Research on the Degradation Model of a Smart Circuit Breaker Based on a Two-Stage Wiener Process

Abstract

As the global energy transition moves towards the goal of low-carbon sustainability, it is crucial to build a new energy power system. The performance and reliability of Smart Circuit Breakers are the key to ensuring safe operation. The control circuit is the key to the reliability of Smart Circuit Breakers, so studying its performance-degradation process is of great significance. This study centers on the development of a degradation model and the performance-degradation-assessment method for the control circuit of Smart Circuit Breakers and proposes a novel approach for lifetime prediction. Firstly, a test platform is established to collect necessary data for developing a performance-degradation model based on the two-stage Wiener process. According to the theory of maximum likelihood estimation and Schwarz information criterion, the estimation method of model distribution parameters in each degradation stage and the degradation ‘turning point’ method are studied. Then, reliability along with residual life serve as evaluation criteria for analyzing the control circuit’s performance deterioration. Taking the degradation characteristic data into the degradation model, for example, analysis, combined with the Arrhenius empirical formula, the reliability function at room temperature and the curve of the residual life probability density function is obtained. Ultimately, the average service life of the Smart Circuit Breaker control circuit at room temperature is 178,100 h (20.3 years), with a degradation turning point at 155,000 h (17.7 years), providing a basis for the lifetime evaluation of low-voltage circuit breakers.

1. Introduction

A low-voltage circuit breaker is an important protective switch electrical appliance in the power distribution system. According to statistics, over 80% of the electric energy is distributed and used through the switch electrical appliance [1]. In 2023, the market size of China’s circuit breakers will reach 12.427 billion yuan, and its application will cover various fields of the national economy, such as national defense and military, aerospace, transportation, and electric power [2,3]. Advancements in electronic technology have led to low-voltage electrical devices exhibiting an extended service life and enhanced reliability, with only minimal performance degradation over time [4,5]. Therefore, the reliability evaluation of low-voltage electrical appliances, particularly low-voltage circuit breakers, remains a significant challenge in the industry. To rapidly obtain life data and degradation process data for reliability assessments, as well as to identify product defects promptly, accelerated degradation testing is a crucial method widely employed in research [6,7]. SCB (Smart Circuit Breaker, SCB) is a key part of the intelligent power system in modern society [8]. It is an emerging product in the low-voltage circuit breaker industry. Its internal structure is complex and compact. It is mainly composed of traditional contact mechanical systems and intelligent electronic systems. It can replace the traditional manual opening and closing mode to realize remote automatic opening and closing [9].

In the advancement of reliability engineering, modeling approaches based on failure physics and performance degradation have become the leading research focus [10]. Among them, the reliability modeling method based on failure physics mainly explains the specific mechanism of electronic product failure through electronic products’ physical and chemical information factors. It obtains the accurate failure information of products [11]. The reliability modeling method based on performance degradation pays attention to the data information in the evolution process of electronic product degradation failure. It uses the appropriate mathematical and statistical model to effectively fit the degradation trend to evaluate electronic products’ reliability [12]. Among these, degradation refers to the gradual and irreversible deterioration of key performance parameters in the SCB control circuit over time, under long-term operation or accelerated stress conditions. It is typically reflected by the measurable changes in electrical characteristics, indicating internal aging mechanisms and forming the basis for reliability assessments and lifetime prediction [13]. Based on the Weibull distribution model of degradation quantity and finite element analysis method, Liu Junzuo studied the degradation process of air tightness of a certain type of sensor shell [14]. Aiming at the distribution of an unknown degradation quantity and bounded degradation quantity, Li Jiang et al. [15] proposed a semi-parametric degradation model for assessing the reliability of breaker products, significantly enhancing the accuracy and credibility of the evaluation process. Yu Rongbin studied the performance degradation modeling method of photovoltaic modules. Without the pre-subjective assumption of the distribution type, it can also effectively fit the distribution of degradation at each moment [16]. Tuo Yantian studied the two-stage degradation process of products and used two drift functions to establish the degradation model for each stage [17]. Tang Xuri et al. [18] constructed a bidirectional multi-point grey prediction model and proposed the concept of equivalence for different breaking currents. The proposed model improves the prediction accuracy of the remaining life of low-voltage DC circuit breakers used in photovoltaic systems. Yang Nuoming et al. [19] proposed a method for predicting the electrical life of DC circuit breakers based on the characteristics of DC arc interruption. In this method, contact quality is used to represent the electrical life condition. A contact quality degradation curve is obtained through linear interpolation and is used as a training label for the life prediction network. Most of the previously proposed models primarily address a constant-rate decline scenario in electronic systems, focusing only on a single-phase degradation trend. In actual engineering contexts, the complex internal faults inside electronic systems combined with fluctuating environmental influences result in the observable wear-out process often presenting as multiple distinct stages rather than a single phase. Different stages of wear display specific statistical properties, featuring a “change point” where the speed of performance loss in electronics changes [20,21,22]. This study constructed a reliability degradation model using a two-stage Wiener process method. According to the theory of maximum likelihood estimation and Schwarz information criterion, the estimation method of model distribution parameters and the degradation ‘turning point’ method in each degradation stage are studied. The performance degradation assessment of the control circuit is carried out. Through the example analysis, the curve of the reliability function and the residual life probability density function at room temperature is obtained, and the maintenance strategy of the SCB control circuit is studied based on the results.

