Introduction

PHM (Prognostics and Health Management) technology is widely applied in the fields of aerospace, transportation, machinery, electronics, and more1,2,3,4,5,6,7,8,9. Through the application of PHM technology can predict the trend of equipment degradation, discover the potential failure of equipment operation, and carry out timely maintenance decision-making work, thus effectively reducing economic losses and improving equipment reliability10,11. Remaining useful life prediction research is a key technology in the application of PHM. Therefore, it is of great practical significance to carry out equipment remaining useful life prediction research.

Currently, the research on the remaining useful life prediction methods of the system mainly includes the methods based on the physical model of failure, the data-driven methods and the fusion of the two methods12,13,14. Physical model of failure and the integration of the two methods require the establishment of an internal failure mechanism model, deeply analyzing the causes of equipment degradation failure, so as to achieve the prediction of the remaining useful life of the equipment15,16. However, different devices have different internal structures and different failure mechanisms, so the remaining useful life prediction methods based on physical models of failure are less generalizable. The data-driven methods analyze degradation data characteristics from a statistical perspective and predict remaining useful life by establishing a degradation model17,18. This method is highly versatile and easy to implement, so the data-driven approach has been widely researched and applied in recent years19,20. Data-driven prediction methods mainly include statistical data-based methods and machine learning methods. Although machine learning methods have strong data learning and prediction functions, the black box problem in its data processing makes it difficult to analyze the characteristics of data variation21,22. The statistical data-based method establishes the equipment degradation model according to the characteristics of the degradation data distribution, which can not only analyze the variability of the degradation data, but also predict the remaining useful life of the equipment. Therefore, the statistical data-based method has been studied and applied in engineering by a large number of scholars23,24. Common statistical data-based methods include the Gamma stochastic degradation model25, the Wiener stochastic degradation model26, the IG stochastic degradation model27, and so on. Although there have been a number of research results showing that Gamma process and IG process models provide good fitting performance in equipment degradation modeling, they are only applicable to equipment with strictly monotonic degradation characteristics28,29. However, the degradation characteristics of most complex systems exhibit non-monotonicity. The Wiener process degradation model has a better fitting effect on the non-monotonic degradation features by adjusting the drift coefficients and diffusion coefficients30. As a result, the Wiener process degradation model has been widely applied in the degradation modeling and remaining useful life prediction of non-monotonic devices, such as rolling bearings31, lithium-ion batteries32, electronic transistors33, and multi-component systems34. The performance parameters degradation of the power driver device studied in this paper presents non-monotonic characteristics. Additionally, the carrier motion and distribution inside the power driver device under multiple stress conditions are similar to the Brownian motion characteristics of the Wiener process. Therefore, using the Wiener process degradation model better aligns with the degradation characteristics of the power driver device.

However, with the improvement of manufacturing process and material performance, electronic products are characterized by “high reliability” and “long life”. Therefore, it is difficult to carry out degradation modeling and remaining useful life prediction research to monitor the long degradation and failure cycle of such products35. Accelerated degradation test accelerates the degradation of equipment performance, shortens the degradation failure time, and reduces the cost of reliability research by applying accelerated stress to the equipment beyond the normal operating conditions36,37. Commonly applied accelerated stresses include temperature stress, humidity stress, vibration stress, electrical stress, etc. To quantitatively describe the relationship between accelerated stress and the life characteristics of the equipment, many scholars have conducted extensive experimental research and obtained some classical acceleration models, such as the Arrhenius model, the Eyring model, the inverse power law model, the exponential model, and other single-stress models38,39,40,41,42. However, the actual operating environment of the majority of equipment is compound stress. Therefore, multi-stress acceleration modeling is a hot issue in current research. Common multi-stress acceleration models include the generalized linear logarithmic acceleration model, the polynomial acceleration model and the proportional risk model43,44. Among the three multi-stress acceleration models, the polynomial acceleration model and the proportional risk model are less applied in engineering due to their lower accuracy and complex reliability functions45. In contrast, the generalized linear logarithmic acceleration model can effectively couple the influences of multiple stress factors and is easy to implement in engineering practice. It is currently the most widely used multi-stress acceleration model, such as the generalized Eyring model and the Peck model46. Luo47 respectively designed single-stress and multi-stress accelerated life tests, proposed a life prediction model integrating the corrosion kinetics model and the multi-stress accelerated model, and ultimately obtained the positions of the weak components of the electronic fuse and the storage life of the fuse components under multi-stress conditions. Zhang48 proposed a generalized multi-stress coupled accelerated life assessment method. Based on the traditional accelerated stress model, this method considered the relationship of multi-stress coupling, combines fuzzy mapping and fuzzy correlation to evaluate product life, and verifies the accuracy and reliability of this method using multi-stress accelerated life test data of low-noise amplifier module. The life prediction standard error is less than 5%, but this accelerated model did not consider the uncertainty and randomness issues of equipment degradation. Pan49 proposed a reliability assessment method of multi-stress coupled accelerated degradation model for electromechanical products with multi-failure modes and coupled competitive characteristics, and carried out a reliability assessment on a certain servo system. The results indicated that the proposed method has a certain superiority in evaluation accuracy. Zhao50 proposed an optimization criterion considering stress weights for the accelerated degradation test optimization design problem, and combined it with the Tabu Search-Particle Swarm Optimization (TS-PSO) algorithm to solve the issue of low estimation accuracy of some stress models parameters. Dong51 proposed a generalized coupled multi-stress accelerated degradation test and life assessment method for the complexity of the LED working environment. Then, they used the maximum likelihood estimation and particle swarm optimization to solve the unknown parameters, and finally verified that the assessment method is more in line with the actual working environment of the LED by using the experimental data. However, the impact of the uncertainty factors associated with LED degradation was not considered during the life assessment process.

