Precipitation Simulation and Dynamic Response of a Transmission Line Subject to Wind-Driven Rain during Super Typhoon Lekima

Abstract

Typhoons bring great damages to transmission line systems located in coastal areas. Strong wind and extreme precipitation are the main sources of damaging effects. Transmission lines suffered from wind-driven rain exhibit more susceptibility to damage due to the coupled effect of wind and rainwater. This paper presents an integrated numerical simulation framework based on mesoscale WRF model, multiphase CFD model and FEM model to analyze the motions of a transmission line subjected to coupled wind and rain loads during typhoon events. A full-scale transmission line in Zhoushan Island is employed to demonstrate the effectiveness of the proposed framework by simulating typhoon evolution in terms of wind fields and rainfall, solving the coupled wind and rain fields around the conductor and predicting the dynamic responses of the transmission line during Super Typhoon Lekima in 2019. The results show that the horizontal displacements of the transmission line under the joint actions of wind and rain increase approximately 17–18% compared to those of wind loads only. It is important to consider the coupled effects of wind-driven rain on conductors in the design of transmission lines under typhoon conditions.

1. Introduction

Typhoons are one of the major natural disasters threatening China’s southeast coasts, causing much damage to buildings and infrastructures, including transmission tower-line systems of the power grid. The transmission tower-line system is a typical wind-sensitive structure with the characteristics of high flexibility, long span, and strong geometric nonlinearity [1]. In 2005, the typhoon “Damrey” caused the collapse of the power system of Hainan [2]; During Hurricane “Sandy” in 2012 and Hurricane “Irene” in 2011, over 200 transmission towers failed [3]; In 2015, the super typhoon “Mujigae” damaged 80 towers, impacting thousands of transmission lines [4]. Anthropogenic climate change has already substantially increased the exposure to extreme climate impact events worldwide, and further global warming is projected to exacerbate the weather hazards as we already see today [5]. Huang [6] indicates that under the backdrop of anthropogenic warming, both the landfall intensity and precipitation of typhoons in the southeastern coastal areas of China are expected to increase. The risk of typhoon-induced disasters on Southeast China is also significantly elevated [6].

The structural damage to transmission tower-line systems during typhoon weather can be attributed to strong winds and heavy rainfall. Burgh and Harton [7], as well as Burgh et al. [8], proposed a model equation to investigate rain–wind-induced vibrations in a simple oscillator. They discovered that variations in the detuning parameter can result in saddle-node and Hopf bifurcations. Deng and Yang conducted numerical simulations [9,10] to study the response of a tower-line system under windy conditions. An’s research [11,12], based on numerical simulations, considered the combined effects of wind and rain. However, their study primarily focused on the structural failure of towers. Some researchers analyzed the windage yaw of insulator strings and wires due to wind and rain loads. Yan et al. [13] introduced a numerical model of a transmission line section to calculate the dynamic swing of the suspension insulator. Zhou [14] proposed a model with a derived stability criterion to analyze the significant swing of the overhead conductor under wind and rain conditions. Fu and Li [15,16] established a calculation method for wind and rain loads on transmission conductors and discussed the effects of wind and rain excitation. Their results indicate that rain loads relative to wind loads could lead to a 22% increase for tower displacement and 19.51% for conductor. Zhou et al. [17,18] developed a two-dimensional model to investigate rain–wind-induced vibration in transmission lines and discussed the aerodynamic instability zone using the Lyapunov stability criterion.

With the rapid progress of high-performance computation technique, numerical weather prediction (NWP) becomes an effective method to analyze wind–rain climates and meteorological variables [19]. The weather research and forecasting model (WRF) is one of the most widely used NWP systems designed for atmospheric research and operational weather forecasting. It is a state-of-the-art modeling system that simulates the dynamics and physics of the atmosphere to generate detailed weather forecasts and conduct meteorological research. WRF has gained popularity due to its flexibility, scalability, and ability to simulate a wide range of meteorological phenomena. The Advanced Research WRF (ARW) is one of the dynamical cores available within the WRF model based on a fully compressible set of atmospheric equations and is an appropriate model for simulating tropical cyclone systems. Many researchers have conducted typhoon-related studies based on the ARW model and found that the ARW model could reproduce atmospheric conditions and agree well with the measurements during typhoons [20,21,22,23,24].