The main contributions of this paper are as follows:

(1)

An in-depth study was conducted on the related degradation mechanisms, and its key functional modules are identified. Based on feature extraction and trend analysis of experimental data, the control circuit is found to exhibit a clear two-stage degradation pattern.

(2)

A degradation model based on a two-stage Wiener process is developed. By integrating the maximum likelihood estimation, the Schwarz information criterion, and reliability analysis, the turning point of performance degradation is accurately identified.

(3)

The quantile plot method is used to test the normality of degradation increments at different stages, and the reliability function along with the failure probability density function curve is obtained by extrapolating from the experimental data. The average service life of the control circuit under normal temperature conditions is predicted to be 178,100 h (20.3 years), and the turning point of degradation is estimated at 155,000 h (17.7 years).

This paper is structured into four primary sections: examination of the deterioration mechanism within the SCB control system; identification of crucial components; analysis of degradation trends and development of a two-stage Wiener process-based degradation model; extrapolation of experimental data to derive the reliability and failure density functions, and prediction of the average service life and turning point of degradation under normal temperature conditions.

2. Research on the Deterioration Process of the Control System in SCB

2.1. Study on the Functional Mechanism of the Control System

The SCB consists of a contact mechanical system and an intelligent electronic system. The smart electronic setup comprises a control unit and a geared motor. The main function of the control circuit is to receive and judge the external input opening/closing control signal; on the other hand, electric power is transmitted to provide the electric power input for the gear motor. The geared motor transforms the electrical energy supplied by the control unit into mechanical energy to actuate the contact mechanism. The control circuit is the ‘heart’ of the SCB and the key link of the circuit breaker system. Its reliability determines whether the SCB can work reliably and accurately. The principle diagram of the control circuit is shown in Figure 1.

Figure 1 shows that the control unit comprises several operational blocks, including the power module, microcontroller unit, position sensing module, motor driver, control signal acquisition, and feedback for the on/off state. The linear module supplies a constant voltage to the MCU and the signal acquisition subsystem, while the switching section employs low-power design techniques to reduce unnecessary energy consumption for the end user. The control signal acquisition module translates the meter’s status information (e.g., payment in arrears or not) into a direct current (DC) signal, which is then transmitted to the microcontroller for analysis and decision-making. Through the feedback line, the switching status detection circuit conveys connection and disconnection signals of the power system to the energy measurement device. The motor-position-detection circuit is used to detect the motor position and to limit the maximum angle of motor rotation.

Before executing the external control command, the control circuit detects the Hall position and external control signals. Then, it performs different action processes according to the signal state. The complete working logic is shown in Figure 2.

Figure 2 illustrates the automatic control process of the circuit breaker’s opening and closing. After the system is powered on, the motor position is detected by the Hall sensor, and the external control signal is monitored. When a rising edge is detected and the circuit breaker is in the closed state, the start-up power supply drives the motor to reverse to the open position and then locks it. If the circuit breaker is already open, it is directly locked without action. When a falling edge is detected and the circuit breaker is open, the motor drives it to the closed position. If it is already closed, no action is taken. The system remains in a low-power sleep mode when the control signal stays steady. After all operations are completed, the power supply is automatically turned off, and the system enters sleep mode, thereby achieving intelligent control with high efficiency and energy savings.

2.2. Determination of Key Modules

Based on the above theoretical analysis, the following assumption is made: the components have only two states of normal and failure; the system works in the rated input state; the reliability of all components obeys exponential distribution. The exponential distribution has limitations. When assuming that its failure rate is constant, it is not suitable for the description of aging or failure of mechanical components such as bearings and gears.

$$R(t)=e^{-\lambda t}$$

(1)

In the formula, λ refers to the failure occurrence rate; R(t) is the reliability function. According to engineering experience statistics, the time-dependent variation in the failure rates of various components is generally classified into three phases: the initial failure stage, the random malfunction period, and loss failure period. Based on the reliability study of the control unit in the SCB, a mathematical model is established.

$$\left\{\begin{array}{l}{\lambda }_{\mathrm{S}}={\displaystyle \sum _{i=1}^n{\lambda }_i}\\ T_{\mathrm{BF}}={\displaystyle \frac{1}{{\lambda }_{\mathrm{S}}}}={\displaystyle {\int }_0^{\infty }R(t)\mathrm{d}t}\end{array}\right.$$

(2)

In this equation, λ_S indicates the control circuit’s failure frequency; λ_i corresponds to the failure frequency of the ith component contained therein; T_BF is the average failure-free time. Determine the fault occurrence rate across all constituent elements and functional units, as shown in Figure 3 and Figure 4.

