Dynamics Performance Research and Calculation of Speed Threshold Curve for High-Speed Trains Under Unsteady Wind Loads

Abstract

Affected by strong wind environments, the vibration of trains will significantly intensify, which will severely impact the running quality of trains. To address such challenges, an improved wind load model is proposed in this paper to simulate the shock of strong wind on trains. The proposed model employs the integral approach to calculate the equivalent wind load on trains and applies it to the body of trains during the dynamics simulation process. Eventually, the two-level running quality threshold curve for passenger and freight trains is acquired through the conditional probability density function and the regularized regression model. This achievement covers train speed restrictions for wind speeds ranging from 0~25 m/s, providing a scientific basis for railway departments to adjust train speeds based on real-time wind speeds. It is of utmost importance for ensuring the safe and efficient operation of trains under strong wind conditions.

1. Introduction

Railway operation environments vary greatly in different regions, with strong wind being one of the most common external complex environments [1,2]. Natural wind, as a random excitation [3], interacts with train wind and then causes violent train vibration, which seriously affects the safety and stability of train running [4,5]. A bridge is a flexible structure. It is prone to deformation in high-wind environments. Among other things, pulsating winds can also directly cause low-frequency vibration of the bridge, which will affect the operational safety of trains [6]. At the same time, lateral wind pressure applied to the car body will also cause the train to have a dynamic effect on the bridge [7]. With the continuous improvement of both the proportion of bridges in the total line and train speeds, the impact of strong wind environments on train running is even more obvious. According to statistics, the maximum wind speed in the four wind areas of China’s Lanzhou-Xinjiang Railway Line is 64 m/s, and altogether there have been more than 30 train accidents caused by strong wind on this line [8,9]. Train accidents caused by strong winds are not uncommon abroad. For example, a train derailment accident occurred in Japan because the train was affected by crosswind when passing over a bridge [10]. Meanwhile, a strong wind in a valley once led to a derailment accident of a train in Switzerland [11]. In addition, similar accidents have also occurred in Belgium, Italy, and the United States [12,13]. The above accidents show the importance to ensure the safe running of trains in strong wind environments.

Currently, the main solution to the problem of safe operation of trains in strong wind environments is to reduce speed or stop services of trains [14]. Therefore, whether the running speed of trains at different wind speeds is controlled within the safe range should be judged firstly [15]. And the dynamic evaluation criterion is often used as the basis to evaluate the safety and stability of train operation. Generally, traditional train dynamics evaluation criteria mainly include derailment coefficient, overturning coefficient, wheel unloading rate, wheel–rail lateral force, Sperling index, etc. [16,17]. For example, Di Gialleonardo [18] analyzed train operation safety by studying the derailment mechanism; Deng [19] judged train running stability by calculating the Sperling index. In fact, natural wind, as an external excitation with a wide variation range, cannot be ignored in the actual running environment because of its impact on train safety and stability. To this end, Giappino [20] assessed the rollover risk of light trains on a wind farm; Dai [21] designed a kind of wind barrier and studied the dynamic performance of trains under the protection of the wind barrier. Liu [22] proposed a method to predict wind speed, which can judge the safety and stability during train running in advance, and solved the problem of time delay in the process of train dynamics analysis. However, the train wind, generated by the relative air movement of the train, can also affect the dynamic characteristics of the running train [23]. Therefore, when studying train running quality in strong wind environments, the law of dynamic characteristics of a train at different speeds also needs to be studied. For example, Yu [24] considered the influence of train movement on wind speed on a train surface in their work and calculated the influence of corresponding wind load on the train operation safety. Xiang [25] considered simultaneously the effects of wind speed, train speeds, heights of wind barriers, and wind directions on wind load. Their work laid a foundation for further study on dynamic response and running quality of trains under various external factors. Gou [26] designed a wind hazard warning system to ensure the running safety and stability of trains. However, this system takes into consideration only the speed limit of trains with a wind speed in the range of 15~26.5 m/s but fails to consider the speed limit of trains with wind speed lower than 15 m/s. In addition, the wind load on trains at different embankment heights changes obviously, hence, the embankment height also needs to be considered when studying the dynamic characteristics of trains in strong wind environments.

