance of radial distribution systems. First, a network consisting of nodes and edges is used to represent a distribution system. The nodes and edges represent the poles and wires, respectively. All paths connecting customers and the substation are identified. Each path is a set of nodes and edges. Second, the protective devices (for example, switches, fuses, reclosers, and circuit breakers) that affect the power flow of each path are identified. If at least one of these devices is open, the path is disconnected. Third, components (poles and wires) that can trip each protective device are identified. If at least one of these components fails, the protective device is open. Finally, the connectivity of each path is determined by the state of protective devices. After the connectivity of all paths is known, the PDS performance, namely, the ratio of customers with power, can be evaluated with Eq. 15:

$$Q\left(t\right)=\frac{{N}_{0}\left(t\right)}{N},$$
(15)

where \(Q(t)\) is the system performance at time \(t\); \({N}_{0}(t)\) is the number of customers with power at the time \(t\); \(N\) is the total number of customers.

Fig. 4
figure 4

Flowchart of connectivity-based method

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2.5 System Restoration Modeling

Rapid restoration of PDS after natural hazard impact can not only reduce the loss caused by power outages but also expedite the restoration of other infrastructures that are dependent on PDS. Three factors must be considered in the post-disaster restoration: restoration resources, restoration time, and restoration sequence. Restoration resources refer to the number of repair teams, consisting of repair crews, equipment, and material. It is assumed that a failure only requires one repair team for the restoration. To consider the uncertainty in component restoration time, the normal distribution is used to describe the restoration time of different failures. In this study, the restoration time (in hours) for a pole failure, a wire failure, and a short circuit has the distribution \({t}_{\mathrm{pole}}\sim N\left(5, 2.5\right)\), \({t}_{\mathrm{conductor}}\sim N\left(4, 2.0\right)\), and \({t}_{\mathrm{short}}\sim N\left(1, 0.5\right)\), respectively (Ouyang and Duenas-Osorio 2014). These distributions have been widely used in previous research on power system restoration and they are considered good representations of restoration time for different failures (Ouyang and Duenas-Osorio 2014; Zou and Chen 2020; Hou et al. 2023). To maximize PDS resilience, restoration priority should be given to the critical components connected to the greatest number of customers. In this study, the restoration sequence is determined by the component criticality, which is measured with a critical index (\(CI\)). The critical index of a component is defined as the change in system performance (that is, the number of customers with power) when the component is removed from the system. Component \(CI\) can be calculated as follows:

$${CI}_{i}=1-Q\left({\delta }_{i}=1\right),$$
(16)

where \({\delta }_{i}\) is the state indicator of component \(i\), if component \(i\) fails, \({\delta }_{i}=1\) or else \({\delta }_{i}=0\); \(Q\left({\delta }_{i}=1\right)\) is the system performance when only component \(i\) fails but others still operate. All components are ranked based on their \(CI\). Typically, components on the mainline and close to the substation have a relatively high \(CI\) ranking and high restoration priority. The resilience of PDS is evaluated as follows:

$$RI=\frac{{\int }_{{t}_{0}}^{{t}_{0}+{t}_{c}}Q(t)dt}{{t}_{c}},$$
(17)

where \(RI\) is the resilience index; \({t}_{0}\) is the time when the power network is hit by wind hazards; \({t}_{c}\) is the control time for the period of interest. The expected energy not supplied (EENS) is calculated as follows:

$$EENS=\sum_{i}\sum_{t}{P}_{i,t},$$
(18)

where \({P}_{i,t}\) is the lost load of customer \(i\) at time \(t\).

2.6 Resilience Enhancement Evaluation

Resilience enhancement strategies of power distribution systems can be classified as hardening strategies and operational strategies. Hardening strategies usually include upgrading poles, vegetation management, burying power lines underground, and elevating substations. Common operational strategies include microgrid island operation, automated protection and control actions, disaster impact assessment and priority setting, and improving emergency and preparedness plans, among others (Amit et al. 2016; Lin and Bie 2016; Panteli and Mancarella 2017). This study focused on evaluating the effect of crown thinning, a common vegetation management measure, on the PDS resilience enhancement. Crown thinning involves the selective removal of stems and branches to produce an evenly spaced branch structure. As a popular vegetation management measure, crown thinning can effectively reduce the windthrow risk without affecting tree health. Through crown thinning, the crown weight is reduced. In addition, because of the reduction in the ventilation ratio of the crown, the wind loads acting on the crown are also reduced. Therefore, crown thinning can reduce the windthrow likelihood of a tree and further reduce the tree-failure risk of PDS.