In summary, although many scholars have conducted extensive studies in the research of multi-stress accelerated degradation models, there are still some research deficiencies as follows: (1) Currently, the modeling of multi-stress accelerated degradation is carried out with simple coupling of a single stress, such as through log-linear models and polynomial models, ignoring the effects of interactions between the stresses. (2) With the increase of stress factors, the number of unknown parameters in the multi-stress acceleration model increases, making parameter estimation more difficult and the accuracy hard to guarantee. (3) There are uncertainties and random effects in conducting accelerated degradation tests, which introduce errors in equipment degradation modeling. In light of the existing research gaps, this paper mainly focuses on the following tasks:

(1) Considering the random effects in the degradation process of electronic device performance parameters, a stochastic degradation model based on the Wiener process is established to describe the degradation trajectory of electronic device performance parameters over time.

(2) Considering the influence of temperature stress and electric stress coupling effect, a multi-stress coupling acceleration model is established. Since the acceleration stress affects the degradation trend of the equipment, the acceleration stress is associated with the drift coefficient of the Wiener process, and a multi-stress coupling accelerated degradation model based on the Wiener process is established.

(3) Aiming at the problem of solving unknown parameters in the multi-stress accelerated degradation model, an optimization algorithm based on MLE-SA (Maximum Likelihood Estimation-Simulated Annealing) is proposed to obtain the global optimal solution for all unknown parameters through a two-step solving method.

(4) To verify the reliability of the model, the multi-stress accelerated degradation test is carried out to obtain the accelerated degradation data of the power driver devices under the combined effects of temperature, humidity, and electrical stress, and use the accelerated degradation data to verify the model.

The structure of this paper is organized as follows: the Sects. 2 to 4 respectively describe the Wiener stochastic degradation process, the multi-stress coupled acceleration model, and the multi-stress coupled accelerated degradation model based on the Wiener process. The Sect. 5 describes the method of solving the parameters of the model. The Sect. 6 carries out the accelerated degradation test of the power driver device and the validation of the model. The Sect. 7 concludes the paper.

Wiener stochastic degradation model

The degradation of equipment performance is a continuous time process, and its degradation increments are independent of each other52. The performance degradation of power driver devices is essentially the effect of cumulative internal material damage. Due to the lack of necessary correlation between these factors, the factors causing internal damage exhibit strong randomness. Therefore, the degradation amount of power driver devices at a certain moment also has strong randomness53. The Wiener process, also known as Brownian motion, was first proposed by the American mathematician Norbert Wiener. The core idea is that it is a continuous-time stochastic process in which the increments are independent of each other and the increments obey a normal distribution. The specific expressions are as follows54:

$$X\left( t \right)=X\left( 0 \right)+\mu \int_{0}^{t} {\Lambda \left( {s,\theta } \right)} ds+\sigma B\left( t \right)$$
(1)

Where, \(X\left( t \right)\)is the degradation process of the power driver device (Hereinafter name as device), \(X\left( 0 \right)\) is the initial degradation state, \(\mu\) is the drift coefficient, indicating the rate of equipment degradation, \(\Lambda \left( {s,\theta } \right)\) is the time scale function, \(\sigma\)is the diffusion coefficient, indicating the fluctuation of equipment degradation, \(B\left( t \right)\)is the standard Brownian motion.