Briefly, based on the wind speed data along the Qinghai-Tibet Railway, this paper proposes an improved wind load model to simulate the shock of strong wind on trains and obtains the two-level running quality threshold curve of passenger and freight trains, which plays a crucial guiding role in the decision making of the safe running speeds of trains in a strong wind environment. The main contributions of this paper are highlighted as follows:

• An improved wind load model is proposed based on the weighted amplitude wave superposition (WAWS) method and the integral method in this paper.
• We have collected a real-world wind speed data set, which encompasses 31 continuously deployed wind sensors along the Qinghai-Tibet Railway in China (91°86′ E, 33°25′ N). The aforementioned data set encompasses almost all common running conditions of trains on the Qinghai-Tibet Railway under a strong wind environment. Based on this data set, we have verified the accuracy of our proposed pulsating wind model.
• Utilizing the conditional probability density function to design the dynamics evaluation criteria of trains.
• The two-level running quality threshold curve of trains is constructed via the regularized regression model. On the premise of ensuring the safe operation of trains, the threshold curve of Grade-I can offer guidance for the smooth running of passenger trains. Likewise, the threshold curve of Grade-II can provide guidance for the efficient running of freight trains.

The remainder of this study is organized as follows. In Section 2, according to the monitoring data along the Qinghai-Tibet Railway, the random fluctuating wind field is simulated, and the corresponding wind load data are obtained. Section 3 is the simulation experiment. Through a large number of simulation experiments, the regression relationship between the over limit probability of high-quality operation of the train and the wind speed, as well as the two-level running quality threshold curve of the train, is obtained. Section 4 is the discussion of our findings and the conclusion.

2. Methodology

Based on the wind speed data measured along the Qinghai-Tibet Railway in China (91°86′ E, 33°25′ N), this paper mainly studies the influence of lateral wind on the running of CRH2 high-speed trains.

2.1. Simulation Method of Random Fluctuating Wind Field

Fluctuating wind, as one of the dynamic components of wind, has the characteristics of randomness and uncertainty in the time domain [27]. Therefore, scholars often use a wind speed power spectrum to simulate fluctuating wind speed [28,29]. In addition, as wind is affected by ground friction, there is a linear correlation between wind speed and the height of trains above the ground [30,31]. Therefore, the Kaimal spectrum and harmonic synthesis method [32,33], given by the International Electrotechnical Commission, are adopted to simulate the time history of random fluctuating wind. A variable, $W_j(p\mathsf{\Delta }t)$, is defined to denote the random time history of simulation point j, which can be formulated as follows:

$$\begin{array}{l}{W}_j(p\mathsf{\Delta }t)=\mathrm{Re}\left\{\sqrt{2\mathsf{\Delta }\omega }{\displaystyle \sum _{m=1}^j{\displaystyle \sum _{l=0}^{N-1}\sqrt{S(f,U_Z)}}{G}_{jm}\left(f,U_Z\right)\exp \left[i\left(\omega \left(p\mathsf{\Delta }t\right)+{\phi }_{ml}\right)\right]}\right\}\\ (j=1\sim n),(p=0\sim 2N\times n-1)\end{array}$$

(1)

where $\mathsf{\Delta }\omega$ is the frequency increment, $\omega$ is the angular frequency. $\omega$ and $\mathsf{\Delta }\omega$ are given as follows:

$$\omega =l\mathsf{\Delta }\omega +m\mathsf{\Delta }\omega /n$$

(2)

$$\mathsf{\Delta }\omega ={\omega }_{up}/N$$

(3)

where ${\omega }_{up}$ is the cut-off frequency, and N is the number of points of the sampling frequency. S represents the Kaimal spectrum. $G_{jm}$ denotes the number of the j-th row and the m-th column of the correlation coefficient matrix. When $i=\sqrt{-1}$, ${\phi }_{ml}$ is the random phase angle uniformly distributed in $\left[0,2\pi \right]$. And n is the number of simulated wind speed points $\left(n=20\right)$.

After confirming the frequency increment, the Kaimal spectrum of lateral fluctuating wind can be formulated as:

$$S(f,U_Z)=4{\sigma }^{2}L/U_Z{\left(1+6fL/U_Z\right)}^{-5/3}$$

(4)

where $\sigma ,L$ are the standard deviation and integral scale of turbulence, f is the fluctuating wind frequency, and $U_Z$ is the wind speed of the lateral average wind at height Z above the ground. As for the calculation method of relevant parameters, we can refer to the methods of IEC 61400-1 [34].