According to the analysis of Figure 3 and Figure 4, electrolytic capacitors and power MOSFETs are the principal components impacting the overall reliability of the circuit system. The main functional elements responsible for system operation are the drive circuits, integrating both the switching power supply and motor drive modules. The above analysis provides a basis for selecting condition-monitoring signals for the accelerated degradation test.

3. Accelerated Degradation Test Design and Characteristic Trend Analysis

3.1. Degradation State Monitoring Signal Selection

The control circuit includes not only device-level degradation monitoring signals, such as capacitance C, equivalent series resistance ESR, etc. It also includes circuit board-level degradation status monitoring signals, such as input voltage signals, output voltage signals, sampling voltage and current signals, and output current signals. Each signal has a coupling redundancy relationship. If these signals are considered for measurement, on the one hand, it will increase the difficulty of data processing and analysis; on the other hand, it will waste test resources. Therefore, by determining the above key modules, combined with the circuit’s topology, the electrical characteristics of the monitoring signal, the difficulty of measurement, and the cost of measurement, the priority of the degradation state monitoring signal is considered. The monitoring signal most representative of the control system’s degradation condition is chosen.

According to the failure statistics of SCB products, most are the failure forms of the closing process [23]. Therefore, the board-level electrical signals related to the control unit during the closing phase of the smart breaker are chosen: motor current, motor terminal voltage, switching power supply voltage, linear power supply voltage, closing and opening control signals, and closing and opening state feedback signals, which are drawn together at the same time, as shown in Figure 5. According to the analysis of the working principle of the control circuit and the characteristics of the electrical signal waveform, the working stages are divided as shown in

Table 1. Control circuit working stage division of intelligent circuit breaker closing process.

Time and Time Period		Meaning of Electrical Characteristics
A		Open state
B	a	Closing signal triggered
	b	Switching power supply startup
	c-d	Motor start
	d-e	Motor unloaded
	f	The dynamic and static contacts start to make contact
	h-i	Motor braking
	j-k	Braking/precise positioning
	k	Close the switch in place
	l	Turn off the switch power supply
C		Closed state

.

The critical functional units within the initial segment are identified, considering the circuit’s topology, the monitoring signal’s electrical characteristics, and the measurement’s difficulty and cost. The monitoring signals that can best characterize the control circuit’s performance degradation state are selected as the switching power supply voltage, the motor terminal voltage, and the motor current.

3.2. Accelerated Test Design

Combined with the above-selected degradation state detection signal, accelerated degradation testing is commonly categorized into three distinct forms. As shown in Figure 6, under the premise of a comprehensive consideration of test cost and feasibility, to improve testing efficiency, minimize the number of test samples, and ensure the reliability of data modeling, this study adopts a constant-stress accelerated testing approach.

Several key elements must be considered when developing a scheme for accelerated degradation testing.

(1)

Stress Type and Levels in Accelerated Testing: Temperature is chosen as the sole accelerated stress factor. Typically, a single-stress temperature model requires test data at a minimum of three distinct temperature levels to enable an accurate parameter estimation. The temperature difference between adjacent levels is maintained at no less than 10 °C.

(2)

Determination of Sample Size for Accelerated Testing: Based on prior studies, a standard practice under single-temperature stress conditions is to include five samples per group. This approach minimizes the impact of individual variability on data interpretation and parameter estimation.

(3)

Accelerated Test Duration and Frequency: The test cycle is defined by taking into account changes in performance parameters, the overall testing workload, cost considerations, and the desired quantity of data. An initial inspection interval is established, and subsequent adjustments are made based on observed trends in performance degradation during testing.

The following are the steps of the experiment:

(1)

At 25 °C, the initial values of key condition monitoring signals are recorded as the reference for the samples to evaluate performance degradation.

(2)

The samples are then sequentially subjected to cyclic accelerated aging tests under thermal stress conditions of 105 °C, 95 °C, and 85 °C.

(3)

Each test cycle lasts for 24 h. The process begins with the sample being stabilized at 25 °C for 0.5 h, followed by a temperature increase to the specified stress level over the next 0.5 h. The sample is then operated with one on–off switching cycle lasting 1 h. Afterward, the temperature is reduced back to 25 °C within 0.5 h. Following a 0.5 h hold at this temperature, the sample is removed for a measurement of its condition-monitoring parameters. This process is repeated for the next cycle.