Assuming the \(X\left( t \right)\) degradation process from zero time to the first time it reaches the failure threshold is T, then the life of the ends. According to the definition of the first arrival time, the remaining useful life of the device is:

$$T=\inf \left\{ {t:X(t) \geqslant w|X(0)<w} \right\}$$
(2)

The probability density function of the first arrival time of the device can be expressed as:

$${f_T}\left( t \right) \cong \frac{1}{{\sqrt {2\pi t} }}\left( {\frac{{{S_B}\left( t \right)}}{t}+\frac{{\mu \Lambda \left( {t,\theta } \right)}}{\sigma }} \right)\exp \left( { - \frac{{{S_B}{{\left( t \right)}^2}}}{{2t}}} \right)$$
(3)
$${S_B}\left( t \right)=\frac{{\left( {w - X\left( 0 \right) - \mu \int_{0}^{t} {\Lambda \left( {s,\theta } \right)ds} } \right)}}{\sigma }$$
(4)

Then, substituting (4) into (3) is:

$${f_T}\left( t \right)=\frac{{w - X\left( 0 \right) - \mu \int_{0}^{t} {\Lambda \left( {s,\theta } \right)ds} +\mu \Lambda \left( {t,\theta } \right)t}}{{\sqrt {2\pi \sigma _{{}}^{2}{t^3}} }}\exp \left[ { - \frac{{{{\left( {w - X\left( 0 \right) - \mu \int_{0}^{t} {\Lambda \left( {s,\theta } \right)ds} } \right)}^2}}}{{2t\sigma _{{}}^{2}}}} \right]$$
(5)

If the degradation of device at \({t_k}\) is \(X\left( {{t_k}} \right)={x_k}\), the first arrival time is t. Then the remaining useful life at any time can be expressed as: \({l_k}=t - {t_k}\). Then, the \({l_k}\) can be defined as:

$${L_k}=\inf \left\{ {{l_k}>0:X\left( {{t_k}+{l_k}} \right) \geqslant w|X\left( {{t_k}} \right)<w} \right\}$$
(6)

The degradation process is:

$$X\left( t \right)={x_k}+\mu \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds+\sigma } \left( {B\left( t \right) - B\left( {{t_k}} \right)} \right)$$
(7)

Then the remaining useful life probability density function at \({t_k}\) can be expressed as:

$$f\left( {{l_k}} \right)=\frac{{w - X\left( {{t_k}} \right) - \mu \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} +\mu \Lambda \left( {{t_k}+{l_k},\theta } \right){l_k}}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }}\exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} } \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right]$$
(8)

Multi-stress coupling acceleration model

The accelerated degradation model refers to the relationship between the life characteristics of the device under accelerated test conditions and the changes in external stress. The common single-stress acceleration models include the Arrhenius model, the inverse power model, the exponential model55. The specific forms are as follows:

$$\begin{gathered} {R_1}=A\exp \left( {\frac{E}{{KS}}} \right) \hfill \\ {R_2}=A \cdot {S^m} \hfill \\ {R_3}=A \cdot \exp \left( {B \cdot S} \right) \hfill \\ \end{gathered}$$
(9)

The common multi-stress acceleration models include the Pecht model and the generalized Eyring model, where the Pecht model characterizing the effect of temperature and humidity stresses on equipment degradation, the specific forms are as follows:

$${R_4}=A \cdot {H^{ - m}}\exp \left( {\frac{E}{{KT}}} \right)$$
(10)

The generalized Eyring model characterizes the impact of temperature and electrical stress on equipment degradation, the specific forms are as follows:

$${R_5}=\frac{A}{T}\exp \left( {\frac{B}{{KT}}} \right)\exp \left( {V\left( {C+\frac{D}{{KT}}} \right)} \right)$$
(11)

In summary, this paper will conduct research on the effects of temperature, humidity and electrical stress on the accelerated degradation of power driver devices. Under high-temperature conditions, the internal temperature of power driver device increases, affecting the carrier migration rate and density distribution. When the electrical stress is applied, the internal electric field of the power driver device strengthens and the carrier movement intensified, which similarly increases the internal temperature of the device. Therefore, there is a coupled effect on the accelerated degradation of the power driver device when the temperature and electric stress act simultaneously. The paper proposes a coupled acceleration model for the combined effects of temperature, humidity, and electrical stress according to the generalized Eyring multi-stress model. This model not only considers the individual impacts of the three stresses on the degradation of power driver device but also takes into account the coupling effects of temperature and electrical stress, the specific forms are as follows:

$$R=\frac{{A \cdot {H^{ - m}}}}{T}\exp \left( {\frac{B}{{KT}}} \right)\exp \left( {V\left( {C+\frac{D}{{KT}}} \right)} \right)$$
(12)

Thus, the acceleration factor model is:

$$AF=A \cdot \frac{T}{{T{}_{0}}}{\left( {\frac{{{H_0}}}{H}} \right)^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right)$$
(13)

Where, \(R,{R_1},{R_2},{R_3},{R_4},{R_5}\)is the life characteristic factor, AF is the acceleration factor, S is the accelerated stress, E, is the activation energy, A, B, C, D and m are constants, K is the Boltzmann constant, K = 8.167 × 10−5 ev/k, H is the humidity stress, H0 is the normal working humidity, T is the absolute temperature, T0 is the normal working temperature, V0 is the normal working electrical stress, and V is the electrical stress.