Based on Davenport coherence function, the correlation coefficient matrix can be formulated as:

$$G_{jm}\left(f,U_Z\right)=\left\{\begin{array}{c}\begin{array}{cc}& \end{array}0\begin{array}{ccc}& \begin{array}{cc}& \end{array}& 1\le j<m\le n\end{array}\\ \begin{array}{cc}\exp \left(-cf\mathsf{\Delta }\left|j-m\right|/U_Z\right)& \begin{array}{cc}& m=1,1\le j\le n\end{array}\end{array}\\ \exp \left(-cf\mathsf{\Delta }\left|j-m\right|/U_Z\right)\sqrt{1-\exp \left(-2\mathsf{\Delta }cf/U_Z\right)}\begin{array}{cc}& 2\le m\le j\le n\end{array}\end{array}\right.$$

(5)

where $c$ is the exponential attenuation coefficient, and $\mathsf{\Delta }$ is the spacing of simulation points. Equations (1)–(5) mean that the simulated fluctuating wind field is mainly related to the average wind speed $U_Z$. As for the calculation method of relevant parameters, we can refer to the methods of IEC 61400-1 [34].

According to the sampling data, the wind speed of the monthly maximum average wind at the track side is 20 m/s. The random time history of transverse fluctuating wind when the embankment height is 3 m [35] is shown in Figure 1. The power spectrum of simulated fluctuating wind is shown in Figure 2. To further evaluate the effectiveness and accuracy of the method, we compared the results from the commonly employed Davenport power spectrum simulation with the method proposed in this paper. This paper adopts the simulation results of the Kaimal power spectrum.

From Figure 2, we can intuitively observe that the simulation results based on the Davenport power spectrum exhibit significant deviations in the high-frequency region. At the same time, we can find that the simulated power spectrum is in good agreement with the target spectrum, indicating that the above random fluctuating wind field simulation method is feasible. This proves the effectiveness and accuracy of the proposed method. Similarly, the above method can be used to simulate the vertical fluctuating wind.

2.2. Calculation Method of Wind Load

Wind load is related to the wind field around the train. The wind load is usually caused by the interaction of average wind and fluctuating wind in transverse $u$ and vertical $w$, wherein a random fluctuating wind field will cause buffeting load $F^{bf}$ of trains and the interaction between the average wind and train wind will cause static wind load $F^{st}$ acting on the train body. Based on the literature [12,31,36], the wind load $F_{\chi }$ can be formulated as follows:

$$F_{\chi }=F_{\chi }^{st}+F_{\chi }^{bf}=\rho A{C}_{\chi }(\alpha )\left(0.5U_R^2+U_R{W}_{\chi }\right)$$

(6)

where $\rho$ is air density. A is the equivalent windward area and the influence of the bogie and wheelset of trains is not considered. $W_{\chi }$ is the wind speed of fluctuating wind. $C_{\chi }\left(\alpha \right)$ is the aerodynamic coefficient, wherein $\chi =x,u,w$. $x$ is the running directions of trains, and $\alpha$ is the shaking angle of the wind direction. $\alpha =\arctan (U_Z/\left|V\right|)$, $\mathrm{w}$ here $V$ is the speed of trains.

The aerodynamic coefficient $C_{\chi }\left(\alpha \right)$ can be formulated as follows [37,38,39]:

$$C_{\chi }\left(\alpha \right)\left\{\begin{array}{c}{C}_u\left(\alpha \right)=2.1\exp \left(-3\right){\alpha }^2+1.44\exp \left(-2\right)\alpha +3.3\exp \left(-3\right)\\ C_w\left(\alpha \right)=2.6\exp \left(-3\right){\alpha }^2+2.02\exp \left(-2\right)\alpha -1.73\exp \left(-2\right)\\ C_{mx}\left(\alpha \right)=-1.5\exp \left(-5\right){\alpha }^3+5\exp \left(-4\right){\alpha }^2+0.09\alpha -0.22\end{array}\right.$$

(7)

where $C_u\left(\alpha \right)$ is the lateral aerodynamic coefficient, $C_w\left(\alpha \right)$ is the vertical aerodynamic coefficient. $C_{mx}\left(\alpha \right)$ is the torsional moment coefficient.

Notably, as shown in Figure 3, $U_R$ represents the relative wind speed of trains, which is the synthetic speed of actual wind speed and train speed.

