Fragility models of electrical conductors in power transmission networks subjected to hurricanes

Structural Safety 82 (2020) 101890

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Fragility models of electrical conductors in power transmission networks subjected to hurricanes

T

Liyang Ma, Paolo Bocchini , Vasileios Christou ⁎

Department of Civil and Environmental Engineering, ATLSS Engineering Research Center, Lehigh University, 117 ATLSS Drive, Bethlehem, PA 18015-4729, USA

ARTICLE INFO

ABSTRACT

Keywords: Transmission conductors Fragility models Hurricane load Modal superposition method Nonlinear FEM Extreme value analysis First order reliability method Power networks

Power transmission conductors work as links in power transmission system. They allow the bulk transportation of energy between power plants and substations. A large number of these conductors may fail when subjected to hurricanes. Failure of these conductors may result in significant disruption in the power network and millions of dollars of replacement cost. This paper presents a framework for the development of fragility models of the transmission conductors. The framework considers the uncertainties in the wind turbulence and conductor capacity. The modal superposition method has been used to model the mechanical response of the conductors efficiently, and is validated by nonlinear finite element analysis. The first order reliability method has been implemented to capture the low failure probability of a single conductor with sufficient accuracy as confirmed by Monte Carlo simulation. The results of this study show that the failure probability of the conductors increases significantly once the wind speed reaches a certain critical value. The wind direction and span length have large influence on the failure probability. Therefore, the variability of span lengths over the transmission network has a substantial effect on the overall system failure. The fragility model provided in this research constitutes a major component for risk assessment of power transmission networks against hurricane hazards. It can be used to help engineers gain fundamental insights on the mechanical and probabilistic performance of power transmission conductors.

1. Introduction, motivation and scope Power transmission systems are among the most critical infrastructure systems, which support the well being of our society and economy. The major components of this system are power plants, transmission line and substations. PRAISys is a platform [1] being developed to model all these components, and this paper focuses on transmission lines. A transmission line in this study is defined as the link between two substations which are often separated by long distance. The function of a transmission line is to bulk transport highvoltage electrical power. Transmission lines with length not exceeding 80 km are classified as short lines. Transmission lines with length ranging from 80 to 240 km are called medium lines [2]. A transmission line usually consists of hundreds or thousands of transmission conductors. A transmission conductor in this study is defined as the segment of conductive material between two adjacent support structures, such as transmission towers. The distance between the two adjacent support structures is called “span length” of the conductor. The span length of a transmission conductor varies depending on the electrical properties of the transmission line, the geographical

⁎

characteristics of the area and the structural properties of the support structures. In general, a typical span length of a transmission conductor ranges from 100 m to 500 m. The transmission conductors, due to their flexible nature, are very sensitive and vulnerable to hurricane winds. During a hurricane, a large number of transmission conductors might be affected. Hurricane Harvey in 2017 caused damages that required the replacement of over 1200 km of transmission and distribution conductors [3]. Hurricane Charley and Hurricane Frances struck Florida in 2004, and resulted in the replacement of 2521 km and 1689 km of conductors, respectively [4]. Because the conductors in a transmission line are connected in series, failure of one conductor trips the whole transmission line. Such failure can cause a sudden and significant disruption of the electrical flow in the power network. Cascading failures may be initiated due to the unbalanced electrical loadings and lead to large scale power loss in the region. It is worth noting that, in addition to the failure of conductors, the failure of support structures is very important, and it is actually the dominant failure mode today for power transmission systems. Therefore, while this paper addresses conductors, the support structures should be separately investigated. Nevertheless, the failure of

Corresponding author. E-mail addresses: [email protected] (L. Ma), [email protected] (P. Bocchini), [email protected] (V. Christou).

https://doi.org/10.1016/j.strusafe.2019.101890 Received 21 February 2019; Received in revised form 2 September 2019; Accepted 3 September 2019 0167-4730/ © 2019 Published by Elsevier Ltd.

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Fig. 1. Flowchart of the procedures for developing fragility models of transmission conductors.

conductors during high wind events is one of the important failure modes which has raised concern in the industry [5]. Fragility curves are a practical tool to assess the probability of failure of classes of structural elements, especially convenient for system analyses. Fragility curves can be developed with different methods [6,7] and relate the probability of failure to the intensity of the hazard, accounting for uncertainty in structural demand and capacity. Once the fragility curve has been assessed for a class of elements or structures, it can be readily used for practical risk and resilience assessment [8,1,9]. Fragility modeling was initially developed in seismic analysis field, and recently has been applied to extreme wind analysis for residential buildings and utility poles [10,11]. However, to the best of our knowledge, the fragility assessment of transmission conductors is still lacking in the literature. Unlike developing fragility models for houses and utility poles, where a static analysis might be sufficient to determine the structural response, to accurately predict the behavior of conductors, which are highly flexible under hurricane loadings, a dynamic analysis is necessary. Substantial research efforts have addressed the response of conductors. Irvine [12] presented the equations of motion of conductors under dynamic wind load, and developed the analytical expressions of natural frequencies and mode shapes of the conductor. Davenport [13,14] studied the gust wind response of conductors based on random vibration theory. Some scholars have focused on developing nonlinear finite element models of the conductor [15–17]. Although the nonlinear Finite Element Method (FEM) is regarded as most accurate, due to its capability of considering the geometrical nonlinearity of the conductor, its repeated solution for probabilistic analysis is very time-consuming, which hinders its application in fragility curve estimation. More recently, Wang et al. [18]

introduced a novel approach to predict wind response of overhead conductors based on the modal superposition method, with closed-form formulations. In this study, the conductor is modeled with fully-hinged supports at the same level. The flexibility of support structures does not affect meaningfully the maximum response of conductors and is not included in the analysis because of three main reasons. First, the conductors are only fully constrained at end towers or tension towers, which are designed to be very heavy and stiff with much higher fundamental frequencies than the conductors, making the dynamic coupling with conductors negligible. In contrast, the suspension towers have lower fundamental frequencies, but the conductors are just hanged by the suspension insulator attached to these towers. The swing of the suspension insulators decouples partially the motion between conductors and suspension towers. Compared to the tension tower, the motion of the suspension tower has much less influence on the conductors [19]. Second, the support structures, even for the suspension towers, normally have natural frequencies greater than 2.0 Hz [20], which is much larger than the conductors’ fundamental frequencies. The first six natural frequencies of the conductors are usually less than 1 Hz and their first and second natural frequencies are found to be around 0.05–0.3 Hz [21–23]. Third, even in the rare cases where there might be some small resonance between tower and conductors, the resonant dynamic response does not appear to be a problem for the conductors [24]. The resonant response is largely damped out because of the very large aerodynamic damping of the conductors [25,26]. The above theories have been validated both by researchers and practitioners and are implemented in the standard for the design of electrical transmission line structures [20]. It is worth noting that there are two cases where the 2

