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Abstract

Remaining life prediction is an effective way to optimize maintenance strategy and improve service life for light-emitting diode driving power in rail vehicle carriage. In this article, a Wiener process–based remaining life prediction method is proposed with the analysis of performance degradation data of light-emitting diode driving power in rail vehicle carriage. First, the temperature and humidity stress accelerated degradation tests are put forward in order to measure the output current of light-emitting diode driving power. Based on the output current, the accelerated degradation model is established. The drift and diffusion coefficients of the Wiener process are then obtained without prior information. Finally, the reliability of light-emitting diode driving power in rail vehicle carriage is assessed and the remaining lifetime is predicted after updating the degradation model parameters with Bayesian inference. The results show that the proposed method can improve the precision of assessment and reduce the uncertainty of prediction significantly. It also provides a potential solution for life prediction of other similar products.

Keywords

Light-emitting diode driving power remaining life prediction Wiener process Bayesian method accelerated degradation test

Introduction

The light-emitting diode (LED) lighting control system is a complex electrical control system in rail vehicle carriage. Its performance affects the normal illumination of the rail vehicle carriage. As the “heart” of the LED lighting control system, the status of LED driving power is related to the stability of the LED lighting control system closely. Its failure may not only break down the lighting control system, but also break down the other systems like air conditioning system, operation control system, and some related systems, which will cause unexpected damage for the whole system. Therefore, research related to reliability assessment and remaining life prediction is essential for designing and optimizing the LED driving power. It is also of great significance to the safe, stable, and efficient operation of the rail vehicle.^1,2

In traditional remaining life prediction experiment, failure data can be used to predict the remaining life of a product. But for high-reliability products, failure data are often difficult to be obtained in a short time.³ Fortunately, some of the characteristics of a product will degrade over time. A large amount of information related to reliability and remaining life can be obtained from the degradation data which can be used to evaluate reliability and predict the remaining life of the LED driving power.⁴ Through the accelerated degradation test, both reliability and life information under different environmental conditions of the LED driving power can be obtained. At present, the commonly used accelerated degradation methods include Arrhenius model,⁵ Eyring model,⁶ inverse power law (IPL),⁷ and Hallberg–Peck model.⁸ Arrhenius and Eyring models are always used for thermal accelerated stress in systems in which temperature is the major factor in aging. IPL is always used for other different kinds of accelerated stresses expect thermal accelerated stress. Hallberg–Peck model which synthesizes both temperature and humidity can describe the aging test more accurately. For the LED driving power in rail vehicle carriage, the factors affecting the performance include temperature, humidity, and vibration. With the continuous improvement of technology, LED driving power packaging technology has been improved greatly. In practical applications, because the running state of the rail vehicle tends to be stable, the temperature and humidity are the main factors affecting the performance of the LED driving power. Therefore, the Hallberg–Peck model is chosen to be the accelerated degradation method of the LED driving power in the LED lighting system of the rail vehicle carriage in this article.

The prediction methods can be generally divided into the following two types, i.e., physics-based prediction method⁹ and data-driven prediction method.¹⁰ With the development of related technologies such as signal acquisition and signal processing, abundant data of the system operation can be obtained. According to these data, the corresponding mathematical model can be utilized, which is mainly composed of two kinds of techniques: artificial intelligence and probability statistics. The artificial intelligence methods have a higher degree of data fitting. The probability statistics can predict the future state better. Wiener process model is a kind of the most commonly used probability statistical models, which can describe the degradation process accurately.^11,12 The degradation model based on Wiener process has obvious advantages in mathematics.¹³ Now, Wiener process has been widely applied in many research fields^14–17 including the degradation process because of its excellent capability of analyzing and describing the process of degradation.^18,19 Zhu et al.²⁰ proposed a reliability assessment method that fused both prior and on-site degradation data. To update the parameters of the Wiener process, the prior data were used and the on-site data were fused by Bayesian method. In Liu et al.,²¹ the degradation of aero-engine was modeled based on multi-stage Wiener process. Based on the historical degradation data and the historical failure time data, the prior distribution of the model parameters was estimated using the expectation maximization (EM) algorithm. In Si et al.,²² the recursive filtering algorithm and EM algorithm were combined to update the parameters of the Wiener process, and the method is applied to the inertial navigation system to predict the remaining life accurately. However, for rail vehicle carriage LED lighting system, there have been very few results on the remaining life prediction and this constitutes the main motivation of our study.