Multi-stress coupled accelerated degradation model based on wiener process

There are two core parameters of the Wiener stochastic process: the drift coefficient and the diffusion coefficient. The drift coefficient affects the degradation rate, while the diffusion coefficient affects the amplitude of the degradation fluctuations. Accelerated degradation testing accelerates the degradation rate of the device by applying stress to the device that exceed normal levels. Therefore, accelerated stress primarily affects the degradation rate of the device. To quantify the effect of acceleration stress on the device performance degradation, this paper establishes a functional relationship between the Wiener process drift coefficient and the acceleration factor, the specific forms are as follows:

$$\mu \left( {T,H,V} \right)=A \cdot \frac{T}{{T{}_{0}}}{\left( {\frac{{{H_0}}}{H}} \right)^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right)$$
(14)

Then the device accelerated degradation model is:

$$X\left( t \right)={x_k}+\mu \left( {T,H,V} \right) \cdot \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds+\sigma } \left( {B\left( t \right) - B\left( {{t_k}} \right)} \right)$$
(15)

Thus, the probability density function of the remaining useful life of the device at the \({t_k}\) can be expressed as follows:

$$\begin{gathered} f\left( {{l_k}} \right)=\frac{{w - X\left( {{t_k}} \right) - \mu \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} +\mu \Lambda \left( {{t_k}+{l_k},\theta } \right){l_k}}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \cdot \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} } \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right] \hfill \\ =\frac{{w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \left( {\int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} - \Lambda \left( {{t_k}+{l_k},\theta } \right){l_k}} \right)}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \hfill \\ \cdot \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} } \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right] \hfill \\ \end{gathered}$$
(16)

If the time scale function in the process of stochastic degradation is linear, \(\int_{0}^{t} {\Lambda \left( {s,\theta } \right)} ds=t\), the accelerated degradation model M1 of the Wiener process in linear time scale can be expressed as follows:

$$X\left( t \right)={x_k}+\mu \left( {T,H,V} \right) \cdot {l_k}+\sigma \left( {B\left( t \right) - B\left( {{t_k}} \right)} \right)$$
(17)

Then the probability density function of the remaining useful life of the device at \({t_k}\) is:

$$f\left( {{l_k}} \right)=\frac{{w - X\left( {{t_k}} \right)}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \cdot \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot {l_k}} \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right]$$
(18)

If the time scale function in the stochastic degradation process is power law relationship, \(\int_{0}^{t} {\Lambda \left( {s,\theta } \right)} ds={t^\theta }\), the accelerated degradation model M2 of the Wiener process in power law time scale is:

$$X\left( t \right)={x_k}+\mu \left( {T,H,V} \right) \cdot \left( {{{\left( {{l_k}+{t_k}} \right)}^\theta } - t_{k}^{\theta }} \right)+\sigma \left( {B\left( t \right) - B\left( {{t_k}} \right)} \right)$$
(19)

Then the probability density function of the remaining useful life of the device at \({t_k}\) is:

$$\begin{gathered} f\left( {{l_k}} \right)=\frac{{w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} +\mu \left( {T,H,V} \right) \cdot \Lambda \left( {{t_k}+{l_k},\theta } \right){l_k}}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \hfill \\ \cdot \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} } \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right] \hfill \\ =\frac{{w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \left( {{{\left( {{l_k}+{t_k}} \right)}^\theta } - t_{k}^{\theta } - \theta \cdot {l_k} \cdot {{\left( {{l_k}+{t_k}} \right)}^{\theta - 1}}} \right)}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \hfill \\ \cdot \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \left( {{{\left( {{l_k}+{t_k}} \right)}^\theta } - t_{k}^{\theta }} \right)} \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right] \hfill \\ \end{gathered}$$
(20)

If the random degradation process shows a composite exponential degradation trend, \(\int_{0}^{t} {\Lambda \left( {s,\theta } \right)} ds=\exp \left( {\theta t} \right) - 1+t\), the accelerated degradation model M3 of the Wiener process on composite exponential time scale is:

$$X\left( t \right)={x_k}+\mu \left( {T,H,V} \right) \cdot \left( {\exp \left( {\theta \cdot \left( {{l_k}+{t_k}} \right)} \right) - \exp \left( {\theta {t_k}} \right)+{l_k}} \right)+\sigma \left( {B\left( t \right) - B\left( {{t_k}} \right)} \right)$$
(21)