The coupled phenomena of wind and rain is referred to as wind-driven rain (WDR), that is, raindrops carried obliquely by strong winds. It occurs when raindrops get horizontal velocity component from wind blows, rather than falling straight down vertically. Wind-driven rain can lead to water intrusion, moisture damage, and potential issues like leaks, rot, or mold growth, affecting the thermal and moisture performance of buildings [25]. Researches have been carried out to study the WDR on building facades with full-scale measurements [26,27], semiempirical methods [28,29], and numerical simulations. Although the numerical simulation of WDR with computational fluid dynamics (CFD) is complex and time-consuming, it can accurately determine the temporal and spatial distribution of WDR on complicated building facades, and the simulation accuracy has been validated with field measurements [30,31,32,33,34]. Most of the WDR researches focused on the amount of rainwater captured by building facades; however, the rain load caused by droplet impingement and water adherence cannot be neglected on large deformation structures like transmission conductors. The CFD model of WDR is a suitable tool to obtain the rain loading input of mechanical analysis of conductors since it provides the information of rain impact at droplet scale.

In this paper, a multiscale numerical framework is developed based on the WRF-CFD and FEM methods to solve the WDR effects on the motion of a span of transmission lines located in Zhoushan Island. Compared to existing researches, the presented method enables the prediction of the dynamic responses of transmission lines under real weather conditions based on multiscale model coupling, and provides an insight into the wind-driven rain effect on conductors by means of multiphase CFD simulation. Section 2 proposes the methodology of the numerical simulation. Section 3 details the configurations of WRF model and presents the simulation results of Typhoon Lekima in 2019. Section 4 presents the CFD simulation based on the Eulerian multiphase (EM) model of the WDR flow field. Section 5 shows the finite element analysis of the transmission line under WDR condition. Section 6 provides discussion and conclusions.

2. Methods

Figure 1 demonstrates the workflow of the numerical simulation of WDR effects on a transmission line during a typhoon event. The framework consists of a mesoscale typhoon simulation through WRF, a CFD model of WDR multiphase flow and a finite element model of transmission line conductors.

2.1. Model Equations in WRF

The compressible, nonhydrostatic governing equations, i.e., time-averaged Reynolds equations, that govern the mesoscale NWP model are cast in flux forms and can be expressed in a terrain-following hydrostatic pressure vertical coordinate, as in the literature [35,36]. Modeling of clouds microphysical process plays an important role in mesoscale convective storm simulations. The WRF double-moment 6-class (WDM6) microphysics scheme [37] is applied in this study. It is a bulk parameterization scheme that contains six water species: water vapor, cloud droplets, cloud ice, snow, rain, and graupel. Cloud particles are represented as particle size distribution (PSD) functions and are characterized by their moments in modeling of physical processes. The cloud raindrop size distribution in WDM6 follows the normalized form given by [37]:

$$n_X(D_X)=N_X\frac{{\alpha }_X}{\Gamma (v_X)}{\lambda }_X^{{\alpha }_X{v}_X}{D}_X^{{\alpha }_X{v}_X-1}\exp [-{({\lambda }_X{D}_X)}^{{\alpha }_X}],$$

(1)

$${\lambda }_X={[\frac{\pi }{6}{\rho }_w\frac{\Gamma (v_X+3/{\alpha }_X)}{\Gamma (v_X)}\frac{N_X}{{\rho }_a{q}_X}]}^{1/3},$$

(2)

where $X=C,R$ refers to clouds or rain, respectively; ${\lambda }_X$ is the corresponding slope parameter; whereas $v_X$ and ${\alpha }_X$ are the two dispersion parameters.

