# Physics-informed Markov chains for remaining useful life prediction of wire bonds in power electronic modules

Mehdi Ghrabli, Mounira Bouarroudj, Ludovic Chamoin, Emanuel Aldea

► To cite this version:

Mehdi Ghrabli, Mounira Bouarroudj, Ludovic Chamoin, Emanuel Aldea. Physics-informed Markov chains for remaining useful life prediction of wire bonds in power electronic modules. Microelectronics Reliability, 2025, 167, pp.115644. 10.1016/j.microrel.2025.115644. hal-04992688

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Submitted on 2 Apr 2025

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Research paper

# Physics-informed Markov chains for remaining useful life prediction of wire bonds in power electronic modules☆

M. Ghrabli<sup>a,b,c,*</sup>, M. Bouarroudj<sup>b,d</sup>, L. Chamoin<sup>a,c</sup>, E. Aldea<sup>a,b</sup>

<figure><img src="image_2.png" /></figure>

## ARTICLE INFO

### Keywords:

- Machine learning

- Remaining useful life

- Power electronic modules

- Wire bonds

## ABSTRACT

This paper presents a new approach to estimate the remaining useful life of a power electronic module where failure is caused by degradation in the wire bonds. The novelty of this work is that estimation is given for each loading cycle as opposed to estimating only the number of cycles to failure. A direct consequence is that one can make predictions on variable loading profiles using the proposed method, whereas classical solutions assume periodic loading, which limits their applicability. Experimental data of failure tests are used alongside finite element simulation to mechanically describe the state of the power module at each cycle. Using these mechanical quantities, we iteratively infer how the degradation evolves using Markov chains until failure. A first machine learning algorithm is used to establish a relationship between the degradation and the health indicator, and a second algorithm is used as a surrogate model for finite element simulations to drastically reduce computational time. Results show high extrapolation and interpolation capabilities of the obtained model, meaning that precise predictions can be obtained from experimental data where loading conditions are significantly different from realistic conditions.

## 1. Introduction

The success of power electronic modules can largely be attributed to their electric, thermal, and mechanical properties, which are achieved by assembling various materials at a small scale. This creates a complex structure, where reliability is difficult to assess.

Under working conditions, power modules are exposed to different types of constraints (thermal, mechanical, moisture). Temperature variation is the most severe form, which creates thermo-mechanical stress. As a result, damage is observed mostly at contact interfaces, leading to degradation in solder joints and wire bonds. We focus in this study on failure caused by wire bond degradation in IGBT power modules, as this form of failure is the most predominant, according to the literature [1,2].

This work presents a pipeline to estimate the health state, and thus

the remaining useful life (RUL) of power electronic modules at each instance of their lifetime, for any arbitrary loading profile, based on experimental runs to failure. This pipeline is intended to be more suited for realistic cases compared to classical approaches, thus tackling the problems of interpolation and, more importantly, extrapolation of predictions on unseen loading conditions in the experimental setup. Machine learning is used in multiple steps of this pipeline to circumvent computational issues in finite element simulations and to compensate for lacks and uncertainties in physical knowledge.

The remainder of the paper is organized as follows: Section 2 gives an overview of the existing literature to highlight similarities and differences between our work and existing solutions. Section 3 presents the pipeline of our prediction scheme by thoroughly explaining each of its steps. Results are presented in Section 4 followed by the limitations in Section 5, before concluding the work in Section 6.

**Abbreviations:** MSE, mean squared error; PCT, power cycling tests; RBT, relative bounding time; RMSBT, relative minimal strict bounding time; RUL, remaining useful life.

* This article is part of a special issue entitled: 'ESREF 2024' published in Microelectronics Reliability.

* Corresponding author: Paris Saclay university, 9 Rue Joliot Curie, 91190 Gif-sur-Yvette, France.

**E-mail addresses:** mehdi.ghrabli1@ens-paris-saclay.fr (M. Ghrabli), mounira.bouarroudj@ens-paris-saclay.fr (M. Bouarroudj), ludovic.chamoin@ens-paris-saclay.fr (L. Chamoin), emanuel.aldea@universite-paris-saclay.fr (E. Aldea).