The calculation of relative wind speed $U_R$ is given as follows:

$$U_R=\sqrt{{\left(U_Z\cos \beta +\left|V\right|\right)}^2+{\left(U_Z\sin \beta \right)}^2}$$

(8)

where $\beta$ is the angle between of the average wind and the running directions of trains.

The values of the average wind speed $U_Z$ along height $\mathrm{Z}$ can be calculated by the following formula:

$$U_Z=\left(\lg Z-\lg {\mathit{Z}}_0\right)/2.3U_{\mathit{Z}s}$$

(9)

where $U_{Zs}$ is the average wind speed at height Zs, the standard height Zs = 10 m [34]. Z₀ is the height when the wind speed decreases to zero, and in this paper Z₀ = 0.05 m [34].

According to (1), (6), and (8), the calculation of wind load is mainly related to the speed $V$ and the average wind speed $U_Z$ of trains. The wind load can be expressed as follows:

$$F_{\chi }={\delta }_{\chi }\left(V,U_Z\right)$$

(10)

To estimate the effect of $V$ and $U_Z$ on wind load, we randomly generate 36 instances for different speeds and average wind speeds. The values of $V$ and $U_Z$ are less than 200 km/h and 30 m/s in the actual running conditions of CRH2 high-speed trains on China’s Qinghai-Tibet Railway. Therefore, the values of $V$ are set to 50, 80, 110, 140, 170, 200, and the values of $U_{Zs}$ are set to 5, 10, 15, 20, 25, 30, respectively, in our simulation experiment.

$$\begin{array}{l}V=\left\{V_k\right\}=\left\{50,80,110,140,170,200\right\}\\ \left(k=1,2,\cdots ,6\right)\end{array}$$

(11)

$$\begin{array}{l}{U}_{Zs}=\left\{U_{Zsd}\right\}=\left\{5,10,15,20,25,30\right\}\\ (d=1,2,\cdots ,6)\end{array}$$

(12)

The length, width, and height of the train model are set according to the size of CRH2 high-speed trains. For sampling, we only consider the wind load applied on the centroid of the train [40]. The calculation results of wind load are illustrated in Figure 4.

$C_{kd}$ in Figure 4 shows different instances which correspond to the k-th train speed $V_k$ and the d-th average wind speed $U_{Zd}$. ${\delta }_u-\max$ represents the maximum value of transverse wind load, and ${\delta }_u-mean$ represents the average value of transverse wind load. ${\delta }_w-\max$ represents the maximum value of vertical wind load, and ${\delta }_w-mean$ represents the average value of vertical wind load.

As observed from Figure 4, the wind load increases rapidly with the increase in average wind speed when the train speed is constant. Therefore, without considering the influence of height on wind speed, the accuracy of simulation results will be affected.

2.3. Multi-Body Dynamics Model

The dynamic performance of trains varies sharply due to the change in wind load. Hence, considering that the random response process of trains is mainly affected by wind load, the dynamic response equation of a multi-body system of trains in absolute coordinates can be expressed as follows:

$$M{X}^{″}+C{X}^{\prime }+KX={\delta }_{\chi }\left(V,U_Z\right)$$

(13)

where $M$, $C$, and $K$ represent the mass, damping, and stiffness matrix of the train system, respectively. $X^{″}$, $X^{\prime }$, and $X$ represent the acceleration, speed, and displacement response of the train system, respectively. In addition, ${\delta }_{\chi }\left(V,U_Z\right)$ represents the wind load related to wind speed and train speed.

With reference to relevant parameters of CRH2 high-speed trains, we build a train model with 38 degrees of freedom (DOF) in SIMPACK 2018 software, as shown in Figure 5. The model consists of multiple rigid bodies including train body, bogie frame, and wheelset [41], in which the vertical DOF of the wheelset is not considered. The suspension characteristics and connection relationship between rigid bodies are determined by the force element, hinge, and constraint.

3. Simulation Experiments

3.1. Experiments

The method of dividing the model body surface into numerous units that are as small as possible and applying wind load on each unit can make the simulation results closer to the actual running environment of the train. However, this method encounters the significant challenge of computational complexity and simulation difficulty. Currently, a much more common method is to apply the wind load to the centroid of the train, that is, collecting the wind speed at the centroid height of the train body, and then calculating the train’s wind load [40]. Nevertheless, as the wind speed will vary obviously with the change in embankment heights, as shown in Figure 6, and different wind loads will produce different rolling moments of the train body, the accuracy of simulation results will be affected.