(4)

Until all five samples under each stress fail, the test ends.

3.3. Experimental Data Feature Extraction and Trend Analysis

Due to noise interference in the experimental data, it is necessary to perform denoising before further analysis. Common methods such as wavelet denoising, moving average filtering, and low-pass filtering are applied. The appropriate denoising method is selected based on the signal-to-noise ratio (SNR), which is calculated using the following formula.

$$SNR=10\log {\displaystyle \frac{\frac{1}{N}{\displaystyle \sum _{i=1}^N{n}^2(i)}}{\frac{1}{N}{\displaystyle \sum _{i=1}^N{(n(i)-s(i))}^2}}}$$

(3)

In the formula, n(i) represents the original degradation state monitoring signal, s(i) denotes the signal after denoising, and N is the total number of signal sampling points. The experimental data were denoised using three different methods, and the signal-to-noise ratio (SNR) was calculated for each. The results are shown in Figure 7.

In general, a higher SNR indicates better signal quality. As shown in Figure 7, wavelet denoising achieves the highest SNR and is therefore selected.

According to the above experimental scheme, the accelerated test is carried out to obtain the sample’s monitoring value under each temperature group. The monitoring data are processed, and the signal-to-noise ratio is calculated and compared. Finally, the wavelet denoising is selected to denoise the data. The extracted feature parameters from the test data include the mean, peak, peak-to-peak, and kurtosis in the time domain; center frequency and standard deviation in the frequency domain; and energy entropy and singular entropy in the wavelet domain. In summary, six feature parameters from the time domain, frequency domain, and wavelet domain can be obtained. Some of these features are highly correlated with performance degradation, while others do not effectively support the evaluation of performance degradation. Moreover, an excessive number of features may impose computational and storage burdens. Therefore, it is necessary to optimize the above characteristic parameters. Therefore, the feature selection method shown in Figure 8 is used to optimize the features.

The feature-selection process shown in Figure 8 is mainly divided into three steps: data acquisition, feature candidate generation, and optimal feature selection. Specifically, the process involves collecting raw data, extracting potential features from the data, and selecting the features that are most relevant to the target task. The core goal is to identify the most critical features from the original data to make the subsequent analysis or modeling more efficient and accurate.

The feature-selection process employs a multi-factor assessment index model to identify the optimal feature parameters. The evaluation considers three dimensions: trend, monotonicity, and robustness. The corresponding formulas are as follows.

$$Tre(F,T)={\displaystyle \frac{\left|N{\displaystyle \sum _i{f}_i{t}_i-{\displaystyle \sum _i{f}_i{\displaystyle \sum _i{t}_i}}}\right|}{\sqrt{\left[N{\displaystyle \sum _i{f}_i^2-{({\displaystyle \sum _i{f}_i})}^2}\right]\left[N{\displaystyle \sum _i{t}_i^2-{({\displaystyle \sum _i{t}_i})}^2}\right]}}}$$

(4)

Here, F = (f₁, f₂, …, f_N) denotes the time series of degradation feature parameters, T = (t₁, t₂, …, t_N) represents the corresponding sampling time points, and N is the total number of samples.

$$Mon(F)={\displaystyle \frac{\left|{\displaystyle \sum _i\delta (f_i-f_{i-1})-{\displaystyle \sum _i\delta (f_{i-1}-f_i)}}\right|}{N-1}}$$

(5)

In the above equation, δ (*) denotes the unit step function.

$$Rob(F)={\displaystyle \frac{1}{N}}{\displaystyle \sum _i\exp (-\frac{f_i-\widetilde{f_i}}{f_i})}$$

(6)

The comprehensive index formula for multi-dimensional feature evaluation and optimization is derived as follows, based on the above equations.

$$\begin{array}{l}\mathrm{Z}=k_{1}Tre(F,T)+k_{2}Mon(F)+k_{3}Rob(F)\\ s.t.\left\{\begin{array}{l}{k}_i>0\\ {\displaystyle \sum _i{k}_i=1}\end{array}\right.\end{array}$$

(7)

The test data are brought into the above process for feature selection, and a three-dimensional map of the scoring effect is obtained, as shown in Figure 9.

Figure 9 shows three key feature indices: correlation, monotonicity, and robustness. The candidate feature parameter with the highest score corresponds to the motor terminal voltage’s peak-to-peak amplitude. The score table is shown in

Table 2. The best degradation characteristic parameter score table.

Characteristic Parameter/V	Trend Score	Monotonicity Score	Robustness Score	Comprehensive Score
Peak-to-peak voltage at motor terminals	0.836	0.329	0.992	0.715

. According to testing and statistical analysis, the failure threshold H for the characteristic parameters is 3 V. The comprehensive score histogram for the corresponding characteristic parameters is shown in Figure 10. By counting the empirical failure date of 15 samples, the failure threshold H of the characteristic parameters can be obtained as 3 V.