3 Illustrative Example

The proposed framework is illustrated through the resilience assessment and improvement of a power distribution network in the state of Oklahoma, United States. Figure 5 shows the layout of the power distribution network, which consists of 770 wood poles and 769 power lines. Bur oaks are the most common trees, accounting for at least 70% of the trees in the area where the power distribution network is located. Hazardous trees, which are close to and taller than power lines, were identified with Google Earth. The height of hazardous trees and the distance to power lines were collected manually using both satellite view and street view on Google Earth with the method used by Ahmed and Evans (2022). Figure 5 also shows the number of hazardous trees along power lines. There are three pole classes: Class 2, 4, and 5, accounting for 30.5%, 64.9%, and 4.6% of the total number of poles, respectively. The substation is located at the northernmost part of the network. Protective devices in the network such as switches, fuses, line sectionalizers, reclosers, and circuit breakers are not shown in the figure. Other network data include the power line phase, power line orientation, and pole class.

Fig. 5
figure 5

A power distribution network in Oklahoma. HT refers to hazardous trees

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In the illustrative example, first of all, the fragility functions of Bur oak trees subjected to winds, as well as equations for estimating the weight and length of falling trees are developed. Next, fragility functions of PDS components under the impact of winds and falling trees are developed. After that, the resilience of the power distribution network subjected to winds is assessed; the effect of wind speed, wind direction, and tree-failure risk on the system resilience is investigated. Finally, the effectiveness of crown thinning on resilience enhancement is evaluated.

3.1 Fragility Functions of Bur Oak Trees

Table 1 shows statistics of parameters of Bur oak trees and wind load, as well as their sources. The crown characteristics of Bur oak trees, such as drag coefficient and weight of the crown, change between seasons. A larger drag coefficient and heavier crown make Bur oak trees more vulnerable to windthrow in the summer than in other seasons. In this study, the crown characteristics in the summer are used to be conservative. Most parameters in Table 1 follow the normal distribution. The triangle shape is used to model the reconfigured frontal crown shape under the impact of wind when calculating the wind loads (Peltola et al. 1999). A stem taper equation developed by Westfall and Scott (2010) is used to predict the stem diameter at a given height. The uniform distribution is often adopted when there is insufficient information about the distribution of the parameters (Balomenos et al. 2020). Therefore, the tree height and wind speed are assumed to have a uniform distribution. A total of 50,000 samples of parameters in Table 1 are generated with Latin hypercube sampling (LHS). For each realization, the binary state (Fail/Not Fail) of a tree is determined by checking the limit state functions of two failure modes: stem breakage and uprooting. The combined data including parameter samples and binary output is divided randomly into training and test sets (70% and 30%, respectively) for logistic regression with the model in Eq. 2. Table 2 provides regression results for the fragility of Bur oak trees, which includes regression coefficients and the area under the receiver operation characteristic curve (AUC). The AUC is an important evaluation metric for calculating the performance of the logistic regression model. It takes values between 0 and 1, where 0 indicates a perfectly inaccurate classification and 1 indicates a perfectly accurate classification (Mandrekar 2010). As shown in Table 2, the AUC for both stem breakage and uprooting is 0.98, indicating the good performance of the predictive models.

Table 1 Statistics of parameters of Bur oak trees and wind load
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Table 2 Regression results for the fragility of Bur oak trees
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As mentioned previously, the weight of falling trees can be used to estimate the load of falling trees on power lines, which is an input parameter of fragility functions of PDS components. It is necessary to estimate the weight of falling trees with other parameters such as tree height. The weight of falling trees is extracted from the stem breakage and uprooting failures out of the 50,000 simulations. It is found that power-law functions can be used to describe the relationship between the weight of falling trees and tree height, which are shown as follows:

$$ {\text{Stem breakage}}:W_{tb} = 2.008H^{3.076} { }\left( {{\text{R}}^{2} = 0.90} \right), $$
(19)
$$ {\text{Uprooting}}:W_{t} = 0.0358H^{5.019} { }\left( {{\text{R}}^{2} = 0.99} \right) $$
(20)

The length of a falling tree, along with other parameters such as the height of the power line and the distance between the tree and the power line, determines whether the tree can fall on the power line. If a tree is uprooted, the falling length is its height. However, the falling length of a broken tree is affected by the breaking point. The data on the length of broken trees are extracted from the stem breakage failures out of the 50,000 simulations. Based on the results of simulated stem breakage failures, the length of broken trees and the tree height is found to have a linear relationship as follows:

$$ H_{b} = 0.946H - 1.827\;\left( {{\text{R}}^{2} = 0.99} \right). $$
(21)