The governing equation of the number concentration for each species is expressed as

$$\frac{\partial N_X}{\partial t}=-V\cdot {\nabla }_3{N}_X-\frac{1}{{\rho }_a}\frac{\partial }{\partial z}({\rho }_a{N}_X{V}_X)+S_X,$$

(3)

where the first and second terms in the RHS represent the 3D advection and sedimentation for $X$, respectively. The term $S_X$ represents the source and sink of number concentration for $X$.

The number-weighted-mean terminal velocity which is responsible for the sedimentation of the rain number concentration can be obtained by integrating the terminal velocity of rainwater:

$${\overline{V}}_{NR}=\frac{{\displaystyle \int V_R(D_R)d{N}_{DR}}}{{\displaystyle \int d{N}_{DR}}}=\frac{a_R}{{\lambda }_R^{b_R}}\Gamma (2+b_R){(\frac{{\rho }_0}{{\rho }_a})}^{1/2},$$

(4)

where $V_R(D_R)$ represents the terminal velocity of rain particles with diameter $D_R$. Based on Locatelli and Hobbs [38], the expression is given by

$$V_R(D_R)=a_R{D}_R^{b_R}{(\frac{{\rho }_0}{{\rho }_a})}^{1/2}.$$

(5)

2.2. WDR Simulation by Eulerian Multiphase Model

Under severe convective weather conditions, the coupled effect of strong wind and extreme rainfall will increase the load on transmission lines, and the rainwater adhering to the conductor surface can significantly change its aerodynamic characteristics. The WDR loads is related to the interaction of raindrops of different sizes with the wind flow. In the EM model, rain is regarded as various continuum and dilute phases with equivalent raindrop sizes, reducing the computation cost compared to Lagrangian particle tracking (LPT) model. Raindrops are divided into N groups according to the diameter D, and the kth rain phase represents the collection of raindrops of which the diameter $D\in \left[D_k-\Delta D/2,D_k+\Delta D/2\right]$, k = 1, 2, …, N, where $D_k$ is the kth diameter and $\Delta D=D_k-D_{k-1}$. The mass and momentum conservation equations for kth rain phase are presented as follows [30]:

$$\frac{\partial {\rho }_{\mathrm{w}}{\alpha }_k}{\partial t}+\frac{\partial ({\rho }_{\mathrm{w}}{\alpha }_k{u}_{kj})}{\partial x_j}=0,$$

(6)

$$\frac{\partial {\rho }_{\mathrm{w}}{\alpha }_k{u}_{ki}}{\partial t}+\frac{\partial ({\rho }_{\mathrm{w}}{\alpha }_k{u}_{ki}{u}_{kj})}{\partial x_j}={\rho }_{\mathrm{w}}{\alpha }_k{g}_i+{\rho }_{\mathrm{w}}{\alpha }_k\frac{3\mu C_{\mathrm{D}}R{e}_{\mathrm{R}}}{4{\rho }_{\mathrm{w}}{D_k}^2}\left(u_i-u_{ki}\right),$$

(7)

$$R{e}_{\mathrm{R}}=\frac{{\rho }_{\mathrm{a}}{D}_k}{\mu }\left|u-u_k\right|,$$

(8)

where ${\rho }_{\mathrm{w}}$ and ${\rho }_a$ are the density of water and air, respectively; ${\alpha }_k$ is the volume fraction of the kth rain phase; $u$ and $u_k$ represent the velocity vector of wind and the kth rain phase, respectively; $g_i$ is the gravity along i direction; $\mu$ is the dynamic viscosity of air; $C_{\mathrm{D}}$ is the drag coefficient; $R{e}_{\mathrm{R}}$ represents the relative Reynolds number.