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interaction between towers and conductors is relevant. The first case is when the tension towers are extremely high and flexible with very low natural frequencies, hence the natural frequencies of towers and conductors become close. For instance, the coupling motions can be recognized for tension towers of the height of 106.5 m from field tests [27]. For these rare cases, the interaction should indeed be considered, with an ad hoc investigation. Fortunately, the majority of the transmission towers are shorter than 55 m [28] and most of the tension towers have much higher fundamental frequency than conductors. Second, if the support structures undergo very high deformation, such as in the conditions of total or partial collapse or extremely large nonlinearity, their influence on the conductors is noticeable. However, in these cases, since the support structures are already failed, the transmission system is failed (series system), and the failure of the conductors become irrelevant for the system failure. In particular, this paper focuses on developing a framework for efficient fragility curve estimation of transmission conductors under hurricanes, considering the failure limit state which corresponds to the ultimate strength of the conductor. The uncertainties on the transmission conductor behavior during a hurricane stem from the random nature of the wind fluctuation as well as the capacity of the conductor. There are two main challenges in deriving the fragility model for transmission conductors: (1) predicting the load response of a conductor with accuracy and efficiency, and (2) computing the probability of failure, which is a rare event, with sufficient accuracy. Fig. 1 presents the conceptual flowchart of the procedures developed in this paper. The wind induced demands on conductors are modeled probabilistically using a multivariate random wind field to capture the uncertainties of the wind fluctuating process and correlations in time and space. Time history analysis based on nonlinear FEM is conducted using the simulated wind field and catenary cable elements. Time history analysis based on the modal superposition method, taking advantage of the assumption on linear behavior of the conductor, is validated by comparing its results to those of the nonlinear FEM. Extreme value analysis is carried out to derive the closed-form distribution of extreme loading on conductors. The distribution of the conductors’ capacity is also modeled, and fragility curves are computed using first order reliability theory and validated by brute-force Monte Carlo simulation. The developed fragility models can be applied in risk and resilience assessment of the regional power network against hurricane hazards. A power network consists of hundreds or thousands of conductors covering hundreds of miles. It requires regular inspection and maintenance, which are expensive and time-consuming. The developed fragility model can be used to identify the most vulnerable conductors in the network and tune the frequency of inspection, maintenance and retrofit strategies accordingly. Therefore, the developed fragility models can contribute to effective management strategies for the power network.

Vz =

1 z u ln k z0

(1)

where Vz is the mean wind speed at the height of z meters above ground; u is the shear velocity of the flow in m / s;k = 0.4 is known as von Karman’s constant; z 0 is the roughness length in m, which is a measure of the roughness of the ground surface. Table 1 describes various types of terrain and corresponding roughness length [31]. As an example, the application section focuses on the calculation of fragilities for conductors in the open terrain, as the majority of the transmission lines are located in this type of terrain. The coefficient of variation of the wind speed decreases with the height above the ground and increases with the roughness length. The variation of wind speed is caused by eddies, which are random in nature. The modulus of the wind velocity is described statistically by a Gaussian probability density function, as confirmed by various measurements [24]. The variation of wind velocity with respect to time can be described by the Kaimal spectrum Sv (f ) [32]:

f · S v (f ) 200fz / V¯z = (1 + 50fz / V¯z )5/3 u2

(2)

where f is the frequency in Hertz. The Kaimal spectrum has been used widely in structural engineering to represent the stochastic properties of strong wind, and its accuracy has been proven through various tests around the globe during high wind events. The correlation of the fluctuating wind velocities at two points on a transmission conductor can be modeled by the coherence function developed by Davenport [33]:

cohV x1, x2 , f = exp

C x1

x2 f Vz

(3)

where C = 16 is the coherence parameter obtained on the basis of wind tunnel measurements [34,35], x1 and x2 are the locations along the conductor measured in meters. The wind field is modeled by a stationary, Gaussian, one-dimensional and multivariate random process. The simulation algorithm used in this study is the one developed by Deodatis [36], based on the spectral representation method and fast Fourier transform technique. It can not only model the wind time history correlation in longitudinal direction but also in the transversal and vertical directions if needed. In this paper, the focus is on the fragility of the individual conductor, but the spatial correlation of the load should be considered for accurate regional analyses [37]. Fig. 3 presents one realization of the wind field simulation correlated in time and space. The simulated wind field covers a distance of 500 m and a time span of 10 min, with a resolution of 1 m and 0.1 s. 2.2. Time history analysis based on nonlinear FEM Fig. 2(b) shows the model of a transmission conductor with hinged supports at the same level. The conductor is characterized by the following parameters: span length L, sag d 0 , uniform mass density m, diameter D, elastic modulus E, longitudinal pretension force H0 and reference height. The conductor height varies along the span due to the sag of the conductor. However, the variation is small and the mean wind speed Vz acting along the conductor span can be approximately 2 taken as the mean wind speed at the reference height which is 3 d 0 below the support level [18]. The wind field simulation model used in this research is capable of taking different heights into consideration and remove this approximation, but the difference would be so small that it is recommended to use this approximation. The FEM model is built in the OpenSees platform [38], and the catenary element [15] is used in this study to model the conductor. This element was chosen because the stiffness matrix and element nodal forces are derived based on exact analytical expressions of elastic catenary structures. The conductor response is computed taking into account its large geometrical

2. Methodology 2.1. Wind field model and simulation In a hurricane event, the effective force acting on a conductor is perpendicular to the conductor’s longitudinal direction. The component of wind parallel to the conductor has little effect on the conductor response and is therefore ignored [20]. Winds whose angle of incidence is not perpendicular to the conductor are called yawed wind. The angle of yaw is designated by and is measured in the horizontal plane. Fig. 2(a) shows an example of yawed wind and the resultant force direction. The logarithmic law is regarded as a superior representation of strong wind profiles in the lower atmosphere than the power law [29]. In terms of hurricanes, according to Simiu et al. [30], the mean wind profiles differ only insignificantly from the logarithmic profiles in the lowest 400 m of the atmospheric boundary layer. Therefore, the logarithmic law is chosen to model the mean wind profile: 3

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Fig. 2. (a) Description of yawed wind on a transmission conductor (b) Finite element model of the conductor subjected to wind field (c) Effective wind force (d) Configuration of the conductor under static mean wind.

force in the conductor. The transmission conductors are often pretensioned, so that the sag of the conductors satisfies the clearance requirement. In order to model the pretension force, the unconstrained length for each element needs to be determined and provided to the FEM program. A typical conductor has a sag to span ratio of 1/30 to 1/50 [18], and the pretension force can be determined from catenary cables theory as:

Table 1 Terrain types and roughness length. Terrain type

Roughness length (m)

Very flat terrain (snow, desert) Open terrain (grassland, few trees) Suburban terrain (buildings 3–5 m) Dense urban (buildings 10–30 m)

0.001–0.005 0.01–0.05 0.1–0.5 1–5

H0 =

nonlinearity. An internal iterative scheme using the Newton-Rhapson method is implemented to facilitate the computation. One of the difficulties in building the FEM is to model the pretension

mgL2 8d 0

(4)

Then, the pretension force can be used to compute the unstrained length L0 of the conductor by solving the following equation using

Fig. 3. Realization of wind field simulation. 4

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Ft =

1 CD D {[(V + Vt ) 2

uz ]2 + (u y ) 2}

(10)

where Vz , Vz, t , V and Vt are the mean wind speed, wind turbulence, projection of mean wind speed perpendicular to the conductor and projection of wind turbulence perpendicular to the conductor, respectively as shown in Fig. 2(a); is the angle of yaw; = 1.226 kg/m3 is the mass density of air [20]; CD is the force coefficient, for instance CD = 1.0 is recommended by guidelines [20] for conductors with diameter larger than 1.27 cm, which is valid for most of the transmission conductors; uz and u y are the nodal velocities in the z and y directions at time instant t, respectively. Projections of the wind and gravity load per unit length in the y and z directions are the following: t

Fig. 4. Comparison of conductor tension response between finite element models with different number of elements.

mgL0

mgL 2H0

=

2EA0

mgL0

(5)

2H0

where A0 is the uniform cross-sectional area of the unstrained cable profile. As shown in Fig. 2(b), the conductor is divided into a number of catenary cable elements equally spaced along its longitudinal direction. The longitudinal coordinate x of each node is related to the Lagrangian coordinate s in the unstrained profile:

x=

H0 s H + 0 sinh EA 0 mg

Fv H0

1

sinh

1

Fv

mgs H0

(6)

where Fv = mgL0 /2 is the vertical reaction at the supports. In this way, the unstrained Lagrangian coordinate s for each node can be computed iteratively. Once the unstrained length is obtained for each element, in theory the initial y coordinates of the nodes can be specified as any arbitrary value. However, it will be faster for the program to converge if the input coordinates are close to the final shape of the conductor. Therefore the initial y coordinate of each node is specified using the catenary equation:

mgL0 EA 0

Fv mgL0

1+

Fv

s 2L 0 mgs H0

+

H0 mg

1+

Fv H0

Vt = Vz , t cos( )

Fy, t = Ft sin( t ) + mg

(12)

Fz , t = Ft cos( t )

(13)

The time-consuming nonlinear FEM is replaced by the modal superposition method in this study. The first step of the method is to determine statically the state of the conductor under mean wind, including the static tension components at the supports and the mean swing angle ¯ , as shown in Fig. 2(d). The mean swing angle can be computed as:

2 1/2

1 2 f¯D = V DCD 2

2 1/2

(7)

Finally, the gravity load is applied to the conductor, and the pretension force is automatically implemented as reaction force at the supports, for the specified unstrained length. The main sources of the fluctuating forces applied on the conductors are the turbulence of the wind flow and the forces due to movement of the conductor itself. Since the conductor diameter is very small relative to the length scales of the turbulence, the fluctuating forces applied on the conductor follow the variations in wind velocity (quasi-steady assumption). Incorporating the movement of the conductor body (aerodynamic damping), the simulated wind velocity field is converted to wind effective force per unit length of the conductor as:

V = Vz cos( )

(11)

uz

2.3. Time history analysis based on modal superposition

y =

V + Vt

where t is the angle between the nodal velocity in the z direction and the effective wind force Ft , as shown in Fig. 2(c), subscript t indicates that these quantities change over time. In Fig. 2(c), the lumped mass of the conductor moves under loadings and forms a swing angle . It is noted that the wind load at time instant t involves the unknown velocities uz and u y . These unknown velocities are approximated by those at the previous time step, given that the time step is small enough. In this study, the time step is set to be 0.01 s, and it was proven to be small enough through various convergence tests performed specifically for this study. Similarly, the number of elements required to model the conductor with sufficient accuracy has been determined through mesh refinement tests. As shown in Fig. 4, the conductor responses converge when the conductor is modeled with at least 80 elements. The nonlinear FEM is used as a reference solution, since it accounts for the geometrical nonlinearity of the conductor and the aerodynamic damping effect. However, completing a time history analysis through nonlinear FEM is very time-consuming. Therefore, this approach is not practical for fragility curve estimation.

numerical iteration [12]:

sinh

uy

= arctan

(14)

f¯D

¯ = arctan

(15)

mg

where f¯D is the mean drag force per unit length. Then, the in-plane and out-of-plane natural frequencies and mode shapes are determined by the solution of the eigenvalue equations [12]. The coupled dynamic equations of motions in terms of modal superposition are expressed as:

q¨iv + 2

iv

+

aiv

iv qiv

2 iv qiv

+

Nw

+

2

aijvw

jw qjw

=

j =1

(8)

q¨iw + 2

(9)

iw

+

aiw

iw qiw

+

2 iw qiw

Nw

+

2 j=1

5

Qiv m

aijwv

jv qjv

=

Qiw m

(16)

(17)

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aiv

=

CD 4

D2 m

2 V iv D

aiw

=

CD 4

D2 m

2 V iw D

vv

(18)

ww

(19)

aijvw

=

CD 4

D2 m

2 V jw D

ijvw vw

(20)

aijwv

=

CD 4

D2 m

2 V jv D

ijwv wv

(21)

ijvw

=

ijwv

=

vv

=

L 0 L 0

iv (x ) jw (x ) dx

(22)

iw (x ) jv (x ) dx

(23)

1 1 + sin2 ( ¯) ; 2

ww

=

1 1 + cos2 ( ¯) ; 2

vw

=

wv

=

1 sin( ¯)cos( ¯) 2

Fig. 5. Comparison of conductor tension response between FEM and modal superposition method.