To address the aforementioned issues, this paper proposes a novel method for calculating the equivalent application position of wind load. This method is primarily divided into three steps.

First, we determine the application position of the equivalent wind load, which is the equivalent action position of the wind load. To make the simulation results more closely resemble the on-site environment, this paper takes into account the characteristic that wind speed changes with the height of the embankment when calculating the equivalent application position of wind load, and the relationship between average wind speed and embankment height is shown in (9). Then, the equivalent average wind speed of the train is calculated according to the proposed method. Finally, the equivalent wind load is calculated using the equivalent average wind speed.

As the primary step, we will calculate the equivalent action position of the wind load, with the detailed steps as follows: firstly, the rolling moment of the train body is calculated according to the average wind speed. Then, by integrating the roll moment along with the height, the equivalent application position of wind load on the train model under different embankment heights can be calculated. The calculation formula is shown in (14).

$${\displaystyle {\int }_{Z_E}^{Z_0+3.5}0.5H^{\prime }\zeta \left(Z\right)}dZ=0.5\rho A{U}_{Rh}^{2}H{C}_u({\alpha }_h)$$

(14)

where $\zeta \left(Z\right)$ is formulated as follows:

$$\zeta \left(Z\right)=L\rho U_R^2{C^{\prime }}_{mx}(\alpha )$$

(15)

where $Z_E$ is the embankment height, $H^{\prime }$ is the distance between the centroid of the train body and embankment, $U_{Rh}$ is the relative wind speed at height $h$, $h=Z_0+H$, the height of train body is set to 3.5 m, $H$ is the equivalent applied height of wind load on the train body, and $H\le 3.5m$. $C_u({\alpha }_h)$ is the lateral aerodynamic coefficient at height $h$, ${\alpha }_h$ represents the shaking angle of wind direction at height $h$. ${C^{\prime }}_{mx}(\alpha )$ is the lateral aerodynamic coefficient at height $h^{\prime }$, and $h^{\prime }=Z_0+H^{\prime }$ [39,40]. $L$ is the train length. Furthermore, the equivalent wind load is calculated according to (6). In the four experimental instances, $C_{22}$, $C_{33}$, $C_{44}$, and $C_{55}$, the calculation results of $H$ under different embankment heights are shown in Figure 7.

Figure 7 shows that the application position of the train’s equivalent wind load is high, even exceeding the train height. This may be due to the influence of ground friction: the closer to the ground, the greater the wind energy loss and the more obvious the change in wind load. Hence, the correction equation is shown as follows:

$${\displaystyle {\int }_{Z_0}^{Z_0+3.5}\zeta \left(Z\right)}dZ=\rho A\left({U^{\prime }}^2+V^2\right){C^{\prime }}_{mx}(\alpha )$$

(16)

where $U^{\prime }$ is the equivalent average wind speed at the centroid of the train.

Finally, the equivalent wind load is calculated according to $U^{\prime }$. The calculation formula is shown in (6).

3.2. Results

This paper uses the measured wind speed data in the plateau tundra area of the Qinghai-Tibet Railway: 91°86′ E, 33°25′ N. Therefore, the simulation experiments were performed according to the actual parameters of CRH2 high-speed trains. In addition, the embankment height is 3 m, and $g=10 \mathrm{m}/{\mathrm{s}}^2$.

The results of the train dynamics response are different at different wind speeds and vehicle speeds. In order to ensure the safety and smoothness of train operation, the dynamic response results should be controlled within a certain range. Therefore, in order to calculate the train speed threshold curve, it is necessary to first obtain the kinetic response results under different operating conditions. The response results of experimental instances on the multi-body dynamic model are shown in Figure 8 and Figure 9.

As can be seen from Figure 8 and Figure 9, simulation results show the non-normal responses of the train under random excitation. Meanwhile, the dynamic performance of the whole train model changes continuously under different wind speeds and train speeds. Especially, the lateral vibration response of the train body changes obviously because of the influence of lateral wind.