Figure 10 shows the comprehensive score of each feature parameter. As shown in the figure, the voltage difference between the highest and lowest points at the motor terminals achieves the highest score, indicating that it is the optimal feature across all dimensions.

The compiled degradation data from samples exposed to various temperature conditions are used to model the characteristic parameter’s degradation trend through least squares fitting. The trend under the three temperature stress levels is consistent, and the fitting trend of 85 °C is shown in Figure 11.

The evolution trend of the degradation characteristics of the five samples is the same. They all experience a relatively stable degradation stage first, pass through the degradation ‘turning point’ at a certain time, and enter the degradation stage with a significantly faster degradation rate. That is, the degradation process of the sample shows a more obvious two-stage nonlinear characteristic. Because the degradation characteristic parameters of the control circuit show obvious two-stage nonlinear characteristics and there is a degradation ‘turning point’, it is difficult to use the traditional linear statistical model to infer. Therefore, this paper proposed a degradation model of the SCB based on the two-stage Wiener process for analysis.

4. Two-Stage Wiener Process Degradation Model

4.1. Degradation Model Construction

Given the research object’s above-mentioned performance-degradation characteristics, a two-stage degradation model should be considered to describe the circuit system’s performance-degradation process. In addition, in the field of reliability engineering, the Wiener process model can describe the inherent randomness of the product degradation path well, and the statistical computational complexity is low, which has a wide range of application research bases. Therefore, this paper uses a two-stage Wiener process model to describe the circuit system’s degradation process. The complete degradation model construction and parameter estimation are based on the flow chart shown in Figure 12.

Define that the first degradation stage obeys the Wiener process X1(t; μ1, σ1), and the second degradation stage obeys the Wiener process X2(t; μ2, σ2), and τ is the time when the turning point of degradation occurs. The control circuit’s two-stage Wiener degradation process model can be expressed as follows.

$$X(t)=\left\{\begin{array}{l}{X}_1(t;{\mu }_1,{\sigma }_1)=x_{01}+{\mu }_{1}t+{\sigma }_{1}B(t)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0<t\le \tau \\ X_2(t;{\mu }_2,{\sigma }_2)=x_{02}+{\mu }_2(t-\tau )+{\sigma }_{2}B(t-\tau )\hspace{1em}\hspace{1em} t>\tau \end{array}\right.$$

(8)

In the expressions above, x₀₁ and x₀₂, are the initial degradation quantities of the first and second degradation stages. μ₁ and μ₂ are the drift parameters of the first and second degradation stages, respectively, representing the degradation rate of each stage. σ₁ and σ₂ are the diffusion parameters of the first and second degradation stages. The Wiener degradation process is continuous, and the performance degradation values of the two degradation stages at the degradation turning point are consistent, that is, x₁(τ) = x₀₂.

4.2. Parameter Estimation and Turning Point Estimation

A set of m + 1 degradation data with a periodic detection interval Δt = t_j − t_j−1 and degradation increment Δx_j = X(t_j) – X(t_j−1) is obtained from the degradation test. Because the degradation process of the control circuit before and after the degradation turning point is different, the probability density function f(t) of the degradation increment can be expressed in the following two cases:

(1)

When t_j < τ, the degradation process is in the first stage. The degradation increment Δx_i obeys the inverse Gaussian distribution with drift parameter μ₁ and diffusion parameter σ₁. The likelihood function of the degradation process increment is as follows:

$$L_1(\Delta x_i;{\mu }_1,{\sigma }_1^2)={\displaystyle \prod _{i=1}^k\frac{1}{\sqrt{2\mathsf{\pi }\Delta t}{\sigma }_1}\exp \left[-\frac{{(\Delta x_i-{\mu }_1\Delta t)}^2}{2{\sigma }_1^2\Delta t}\right]}$$

(9)

(2)

When τ < t_j−1, the degradation process is in the second stage. The degradation increment Δx_i obeys the inverse Gaussian distribution with drift parameter μ₂ and diffusion parameter σ₂. The likelihood function of the degradation process increment is as follows:

$$L_2(\Delta x_i;{\mu }_2,{\sigma }_2^2)={\displaystyle \prod _{i=k+1}^m\frac{1}{\sqrt{2\mathsf{\pi }\Delta t}{\sigma }_2}\exp \left[-\frac{{(\Delta x_i-{\mu }_2\Delta t)}^2}{2{\sigma }_2^2\Delta t}\right]}$$

(10)