In this article, we pay special attention to the remaining life prediction for the rail vehicle carriage LED lighting system. We endeavor to handle the following two fundamental questions: (1) how to develop a recursive algorithm suitable for online applications in response to accelerated degradation test of LED driving power under consideration and (2) how to reasonably improve the accelerated degradation test precision in the presence of temperature and humidity phenomena? To solve the two questions, we employ the Wiener process and Bayesian method. Therefore, we augment the degradation process for LED driving power of rail vehicle carriage under the frame of Wiener process. Different temperature and humidity gradients are investigated to accelerate the degradation process, and the accelerated degradation model is then presented based on the measured electric data of the LED driving power. Hallberg–Peck model is used to model the relationship between the drift parameters $μ$ and the temperature and humidity in the Wiener process. We conduct simulation studies on the LED driving power and the results illustrate that our proposed approach is effective.

This article is organized as follows. In section “Degradation model of LED driving power based on Wiener process,” the degradation model of LED driving power based on Wiener process is formulated. In section “The estimation and updating of parameters,” the estimation and updating method of parameters is described. Section “Remaining life prediction of the LED driving power based on the Wiener process” presents a case study for the remaining life prediction of the LED driving power. Finally, the conclusions are given in section “Conclusion.”

Degradation model of LED driving power based on Wiener process

If $Z (t)$ is the degradation measure of the LED driving power at time t, the degradation model of Wiener process based on the drift parameter is given by

Δ Z (t) = Z (t + Δ t) - Z (t) = θ Δ t + ε Δ W (t)

(1)

where $θ$ denotes the drift coefficient, $ε$ denotes the diffusion coefficient, $Δ t$ is a short time interval, and $W (t)$ is the standard Brownian motion which has two properties as follows:

Property 1. $W (t) ~ N (0, t)$ ;

Property 2. For any two different time points $t_{1}$ and $t_{2}$ , $t_{1} < t_{2}$ , $t_{1} + Δ t < t_{2}$ , and $Δ W (t_{1})$ and $Δ W (t_{2})$ are independent.

According to the characteristics of $W (t)$ , it can be assumed that $Δ Z (t) ~ N (θ Δ t, ε^{2} Δ t)$ , so its degradation amount mean is $E (Δ Z (t)) = θ Δ t$ and the degradation amount variance is $D (Δ Z (t)) = ε^{2} Δ t$ . The mean reflects the trend of the degradation data affected by the accelerated stress. Therefore, it is considered that the drift coefficient $θ$ is related to the acceleration stress. The variance reflects the degree of deviation between the degenerate data and the mean, which is not usually related to the acceleration stress. Therefore, it is considered that the diffusion coefficient $ε$ is not related to the acceleration stress. In this article, as the random parameters, $θ$ and $ε^{2}$ are used to describe the individual difference of the LED driving power in a rail vehicle. Since there is no sufficient prior information of the parameters to be estimated, the prior distribution without information is adopted.

The failure threshold of the LED driving power of the rail vehicle carriage is a fixed value, which is set as a constant C. When the electric data reach the threshold value for the first time, the LED driver power has failed. Therefore, its remaining useful life can be defined as

γ = inf {t | Δ Z (t) ⩾ C, t > 0}

(2)

It is proved that its first failure threshold time $γ$ is subject to inverse Gaussian distribution,^23,24 while the probability density function (PDF) is expressed as

f (t | θ, ε) = \frac{C - Δ Z (t)}{\sqrt{2 π ε^{2} t^{3}}} \exp [- \frac{{(C - Δ Z (t) - θ t)}^{2}}{2 ε^{2} t}]

(3)

and the cumulative distribution function is expressed as

\begin{matrix} F (t) = ϕ (\frac{θ t - Δ Z (t) - C}{ε \sqrt{t}}) + \exp (\frac{2 θ C}{ε^{2}}) \\ ϕ [\frac{- (θ t + Δ Z (t) + C)}{ε \sqrt{t}}] \end{matrix}

(4)

where $ϕ (\cdot)$ is PDF of the standard normal distribution.