(24) where qiv and qiw are the ith in-plane and out-of-plan modal displacements, respectively; iv and iw are the ith in-plane and out-of-plane natural frequencies, respectively; iv and iw are structural modal damping ratios which are assumed to be zero in this study; aiv is the ith in-plane aerodynamic damping ratio and aiw is the ith out-of-plane aerodynamic damping ratio; aijvw and aijwv are the coupling aerodynamic damping terms for in-plane and out-of-plane vibrations, respectively; iv and iw are the ith in-plane and out-of-plane mode shapes, respectively; Qiv and Qiw are generalized forces. Finally, it is assumed that the dynamic deformation around the static mean state of the conductor is small, and the conductor can be characterized as a linear system with dynamic properties computed under the static mean wind. For the sake of completeness, the coupled dynamic equations of motions considering the aerodynamic effects are expressed in matrix form:

Mq¨ + Cq + Kq = Q

significantly. However, under low wind loadings, the conductor becomes less tight and its dynamic properties deviate from the linear estimation. A static test is conducted using FEM where the conductor is subjected to a uniform load, and the displacement of the midpoint is measured. As shown in Fig. 6, the stiffness of the conductor changes very little from mean wind state to large loadings, but it changes substantially from mean wind state to small loadings. Since the assessment of fragility curves requires to determine only the peak response, the modal superposition method is considered sufficiently accurate to be used to evaluate the maximum response of conductors during a hurricane event. The modal superposition method has two main advantages compared to the nonlinear FEM. In the time domain, time history analysis using modal superposition is much faster than nonlinear FEM. This is because for modal superposition, the number of degrees of freedom equals the number of mode shapes considered, which is much lower than the number of degrees of freedom required for nonlinear FEM. Even more importantly, neglecting nonlinearities allows to avoid iterative solutions. Moreover, in the frequency domain, the modal superposition method enables directly the spectral analysis, in which the probabilistic properties of the responses can be computed without the need of any wind field simulation and time history analysis.

(25)

where q is the modal displacements vector, Q is the generalized forces vector. The mass matrix M, damping matrix C and stiffness matrix K can be computed using the coupled dynamic equations of motions as shown in Eqs. (16) and (17). The modal displacements can be computed using the classical Newmark method with = 1/2 and = 1/4 . Once the modal displacements are obtained, the response components R(t ) can be easily determined:

R (t ) = rT q (t )

(26)

2.4. Extreme value analysis

where r is the vector of modal participation coefficients. More details on the modal superposition method can be found in Wang’s paper [18] or any book on structural dynamics. To validate the linearity assumption required by the modal superposition method, its calculated response is compared to the response assessed by nonlinear FEM time history analysis as shown in Fig. 5, where geometrical nonlinearity is considered. Table 2 shows multiple tests with different cables and wind velocities. The physical properties of these conductors are described in Table 3. As shown in Table 2, the modal superposition method tends to underestimate the conductor response. It captures well the peak response with percentage of difference around 1% , but captures less accurately the minimum response with percentage of difference around 3.5% . This is probably due to the fact that the conductor’s dynamic properties under mean wind speed are similar to the dynamic properties under extreme loadings, but different from the dynamic properties under relatively small loadings. In fact, at the mean wind state the conductor is already stretched tight, similarly to what happens under strong wind, therefore the dynamic properties do not differ

Based on classic random vibrations theory, the cross-spectral density matrix of the modal displacement vector can be calculated as: (27)

Sq (f ) = H (f ) SQ (f ) H (f )

where H(f ) is the transfer matrix and the asterisk indicates conjugate transpose; SQ (f ) is the cross-spectral density matrix of the generalized forces vector and is a function of the wind spectrum and coherence function [18]:

SQij (f ) = 2

Jij (f )

=

4(f¯D L)2Sv (f ) Jij (f ) V2 1 L2

L 0

L 0

2 ij

cohV x1, x2 , f

(28) i (x1) j (x2 ) dx1 dx2

(29)

where Jij (f ) 2 is called the ‘joint acceptance function’. It defines the sensitivity of the interaction between the wind turbulence structure and the modes of vibration. ij is the angle coefficient for coupled mode 6

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Table 2 Comparison of conductor response computed by modal superposition method and finite element method under different maximum sustained wind speed (highest average wind speed over a one-minute time span measured at 10 m height) 1maximum sustained wind speed 2maximum axial load computed by modal superposition method (unit kN). 3maximum axial load computed by FEM (unit kN). 4percentage of difference. 5peak response. 6minimum response. V1

Coot 2

(m/s) 5

40+ 40−6 50+ 50− 60+ 60− 70+ 70− 80+ 80−

Drake

3

Mod

FEM

68.9 41.9 91.9 53.2 117.1 65.7 144.1 79.1 173.4 93.3

69.3 43.9 93.1 54.9 118.7 67.8 146.5 81.1 176.4 95.7

D

4

−0.7 −4.5 −1.3 −3.1 −1.3 −3.1 −1.6 −2.5 −1.7 −2.5

Tern

Coot Drake Tern Condor Mallard

Size (kcmil)1

795 795 795 795 795

Stranding (Al/Stl)

FEM

D

Mod

FEM

D

Mod

FEM

D

Mod

FEM

D

81.5 50.7 108.0 62.9 137.4 77.1 168.7 92.2 203.7 108.1

81.6 53.0 109.1 65.9 138.8 79.7 170.9 95.2 205.3 111.3

0.0 −4.5 −1.0 −4.6 −1.0 −3.2 −1.3 −3.1 −0.8 −2.9

73.0 44.7 97.3 56.4 123.7 69.5 152.3 83.4 183.5 98.2

73.5 46.8 98.3 58.3 125.3 71.7 154.7 86.0 186.3 100.9

−0.6 −4.6 −1.0 −3.3 −1.2 −3.2 −1.5 −2.9 −1.5 −2.7

78.6 48.6 104.3 60.7 132.7 74.4 163.1 89.2 196.8 104.6

78.7 50.9 105.5 63.1 134.0 76.9 165.1 91.8 198.6 107.6

−0.2 −4.5 −1.1 −3.9 −1.0 −3.2 −1.2 −2.8 −0.9 −2.8

87.7 55.0 115.4 67.5 147.1 82.5 180.0 98.4 217.6 115.2

87.4 57.7 116.8 70.8 148.5 85.4 182.9 102.0 220.0 118.4

0.3 −4.6 −1.2 −4.7 −0.9 −3.4 −1.6 −3.5 −1.1 −2.7

Diameter (m) Al wires

Stl wires

0.0038 0.0044 0.0034 0.0031 0.0041

0.0038 0.0035 0.0023 0.0031 0.0025

36/1 26/7 45/7 54/7 30/19

Mallard

Mod

where Rx , Ry , Rz are reactions in the x , y and z direction respectively. In principle, the conductor moves and changes shape, so the contribution of Rx , Ry , and Rz towards the axial force N changes over time. However, such variation is small, and the contribution of the longitudinal force (Rx ) is always much larger compared to the other two directions. Therefore, the angles between these forces and N are assumed constant and approximated by the angles at the state of the conductor under static mean wind. As a result of this assumption, the axial reaction can be approximated as:

Table 3 Properties of typical ACSR. 1In the United States electrical industry, transmission conductors are often identified by the area in thousands of circular mils (kcmil), where 1 kcmil = 0.5067 mm2 . Code