Then the probability density function of the remaining useful life of the device at \({t_k}\) is:

$$\begin{gathered} f\left( {{l_k}} \right)=\frac{{w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} +\mu \left( {T,H,V} \right) \cdot \Lambda \left( {{t_k}+{l_k},\theta } \right){l_k}}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \hfill \\ \cdot \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \int_{{{t_k}}}^{{{t_k}+{l_k}}} {\Lambda \left( {s,\theta } \right)ds} } \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right] \hfill \\ =\frac{{w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \left( {\exp \left( {\theta \cdot \left( {{l_k}+{t_k}} \right)} \right) - \exp \left( {\theta {t_k}} \right) - \theta \cdot {l_k} \cdot \exp \left( {\theta \cdot \left( {{l_k}+{t_k}} \right)} \right)} \right)}}{{\sqrt {2\pi \sigma _{{}}^{2}{l_k}^{3}} }} \cdot \hfill \\ \exp \left[ { - \frac{{{{\left( {w - X\left( {{t_k}} \right) - \mu \left( {T,H,V} \right) \cdot \left( {\exp \left( {\theta \cdot \left( {{l_k}+{t_k}} \right)} \right) - \exp \left( {\theta {t_k}} \right)+{l_k}} \right)} \right)}^2}}}{{2\sigma _{{}}^{2}{l_k}}}} \right] \hfill \\ \end{gathered}$$
(22)

Parameter estimation

In terms of parameter estimation, the maximum likelihood estimation method is firstly used to estimate partial parameter values, and then the simulated annealing algorithm is employed to optimize the rest of the parameters to obtain the globally optimal parameter values. The maximum likelihood method is the most commonly used parameter estimation method, which is simple and easy to calculate. However, when there are more unknown parameters and the likelihood function is too complex, it is difficult to directly estimate accurate parameter values. The simulated annealing algorithm through the simulation cooling of temperature mode performs optimization calculations over the entire range, ultimately obtaining the global optimal solution. This method can solve the difficulties of multi-parameter estimation and the issue of insufficient estimation accuracy56. Taking the M1 model parameter solution as an example, the unknown parameters in the probability density function of the remaining useful life of the device include: \(\sigma\), A, B, C, D, m, the unknown parameter is \(\Theta \left( {\sigma ,A,B,C,D,m} \right)\). The \({X_i}\)is the degradation at \({t_i}\) in the equipment degradation test sample, where, \(i=1,2 \ldots ,n\), n is the number of test samples. Since the\(X\left( t \right)\) is the Wiener random process, and the degradation increment is:

$$\begin{gathered} X\left( {{t_{i+1}}} \right)=X\left( {{t_i}} \right)+\mu \left( {T,H,V} \right)\int_{{{t_i}}}^{{{t_{i+1}}}} {\Lambda \left( {s,\theta } \right)ds+\sigma } \left( {B\left( {{t_{i+1}}} \right) - B\left( {{t_i}} \right)} \right) \hfill \\ \Delta X\left( {{t_i}} \right)=X\left( {{t_{i+1}}} \right) - X\left( {{t_i}} \right)=\mu \left( {T,H,V} \right)\Delta {t_i}+\sigma \left( {B\left( {{t_{i+1}}} \right) - B\left( {{t_i}} \right)} \right) \hfill \\ \end{gathered}$$
(23)

Since the \(B\left( t \right)\) is a standard Brownian motion, the degenerate increment follows a normal distribution, \(\Delta X\left( {{t_i}} \right)\sim N\left( {\mu \Delta {t_i},\sigma _{{}}^{2}\Delta {t_i}} \right)\), The degenerate incremental probability density function is:

$$f\left( {\Delta X\left( {{t_i}} \right)} \right)=\frac{1}{{\sqrt {2\pi \sigma _{{}}^{2}\Delta {t_i}} }}\exp \left( { - \frac{{{{\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right)\Delta {t_i}} \right)}^2}}}{{2\sigma _{{}}^{2}\Delta {t_i}}}} \right)$$
(24)

According to the principle of maximum likelihood estimation, the likelihood function of unknown parameters can be expressed as follows:

$$L\left( {\sigma ,A,B,C,D,m} \right)=\prod\limits_{{i=1}}^{n} {\frac{1}{{\sqrt {2\pi \sigma _{{}}^{2}\vartriangle {t_i}} }} \cdot \exp \left[ { - \frac{{{{\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right)}^2}}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}} \right]}$$
(25)