3.2 Fragility Functions of Power Distribution System Components

Fragility analyses are performed for distribution components including 10,060 poles in Oklahoma. Two commonly used pole types in this area are studied, namely, 1-phase and 3-phase line poles, which are shown in Fig. 6. Because classes 2, 4, and 5 are much more prevalent than other classes, fragility analysis is performed for these three pole classes. Table 3 gives the statistics of parameters of PDS components, wind load, and tree load. For most stochastic parameters related to geometry and material property, such as span, density, diameter, and breaking strength of aluminum conductor steel reinforced (ACSR), lognormal or normal distribution functions are fitted to the field data. Based on the height data, pole height for a certain class can be classified into 3−4 groups; therefore, the discrete probability distribution is used to describe the height. The pole diameter for a given height and class is then determined based on ANSI-O5.1 (ANSI 2017). The poles are made of Southern Pine. The span length of power lines follows a normal distribution. Aluminum conductor steel reinforced is used for both phase and neutral lines. The density, diameter, and breaking strength of ACSR wires follow lognormal distributions, with their coefficient of variation (COV) greater than 0.2. A deterministic elastic modulus of 81 Gpa is used for ACSR wires (Trefinasa 2020). Due to the lack of design details of crossarms, fiberglass crossarms used in the work of Yuan et al. (2018) are used in this study. Parameters related to wind pressure calculation in Eq. 3 used by Yuan et al. (2018) are adopted.

Fig. 6
figure 6

Distribution pole layout (unit: cm). a 1-phase line pole and b 3-phase line pole

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Table 3 Statistics of parameters of power distribution system (PDS) components, wind load, and tree load
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Logistic regression is performed to develop fragility models of PDS components with different pole phases (that is, 1-phase and 3-phase), pole classes (that is, classes 2, 4, and 5), and load scenarios (that is, wind and falling trees). For each case, 10,000 samples of parameters in Table 3 are generated with LHS. For each realization, structural analysis is performed with ANSYS to obtain the response of poles and wires; the binary state (Fail/Not Fail) of PDS components is determined by checking the limit state functions shown in Eqs. 4, 5, 6. The total data including parameter samples and binary output are divided into training and testing sets: 70% of the data are used for training the predictive models in Eqs. 7 and 8 and the remaining 30% are used for testing. Tables 4 and 5 provide the regression results for PDS components subjected to winds and falling trees, respectively. The evaluation metric AUC is higher than 0.9 for the wind scenario and equal to or higher than 0.8 for the falling-tree scenario, indicating a good fit of the predictive models.

Table 4 Regression results for the fragility of power distribution system (PDS) components subjected to winds
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Table 5 Regression results for the fragility of 1-phase power distribution system (PDS) components subjected to falling trees
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3.3 Resilience Assessment of Power Distribution Systems Subjected to Extreme Winds

The resilience of the power distribution network under different wind scenarios is assessed in this subsection. Considering that the geographical area covered by the studied distribution network is relatively small, it is assumed that the wind speed and wind direction are the same throughout this area during a wind event. For simplicity, the peak demand of customers during the summer is used to calculate EENS using Eq. 18. To accurately simulate the restoration process, a simulation time step of 0.5 hour is used. It is assumed that the wind event occurs at the tenth hour of the simulation. In addition, it is assumed that two repair teams will be assigned to restore the PDS immediately after it is hit by winds. Monte Carlo simulation is run 480 times so that the average PDS performance index converges. Figure 7 shows average restoration curves for scenarios with different wind speeds but the same wind direction θ = 0° (Northwind). As the wind speed increases, the system performance decreases, and the recovery time increases. The change is nonlinear, and the rate of change becomes larger with increasing wind speed. This indicates that much more resources are needed to restore the PDS in a given recovery time window for very strong-wind events, compared to other wind events.

Fig. 7
figure 7

Average power distribution system (PDS) restoration curves for scenarios with different wind speeds (\(\theta \) = 0°)

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To understand how the wind direction affects the power distribution network resilience, scenarios with different wind directions (\(\theta \) = 0°, 45°, 90°, 135°, 180°) and the same wind speed (\(U\) = 60 m/s) are studied. Figure 8 presents average restoration curves for scenarios with different wind directions. It is found that both the system performance and recovery time vary with the wind direction. By setting the control time for the period of interest \({t}_{c}\) as 120 hours, the resilience index \(RI\) for the five scenarios is 0.84, 0.64, 0.74, 0.60, and 0.81. There is no obvious pattern between the resilience index and the wind direction. This can be explained by the fact that the PDS resilience also largely depends on the network topology and the tree distribution, besides the hazard parameters (for example, wind speed and wind direction). It should also be realized that the wind direction must be considered when assessing the PDS resilience. Otherwise, the PDS resilience could be overestimated or underestimated, providing misleading information for resilience planning.