Condor

Weight (N/m)

Rated Strength (kN)

11.73 15.95 13.06 14.93 18.01

74.36 140.18 98.35 125.49 170.88

N (t )

Rx (t )cos( x ) + Ry (t )cos( y ) + Rz (t )cos( z ) = rTx q (t )cos( x ) + rTy q (t )cos( y ) + rTz q (t )cos( z )

(32)

where x , y, z are the angles between axial reaction and its components at x , y and z direction respectively under static mean wind; rx , ry, rz are the modal participation coefficients for the reaction in the x , y and z direction respectively, and their expression can be found in Irvine’s book [12]. Also this assumption has been validated in this study using the nonlinear FEM results as reference solutions, and multiple tests with different conductor and wind load samples have shown that the approximation is excellent. Hence, the spectrum of the axial reaction can be determined as:

SN (f ) = rTN Sq (f ) rN

(33)

rN = rxcos( x ) + rycos( y ) + rzcos( z )

(34)

The spectrum of the derivative in time of the axial reaction N is computed as: (35)

S N (f ) = f 2 SN (f )

Fig. 6. Relationship between force and displacement under static uniform loadings.

The variance of N is then computed by integration of SN (f ) and the variance of N can be computed by integration of S N (f ) over the frequency domain. Then, the problem falls under the category of ‘firstpassage problems’ in the theory of random vibrations [39]. The expected number of up-crossings of level a per unit time is:

shapes:

va+ = sin2 ( ¯),

ij

cos2 ( ¯),

=

if both ifboth

i

i

and

and

sin( ¯) cos ( ¯),

a2 2 N2

(36)

otherwise.

P (T0) = 1

(37)

exp ( va+T0)

For a fixed T0 , Eqs. (36) and (37) provide the distribution of in turn, of the peaks N¯ + a :

The maximum load on the conductor is the axial force at supports which can be calculated as:

R x2 (t ) + R y2 (t ) + Rz2 (t )

N

exp

It is well known that the probability of up-crossing level a in the interval 0 < t < T0 is:

(30)

N (t ) =

N

are out-of-plane mode shapes.

j j

are in-plane mode shapes.

1 2

DSi = N + a = N +

(31) 7

2

2 N ln

2 (va+)i N

N

va+,

a , and,

0.5

(38)

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2.6. Fragility models The fragility curves of transmission conductors are derived based on the demand and capacity distributions. A transmission conductor is often designed to have very large safety margin against rupture failure. In traditional structural reliability analysis, when the probability of failure is lower than 10 6 , it is common to consider the likelihood of failure negligible. However, even if the rupture of a single conductor can be considered a rare event, the failure risk of the whole transmission line cannot be ignored. This is because the conductors are connected in series. Failure of a single conductor will trip the whole transmission line, and may result in significant damage to the power network. A medium transmission line ranges from 80 to 240 km [2]. Each transmission line consists of hundreds or thousands of conductors. The probability of failure for a three phases transmission line can be determined as:

Fig. 7. Schematic energy spectrum of near-ground wind speed adapted from Van der Hoven [40].

where N is the mean axial reaction under mean wind speed, N and N are the standard deviations of N and N respectively. (va+)i is the ith realization of va+, treated as a random variable with exponential distribution from Eq. (37). DSi is the corresponding ith realization of the maximum axial reaction which is the demand of the conductor. The maximum sustained wind speed, which is the highest average wind over a one-minute time span, is chosen as the intensity measure in this study. The maximum sustained wind speed is then converted to the average wind speed of 10 min, and is implemented as mean wind speed for the analysis. The first reason for this conversion is that the Kaimal spectrum requires the wind speed averaging period to be at least 10 min. The second reason lies in the concept of spectral gap. According to Van der Hoven [40], there exists a spectral gap that ranges approximately from 2 h to 10 min as shown in Fig. 7. The variability is much less for the mean wind speed in the gap than the mean wind speed outside of the gap. Therefore, averages taken in the range of the spectral gap are better representations of the wind field. The approach developed by Engineering Sciences Data Unit (ESDU) [41] is adopted as the procedure to convert the maximum sustained wind speed to the mean speed over a 10 min time span. The conversion procedure requires the Coriolis parameter, which is a function of the latitude of the site.

n

Pline = 1

Aluminum conductor steel-reinforced cables (ACSR) are used worldwide on high-voltage transmission lines, therefore are the focus of the application of this study. In this case, the conductor is made of aluminum and steel strands. The diameters and numbers of the strands vary for different types of ACSR conductors. In general, steel adds tensile strength to the conductor, but also increases the conductor’s weight. Therefore conductors with larger strength are more expensive and require larger support structures. Most conductor producers specify the conductor strength as a single value: the rated strength. The rated strength represents the minimum value to pass quality control and is calculated in accordance with specification requirements. Therefore, the rated strength cannot be directly used to represent the capacity of the conductor which refers to the axial breaking force. The capacity of the conductor is obtained as the sum of the capacity of its wires. So in this study, conductor capacity is modeled by Monte Carlo simulation with the ASTM rule as:

d (2aw) 4

+ nsw s (sw .1%)

d (2sw) 4

Pi )3

(40)

where Pi is the probability of failure for the ith conductor, n is the number of spans in the transmission line. For instance, considering a 200 km transmission line made of conductors with 200 m span length and Pi = 10 6, then the probability of failure for the transmission line is 0.3% , which is not negligible. Therefore, even if the probability of failure of a conductor is very small, its value needs to be properly evaluated, at least in terms of its order of magnitude. It is worth noting that Eq. (40) ignores the correlation in the capacity and the wind loads of the various conductor segments, which makes their failure not independent. The assumption of independent events enables a simple assessment of the upper bound of the probability of system failure, which allows us to demonstrate the importance of assessing the conductor’s fragility. Brute force Monte Carlo simulation can be utilized to compute the probability of failure of a transmission conductor. However, since the failure of a conductor is such a rare event, the calculation of its probability of occurrence requires a very large amount of realizations, therefore making the brute force Monte Carlo simulation very timeconsuming. The first order reliability method (FORM) is an approximate method based on local linear approximation of the limit-state function. It computes the so-called ‘design point’ using a gradient-based search algorithm. FORM is well-known for its efficiency, unlike Monte Carlo simulation, only a small number of model evaluations is needed to calculate the probability of failure. The accuracy of FORM depends on the properties of the limit-state functions. In this study, large scale Monte Carlo simulations with 1012 realizations were conducted for multiple cases of 300 m conductors using parallel computing. Convergence tests have been carried out to ensure the convergence of the Monte Carlo simulations. In particular, the analysis was stopped at 10 n samples, when for the last 10 n 1 samples the normalized change in the probability of failure was within 1% :

2.5. Capacity of transmission conductors

CRi = naw s (aw )

(1 i=1

Pfmax n Pf n

Pfmin n

< 0.01

(41)

where Pfmax n is the maximum value of the probability of failure in the convergence plot over the last 10 n 1 samples, Pfmin n is the minimum value of the probability of failure in the convergence plot over the last 10 n 1 samples, and Pf n is the value of the probability of failure assessed with 10 n samples. The results of the Monte Carlo simulations are compared against the results obtained by FORM as shown in Table 4. The comparison shows that even for the cases in which the probability of failure is as low as 10 9 , FORM is capable of deriving results with sufficient accuracy (maximum difference: 1.66·10 9 ). Therefore, FORM is chosen to replace the brute force Monte Carlo simulation to efficiently compute the probability of failure of a conductor. To perform FORM, in this study we used the routines provided in the UQlab package [43].