In order to facilitate calculation, taking the logarithm for the likelihood function yields the log-likelihood function as:

$$\ln L\left( {\sigma ,A,B,C,D,m} \right)=\sum\limits_{{i=1}}^{n} {\ln \frac{1}{{\sqrt {2\pi \sigma _{{}}^{2}\vartriangle {t_i}} }} - } \sum\limits_{{i=1}}^{n} {\frac{{{{\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right)}^2}}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}}$$
(26)

To estimate unknown parameters: \(\Theta \left( {\sigma ,A,B,C,D,m} \right)\), maximizing the log-likelihood function:

$$\hat\Theta( {\sigma ,A,B,C,D,m} )={\rm arg\: max \ln} L({\sigma ,A,B,C,D,m} )$$
(27)

Taking the partial derivative of the likelihood function,

$$\begin{gathered} \frac{{\partial \ln L}}{{\partial \sigma }}=\sum\limits_{{i=1}}^{n} {\frac{{{{\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right)}^2}}}{{\sigma _{{}}^{3}\vartriangle {t_i}}} - \sum\limits_{{i=1}}^{n} {\frac{1}{\sigma }} } \hfill \\ \frac{{\partial \ln L}}{{\partial A}}=\sum\limits_{{i=1}}^{n} {\frac{{2\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right) \cdot \left( {\frac{T}{{T{}_{0}}}{{\left( {\frac{{{H_0}}}{H}} \right)}^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right) \cdot \vartriangle {t_i}} \right)}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}} \hfill \\ \frac{{\partial \ln L}}{{\partial B}}=\sum\limits_{{i=1}}^{n} {\frac{{2\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right) \cdot \left( {\frac{1}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)\frac{T}{{T{}_{0}}}{{\left( {\frac{{{H_0}}}{H}} \right)}^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right) \cdot \vartriangle {t_i}} \right)}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}} \hfill \\ \frac{{\partial \ln L}}{{\partial C}}=\sum\limits_{{i=1}}^{n} {\frac{{2\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right) \cdot \left( {\left( {{V_0} - V} \right)\frac{T}{{T{}_{0}}}{{\left( {\frac{{{H_0}}}{H}} \right)}^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right) \cdot \vartriangle {t_i}} \right)}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}} \hfill \\ \frac{{\partial \ln L}}{{\partial D}}=\sum\limits_{{i=1}}^{n} {\frac{{2\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right) \cdot \left( {\frac{1}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)\frac{T}{{T{}_{0}}}{{\left( {\frac{{{H_0}}}{H}} \right)}^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right) \cdot \vartriangle {t_i}} \right)}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}} \hfill \\ \frac{{\partial \ln L}}{{\partial m}}=\sum\limits_{{i=1}}^{n} {\frac{{2\left( {\Delta X\left( {{t_i}} \right) - \mu \left( {T,H,V} \right) \cdot \vartriangle {t_i}} \right) \cdot \left( { - \ln H \cdot \frac{T}{{T{}_{0}}}{{\left( {\frac{{{H_0}}}{H}} \right)}^{ - m}}\exp \left( {\frac{B}{K}\left( {\frac{1}{{{T_0}}} - \frac{1}{T}} \right)} \right)\exp \left( {C\left( {{V_0} - V} \right)} \right)\exp \left( {\frac{D}{K}\left( {\frac{{{V_0}}}{{{T_0}}} - \frac{V}{T}} \right)} \right) \cdot \vartriangle {t_i}} \right)}}{{2\sigma _{{}}^{2}\vartriangle {t_i}}}} \hfill \\ \end{gathered}$$
(28)

Here, the \(\widehat {\sigma }\) value can be estimated directly by making the partial derivative to zero, but it is difficult to estimate the exact solution for the other unknown parameters. When estimating the remaining unknown parameters, substituting the \(\widehat {\sigma }\) values into the log-likelihood function, and then use the simulated annealing algorithm to compute the maximum value optimization on the log-likelihood function to obtain the optimal parameter solution: \(\widehat {A},\widehat {B},\widehat {C},\widehat {D},\widehat {m}\). The specific algorithm process is as follows:

figure a

MLE-SA algorithm parameters estimation: \(\hat\Theta( {\sigma ,A,B,C,D,m} )\)

Experimental verification

Accelerated degradation test

To verify the reliability of the model in paper, an accelerated degradation test under the combined effect of temperature, humidity, and electrical stress was carried out with a certain type of the airborne power driver device as the test object. Consulting the power driver device manual, the most suitable working environment is: temperature 20 ℃ ~40 ℃, humidity 40%~60%. To ensure the power driver device accelerates degradation speed under normal operating conditions, the conditions for the accelerated stress are as follows: temperature 80 ℃, relative humidity 90%, and electric stress 24 V DC voltage.