The second term on the RHS of Equation (7) stands for the dragging of the wind flow. A raindrop in equilibrium state gives

$$g_i+\frac{3\mu C_{\mathrm{D}}R{e}_{\mathrm{R}}}{4{\rho }_{\mathrm{w}}{D_k}^2}{V}_{\mathrm{t}}\left(D\right)=0,$$

(9)

where $V_{\mathrm{t}}\left(D\right)$ is the terminal velocity of raindrops with diameter D, and can be calculated by [39]:

$$V_{\mathrm{t}}\left(D\right)=\left\{\begin{array}{ll}0,\hfill & D\le 0.03\mathrm{mm};\hfill \\ 4.323\left(D-0.03\right),\hfill & 0.03\le D\le 0.6\mathrm{mm};\hfill \\ 9.65-10.3\exp \left(-0.6D\right),\hfill & D\ge 0.6\mathrm{mm}\hfill \end{array}\right.,$$

(10)

Then, the drag coefficient $C_{\mathrm{D}}$ is constructed in the following form to hold Equation (9):

$$C_{\mathrm{D}}=\frac{24}{R{e}_{\mathrm{R}}}\exp \left(-0.1FN+0.201F{N}^2-0.03537F{N}^3+0.002537F{N}^4\right),$$

(11)

$$FN=\ln \left(1+R{e}_{\mathrm{R}}\right).$$

(12)

2.3. Segment Model of Overhead Conductors under WDR Condition

The overhead conductor is hung with suspension insulator strings between transmission towers. A segment model of the conductor is depicted in Figure 2. In this model, the conductor segment is simplified as a cylinder, and $\alpha$ is the inclination of the segment; $\beta$ is the wind angle towards the segment; $U$ is the inflow wind velocity. The impact of WDR on conductor loads can be divided into two parts: the effect of raindrop impingement and the influence of rivulets on aerodynamic characteristic. According to Zhou [14], the wind-driven rain is sufficient to create upper and lower rivulets along the conductor, while the lower rivulet is assumed to make little difference to the aerodynamic coefficients of the conductor and is neglected in this paper.

Figure 3 shows the cross section of the conductor. Under a certain rain–wind condition, upper rivulet occurs at the surface of the cylinder. Relative size of the height and the width to the diameter of the conductor are about 0.10 and 0.17 [17]. The balance angle of the upper rivulet is ${\theta }_0$, determined by the coupled actions of gravity force, surface tension, and rain-wind loads. The unstable angle of the upper rivulet $\theta$ oscillates around ${\theta }_0$ and the vibration of the rivulet can be considered harmonic [17,40]. Yet, for simplicity, we assumed that the relative position of upper rivulet and conductor surface is fixed, that is, $\theta \equiv {\theta }_0$.

The component of the wind velocity $U_0$, perpendicular to the cylinder, can be expressed as

$$U_0=U\sqrt{1-{\cos }^2\alpha {\sin }^2\beta }.$$

(13)

The initial attack angle is defined as

$${\phi }_0=\arctan (\sin \alpha \tan \beta ).$$

(14)

The along-wind direction aerodynamic force per unit length of the cylinder can be expressed as

$$F(\lambda )=\frac{1}{2}\rho U_0^{2}d{C}_F(\lambda ),$$

(15)

where $C_F(\lambda )$ is the aerodynamic force coefficient and $\lambda =\theta +\phi$; $\rho$ is the air density; $d$ is the diameter of the cylinder.

The aerodynamic force coefficient $C_F(\lambda )$ is derived as

$$C_F(\lambda )=U_r^2(C_{drag}(\lambda )\cos \phi -C_{lift}(\lambda )\sin \phi )/U_0^2,$$

(16)

where $C_{drag}$ and $C_{lift}$ are the aerodynamic drag and lift force coefficients, respectively. $U_r$ is the instantaneous relative wind velocity, and its horizontal and vertical component can be computed as

$$U_{r,z}=U_0\cos {\phi }_0-\dot{z},$$

(17)

$$U_{r,y}=U_0\sin {\phi }_0-\dot{y},$$

(18)

where $\dot{z}$ and $\dot{y}$ are the horizontal and vertical velocity of the cylinder, respectively.