In the accelerated degradation test, temperature and humidity are the most common accelerated stresses that can aggravate the reaction to degenerate the product. Temperature and humidity are also the most important factors for reliability of LED driving power in rail vehicle carriage. A high-temperature and humid environment may make the service life of the internal electrolytic capacitor in the power supply decrease, and therefore the entire life of the driving power is affected. Therefore, the Hallberg–Peck acceleration model is used to construct the relationship between the drift coefficient and the temperature and humidity stresses. Considering the influence of temperature and humidity comprehensively, the Hallberg–Peck model⁸ can describe the accelerated degradation test of the product under the condition of temperature and humidity accurately. It can be achieved from the following equation

θ = A {(H_{R})}^{- 3} \exp [\frac{E}{K (273 + T)}]

(5)

where A > 0 is a constant, $H_{R}$ is the relative humidity, E is the activation energy (eV), K is the Boltzmann constant, and its value is chosen as 8.617 × 10⁵ eV/°C. Therefore, the unit of $E / K$ is °C and T is the relative temperature. The degradation model can be obtained through substituting equation (5) into equation (1) as follows

Δ Z (t, T, H_{R}) = A {(H_{R})}^{- 3} \exp [\frac{E}{K (273 + T)}] Δ t + ε Δ W (t)

(6)

The estimation and updating of parameters

The accelerated stresses in the accelerated degradation test are set to $T_{1}, T_{2}, \dots, T_{l}$ . There are m samples for each acceleration stress, and the measured time gradient of each sample is $t_{1}, t_{2}, t_{3}, \dots, t_{n}$ . Under the $T_{i}$ stress condition in $t_{k}$ , $Z_{ijk}$ is the degradation amount of the jth sample, the performance degradation is 0 at the initial time point, and $Δ Z_{ijk} = Z_{ijk} - Z_{ij (k - 1)}$ is the increment of degradation. By the nature of $W (t)$ , $Δ Z_{ijk} ~ N (θ_{ij} Δ t_{ijk}, ε^{2} Δ t_{ijk})$ , we have

E (Δ Z_{ijk}) = θ_{ij} Δ t_{ijk} = A {(H_{R})}^{- 3} \exp [\frac{E}{K (273 + T)}] Δ t_{ijk}

(7)

D (Δ Z_{ijk}) = ε^{2} Δ t_{ijk}

(8)

In the period of $Δ t_{k}$ , the PDF of the performance degradation data is

f_{ijk} = \frac{1}{\sqrt{2 π ε^{2} Δ t_{ijk}}} \exp [- \frac{{(Δ Z_{ijk} - θ_{ij} Δ t_{ijk})}^{2}}{2 ε^{2} Δ t_{ijk}}]

(9)

The likelihood function of $θ$ and $ε^{2}$ can be obtained using equation (10)

\begin{matrix} L (A, E, ε) = Π_{i = 1}^{l} Π_{j = 1}^{m} Π_{k = 1}^{n} \frac{1}{\sqrt{2 π ε^{2} Δ t_{ijk}}} \exp [- \frac{{(Δ Z_{ijk} - θ Δ t_{ijk})}^{2}}{2 ε^{2} Δ t_{ijk}}] \end{matrix}

(10)

The maximum likelihood estimation can be used to estimate the values of $θ$ and $ε^{2}$ in equation (10). However, the solving process is very complicated. Therefore, the first step is to determine the prior distribution of the parameters, and the second step is to update the parameters via the Bayesian method; finally, the distribution of the parameters can be obtained using equation (11). According to the Bayesian theorem, the posterior distribution can be described as

p (θ | y) \propto f (y | θ) p (θ)

(11)

where $p (θ | y)$ is the PDF of posterior distribution, $f (y | θ)$ is the likelihood function, and $p (θ)$ is the PDF of prior distribution.

If the distribution of the measured data of the first set sample in stress $T_{1}$ is a normal distribution, where the parameters $θ_{1}$ and $ε_{1}^{2}$ show non-informative prior distribution, the prior distribution PDF can be generalized as

p (θ, ε^{2}) \propto \frac{1}{ε^{2}}

(12)

Let $θ_{a} = θ_{1} Δ t_{11 k}$ , $ε_{a}^{2} = ε_{1}^{2} Δ t_{11 k}$ , then $p (θ_{a}, ε_{a}^{2}) \propto 1 / ε_{a}^{2}$ , and the joint posterior distribution of $(θ_{a}, ε_{a}^{2})$ is

p (θ_{a}, ε_{a}^{2} | Δ Z_{11 k}) \propto (ε_{a}^{2})^{- \frac{n}{2} - 1} \exp [\sum_{k = 1}^{n} - \frac{1}{2 ε_{a}^{2}} {(Δ z_{11 k} - θ_{a})}^{2}]