The joint likelihood function of the first-stage and second-stage degradation process increments is expressed as follows:

$$\begin{array}{l}L=\left[{\displaystyle \prod _{i=1}^k\frac{1}{\sqrt{2\mathsf{\pi }\Delta t}{\sigma }_1}\exp \left[-\frac{{(\Delta x_i-{\mu }_1\Delta t)}^2}{2{\sigma }_1^2\Delta t}\right]}\right]\cdot \\ \left[{\displaystyle \prod _{i=k+1}^m\frac{1}{\sqrt{2\mathsf{\pi }\Delta t}{\sigma }_2}\exp \left[-\frac{{(\Delta x_i-{\mu }_2\Delta t)}^2}{2{\sigma }_2^2\Delta t}\right]}\right]\end{array}$$

(11)

Further, the maximum likelihood estimation of parameters ${\mu }_1$, ${\sigma }_1{}^2$, ${\mu }_2$, and ${\sigma }_2{}^2$ is obtained as follows:

$$\left\{\begin{array}{l}{\widehat{\mu }}_1={\displaystyle \frac{1}{k}}{\displaystyle \sum _{i=1}^k\frac{\Delta x_i}{\Delta t}}\\ {\widehat{\sigma }}_1^2={\displaystyle \frac{1}{k}}{\displaystyle \sum _{i=1}^k\frac{{(\Delta x_i-{\widehat{\mu }}_1\Delta t)}^2}{\Delta t}}\\ {\widehat{\mu }}_2={\displaystyle \frac{1}{m-k}}{\displaystyle \sum _{i=k+1}^m\frac{\Delta x_i}{\Delta t}}\\ {\widehat{\sigma }}_2^2={\displaystyle \frac{1}{m-k}}{\displaystyle \sum _{i=k+1}^m\frac{{(\Delta x_i-{\widehat{\mu }}_2\Delta t)}^2}{\Delta t}}\end{array}\right.$$

(12)

Due to the existence of a degenerate ‘turning point’, the k in the above formula is unknown. Here, the Schwarz (SIC) information criterion is introduced to estimate it, as follows:

$$SIC=-2\ln L(\widehat{\theta })+p\ln m$$

(13)

In the formula, $L(\widehat{\theta })$ is the maximum likelihood function; $\widehat{\theta }$ is the parameter estimation; p is the number of free parameters; m is the sample size. Using SIC to determine the interval estimation $\widehat{k}$ of the ‘turning point’ is

$$SIC\left(\widehat{k}\right)=\underset{2\le k\le m-2}{\min }SIC(k)$$

(14)

The interval $[t_{\widehat{k}},t_{\widehat{k}+1}]$ of each sample’s performance degradation ‘turning point’ is determined using the above formula, and the estimated values of the distribution parameters ${\mu }_1$, ${\sigma }_1{}^2$, ${\mu }_2$, and ${\sigma }_2{}^2$ can be obtained by bringing $\widehat{k}$ into Formula (12). Finally, the maximum likelihood estimation gives the estimated value of the ‘turning point’ distribution parameter, as follows:

$$\left\{\begin{array}{l}{\widehat{\mu }}_{\tau }={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\widehat{\tau }}_i=\overline{{\widehat{\tau }}_i}}\\ {\widehat{\sigma }}_{\tau }^2={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{({\widehat{\tau }}_i-\overline{{\widehat{\tau }}_i})}^2}\end{array}\right.$$

(15)

4.3. Reliability Estimation

When the degradation quantity X(t) changes over time for the first time and exceeds the specified failure threshold, the product can be considered a failure. At this time, the corresponding time is defined as the life T of the product, and its expression is as follows:

$$T=\inf \left\{t:X(t)\ge H;t\ge 0\right\}$$

(16)

Among them, the failure threshold H can be a fixed product design standard value or an empirical random value according to the working conditions. According to the accelerated test’s product failure and the second section’s analysis results, the failure threshold is set as a constant.

The reliability function of the product can be expressed as follows.

$$R(t)=P\left\{T>t\right\}=P\left\{X(t)\le H\right\}=1-F(t)$$

(17)

In the formula, F(t) is the cumulative failure distribution function, which represents the failure probability of the product under the specified conditions and time. The cumulative failure distribution function and reliability function of the product are studied step by step, and the reliability function R(t) of the two-stage Wiener process is obtained as follows:

The reliability function of the product can be expressed as follows.

$$R(t)=\left\{\begin{array}{l}{R}_1(t;x_{01},{\mu }_1,{\sigma }_1)\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}0\le t\le \tau \\ R_2(t-\tau ;x_{02},{\mu }_2,{\sigma }_2)R_1(\tau )\hspace{1em} \tau <t\end{array}\right.$$

(18)

4.4. Establishment of Residual Life Prediction Model

Under the premise that the time t_k is still not invalid, the performance degradation amount at this time is x_k. According to the independent and identically distributed characteristics of the degradation increment of the Wiener process, the remaining life T_l of the product is defined as Formula (9).