3. Simulation of Typhoon Lekima

3.1. Overview of Lekima and WRF Model Settings

Super Typhoon Lekima in 2019, designated as No. 1909, was a powerful and destructive tropical cyclone that struck East Asia in August 2019. It was the ninth named storm and the fourth typhoon of the 2019 Pacific typhoon season. Lekima originated from a tropical depression that formed east of the Northern Mariana Islands on 30 July 2019. The storm gradually intensified as it moved west-northwestward, reaching typhoon status on 6 August. It continued to strengthen rapidly and became a super typhoon on 7 August with maximum sustained winds of 62 m/s. Lekima then made landfall in Wenling, Zhejiang Province, China, on August 10. The impact of Typhoon Lekima was severe, causing widespread destruction and loss of life. It triggered torrential rainfall and strong winds, resulting in widespread flooding, landslides, and mudslides. The storm caused significant damage to infrastructure, including buildings, bridges, and roads. It also disrupted transportation and led to power outages in several areas. According to the news report, Lekima has affected over 4 million people in Zhejiang Province. Crop damage has occurred on 103,000 hectares of land, with 13,000 hectares suffering complete crop loss. Additionally, over 14,000 houses have been damaged or collapsed due to the disaster, resulting in a direct economic loss of CNY 7.44 billion.

As demonstrated in Figure 4, two-way nested static downscaling domains are used in WRF simulation and the simulation time period is from 0000 UTC 07 Aug to 0000 UTC 13 Aug in 2019. For the initial and boundary conditions of the outermost domain, 6-hourly Final Operational Global Analysis data of 1° resolution from the National Centers for Environmental Prediction (NCEP) are used, while the boundary conditions of the inner domains are provided by the domain above. The spatial and temporal resolutions of each domain are listed in

Table 1. Spatial and temporal resolutions of WRF domains.

Nested Domain	Mesh Size (m)	Horizontal Scale (km)	Time Step (s)
d01	12,000	211 × 229	60
d02	4000	187 × 172	20
d03	1333.3	235 × 193	6.67
d04	444.4	220 × 178	2.22

. The schemes and parameterizations of physical processes of the simulation are listed in

Table 2. Parameterizations of physical processes.

Model Parameter	Description
Dynamics	Advanced research WRF
Simulation period	7–13 August 2019
Microphysics	WRF double-moment 6-class
Radiation	Dudhia scheme and RRTM scheme
Land surface	Unified Noah LSM
Surface layer	Revised MM5 Monin–Obukhov similarity scheme
PBL scheme	YSU
Ocean model	A simple OML model
Air–sea frictional formulation	Donelan scheme

.

3.2. Simulated Tracks and Intensity

Figure 5 shows the result of the simulated track and CMA best track records [41,42] (https://tcdata.typhoon.org.cn, accessed on 12 May 2024). As we can see, the ARW model fully reproduces the movement trajectory of the typhoon’s first landfall, reentry into the sea, and subsequent landfall and the simulated track closely matches the observation. Figure 6 presents the time series of the minimum sea level pressure (SLP) and the maximum sustained wind (MSW) velocity derived from simulation and CMA observation. Before making landfall in Zhejiang, the simulation underestimates the intensity of Lekima with lower MSW and higher minimum SLP, while the intensity results are in good agreement with observations after landfall with maximum deviation of MSW less than 20% and 3% for SLP. The misalignment before landfall might be due to the sea surface flux scheme, assigned by parameter “isftcflx” in WRF model. In this paper, the default sea surface flux scheme is applied, leading to the underestimation of latent heat flux and sensible heat flux, and is not conducive to the rapid intensification of Lekima near the coast.