(13)

In order to update the parameters, the estimated parameters $θ_{a}$ and $ε_{a}^{2}$ can be regarded as redundant parameters. Due to integrating the posterior distribution of the integral, the excess parameters are removed as shown in equation (14)

p (θ_{a} | Δ Z_{11 k}) \propto \int_{0}^{\infty} {(ε_{a}^{2})}^{- \frac{n}{2} - 1} \exp [\sum_{k = 1}^{n} - \frac{1}{2 ε_{a}^{2}} {(Δ Z_{11 k} - θ_{a})}^{2}] d ε_{a}^{2}

(14)

The equation (14) finally becomes the following formula (15)

\begin{matrix} p (θ_{a} | Δ Z_{11 k}) \propto \frac{Γ (n / 2)}{{[\sum_{k = 1}^{n} \frac{{(Δ Z_{11 k} - θ_{a})}^{2}}{2}]}^{n / 2}} \\ \propto \int_{0}^{\infty} \begin{matrix} \frac{{[\sum_{k = 1}^{n} \frac{{(Δ Z_{11 k} - θ_{a})}^{2}}{2}]}^{n / 2}}{Γ (n / 2)} {(ε_{a}^{2})}^{- \frac{n}{2} - 1} \\ \exp [\sum_{k = 1}^{n} - \frac{1}{2 ε_{a}^{2}} {(Δ z_{11 k} - θ_{a})}^{2}] d ε_{a}^{2} \end{matrix} \\ = \frac{Γ (n / 2)}{{[\sum_{k = 1}^{n} \frac{{(Δ Z_{11 k} - θ_{a})}^{2}}{2}]}^{n / 2}} \end{matrix}

(15)

$\sum_{i = 1}^{n} {(Δ Z_{11 k} - θ_{a})}^{2}$ is transformed as $n {(Δ Z_{11 k} - θ_{a})}^{2} + \sum_{k = 1}^{n} {(Δ Z_{11 k} - Δ {\bar{Z}}_{11 k})}^{2}$ and then substituted into equation (15)

\begin{matrix} p (θ_{a} | Δ Z_{11 k}) \propto \frac{Γ (n / 2)}{{[\sum_{k = 1}^{n} \frac{{(Δ Z_{11 k} - θ_{a})}^{2}}{2}]}^{\frac{n}{2}}} \\ \propto \frac{Γ (n / 2)}{{[\frac{1}{2} \sum_{k = 1}^{n} {(Δ Z_{11 k} - Δ {\bar{Z}}_{11 k})}^{2}]}^{\frac{n}{2}} \cdot {[1 + \frac{n {(Δ {\bar{Z}}_{11 k} - θ_{a})}^{2}}{\sum_{k = 1}^{n} {(Δ Z_{11 k} - Δ {\bar{Z}}_{11 k})}^{2}}]}^{\frac{n}{2}}} \\ \propto {[1 + \frac{n {(Δ {\bar{Z}}_{11 k} - θ_{a})}^{2}}{\sum_{k = 1}^{n} {(Δ Z_{11 k} - Δ {\bar{Z}}_{11 k})}^{2}}]}^{- \frac{n}{2}} \\ \propto {[1 + \frac{{(θ_{a} - Δ {\bar{Z}}_{11 k})}^{2}}{(n - 1) \frac{S^{2}}{n}}]}^{- \frac{n}{2}} \end{matrix}

(16)

where $Δ {\bar{Z}}_{11 k} = (1 / n) \sum_{k = 1}^{n} Δ Z_{11 k}$ and $S^{2} = (1 / (n - 1)) \sum_{k = 1}^{n} {(Δ Z_{11 k} - Δ {\bar{Z}}_{11 k})}^{2}$ . As shown in equation (16), it is easy to say that the marginal posterior distribution of $θ_{1}$ follows the mean value of $Δ {\bar{Z}}_{11 k}$ , and the scale parameter fits the normal distribution of $S^{2} / n$ .