$$\begin{array}{l}{T}_l=\inf \left\{t_l:X(t_k+t_l)\ge \left.H\right|X(t_k)\le H\right\}\\ =\inf \left\{t_l:X(t_l)\ge \left.H-x_k\right|X(t_k)\le H\right\}\end{array}$$

(19)

(1)

The failure of the product occurs in the first degradation stage, that is, t_k < τ and t_h < τ. The degradation of the product can be regarded as a single-stage Wiener degradation process, and the probability density function of the residual life can be expressed as

$$f_1(t)={\displaystyle \frac{H-x_{01}}{\sqrt{2\mathsf{\pi }{\sigma }_1^2{t}^3}}}\exp \left[-{\displaystyle \frac{{(H-x_{01}-{\mu }_{1}t)}^2}{2{\sigma }_1^{2}t}}\right] 0\le t\le \tau$$

(20)

(2)

When the product fails in the second degradation stage, that is, t_k > τ and t_h > τ, and the product is reliable in the first degradation stage, the residual life probability density function can be expressed as

$$f_2(t)=R_1(\tau )\cdot {\displaystyle \frac{H-x_1(\tau )}{\sqrt{2\mathsf{\pi }{\sigma }_2^2{(t-\tau )}^3}}}\exp \left[-{\displaystyle \frac{{(H-x_1(\tau )-{\mu }_2(t-\tau ))}^2}{2{\sigma }_2^2(t-\tau )}}\right]\tau <t$$

(21)

So far, the two-stage Wiener process degradation model of SCB has been established.

5. Life Prediction Degradation Model Test and Result Analysis

5.1. Degradation Model Test

The degradation test data were brought into the above degradation model for a case analysis of degradation assessment.

Table 3. The SIC values of each detection interval of sample 1 at 85 °C.

k	SIC(k)	k	SIC(k)	k	SIC(k)
2	34.891	9	55.500	16	99.162
3	40.756	10	73.262	17	24.443
4	37.483	11	62.616	18	81.507
5	39.588	12	55.441	19	77.703
6	39.884	13	94.975	20	78.662
7	35.744	14	91.962	21	84.156
8	68.757	15	79.139	22	96.358

shows the SIC values of each detection interval of sample 1 at 85 °C.

According to Table 3, SIC(17) = 24.443 is the smallest, and SIC(22) > SIC(17). Hence, the sample has a degradation ‘turning point’, and the turning point τ occurs within [t17, t18), that is $\widehat{k}$ = 17, and the degradation data sequence can be divided into two stages. The quantile–quantile plot (QQ) method performs a phased normal distribution test on the degradation increment of degradation characteristics. The test results are shown in Figure 13 and Figure 14.

According to Figure 13 and Figure 14, the QQ chart is a scatter plot of the functional relationship between actual sample data and standard normal distribution data, with the y-axis representing the actual input sample quantile and the x-axis representing the standard normal distribution quantile. Most blue data points are distributed on the red straight line, and only a few points fall outside the straight line. Through probabilistic statistical inference, it is considered that the degradation increment Δx_i of the degradation characteristic quantity in each degradation stage conforms to the normal distribution, and the two-stage Wiener process model can be used for degradation modeling.

Similarly, there are also degradation change points in the degradation process of other samples. The Δxi of each degradation stage obeys the normal distribution, and the estimated values k of the corresponding samples are calculated. By bringing k into Equations (7) and (10), the estimated values of the model distribution parameters and the values of the change points at each temperature stress level can be obtained, as shown in

Table 4. Estimation results of model distribution parameters.

	Parameter	${\widehat{\mathit{\mu }}}_1$	${\widehat{\mathit{\sigma }}}_1^2$	${\widehat{\mathit{\mu }}}_2$	${\widehat{\mathit{\sigma }}}_2^2$	$\widehat{\mathit{\tau }}$/h
Temperature/°C
105		3.38 × 10−3	3.47 × 10−2	1.26 × 10−2	3.59 × 10−2	410.60
95		2.79 × 10−3	2.36 × 10−2	0.99 × 10−2	2.37 × 10−2	601.34
85		1.82 × 10−3	1.76 × 10−2	0.67 × 10−2	1.26 × 10−2	916.51

.