To ensure the safety and stability of trains during running, parameter limits are set in the train model, as shown in [42]. The maximum lateral acceleration $\epsilon 1$ of the train body for Grade-I stability is 0.6 m/s², and the maximum lateral acceleration $\epsilon 2$ of the train body for Grade-II stability is 0.9 m/s²; the maximum vertical acceleration of the train body is 1.0 m/s², the maximum wheel–rail lateral force is 56 KN, and the maximum wheel–rail vertical force is 170 KN. From the simulation results in Figure 8 and Figure 9, it can be found that only the lateral acceleration of the train body exceeds the maximum value among the four typical response results. Therefore, we will further explore the lateral acceleration response of the train body.

It can be found from Figure 10 that when the train speed remains the same, the discreteness of the train’s lateral acceleration response became large with the increase in standard wind speed, and the number of transfinite individuals in the sample is also increasing. In addition, with the increase in train speed, the lateral acceleration of the train also tends to increase under the same wind speed.

The vibration response of the train body has randomness under the influence of external random wind load. Considering the interference of instantaneous wind load peak in a random fluctuating wind field in the experimental results, the method of probability and statistics is introduced into the simulation experiment. Firstly, the average and maximum lateral accelerations of the train body under different running conditions are calculated. Then the overrun probability is used to describe the probability that the lateral acceleration exceeds the maximum value required for stationarity of trains. Meanwhile, the simulations are run thirty times independently to reduce the impact of noisy data on results. The lateral acceleration function set is established for the carbody.

$$\left\{{\psi }_s\left(A^{\left(C_{kd}\right)}\right)={\psi }_1\left(A^{\left(C_{kd}\right)}\right),{\psi }_2\left(A^{\left(C_{kd}\right)}\right),\cdots ,{\psi }_{30}\left(A^{\left(C_{kd}\right)}\right)\right\}$$

(17)

where $C_{kd}$ represents different instances.

The probability distribution function for the carbody is as follows.

$$P_{\epsilon }(\left|A^{\left(C_{kd}\right)}\right|<\epsilon )=\left(1/30\right){\displaystyle \sum _{i=1}^{30}{\displaystyle {\int }_{-\epsilon }^{\epsilon }{\psi }_s\left(A^{\left(C_{kd}\right)}\right)d{A}^{\left(C_{kd}\right)}}}$$

(18)

where ε represents the maximum lateral acceleration, $A^{\left(C_{kd}\right)}$ represents the lateral acceleration of carbody under instances $C_{kd}$.

The experimental results are presented in

Table 1. Response of lateral acceleration under different running conditions.

Train Speed (Km/h)	Response	Average Speed of the Winds (m/s)
		0	5	10	15	20	25	30
50	Amean	0.0583	0.0623	0.1114	0.2118	0.2927	0.4902	0.7150
	Amax	0.1904	0.1959	0.3900	1.0804	2.2092	2.6112	3.3500
	Pε1	0	0	0	0	0.0960	0.2460	0.4500
	Pε2	0	0	0	0	0.0300	0.1260	0.3000
80	Amean	0.1135	0.1182	0.1325	0.1668	0.3048	0.5090	0.7514
	Amax	0.4167	0.4160	0.4470	0.5028	0.9095	2.1728	4.4824
	Pε1	0	0	0	0.0100	0.1100	0.2900	0.4200
	Pε2	0	0	0	0	0.0300	0.0300	0.2800
110	Amean	0.1745	0.1760	0.1876	0.2175	0.3601	0.5147	0.7490
	Amax	0.6041	0.6131	0.6302	0.7140	0.9543	1.8032	5.2800
	Pε1	0.0200	0.0200	0.0300	0.0600	0.1600	0.3300	0.4200
	Pε2	0	0	0	0	0.0200	0.1600	0.2800
140	Amean	0.2433	0.2463	0.2470	0.2584	0.3613	0.5316	0.7004
	Amax	0.7863	0.8178	0.7831	0.8742	1.4853	1.8082	2.8564
	Pε1	0.0500	0.0500	0.0500	0.0500	0.1900	0.3300	0.4700
	Pε2	0.0100	0.0100	0.0100	0.0100	0.0400	0.2100	0.3100
170	Amean	0.2510	0.2507	0.2620	0.2896	0.3255	0.4619	0.6239
	Amax	0.8087	0.8035	0.9345	0.8677	1.0717	1.3770	2.3109
	Pε1	0.0700	0.0600	0.0700	0.1200	0.1700	0.2700	0.4300
	Pε2	0.0100	0.0100	0.0120	0.0100	0.0400	0.0900	0.2900
200	Amean	0.3262	0.3268	0.3291	0.3305	0.3664	0.4573	0.7416
	Amax	1.0067	1.0138	1.0147	0.9636	1.1910	1.6318	2.9297
	Pε1	0.1700	0.1700	0.1700	0.1700	0.2100	0.2900	0.5200
	Pε2	0.0300	0.0300	0.0300	0.0400	0.0800	0.1700	0.3100