The estimated parameter $ε_{a}^{2}$ in equation (17) is updated as follows

\begin{matrix} p (ε_{a}^{2} | Δ Z_{11 k}) \propto \int_{- \infty}^{+ \infty} {(ε_{a}^{2})}^{- \frac{n}{2} - 1} \exp [- \frac{1}{2 {ε_{a}}^{2}} \sum_{i = 1}^{n} {(Δ Z_{11 k} - θ_{a})}^{2}] d θ_{a} \\ \propto (ε_{a}^{2})^{- \frac{n}{2} - 1} [\int_{- \infty}^{+ \infty} \exp [- \frac{1}{2 ε_{a}^{2}} (n (Δ Z_{11 k} - θ_{a})^{2} \\ + \sum_{k = 1}^{n} (Δ Z_{11 k} - Δ {\bar{Z}}_{11 k})^{2})]] d θ_{a} \\ \propto (ε_{a}^{2})^{- \frac{n}{2} - 1} \int_{- \infty}^{+ \infty} \exp [- \frac{{(Δ {\bar{Z}}_{11 k} - θ_{a})}^{2}}{2 ε_{a}^{2} / n} - \frac{S^{2}}{2 ε_{a}^{2} / (n - 1)}] d θ_{a} \\ \propto (ε_{a}^{2})^{- \frac{n}{2} - 1} \cdot \exp [- \frac{(n - 1) S^{2}}{2 ε_{a}^{2}}] \cdot \int_{- \infty}^{+ \infty} \exp [- \frac{{(Δ {\bar{Z}}_{11 k} - θ_{a})}^{2}}{2 ε_{a}^{2} / n}] d θ_{a} \\ \propto (ε_{a}^{2})^{- \frac{n - 1}{2} - 1} \cdot \exp [- \frac{(n - 1) S^{2}}{2 ε_{a}^{2}}] \end{matrix}

(17)

From equation (17), it can be observed that the posterior distribution of $ε_{a}^{2}$ is proportional to the PDF of the inverse gamma distribution. The shape parameter is $(n - 1) / 2$ and the scale parameter is $((n - 1) S^{2}) / 2$ . Combining equations (16) and (17), the mean value of the posterior distribution of $θ_{1}$ is $Δ {\bar{Z}}_{11 k} / Δ t_{11 k}$ , and the scale parameter is $s^{2} Δ t_{11 k} / n$ . The shape parameter of the posterior distribution of $ε_{1}^{2}$ is $(n - 1) / 2$ , and the scale parameter is $((n - 1) S^{2}) / 2 Δ t_{11 k}$ . The estimated values of the parameters ${\hat{θ}}_{1}$ and ${\hat{ε}}_{1}^{2}$ under the stress of $T_{1}$ are

{\hat{θ}}_{1} = \frac{1}{m} \sum_{j = 1}^{m} θ_{1} and {\hat{ε}}_{1}^{2} = \frac{1}{m} \sum_{j = 1}^{m} ε_{1}^{2}

(18)

In the same way, the estimated parameters can be obtained under $T_{2}, T_{3}, \dots, T_{l}$ .

In order to obtain $T_{l}$ , the logarithm of equation (7) can be evaluated as follows

\ln θ_{i} = \ln A - 3 \ln (H_{Ri}) + \frac{E}{K (273 + T_{i})}

(19)

Therefore, we have

\begin{matrix} \ln θ_{i} - \ln θ_{j} = 3 [\ln (H_{Rj}) - \ln (H_{Ri})] \\ + [\frac{1}{K (273 + T_{i})} - \frac{1}{K (273 + T_{j})}] E \end{matrix}

(20)

Substitute the estimated value of equation (18) into equation (20)

{\hat{E}}_{ij} = \frac{(\ln {\hat{θ}}_{i} - \ln {\hat{θ}}_{j}) - 3 [\ln (H_{Rj}) - \ln (H_{Ri})]}{\frac{1}{K (273 + T_{i})} - \frac{1}{K (273 + T_{j})}}

(21)

The estimated value of E can be calculated as

\hat{E} = \frac{1}{l (l - 1)} \sum_{i = 1}^{l} \sum_{j = 1, i \neq j}^{l} {\hat{E}}_{ij}

(22)

Based on equation (5), the estimated value of A is

\hat{A} = \frac{1}{l} \sum_{i = 1}^{l} \frac{{\hat{θ}}_{i}}{{(H_{Ri})}^{- 3} \exp [\frac{\hat{E}}{K (273 + T_{i})}]}

(23)

Because the diffusion coefficient $ε$ is independent of the acceleration stress, its estimated value is

{\hat{ε}}^{2} = \frac{1}{l} \sum_{i = 1}^{l} {\hat{ε}}_{i}^{2}

(24)

Therefore, the mean and variance of the degradation data under normal stress level can be obtained according to the accelerated degradation model. The values of the parameters in the degradation model are acquired. The corresponding failure distribution can be gained by substituting them into equation (3).