(39)

where CRi is the ith realization of the conductor capacity; daw and dsw represent the aluminum wire diameter and steel wire diameter respectively; s (aw) is the breaking stress of individual aluminum strands; and s (sw .1%) is the stress in the steel strands at 1% extension. daw , dsw , s (aw), s(sw .1%) are all random variables with truncated normal distribution. The parameters of these distributions were estimated empirically by Farzaneh and Savadjiev [42]. 8

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Table 4 Comparison of probability of failure computed by MCS and FORM.

Case Case Case Case Case Case

Maixmum sustained wind speed (m/s)

Conductor type

Angle of yaw (°)

(10 9 )

MCS

FORM

Difference

60 70 55 52 65 60

Drake Drake Coot Tern Mallard Condor

15 35 40 20 10 25

6.1890 1.1330 1.6020 4.2910 1.9850 1.2900

7.8239 1.4793 2.0586 5.9531 2.5899 1.5520

1.6349 0.3463 0.4566 1.6621 0.6049 0.262

1 2 3 4 5 6

(10

9)

Site-specific information

Conductor properties Span length Sag to span ratio Height Conductor type (mass per unit length, diameter, rated strength, number of aluminum and steel strands and diameters)

(10 9 )

Drag coefficient Air density Terrain roughness Coriolis parameter Wind model Coherence parameter

Fig. 8. Summary of input parameters for fragility analysis.

Given the demand and capacity distribution parameters, the computation time for generating a fragility curve using FORM is about 10 s using an i7 16 GB RAM desktop. Compared to other methods, the proposed framework is able to generate fragility curves with great computational efficiency. For instance, it takes about 20 h to compute one fragility curve with brute force Monte Carlo simulation. Moreover, the other methods such as Monte Carlo simulation with time history analysis are too time-consuming for computing fragility curves. A single run of the time history analysis takes about 30 min with nonlinear finite element method and 1 min with modal superposition method, however billions of such analyses are required to compute a fragility curve. Table 5 shows five categories of hurricanes and tropical storms and their corresponding one-minute maximum sustained wind speeds based on the Saffir-Simpson hurricane scale. In recent years, multiple major hurricanes (Category 3 and above) have formed in the Atlantic Basins and made landfall: Otto (Category 3, 2016), Harvey (Category 4, 2018), Michael (Category 4, 2018), Irma (Category 5, 2017) and Maria (Category 5, 2017). These hurricanes, along with many smaller ones, have caused significant damages to the power system. Therefore, in this study, the hurricane intensity measure considered ranges from 33 m/s to 80 m/s with the resolution of 1 m/s, which covers all five categories of hurricane.

(a)

(b)

3. Application To demonstrate the proposed methodology and provide an initial set of results for the scientific community, fragility curves are assessed for five types of widely used ACSR with the same size as shown in Table 3. All the input parameters required for the fragility analysis are summarized in Fig. 8. The sag and height of the conductor are assumed to be known by the utility company. The coherence parameter is obtained on the basis of wind tunnel measurements [34,35]. The other parameters are region-specific and should be obtained for the region of interest through specifications [20] or regional test data. While there is some uncertainty in these parameters, their variability is assumed to be negligible for the development of fragility curves, compared to the large uncertainty in the load. The parameters assumed for the following analysis are: sag to span ratio d 0/L = 1/50 , conductor height z = 40 m, drag coefficient CD = 1.0 , air density = 1.226 kg/m3 , terrain roughness z 0 = 0.05 m, Coriolis parameter fcoriolis = 9.375 × 10 5 and coherence parameter C = 16.

(c) Fig. 9. (a) Demand distribution of five types of conductor with the maximum sustained wind speed of 50 m/s and 0° of angle of yaw (b) Conductor ‘Drake’ demand distribution with different maximum sustained wind speeds (c) Conductor ‘Drake’ demand distribution with different span lengths.

3.1. Demand and capacity The results of the demand distribution of the conductors are presented in Fig. 9. Fig. 9(a) presents the demand PDFs of the five types of conductors with the same span length of 300 m and subjected to the same maximum sustained wind speed of 50 m/s with 0° of angle of yaw. It is noted that there exists a positive correlation between the conductor demand and weight. Fig. 9(b) shows the demand PDFs of the conductor ‘Drake’ with span length of 300 m subjected to different maximum sustained wind speeds with 0° of angle of yaw. The probability distributions of the demand are sharp when the wind speed is relatively low, but under large wind speed the probability distribution becomes wide and the variability increases significantly. This is because when the wind speed is larger, the wind fluctuation becomes more violent resulting in wider variability of the wind load. It is noted that most building codes are developed based on the work by Davenport [14], assuming extremely narrow probability distributions of the demand. The presented results show that the assumption is valid for relatively

Table 5 Categories of hurricane and tropical storm. Category

m/s

mph

Five Four Three Two One Tropical storm

70 58–70 50–57 43–49 33–42 18–32

157 130–156 111–129 96–110 74–95 39–73

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difficult to obtain appropriate information on hurricane wind direction. When this is the case, the probability distribution of the angle of yaw should be assumed to be uniform. In this study, the fragility curve without knowledge on wind direction during a hurricane is derived based on 18 fragility curves computed with angle of yaw ranging from 0° to 90°, with 5° resolution. Fig. 14 shows the fragility curves derived without information on wind direction. These fragility curves are flatter than the fragility curves computed given the angle of yaw, as shown in Fig. 11. Even under extremely large wind speeds, such as 80 m/s, the profitability of failure for the weakest conductor is smaller than 60% . This is because there are chances in which the angle of yaw is close to 90°, and the probability of failure in such conditions is negligible. The fragilities presented in the discussed figures have been computed numerically, point-by-point. This approach is preferable over fitting a parametric analytical curve, because the fitting would add an error that nowadays is unnecessary. Some papers have discussed this topic and provided arguments against the use of curve fitting for the cases of building and bridge fragilities [7,44]. However, in the spirit of providing a resource that can be used by planners, practitioners and researchers for simplified, approximate analyses, parametric models of conductor fragility curves are presented in the Appendix of the paper.