<footnote><sup>a</sup> Paris Saclay University, Gif-sur-Yvette, France</footnote>

<footnote><sup>b</sup> SATIE Laboratory, CNRS, ENS Paris Saclay, Gif-sur-Yvette, France</footnote>

<footnote><sup>c</sup> LMPS Laboratory Université Paris-Saclay, Centrale Supélec, ENS Paris Saclay, CNRS, Gif-sur-Yvette, France</footnote>

<footnote><sup>d</sup> Paris EST Créteil University, Créteil, France</footnote>
## 2. Related work

The problem of power electronic modules' reliability can be tackled using different approaches from different backgrounds. We will briefly present three types of models that are closely related to our work.

### 2.1. Physics-based models

According to our literature research, this type of model uses physical equations and formulae to determine failure. It requires precise knowledge of the degradation process and the physical properties to ensure accurate predictions. The widely used Coffin-Manson law was modified in [3] to obtain the number of cycles to failure given the values of the plastic and elastic strain, whereas a physicochemical approach was taken in [4] to predict the crack path by studying the microstructural behaviour of the top side interconnections. Finally, a cohesive zone model was presented in [5] to mechanically describe the behaviour of the contact in the presence of a crack.

These models require detailed knowledge of the physics applied, and numerical simulations to infer the quantities needed by the model. This usually leads to a divergence in the predictions as the physical behaviour of power modules remains unclear, based on our research.

### 2.2. Data-driven models

Literature shows that data-driven models for RUL estimation require little to no prior knowledge of the underlying physics. They rely on mathematical algorithms to make predictions based on observations, also called training data. Examples of these algorithms include neural networks [6], Restricted Boltzmann Machines [7], and Particle Filters [8].

Data-driven approaches have found major success in various domains thanks to the expressive power of these algorithms. This expressive power comes at the cost of complexity, where large databases are required to achieve precise predictions. The problem of data availability is prominent in power electronic modules' failure analysis, which explains the lack of success that data-driven models have found in this area of application.

### 2.3. Hybrid models

Hybrid models combine techniques from the aforementioned models in order to increase precision and circumvent model-specific issues such as data availability, computational time and interpretability. As an

example, Paris' law has been previously used as the physical law that governs degradation, which was combined with polynomial interpolation in [9] and neural networks in [10]. Another hybrid approach consists of using physics to generate synthetic data to overcome the availability of data issue, which was illustrated in [11].

Hybrid models are more and more used in this RUL estimation problem, as classical approaches do not seem to be efficient; our work is classified under this type of models.

## 3. Methodology

### 3.1. Health indicator

The state of the power module can be monitored using several parameters called health indicators. A straightforward indicator would be to measure the crack length $l_c$ in the wire, illustrated in Fig. 1, over the course of the module's lifetime. However, this indicator cannot be utilized directly, as precise measurements of the crack length would require the destruction of the module. The monitoring of the collector-emitter voltage $V_{ce}$ is a common practice to estimate damage located in the top interconnections [14].

### 3.2. Use case

The generic use case for this work is when observations of a number of the previous states are provided. Assuming that the number of observations is $n$, the user would provide as input $(N_1, ..., N_n)$ which corresponds to the cycle indices where measurements were taken, as well as $(V_{ce}^1, ..., V_{ce}^n)$, which is the health indicator at said cycles. Our goal is then to predict the health indicator values until failure given a loading profile and a failure criterion, which is usually an increase of $V_{ce}$ that exceeds 5 % compared to the initial value. The output of our model is the subsequent values of $V_{ce}$ $(V_{ce}^{n+1}, V_{ce}^{n+2}, ..., V_{ce}^{nf})$, which correspond to cycles $(N_n + 1, ..., nf)$, with $nf$ denoting the cycle where failure occurs.

It is worth mentioning that our method can also be used when no observation is provided, but this requires generating estimations of the unobserved $V_{ce}$ values, whose quality depends on the knowledge of the previous loading as well as the available data. This, however, is beyond the scope of this paper.

### 3.3. Approach

Assuming that the power module is currently at a state $i$ defined by the cycle index $N_i$ and an indicator $V_{ce}^i$, the first task is to estimate the