Remaining life prediction of the LED driving power based on the Wiener process

LED driving power plays a role in electric energy conversion in the rail vehicle carriage LED lighting control system, which is the key link affecting the entire lighting control system life. As the LED drive current is an important factor affecting the life of LED, the luminous efficiency will change with the driving current. Thus, the electric current degradation of the LED driving power is used to measure the reliability and the remaining life of the LED driving power in this article (Figure 1).

Figure 1.

LED driving power performance testing.

Temperature and humidity are selected as the accelerated stresses, and the constant stress accelerated degradation test is used to accelerate the degradation test of LED driving power. Three groups of stresses are chosen under the condition of combination stress T including $T_{1}$ , $T_{2}$ , and $T_{3}$ listed in Table 1. Then, through measuring the current values every 1 h, the degradation values of the current are obtained. In order to obtain the accurate test, the test is repeated once. Figure 2 shows the variation of the output electric current degradation with time of the six samples under different stress conditions.

Table 1.

Values of combination stress T.

Stress	Temperature (°C)	Humidity (%)
$T_{1}$	55	55
$T_{2}$	65	65
$T_{3}$	75	75

Figure 2.

Electric current degradation under three sets of stress conditions.

As shown in Figure 2, the degradation data of the three sets present linear trajectory, random process, and its increment is not strict. It can be determined that the degradation process can be described in the Wiener process initially. According to the characteristics of the Wiener process, $Δ Z (t) ~ N (θ Δ t, ε^{2} Δ t)$ , the Wiener process can be determined by fitting the probability distribution of $Δ Z (t)$ . One sample is taken from each stress condition, and the toolbox of MATLAB is used to fit the degenerate data. The fitting result is shown in Figure 3. As shown in Figure 3, we can observe that the degradation process of LED driving power is a Wiener process.

Figure 3.

Fitting of degradation quantity.

Using the above method, three groups of degenerate data are updated to obtain the normal inverse gamma posterior distribution parameters $θ$ and $ε^{2}$ , as shown in Table 2. The estimated value of ${\hat{E}}_{ij}$ can be obtained by substituting the resulting parameter estimates into equation (21) which is shown in Table 3.

Table 2.

Parameter estimation under different stress conditions.

Stress	$\hat{θ}$	$ε^{2}$
50°C/50%	$2.853 \times 10^{- 4}$	$9.83 \times 10^{- 5}$
70°C/70%	$1.365 \times 10^{- 4}$	$1.365 \times 10^{- 4}$
90°C/90%	$4.997 \times 10^{- 4}$	$3.777 \times 10^{- 4}$

Table 3.

Estimated value of ${\hat{E}}_{ij}$ .

	50°C/50%	70°C/70%	90°C/90%
50°C/50%	–	$- 6.301 \times 10^{3} K$	$- 6.811 \times 10^{3} K$
70°C/70%	$- 6.301 \times 10^{3} K$	–	$- 7.385 \times 10^{3} K$
90°C/90%	$- 6.811 \times 10^{3} K$	$- 7.385 \times 10^{3} K$	–

The estimated value of ${\hat{E}}_{ij}$ is substituted into equations (22) and (23), respectively, and thus we can obtain

E = - 6.832 \times 10^{3} K, A = 5.294 \times 10^{3}, ε^{2} = 2.042 \times 10^{- 4}

As a result, the relationship between the drift coefficient $θ$ with the temperature and humidity stress is

θ = 5.294 \times 10^{3} {(H_{R})}^{3} e^{({\frac{- 6.832 \times 10}{273 + T}}^{3})}

(25)