Fig. 10. Distribution of the conductor capacity.

low wind intensity, but may not be valid when the intensity measure is large. Therefore, the current codes may underestimate the risks of the conductor under strong hurricanes. Fig. 9(c) presents the demand PDFs of the conductor ‘Drake’ with different span lengths subjected to the same maximum sustained wind speeds of 50 m/s with 0° of angle of yaw. The conductor demand and the variability of the demand distribution increase with the span length, as expected. Fig. 10 shows the distributions obtained by the Monte Carlo simulation for the conductor capacity in terms of axial force. The vertical lines in the figure represent the rated strength of the corresponding conductors. It is noted that the rated strength always locates in the left tail of the conductor capacity distribution, as expected.

3.3. Effect of the span length The performance of conductors is heavily influenced by their span lengths. Fig. 15 shows the fragility curves of conductor ‘Coot’ with four span lengths in normal and semi-log scale assuming uniform distribution of wind direction. As shown in Fig. 15, under a hurricane with maximum wind speed of 40 m/s, conductors with a span length of 500 m have high probability of failure, but when the span length is less than 300 m, the probability of failure is negligible. The relationship between wind speed (V ) , span length (L) and failure probability (Pf ) can be approximated using an exponential function with five parameters as:

3.2. Effect of the wind direction Two intensity measures are considered in this study: maximum sustained wind speed and its corresponding wind direction. Fig. 11 shows the fragility curves of five types of conductor with span length of 300 m under multiple angels of yaw in normal and semi-log scale. As shown in Fig. 11 from top to bottom, the conductor performs better and better due to the increase of rated strength. In general, the failure probability of the conductor is low, especially for conductors ‘Drake’ and ‘Mallard’. In practice, those types of conductors may fail only if subjected to hurricanes of category four and above. However, failure may occur for other (weaker) types of conductors even under hurricanes of category two and three. Considering a transmission line that consists of thousands of conductors, the risk of failure may become significant. The fragility curves are very steep, indicating that the performance of the conductors is very sensitive to the maximum sustained wind speed. Once the hurricane intensity reaches the critical range, the failure of many spans of conductors can be expected. The wind direction has a relatively small impact on the fragility if the angle of yaw is close to 0°, which is the worst case scenario. However, as the angle of yaw ( ) increases, the fragility curves shift quickly to the right, due to the fact that the wind force is proportional to cos2 ( ) . For the cases in which probability of failure is less than 10 9 , it is assumed that the risk is negligible and therefore is not shown in the figure. Fig. 12 shows the fragility surfaces of two types of conductors considering two intensity measures: maximum sustained wind speed and its corresponding angle of yaw. As shown in Fig. 12, both intensity measures have significant impact on the probability of failure of conductors. The wind speed and angle of yaw have a certain effective range within which the probability of failure is not negligible (e.g., larger than 10 9 ). In Fig. 13, each curve indicates the boundary for its corresponding conductor above which the probability of failure is not negligible. Conductors with higher rated strength have smaller effective range than those with lower rated strength. Unfortunately, though the angle of yaw has significant impact on conductor fragilities, its impact on other structures is not as critical and is often neglected in fragility modeling [11,10]. As a result, it may be

log10 (Pf ) = p1 exp (p2 V + p3 L + p4 V L) + p5

(42)

The parameters can be calibrated easily through regression analysis. For the purpose of demonstration, the five parameters of conductor ‘Coot’ are calibrated: p1 = 50280;p2 = 0.08745;p3 = 0.004931;p4 = 0.0004975;p5 = 0.2383. The R-square value of the regression model in this case is 0.9953. A transmission line consists of thousands of conductors, which may have different span lengths. The variation of the span lengths is due to the geomorphology of the land. For example, it is common for transmission lines to cross multiple rivers and highways. In these cases, large span lengths have to be used. Therefore, it is necessary to assess the influence of the variation of conductor span length on the probability of failure for a transmission line. As an example, a medium three phases transmission line whose length is 240 km is considered. It consists of conductors with average span length of 300 m. The transmission line is subjected to a hurricane with intensity of 55 m/s. The distribution of the span length is assumed to be a truncated Gaussian, with the minimum and maximum of 100 m and 500 m respectively. Fig. 16 shows the probability of failure for the transmission line calculated using Eq. (40) versus the coefficient of variation of the conductor span length keeping minimum, mean, and maximum span constant. Even small changes in the coefficient of variation of the span length (from 2% to 10%) can have dramatic effects on the probability of failure (from negligible to 50%). This is due to the fact that with larger coefficients of variation, there exists more conductors with large span length. These conductors are disproportionately more vulnerable than conductors with average span length, and are driving the failure of the entire transmission line because the conductors are connected in series. 4. Discussion and confidence in the results A number of simplifications and assumptions have been made in 10

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Fig. 11. Fragility curves of five types of conductor with span length of 300 m in linear scale (left figures) and in logarithmic scale (right figures).

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Fig. 12. Fragility surfaces of two types of conductor with span length of 300 m.

context, it seems acceptable to replace the FEM with the modal superposition method. However, in this study very small probabilities are assessed, thus the propagation of this error to the probability of failure needs to be investigated. Fig. 17 shows three fragility curves for conductor ‘Drake’ with a 300 m span length. One is the original fragility curve, and the other two are computed with a 1% or 2% increase of the conductor peak axial response, respectively. Even though the three fragility curves share the same trend, significant differences exist, especially in the portion of the curves corresponding to very low probability (i.e., less than 10 7 ). Therefore, practitioners and analysts who use fragility curves developed with the proposed method cannot expect to have high accuracy for very low probability of failure. However, the results can be used with confidence to assess the order of magnitude of such probability, even for very rare events. In the common practice, very low probabilities of failure are assessed with high good accuracy only for very specific structures (e.g., nuclear power plants) built with extremely high accuracy and well monitored and maintained. Since this research studies a class of structures which are not monitored and sometimes not well maintained, the accuracy of the proposed fragility models is considered sufficient.

Fig. 13. Effective range of wind speed and angle of yaw for five types of conductors with span length of 300 m.