5.2. Analysis of Life Prediction Results

The accelerated degradation stress in this paper is temperature. Without changing the failure mechanism of the product, the higher the temperature, the faster the degradation rate of the electronic product. As early as the 19th century, Arrhenius obtained the relationship model between temperature and the material reaction rate based on experience, that is, the Arrhenius reflection theory model, which was also used as an empirical model of the accelerated life of electronic products. The Arrhenius model is suitable for the scenario where temperature is the dominant failure stress factor, and the results of multi-stress coupling are not considered. The model is defined as follows.

$${\displaystyle \frac{dM}{dt}}=A{e}^{{\displaystyle \frac{-E}{kT}}}$$

(22)

In the formula, M is the amount of product degradation; t is the test time, that is, the product life; A is a constant; E is the activation energy; k is the Boltzmann constant; and T is the absolute temperature (K) of the test stress level. The following formula is obtained based on the integral and logarithmic formula operation (10).

$$\ln t=\ln {\displaystyle \frac{M-M_0}{A}}+{\displaystyle \frac{E}{k}}{\displaystyle \frac{1}{T}}=a+{\displaystyle \frac{b}{T}}$$

(23)

By using this model and combining the accelerated degradation data for extrapolation estimation, the reliability of the statistical inference of electronic products under a normal working environment can be obtained. The reliability function curve and the failure probability density function curve of the SCB in the normal-use environment can be obtained by bringing the experimental data into Formulas (18)–(21), as shown in Figure 15 and Figure 16.

According to Figure 16, in the distribution interval before and after the turning point of the first and second stages of degradation, 155,000 h (17.7 years), the value of the re-liability function will first undergo a gentle transition period. Then, the decline rate will be significantly accelerated, and it will be much larger than the decline rate at the end of the first stage. The failure probability density function curve first shows a slight downward trend, and then, its rise rate is significantly accelerated, and it is much larger than the rise rate at the end of the first stage. The difference between the variation trend of the reliability function curve and the failure probability density function curve in the interval before and after the ‘turning point’ of degradation shows that under a normal working environment, the SCB has high working reliability in the first degradation stage, a slow performance degradation speed, and extremely low failure probability caused by degradation. Then, after years of operation loss, aging, and fatigue accumulation, after the ‘turning point’ of degradation, it enters the second degradation stage. Compared with the first degradation stage, the working reliability of the SCB is significantly reduced, the performance degradation speed is significantly accelerated, and the failure probability caused by degradation is significantly increased. The control circuit of the SCB presents two-stage degradation characteristics on the degradation model, which is consistent with the fitting trend of degradation data and the analysis results of degradation characteristics in the second section.

As shown in

Table 5. Comparison of different degradation models.

Degradation Model	Input Data Volume	The Resources Used for Calculation	Subjective Risk	Mechanism Interpretability
Two-stage Wiener process model	moderate	moderate	small	large
Classical Wiener model [24]	small	small	small	moderate
Bayesian statistics [25]	mid	high	high	dependent on prior distribution selection
Finite element simulation model [26]	high	high	low	high

, the degradation model proposed in this paper is compared with other models. The two-stage Wiener process degradation model proposed in this paper more accurately reflects the two-stage or multi-stage degradation behavior of circuit breakers under real-world operating conditions than the traditional single-stage Wiener model. Although the Bayesian statistical method offers flexibility, its reliance on prior distribution selection can introduce subjective bias into the results. However, the two-stage Wiener model, driven by real-world data, offers greater objectivity. Compared to the time-consuming finite element simulations, it enables faster parameter estimation and demonstrates strong practical applicability.

6. Conclusions

This paper mainly studies performance degradation modeling and evaluation methods for SCB control circuits. The details are as follows:

(1)

The two-stage nonlinear degradation characteristics of the SCB. The distribution parameters and the ‘turning point’ of the degradation model are estimated using maximum likelihood estimation (MLE) and the Schwarz Information Criterion (SIC).

(2)

Reliability estimation and residual life are proposed as key reliability metrics for evaluating the performance decline in SCB. A degradation model, based on a two-stage Wiener process, is developed, and its accuracy is validated through a case analysis.

(3)

By applying the Arrhenius empirical acceleration model, the distribution parameters of the degradation model at room temperature are converted, yielding probability density function curves for reliability and residual life. The prediction results indicate that the average service life of the control circuit at room temperature is 178,100 h (20.3 years), with the degradation ‘turning point’ occurring at 155,000 h (17.7 years). This allows for a reliable evaluation of the decline in SCB performance.

The model and evaluation method proposed in this paper positively impact the safety, efficiency, and sustainability of the modern power infrastructure. They also provide valuable insights for the reliable assessment of performance degradation in SCB control circuits. This paper has two main limitations: first, the accelerated degradation testing scheme only considers constant temperature stress. Second, it focuses on individual components and a single degradation failure mode, without fully addressing the randomness and suddenness of the process. In the future, we will focus on studying the effects of multi-dimensional time-varying stress coupling, such as electrical and mechanical stresses, on the deterioration of SCB functionality. Additionally, we aim to further optimize the degradation testing scheme and investigate the competitive failure model, which includes control circuit degradation, random failure, and sudden failure.