Figure 1 shows the design principle of the accelerated degradation test circuit, in which the C terminal of the power driver device is connected to the positive terminal of the current sensor, the E terminal is connected to the negative terminal of the current sensor, and the B terminal is connected to the timer control. The current sensor is mainly used to collect the current value of the power driver device and transmit it to the computer through the data acquisition card. The duration of this accelerated degradation test was 912 h, during which a total of 16,700 data samples were collected at uniform time intervals.

Fig. 1
figure 1

Accelerated degradation test.

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Fig. 2
figure 2

Accelerated degradation process of power driver devices.

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The Fig. 2 shows the power driver device current gain amplification degradation process, which is the most important life characteristics of the power driver device and characterizes the power driver device current amplification characteristics57. In the accelerated degradation test device, the current sensor mainly collects the collector current and base current of the power driver device. The current gain amplification factor is the ratio of the collector current to the base current. According to the degradation test data, the current gain amplification of the power driver device under accelerated stress gradually increases.

Degradation model solving

The accelerated degradation model based on the Wiener process established in this paper follows a normal distribution. To verify the distribution characteristics of the accelerated degradation test data in this paper, the degradation increments of the power driver device were compared with the standard normal distribution, resulting in the comparison quantile plot shown in Fig. 3. When solving the parameters of the accelerated degradation model, partial parameter estimates are obtained using the maximum likelihood method. Then, the simulated annealing algorithm is used to maximize the log-likelihood function of the degradation increment probability density, obtaining the optimal solution for the remaining unknown parameters. Figure 4 shows the iterative convergence process of the simulated annealing algorithm. Table 1 shows the model parameter estimation results.

Fig. 3
figure 3

Comparison of degradation increments with standard normal distribution quantile plot.

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Fig. 4
figure 4

Parameter iteration calculation results.

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Figure 3 shows the comparison of degradation increments with standard normal distribution quantile plot, in which the red straight line is the standard normal distribution and the blue discrete points are the distribution of the degradation increment. The quantile plot results show that the discrete points closely overlap with the line, indicating that the degradation increment of the power driver device follows a normal distribution obtained in this paper, which satisfies the basic properties of the Wiener process. Figure 4 shows the convergence results of the parameters iterative computation of the simulated annealing algorithm, where (a) is the value of the log-likelihood function, and (b)-(f) are the convergence values of the parameters: A, B, C, D, and m. The results show that the parameter optimization calculation of the simulated annealing algorithm converges faster, reaching convergence after approximately 800 iterations. Additionally, there are noticeable irregular jumps before the convergence. This is because the simulated annealing algorithm jumps out of the local optimal region after reaching local convergence to continue the optimization iterations until the global optimal solution is found.

Table 1 Estimation results of model parameters.
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Result analysis

To verify the accuracy of the degradation model in this paper, setting the current gain amplification of 175 as the pseudo-failure threshold to validate the error between the predicted life span and the true lifespan with the limited current gain amplification degradation data from the accelerated degradation tests. In order to describe the reliability of the model’s remaining useful life prediction at different moments, the data monitoring points before the failure threshold are randomly and uniformly extracted, resulting in the remaining life prediction and its failure probability density function distribution for power driver devices at different times, as shown in Fig. 5. In addition, in order to analyze the reliability of the degradation model at different moments, the reliability function distribution of the different degradation models at different times were obtained, as shown in Fig. 6.

Fig. 5
figure 5

Probability density function and remaining useful life prediction based on Wiener process multi-stress coupled accelerated degradation model: (a)M1, (b)M2, (c)M3.

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Fig. 6
figure 6

Reliability function of the accelerated degradation model: (a)M1, (b)M2, (c)M3.

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Figure 5 shows the probability density function distributions and remaining useful life predictions for different degradation models, where (a) is the prediction result of the linear time scale model M1, (b) is the prediction result of law time scale model M2, and (c) is the prediction result of compound exponential time scale model M3. The results indicate that the failure probability density function distribution curves of the three degradation models are similar, and the actual lifespan distribution of the power driver device is near the mean value of the failure probability density function. The failure probability density curves of the three degradation models at the initial time have smaller peaks and more dispersed probability values compared to those at the end time. Furthermore, the failure probability density curves of the M2 model at different times have higher peaks compared to the corresponding curves of M1 and M3, and the probability value is more concentrated around the peak value. Figure 6 shows the reliability functions of three types of the accelerated degradation models. The results indicate that the reliability function of the three types of the degradation models changes from 1 to 0 with the remaining useful life. Notably, the rate of change in reliability value at the initial moment is less than the rate of change in reliability value at the final monitoring moment, which indicates that the reduction in reliability in the later stages of degradation is much greater than in the early stages under the same degradation increment. Moreover, the change rate of the reliability curve at different monitoring times for the M2 model is greater than that of the M1 and M3 models, which indicates that the reliability of the M2 model is higher than that of the M1 and M3 models under the same lifespan prediction conditions, and the reliability of the M2 model’s prediction is higher.