3.3. Simulated Wind Field and Precipitation

The evolution of the typhoon wind field from 0000 UTC 09 Aug to 0060 UTC 10 Aug 2019 at 30 m above the ground level in WRF d02 domain is plotted with a time interval of 6 h in Figure 7 (Zhoushan Island is marked with a red dot). It shows that the ARW model successfully captured the process of Lekima gradually approaching Zhejiang and reducing its intensity immediately when reaching the coastal area before making landfall due to the mechanical dragging. During this process, the local wind velocity in Zhoushan Island was greatly influenced by the mesoscale spiral structure of Lekima. Figure 8 marks the location of six ground meteorological stations near the east coast of China and Figure 9 compares the simulated 6 h averaged rainfall intensity at these stations with the observation data obtained from China Meteorological Data Service Centre (https://data.cma.cn/, accessed on 12 May 2024). It shows that extreme heavy rainfall occurs from 0000 UTC 09 Aug to 1200 UTC 10 Aug and the ARW model reproduced the rainfall event successfully. The simulated rainfall intensity at coastal stations (Shangyu, Hongjia, and Yuhuan) match the observation more closely, compared to the inland areas (Huzhou, Linan, Hangzhou), because the complex surface condition increases the uncertainty of precipitation simulation. The simulated results indicate that the proposed WRF method could regenerate the whole process of typhoon events and reproduce the strong wind fields and local extreme heavy rainfall with a reasonable accuracy.

4. Wind-Driven Rain Load on Conductor

After the mesoscale WRF simulation of Lekima, local wind field and rainfall intensity in Zhoushan Island was obtained, and the subsequent procedure of our simulation system was to solve the coupled wind and rain field around the transmission line and find out the WDR load on conductors. The WDR simulation was carried out at 0000 UTC on 10 Aug with a mean wind speed of $U=20.3\mathrm{m}/\mathrm{s}$, a wind angle of $\beta ={12.1}^{\circ }$ and a rainfall intensity of $R_{\mathrm{h}}=39\mathrm{mm}/\mathrm{h}$ for the targeted transmission line in Zhoushan Island, as shown in Figure 10. Two-dimensional computational domains and meshes of the fluid region around the cross section of the conductor are established, as presented in Figure 11. Based on the drag coupling between rain phases and wind phase, the horizontal raindrop velocities are assumed to be equal to the inflow wind velocity, and the vertical velocities of rain phases are set by Equation (10). The phase volume fraction ${\mathsf{\alpha }}_k$ at the inlet boundary can be calculated as

$${\mathsf{\alpha }}_k=\frac{R_{\mathrm{h}}{f}_{\mathrm{h}}\left(R_{\mathrm{h}},D\right)}{V_{\mathrm{t}}\left(D\right)},$$

(19)

where $R_{\mathrm{h}}$ is the rainfall intensity; $f_{\mathrm{h}}\left(R_{\mathrm{h}},D\right)$ represents the volume fraction of raindrops with diameter D in all raindrops passing through a unit area horizontal plane in unit time, i.e., flux fraction, can be calculated as

$$f_{\mathrm{h}}\left(R_{\mathrm{h}},D\right)=\frac{D^{3}N\left(R_{\mathrm{h}},D\right)V_{\mathrm{t}}\left(D\right)}{{{\displaystyle \int }}_0^{\infty }{D}^{3}N\left(R_{\mathrm{h}},D\right)V_{\mathrm{t}}\left(D\right)dD},$$

(20)

where $N\left(R_{\mathrm{h}},D\right)$ is the raindrop size distribution function which follows the modified $\Lambda$ distribution proposed by de Wolf [43], and is expressed as

$$N\left(R_{\mathrm{h}},D\right)=N_0\left(R_{\mathrm{h}}\right)D^{2.93}\exp \left(-\Lambda \left(R_{\mathrm{h}}\right)D\right),$$

(21)

$$N_0\left(R_{\mathrm{h}}\right)=1.98\times {10}^{-5}{R_{\mathrm{h}}}^{-0.384}$$

(22)

$$\Lambda \left(R_{\mathrm{h}}\right)=5.38{R_{\mathrm{h}}}^{-0.186}$$

(23)

Figure 12 presents the rain phase flux fraction with different rainfall intensities. According to the illustrated raindrop size distribution, the discretized raindrop sizes are set to be $D=0.5~6.0\mathrm{mm}$ with the increment $\Delta D=0.5\mathrm{mm}$ as inlet rain condition, for the WDR simulation of rainfall intensity $R_{\mathrm{h}}=39\mathrm{mm}/\mathrm{h}$.