. In Table 1, $A_{mean}$ means the average value of lateral acceleration, and $A_{\max }$ means the maximum value of lateral acceleration. $P_{\epsilon 1}$ and $P_{\epsilon 2}$ are the overrun probability of the train’s Grade-I stationarity and Grade-II stationarity, respectively.

From Table 1, we can find that the values of $A_{mean}$, $A_{\max }$, $P_{\epsilon 1}$, and $P_{\epsilon 2}$ grow with the increase in train speed and wind speed. In addition, the lateral acceleration of the train body does not exceed the limit, and the running quality of the train is high when the train speed and wind speed are low.

Considering the uncontrollability of wind speed, we can obtain the function between wind speed and train overrun probability at different train speeds by linear regression of each group of data in Table 1.

$$P_{\theta }\left(x\right)={\theta }_1+{\theta }_{2}x+{\theta }_3{x}^2+\cdots +{\theta }_{\upsilon }{x}^{\upsilon -1}$$

(19)

Larger parameters in linear regression models may lead to overfitting, resulting in poor generalization performance of linear regression results. Therefore, we introduce the regularization method to address this issue. The regularization method is as follows [43]:

$$\Omega \left(\theta \right)=\underset{\theta }{\min }\left\{\left(1/2r\right)\left[{\displaystyle \sum _{i=1}^r{\left(P_{\theta }\left(x^{\left(\nu \right)}\right)-P^{\left(\nu \right)}\right)}^2}+\lambda {\displaystyle \sum _{j=1}^5{\theta }_{\upsilon }^2}\right]\right\}$$

(20)

where $r=30$, and $P^{\left(\nu \right)}$ represents the validation set. Additionally, $\lambda$ is the regularization parameter, which is obtained through a cross-validation method [44,45].

$$C_V\left({\lambda }^{\left(\kappa \right)}\right)=\underset{{\lambda }^{\left(\kappa \right)}}{\min }{{\displaystyle \sum _{\nu =1}^r\left(P_{\theta }\left(x^{\left(\nu \right)}\right)-P^{\left(\nu \right)}\right)}}^2$$

(21)

where the values of $\lambda$ were set to 0, 0.01, 0.02, 0.04, 0.08, 0.16, 0.32, 0.64, 1.28, 2.56, 5.12, 10 [44,45]. The results of functions are shown in

Table 2. Overrun probability of train’s Grade-I stationarity.

Train Speed (Km/h)	Linear Regression Function	R-Square
50	$P=-2.2e^{-6}{x}^4+0.00016x^3-0.0026x^2+0.012x-0.0018$	0.90
80	$P=-3.4e^{-6}{x}^4+0.00022x^3-0.0034x^2+0.014x-0.0024$	0.90
110	$P=-3e^{-6}{x}^4+0.00019x^3-0.0028x^2+0.012x+0.018$	0.87
140	$P=0.0006x^2-0.008x+0.06$	0.86
170	$P=0.0005x^2-0.002x+0.07595$	0.88
200	$P=1.3e^{-6}{x}^4-4e^{-5}{x}^3+0.00056x^2-0.002x+0.1706$	0.90

and

Table 3. Overrun probability of train’s Grade-II stationarity.

Train Speed (Km/h)	Linear Regression Function	R-Square
50	$P=2e^{-7}{x}^4+1.3e^{-5}{x}^3-0.0004x^2+0.0025x-0.00064$	0.92
80	$P=1.9e^{-5}{x}^3-0.0004x^2+0.0027x-0.0002381$	0.85
110	$P=1.8e^{-5}{x}^3-0.0003x^2+0.0004x+0.0024$	0.92
140	$P=1.3e^{-5}{x}^3-2.6e^{-5}{x}^2-0.0029x+0.015$	0.89
170	$P=0.0006x^2-0.01x+0.034$	0.86
200	$P=-1.1e^{-7}{x}^4+2.6e^{-5}{x}^3-0.0005x^2+0.002x+0.03$	0.92

. In them, R-square indicates the degree of fit of the linear regression function.