In general, the temperature and humidity of the LED driving power working environment are 25°C and 30%, respectively, in the rail vehicle carriage. So the drift coefficient $θ$ and the diffusion coefficient $ε$ can be calculated as $2.166 \times 10^{- 4}$ and 0.0143, respectively. The failure threshold of the output current of the LED driving power is 15 mA. The PDF of the remaining life of the LED driving power is

\begin{matrix} f (t) = \frac{C - Δ Z (t)}{\sqrt{2 π ε^{2} t^{3}}} e^{[- \frac{{(C - Δ Z (t) - θ t)}^{2}}{2 ε^{2} t}]} \\ = \frac{15}{\sqrt{4.084 \times 10^{- 4} π t^{3}}} e^{[- \frac{{(15 - 2.166 \times 10^{- 4} t)}^{2}}{4.084 \times 10^{- 4} t}]} \end{matrix}

(26)

where $Δ Z (t)$ is the degradation increment at time t. Because the degradation of the initial time is measured here, the value of $Δ Z$ is 0. The reliability function is defined as follows

\begin{matrix} R (t) = ϕ (\frac{C - θ t}{ε \sqrt{t}}) + e^{(\frac{2 θ C}{ε^{2}})} ϕ [\frac{- (C + θ t)}{ε \sqrt{t}}] \\ = ϕ (\frac{15 - 2.166 \times 10^{- 4} t}{0.0143 \sqrt{t}}) \\ + e^{(31.82)} ϕ [\frac{- (15 + 2.166 \times 10^{- 4} t)}{0.0143 \sqrt{t}}] \end{matrix}

(27)

The relationship between reliability and remaining life under different stress conditions is shown in Figure 4.

Figure 4.

Reliability curve of remaining life (h).

In order to verify the accuracy of the updated data, the PDF curves of the data updated based on the Bayesian method and the non-updated data are shown in Figure 5. As can be seen from the graph, the remaining life distribution predicted by the Bayesian method is higher and narrower than that of the non-updated method, which indicates that the variance of the updated method is smaller, in others words, the uncertainty of the updated method is smaller. By contrasting the updated data based on the Bayesian method and the non-updated data, it can be concluded that the prediction accuracy is improved after the Bayesian updating.⁵ At the same time, to validate the method can be improved by the accuracy of the remaining life estimation, the relative error of the mean remaining life and the estimation about Bayesian updated and non-updated in 70% and 90% lifetime is calculated and the relative error expression is

δ = \frac{Δ}{L} \times 100 %

(28)

where Δ is the absolute error and L is the theoretic service life of LED driving power in railway vehicles, the value of which is 1 × 10⁵. The calculated results are shown in Table 4. From Table 4, it can be seen that the proposed Bayesian method has a higher prediction accuracy and a small relative error. It validates the effectiveness of this method.

Figure 5.

Contrast of distribution of remaining life probability density.

Table 4.

The average remaining life of different lifespans of LED driving power and the relative error.

	70%/h	90%/h
The updated predictionvalue based on Bayesian	$6.999 \times 10^{4}$	$9.003 \times 10^{4}$
Prediction value	$7.641 \times 10^{4}$	$9.170 \times 10^{4}$
Relative error of Bayesian updating	0.143%	0.030%
Relative error	9.157%	1.700%

LED: light-emitting diode.

The results of the case show that the remaining life can be effectively predicted based on Wiener model for LED driving power. The proposed parameter updating method based on the Bayesian method can improve the precision of assessment and reduce the uncertainty of prediction significantly.

Conclusion

In this article, a method has been proposed to provide effective decision support for designing a maintenance strategy of the LED lighting control system for estimating the remaining life based on Wiener process with drift parameters. The parameters are updated by the Bayesian method. Moreover, the detailed process of parameter estimation and the PDF of remaining life have been explicitly derived by means of the above methods. Then, the reliability evaluation and remaining life prediction of LED driving power have been obtained under different temperature and humidity conditions, which provide an important reference for the reliability optimization and improvement of LED driving power. Finally, a simulation example about the remaining life prediction problem for the LED driving power has been provided to show the effectiveness of the proposed method.

Footnotes

Handling Editor: Guian Qian

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Nature Science Foundation under Grant No. 61751304, Achievements transformation project of Jilin science and Technology Department under Grant No. 20160307003GX, and Key R&D Projects of Jilin Science and Technology Department under Grant No. 20180201125GX.

ORCID iD

Zhi Gao

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A Wiener process–based remaining life prediction method for light-emitting diode driving power in rail vehicle carriage

Abstract

Keywords

Introduction

Degradation model of LED driving power based on Wiener process

The estimation and updating of parameters

Remaining life prediction of the LED driving power based on the Wiener process

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References