The volume fraction and raindrop velocities at the top boundary are defined in the same way as the inlet boundary condition. The boundary conditions at the bottom, outflow boundary, and the conductor surface are defined as

$$\left\{\begin{array}{ccc}\frac{\partial {\alpha }_k}{\partial n}=0,& \frac{\partial u_k}{\partial n}=0& \mathrm{for}{u}_k\cdot n>0\\ {\alpha }_k=0,& u_k=0& \mathrm{for}{u}_k\cdot n<0\end{array}\right.,$$

(24)

where $n$ is the unit normal vector pointing out of the computational domain. The first expression in Equation (24) indicates that the raindrops will impact the solid surface just like no blocking. Therefore, based on the momentum theorem, the rain load caused by raindrop impact can be calculated as [44]

$$p_{rain}={\displaystyle {\int }_0^{\infty }{\rho }_l{\alpha }_k{V}_n{(D_k)}^{2}d{D}_k},$$

(25)

$${\tau }_{rain}={\displaystyle {\int }_0^{\infty }{\rho }_l{\alpha }_k{V}_n(D_k)V_{\tau }(D_k)}d{D}_k$$

(26)

where $p_{rain}$ and ${\tau }_{rain}$ refers to the rain phase pressure produced by the momentum loss in the normal and tangential direction, respectively. $V_n(D_k)$ and $V_{\tau }(D_k)$ are the velocity of rain phase with diameter $D_k$ towards the solid surface along normal and tangential directions, respectively. Then, the $C_{drag}$ and $C_{lift}$ under WDR conditions can be derived from the summation of wind pressure and rain phase pressure as

$$C_{drag}=\frac{{\displaystyle \oint (p_{\mathrm{rain}}+p_{\mathrm{wind}})d{A}_z+({\tau }_{\mathrm{rain}}+{\tau }_{\mathrm{wind}})d{A}_y}}{\frac{1}{2}{\rho }_a{U}_0^{2}d},$$

(27)

$$C_{lift}=\frac{{\displaystyle \oint (p_{\mathrm{rain}}+p_{\mathrm{wind}})d{A}_y+({\tau }_{\mathrm{rain}}+{\tau }_{\mathrm{wind}})d{A}_z}}{\frac{1}{2}{\rho }_a{U}_0^{2}d}$$

(28)

Figure 13 shows the simulated drag and lift coefficients of the conductor with upper rivulet under coupled wind and rain condition. Compared to cases without rain, both the drag and lift coefficients exhibit obvious variation from the constant results without rain. The result shows that the derivative of drag coefficients with respect to angle suddenly changed from negative to positive, and the derivative of lift coefficients changed from positive to negative when the attack angle $\lambda \approx {39}^{\circ }$, which is the critical angle causing the boundary layer to trip and influencing the location of the separation point [14].

5. Motion of Conductor under WDR

The transmission line in Zhoushan Island consists of two spans of 160 m and 260 m. Each subconductor of the quad-bundled conductor is made of JL/G1A-300/25, the physical parameters of which are listed in

Table 3. Physical parameters of single conductor (JL/G1A-300/25).

${\mathit{A}}_{\mathit{c}}(\mathbf{m}{\mathbf{m}}^2)$	${\mathit{D}}_{\mathit{c}}\left(\mathbf{mm}\right)$	${\mathit{\rho }}_{\mathit{L}}(\mathbf{kg}\cdot {\mathbf{m}}^{-1})$	$\mathit{E}\left(\mathbf{GPa}\right)$	${\mathit{T}}_{\mathit{Y}}\left(\mathbf{kN}\right)$
333.31	23.8	1.057	65	83.76

. In the table, $A_c$ is the calculated cross-sectional area of the conductor; $D_c$ is the outer diameter; ${\rho }_L$ is the mass per unit length; $E$ is the elastic modulus; $T_Y$ is the tensile strength of the conductor. To improve computational efficiency, the four subconductors of the quad-bundled conductor are modeled as a single equivalent conductor. The cross-sectional area, mass per unit length, and other physical parameters are adjusted accordingly for the single equivalent conductor to create a simplified two-span transmission line model, as shown in Figure 14.