5. Conclusion

deriving the fragility curves. For example, though the Kaimal spectrum has been validated through various tests around the globe during high wind events, for hurricanes recent tests have shown that this spectrum may underestimate the energy at lower frequencies [45]. Nevertheless, due to the lack of a universally validated wind spectrum for hurricanes, the widely accepted Kaimal spectrum is adopted in this study. As mentioned in Section 2.3, compared to the FEM, the modal superposition method tends to underestimate the peak axial response by about 1% . In terms of accuracy of a numerical model for nonlinear structural dynamics, a 1% discrepancy from an experimental or analytical reference solution is considered perfectly acceptable, and not neglecting that the FEM itself is an approximation. Therefore, in this

This paper presents a probabilistic framework for the assessment of fragility curves of electrical conductors in power transmission networks prone to hurricane hazards. The proposed framework properly considers the uncertainties associated with the conductor strength and wind field turbulence. It was shown that the conductor’s behavior can be modeled using modal superposition with linear dynamics around the static mean wind state. This is validated by comparing the time history analysis results with the results computed from nonlinear FEM. The modal superposition method facilitates spectral analysis in the frequency domain, and the statistical properties of the conductor demand under hurricane can be determined through extreme value analysis and random vibration theory. The capacity of the conductor is obtained

Fig. 14. Fragility curves derived without information on wind direction (assumption of uniformly distributed angle of yaw) for five types of conductors with span length of 300 m in linear scale (left figure) and in logarithmic scale (right figure). 12

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Fig. 15. Fragility curves for conductor ‘Coot’ with span length of 200, 300, 400 and 500 m in linear scale (left figure) and in logarithmic scale (right figure).

accuracy (at least in terms of order of magnitude) even for a rare event in which the probability of failure is as lows as 10 9 , which is relevant for system failure. The results show that the fragility curves are very steep and the performance of conductors is very sensitive to the maximum sustained wind speed. Large scale failure may be expected once the hurricane reaches a certain intensity. In general, the probability of failure is low for transmission conductors with sufficient steel reinforcement, such as conductor ‘Drake’ and conductor ‘Mallard’ subjected to hurricanes of category 1 and category 2. However, the probability of failure may increases significantly for major hurricanes whose category is 3 and above. Wind direction and span length also have a substantial impact on the performance of conductors. Since the information on wind direction is not easy to obtain (to the best of our knowledge, there are wind hazard curves including wind speed, but not directionality), a uniform distribution of the wind direction can be assumed, and the fragility curves can be derived accordingly. A sensitivity analysis suggests that for a transmission line with hundreds of conductors, its probability of failure increases with the variability of the conductor span length since the failure is driven by conductors with large span length. Therefore, in order to assess the reliability of a transmission line, the span lengths of conductors need to be obtained. The fragility models of transmission conductors developed in this study constitute an essential component in quantitative risk and resilience assessment for regional power transmission networks. The proposed fragility models can be used with historical and synthetic hurricane hazard maps based on hurricane data archives and simulators [46–48]. They can help engineers gain fundamental insights on the mechanical and probabilistic performance of transmission conductors. The developed framework can be applied to help decision makers plan effective inspection, maintenance and retrofit strategies for transmission conductors in order to minimize the risk of the power transmission systems under hurricane hazards.

Fig. 16. Probability of failure for the transmission line versus the coefficient of variation of conductor span length.

Acknowledgments This work is part of the “Probabilistic Resilience Assessment of Interdependent Systems (PRAISys)” project ( www.praisys.org). The support from the National Science Foundation through grant CMS1541177 and from the Pennsylvania Department of Community & Economic Development through grant PIT-19-02 is gratefully acknowledged. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organizations.

Fig. 17. The effect of error in peak axial response on fragility curves.

through Monte Carlo simulation, and the probabilities of conductor damage and failure are computed by FORM, which was shown to provide good accuracy for this class of problems. The proposed framework is able to assess the fragility curves with great computational efficiency. For instance, generating a single fragility curve costs about 10 s. The proposed methodology can derive the fragility with sufficient

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Appendix A The fragility curves showed in Fig. 11 were computed point-by-point, which is the preferable approach. Herein, also parametric fragility models are provided in the form of piece-wise log-normal cumulative distribution functions. This functional form was chosen because it is the most popular to fit fragility curves. However, it was not possible to determine a single set of parameters that could fit accurately the data for low and high wind speeds. For this reason, two models are provided for each conductor type and wind direction: one that captures well the fragility at low wind speeds, and one that describes accurately the fragility at high wind speeds. For each model, the parameter Vcut determines the recommended transition point from one model to the other. All the curves were fitted using the least-square method. The transition point was selected as the value of the wind speed at which the two models intersect. 1 2

+ 2 erf

(ln(Vin )

1 2

+ 2 erf

1

(ln(Vin )

1

Pf =

µ1)

, if Vin

µ2)

, if Vin < Vcut

2 1

2 2

Vcut

(43)

where Vin is the maximum sustained wind speed; erf is the error function; µ1, µ 2 , 1 and 2 are the parameters of the log-normal model, collected in Table 6 for a set of popular conductor types and wind directions. The fragility curves for additional conductor types can be computed with the methodology described in the paper.

Table 6 Parametric model of the fragility curves. N/A indicates that a single model can be used for the entire wind velocity range. Conductor type

Angle of yaw

µ1

1

µ2

2

Vcut

Coot Coot Coot Coot Coot Coot Tern Tern Tern Tern Tern Condor Condor Condor Condor Condor Drake Drake Drake Drake Drake Mallard Mallard Mallard Mallard

0 10 20 30 40 50 0 10 20 30 40 0 10 20 30 40 0 10 20 30 40 0 10 20 30

3.9319 3.947277715 3.994158629 4.075795731 4.198460381 4.374529350 4.085400325 4.100704588 4.147605636 4.229243345 4.351989553 4.202302065 4.217615683 4.264504819 4.346181231 4.493 4.251219850 4.266512682 4.313416600 4.397681147 4.544 4.359954155 4.375773045 4.429781844 4.531

0.02324 0.023248017 0.023241668 0.023249756 0.023236345 0.024823926 0.025953952 0.025949620 0.025949287 0.025960879 0.026244127 0.025816629 0.025806551 0.025819153 0.025974058 0.03884 0.024495567 0.024492432 0.024491503 0.028333746 0.03849 0.025397388 0.026406430 0.032079743 0.03900

3.942 3.962 4.009 4.090 4.213 4.390 4.100 4.115 4.162 4.244 4.366 4.217 4.232 4.280 4.362

0.03282 0.03480 0.03482 0.03474 0.03485 0.03495 0.03667 0.03656 0.03671 0.03683 0.03668 0.03698 0.03691 0.03717 0.03719

4.267 4.282 4.330 4.414

0.03633 0.03628 0.03659 0.03692

4.377 4.393 4.444

0.03701 0.03715 0.03799

50 50 53 57 65 77 58 59 61 66 75 65 66 69 75 N/A 68 69 72 77 N/A 76 76 78 N/A

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