Considering the random response of the train body and the economical efficiency of train services, the allowable error of the train’s overrun probability P is set to 4.55% [46]. According to the obtained linear regression function and allowable error, we obtain the two-level running quality threshold curve of trains under different running conditions. Under the premise of ensuring the safety of train operation, Grade-I running quality requires higher train dynamics response results. The train runs more smoothly under this requirement, which can be used as the speed threshold curve for passenger trains. In contrast to Grade-I running quality, Grade-II running quality allows the train to run faster. The train’s transportation efficiency is higher under this requirement, and it can be used as the speed threshold curve for freight trains.

As shown in Figure 11, when a train runs at a speed of 50 km/h and the standard wind speed does not exceed 17.64 m/s, the lateral acceleration of the train body is mainly below 0.6 m/s², which meets the requirements of Grade-Ι stability, indicating that the train’s running quality is high at this time. Meanwhile, the lateral acceleration of the train body is mainly below 0.9 m/s² when the standard wind speed does not exceed 20.97 m/s, which meets the requirements of Grade-ΙΙ stability. However, in the windless environment, the lateral acceleration of the train body will exceed 0.6 m/s² when the train speed exceeds 140 km/h, and the train’s running quality will decline. In addition, when the train speed does not exceed 200 km/h, the lateral acceleration of the train body is mainly below 0.9 m/s², which meets the requirements of Grade-ΙΙ stability. Notably, when the wind speed and train speed exceed 20.97 m/s and 50 km/h, respectively, the probability of the lateral acceleration of the train body above 0.9 m/s² increases and the running quality of the train is low at this time.

4. Conclusions and Future Work

In this paper, we have studied the running quality of trains along the Qinghai-Tibet Railway in strong wind environments. As the measured wind speed data along the Qinghai-Tibet Railway are the average wind speed after treatment, we use the weighted amplitude wave superposition (WAWS) method and fast Fourier transform (FFT) to generate the random fluctuating wind field. Meanwhile, to make the established wind load model more in line with the actual wind field, the correlation between the height of a train above the ground and the wind speed has been considered in the modeling. In addition, considering the randomness of wind speed of the established fluctuating wind field, we have proposed the method of overrun probability to avoid the interference of instantaneous wind speed peak in the experimental results. Finally, a CRH2 train simulation model has been established in the simulation environment, and the simulation experiment is carried out combined with the wind load model. According to the simulation experiments, we have obtained the regularized regression model between the overrun probability of train stability and wind speed at different speeds, as well as the two-level running quality threshold curve of the train.

Based on the two-level running quality threshold curve of trains, we can draw the following conclusions: firstly, the above threshold curves contain almost all common running conditions of trains on the Qinghai-Tibet Railway under strong wind environments, which can provide train speed control strategies with different wind speeds for trains running on the Qinghai-Tibet Railway. Secondly, in the above two threshold curves, the threshold curve of Grade-Ι can be used to meet the higher stability requirements of passenger trains, and the threshold curve of Grade-ΙΙ can be used to ensure higher transportation efficiency of freight trains. Therefore, the speed control strategies of trains can be formulated by railway management departments according to the above two-level running quality threshold curve and their transportation needs.

The wind load model proposed in this paper is not only suitable for windy areas such as the Qinghai-Tibet Plateau but also for other regions with significant wind speeds, such as coastal areas and canyons. However, during the train dynamics simulation, this paper primarily analyzed the CRH2 type train. Consequently, the threshold curves derived in this paper are not applicable to all train types. Nevertheless, based on the methods put forward in this paper, once we acquire the relevant parameters of other trains, we can obtain the corresponding speed threshold curves for those trains.

The train speed control strategy under strong wind environments proposed in this paper is an important method to ensure the safe running of trains. Meanwhile, this research can be easily extended to the active control of lateral attitude for trains under strong wind environments. However, there are still many areas worthy of in-depth study on the basis of this article. For example, besides the strong wind environment, other natural disasters encountered during the running of trains, including earthquake, debris flow, rain, snow, and so on, can be considered to ensure the safety of trains in complex environments.