For applying wind or WDR loads, a total of 89 loading nodes are defined along the transmission line in Figure 14. The mean wind velocity and rainfall intensity affecting the transmission line during Lekima in terms of WDR loads are derived from the WDR simulation as presented in Section 4. Local turbulence wind fields are numerically generated by the spectral representation method [45]. Figure 15 shows the generated fluctuating wind velocity at node 34.

According with the derived wind-driven rain meteorological condition and the aerodynamic force coefficients presented in Figure 13, the wind-driven rain loads are applied to the finite element model of the transmission line, and dynamic response analysis of the transmission line is carried out. The initial geometric configuration of the conductor under gravity can be precisely described using the catenary equation, which could be approximated by the parabolic equation for a small ratio of sag to span, i.e., less than 1/8. Figure 16 compares the displacements of node 34 subjected to wind loads and WDR loads. The result shows that the WDR loads would produce more significant horizontal displacements than those by wind loads only. The greater WDR load effects could be attributed to the occurrence of the upper rivulet and the impingement of raindrops. Figure 17 presents the mean and standard deviation of horizontal displacements along the transmission line with node number. It is found that the standard deviation of horizontal displacement shows little difference between two cases of wind loads and WDR loads, while the mean values under the WDR loads are significantly greater than those of wind loads with increase rates of around 17–18%. Figure 18 shows the time history of reaction force at node 88 in X direction, and it is found that WDR exerted larger forces on line support than wind condition, which means that WDR load increases the risk of tower compared to wind load. Such larger swing behavior of the overhead conductor due to WDR loads indicates that it is necessary to take account of the WDR loads in the design of transmission lines for extreme weather conditions of typhoons.

6. Discussion and Conclusions

In this paper, a multiscale WRF-CFD-FEM numerical framework is developed to analyze the motions of the transmission line subjected to coupled wind and rain loads during typhoon events. A case study was presented in detail by solving the dynamic responses of a full-scale transmission line located in Zhoushan Island during Super Typhoon Lekima in 2019. The proposed framework firstly reproduced the evolution of Super Typhoon Lekima, including the wind field and precipitation distribution at the coastal area using mesoscale weather model WRF. Then, the coupled wind and rain fields around the transmission line were simulated by the Eulerian multiphase CFD model. Finally, the motion of the conductor under joint actions of wind and rain, i.e., WDR loads, were solved by the finite element method. Some conclusions are summarized as follows:

(1)

The ARW model is capable of simulating the track and intensity evolution of typhoon events. It can also explicitly resolve the typhoon wind field and the associated precipitation for a target region. The simulated wind and rain results of Super Typhoon Lekima show good agreements with observations.

(2)

The coupled effect of wind and rain load consists of the upper rivulet attached to the conductor surface and the direct impingement of raindrops. The simulated drag and lift coefficients of the overhead conductor under WDR conditions exhibit noticeable variations with the wind angle of attack.

(3)

The proposed multiscale WRF-CFD-FEM numerical framework could be used to analyze and predict the dynamic responses of transmission lines under typhoon conditions by considering coupled wind and rain effects. In the case study of Super Typhoon Lekima, the horizontal displacements of the transmission line under the coupling of wind and rain are significantly larger than those of wind loads only. The increasing rates of 17–18% were found for the mean displacements under WDR loads compared to the wind loads. It is also found that WDR would produce larger conductor tension to the supporting tower than only wind condition. Such larger swing behavior of the overhead conductor due to WDR loads indicates the importance of coupled effects of wind and rain during typhoon conditions for the design of transmission lines.