In order to further evaluate the accuracy of the accelerated degradation model in this paper, the remaining useful life predictions were conducted at different degradation times and compared with the actual remaining life values at the corresponding times. Figure 7 shows the comparison between the true life and predicted life errors at different monitoring times. Furthermore, the model accuracy is further quantified and evaluated by calculating the root mean square error, goodness-of-fit value, and AIC value of the accelerated degradation model. Table 2 shows the specific values.

Fig. 7
figure 7

Comparison of the error between true life and predicted life: (a)M1, (b)M2, (c)M3.

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Table 2 Model reliability assessment values.
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Figure 7 compares the true life and predicted life errors of the power driver devices at different degradation times, the Table 2 shows the model reliability assessment values. The results show that the life prediction errors of the three types of degradation models are smaller at the initial and final moments, but larger at the intermediate moments. This is due to the large fluctuations in the current gain amplification factor during the degradation test of the power driver device. Moreover, the M2 model has the smallest root mean square error of 131.33, the highest goodness of fit of 0.517, and the smallest AIC value of −8.2219 among the three types of degradation models. From this, it can be seen that M2 degradation model has superior reliability.

Comparison and verification of reference data

To compare and verify the accuracy of the multi-stress coupled accelerated degradation model based on the Wiener process in this paper, the multi-stress accelerated degradation test data of a certain model of aviation LED chip from the reference58 is used for comparative verification analysis. The multi-stress accelerated degradation tests conducted in the reference, which consider temperature stress, humidity stress, and electrical stress, are the same stress conditions considered in this paper. Therefore, the experimental data from this reference has certain comparative validation significance. The literature defines that the LED chip reach the failure threshold when the light brightness drops to 50% of the initial brightness. In order to verify the accuracy of the model, the second group of stress design tests in the experiment was selected, which is a temperature of 110 ℃, a humidity of 80%, and a current of 650 mA. During this test, the LED degradation data reached the failure threshold. The degradation data of LED1 and LED2 samples were taken as examples to carry out simulation calculation.

Using the M2 model simulate the LED degradation data and obtain the failure probability density of the LED degradation model and the remaining useful life prediction distribution, as shown in Fig. 8. The Fig. 9 shows the error comparison of the true life and the predicted life.

Fig. 8
figure 8

Probability density function and remaining useful life prediction of M2 model: (a) LED1, (b) LED2

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Fig. 9
figure 9

Error comparison between true life and predicted life: (a) LED1, (b) LED2.

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Figure 8 shows the failure probability density function of M2 model and the predicted distribution of remaining useful life. The results show that the true life of the two groups of LED samples is near the central value of the failure probability density distribution, and the peak value of the failure probability density function curve of the two groups of samples is high and the width is narrow, indicating that the prediction reliability of the model is high. Figure 9 shows the error comparison between the true life of LED and the predicted life. The results show that the M2 model performs well in the degradation prediction of the two groups of LED samples, and the goodness of fit value is above 0.9. Specifically, the root-mean-square error of the predicted life for the LED2 samples is 0.67, and the root-mean-square error of LED1 samples is 0.1524, indicating that there are differences in the degradation of different samples.

Conclusion

Aiming at the research problem of the multi-stress accelerated degradation and remaining useful life prediction for high-reliability products. The paper firstly considers the issue of random effects in the degradation process of the device and establishes a stochastic degradation model based on the Wiener process. Secondly, a new multi-stress coupling model is proposed, associating the accelerated stress with the drift coefficient of the Wiener process, so as to establish a multi-stress coupling accelerated degradation model based on the Wiener process. The global optimal solution for all unknown parameters of the model is estimated by using the MLE-SA optimization search algorithm. Finally, the reliability of the model in this paper is verified by the multi-stress accelerated degradation test data of a certain power driver device. The results show that the multi-stress coupled accelerated degradation model based on the Wiener process can better fit the accelerated degradation test data of power driver device, and lay a theoretical foundation for the accelerated degradation research of high-reliability devices. However, there are still several challenges regarding the accelerated degradation research of high-reliability equipment as follows: (1) There are degradation and regeneration phenomena in the degradation process of some equipment, and the degree of degradation and regeneration and the time of emergence have greater uncertainty, which brings greater challenges to the degradation modeling. (2) The multi-stress degradation model established in this paper is a single characteristic degradation model, and the multi-stress accelerated degradation model for multiple degradation characteristics requires further research.