Fig. 8
figure 8

Average power distribution system (PDS) restoration curves for scenarios with different wind directions (\(U\) = 60 m/s)

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To understand tree failures on the system resilience, scenarios with and without trees are studied and compared. Figure 9 presents average restoration curves for four scenarios under a wind speed of 60 m/s, namely, scenario 1, \(\theta \) = 45°, with trees; scenario 2, \(\theta \) = 45°, without trees; scenario 3, \(\theta \) = 225°, with trees; scenario 4, \(\theta \) = 225°, without trees. Figure 9 shows that the restoration curves of scenarios 2 and 4 overlap almost perfectly. This is because the PDS under the two scenarios suffers the same damage due to symmetry when tree failures are not considered. By comparing scenarios with and without trees, it is found that the network performance decreases, and the recovery time increases when tree failures are considered in scenarios 1 and 3. For example, when the wind direction is 225°, if tree failures are considered, the resilience index \(RI\) reduces by 14.4% (\({t}_{c}\) = 120 hours) and the recovery time increases by 17.5 hours. This implies that neglecting tree failures can lead to an overestimation of the resilience of PDS subjected to winds. The percentages of different PDS component failure modes under scenario 3 are as follows: wind-induced pole failure (17%), wind-induced wire breakage (36%), tree-induced wire breakage (1%), and tree-induced short circuit (46%). Short circuit caused by falling trees is the most common failure mode, which accounts for nearly half of the component failures. In contrast, wire breakage caused by falling trees is the least common failure mode.

Fig. 9
figure 9

Average power distribution system (PDS) restoration curves for scenarios with and without considering trees (\(U\) = 60 m/s)

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3.4 Resilience Enhancement with Vegetation Management

Crown thinning is used to improve the PDS resilience against extreme winds. Normally, the live foliage that can be thinned from a tree must be less than 25% to avoid damaging its health. For the crown thinning method in this study, it is assumed that 25% of the crown is removed from all identified hazardous trees in the network. It is also assumed that the crown density and effective crown area will be reduced by 25% and other tree parameters will keep the same after crown thinning. This will lead to a 25% reduction in both crown weight and wind loads acting on the crown. Figure 10 shows the fragility curves of Bur oak trees with a height of 14 m before and after crown thinning. Both breakage and uprooting probabilities reduce after crown thinning. The weight of falling trees reduces after crown thinning. For example, for a 14 m-tall tree, the weight of the falling tree reduces by 16.6% when its stem is broken by winds. As a result of the reduction in the tree failure probability and weight of falling trees, the tree-failure risk to PDS is reduced. Figures 11 and 12 present the average reduction in EENS and system restoration time with crown thinning, respectively. As shown in Figs. 11 and 12, crown thinning can effectively enhance the system resilience by reducing the EENS and restoration time for scenarios with different wind speeds and directions. It is found that the effectiveness of crown thinning varies with the wind direction. When the wind direction is in the range of 135°−225°, the most reduction in EENS and restoration time can be achieved. The reduction in EENS and restoration time increases with the wind speed and then decreases after reaching a peak at around U = 60−70 m/s. This indicates that crown thinning will become less effective in enhancing the system resilience when the wind speed is very low or high for a given wind direction.

Fig. 10
figure 10

Fragility curves of Bur oak trees before and after crown thinning (CT)

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Fig. 11
figure 11

Average reduction in expected energy not supplied (EENS) with crown thinning

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Fig. 12
figure 12

Average reduction in power distribution system (PDS) restoration time with crown thinning

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4 Conclusion and Future Work

This study proposed an integrated framework for modeling the resilience of PDS against extreme winds, which can simultaneously consider tree failures, PDS component failures induced by falling trees, resilience assessment, and evaluation of resilience enhancement with vegetation management. To consider the impact of tree failures, we developed parameterized fragility models for tree failures and tree-induced PDS component failures. Crown thinning, which focuses on hazardous trees along power lines, was used as a vegetation management method to reduce the windthrow probability of trees and enhance the system’s resilience. An example was presented to demonstrate the applicability of the framework. In the example, the resilience of a power distribution network in Oklahoma was assessed, and the effectiveness of crown thinning in enhancing the system resilience was evaluated. Some conclusions have been drawn as follows:

  1. (1)

    As the wind speed increases, the system performance decreases, and the recovery time increases nonlinearly.

  2. (2)

    The system resilience varies with the wind direction, but the variation depends on the topology of the network and the distribution of the trees in the network.

  3. (3)

    Neglecting tree failures can lead to an overestimation of PDS resilience.

  4. (4)

    Crown thinning can effectively enhance the system resilience by reducing the EENS and restoration time, but the effectiveness is affected by both wind speed and direction.

This study can be improved with future work. For example, failures of branches and limbs can be considered with a refined finite element tree model. In addition, the PDS component failure estimation model can be improved by considering the dynamic effect of falling trees.