<figure><img src="image_3.png" /><figcaption>Fig. 1. Architecture of a power module highlighting the bond wire degradation [12,13].</figcaption></figure>
corresponding degradation $l_c$. This is done using a machine learning regression algorithm called Gaussian Process Regression [15] that exploits destructive tests providing values for $l_c$ with respect to $V_{ce}$. Given the current estimation of the damage, a finite element model presenting the estimated crack length has been created, to which we apply the loading conditions of the following cycle, $N_{i+1}$. This would give us a rough mechanical description of the damaged power module during loading. This mechanical description is used to infer the damage evolution by comparing it to previously processed data from runs to failure. This is accomplished using a Markov chain approach to estimate the crack length variation $\delta l_c^i$ and is detailed in Section 3.6. Finally, we can infer the new degradation value $l_c^{i+1} = l_c^i + \delta l_c^i$ for the cycle $N_{i+1}$ as well as the corresponding indicator value $V_{ce}^{i+1} = V_{ce}^i + \delta V_{ce}^i$ using the same Gaussian Process Regression mentioned previously. This process is repeated for cycles $N_{i+2}$ to $nf$, which is the cycle indicating failure. This iterative model is illustrated in Fig. 2.

### 3.4. Experimental data

Experimental data from power cycling tests (PCT) of IGBT modules were used for this work [3].

#### 3.4.1. Runs to failure

The test conditions of the first type of data are: $T_{min} = 55^\circ C$ to $T_{max} = T_{min} + \Delta T$ with a power injection duration of $t_{on} = 3s$, followed by a cooling phase with a temperature swing from $T_{max}$ to $T_{min}$ for a duration $t_{off} = 6s$. Three campaigns were carried out for $\Delta T$ values of $70^\circ C, 90^\circ C$, and $110^\circ C$. Six power modules were tested for each campaign. A total of 572, 182, and 144 observations were made per campaign, respectively, and are represented in Appendix A, where the relative $V_{ce}$ variation was plotted, computed as $\frac{V_{ce,i} - V_{ce,0}}{V_{ce,0}}$ for cycle number $i$.

$V_{ce}$ is measured at different cycles to capture the health state evolution. As measurements are not taken at regular intervals, we first modify the dataset by interpolating values linearly and then resampling values at a frequency equal to the average interval size, that is $p = 5339$ cycles for this data. This processing is used to have more uniformity and consistency in the data to facilitate convergence of machine learning algorithms. Using this processing, we predict $V_{ce}$ values every $p$ cycles instead.

As this data represents the evolution of the health indicator, it will be used to estimate the damage growth in the Markov chain prediction scheme. Despite data resulting from tests where loading profiles are regular and periodic, they are exploitable for variable profiles, as we utilise mechanical descriptors, which are varying, instead of thermal descriptors, which are static.

#### 3.4.2. Destructive tests

Two campaigns of destructive tests were carried out in the same study to give $l_c$ values with respect to $V_{ce}$. Ninety-six wires were tested in each campaign, and a total of 176 data points were retrieved to establish a relationship between the degradation and the health indicator, as shown in Fig. 3.

Establishing the relationship between $l_c$ and $V_{ce}$ can be seen as a regression task, which is a typical task in machine learning. To choose the best algorithm to use, we need to assess the quality of the data.

The small size of the dataset suggests using either parametric algorithms with a small number of parameters, such as linear or polynomial regression, or non-parametric algorithms, such as support vector regression. Since the dataset also presents high variance, we prioritize probabilistic models over deterministic ones. Examining these two criteria, we chose to use Gaussian Process Regression. This method provides the mean and the standard deviation of $l_c$, given the corresponding $V_{ce}$, which allows us to quantify uncertainties in our predictions. We chose to work with relative values for better generalization and to impose normalization on the data.

The implementation of this algorithm was made using Python and its Scikit-learn library, where the Radial Basis Function was taken as the covariance function and the noise standard deviation was set to $\sigma = 0.2$, which results in the regression shown in Fig. 4.

Using these results, we can not only infer damage from the indicator, which is crucial to estimating its evolution, but we can also use this process inversely to predict the indicator value from each damage level

<figure><img src="image_4.png" /><figcaption>Fig. 3. Relative crack length with respect to the relative $V_{ce}$ variation.</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Fig. 2. Graphical representation of the iterative RUL estimation.</figcaption></figure>
<figure><img src="image_6.png" /><figcaption>Fig. 4. Gaussian Process Regression results with a 95 % confidence interval.</figcaption></figure>

represented by the crack length. Finally, we use this algorithm to generate synthetic data of $l_c$, by transforming $V_{ce}$ values of runs to failure as illustrated in Fig. 5.

For further details on the experimental data used, we refer the reader to the work in [3].

### 3.5. Simulated data

We can determine the state of the power module at a cycle $i$ by defining the crack length at that cycle $l_c^i$ as well as the corresponding temperature variation. To provide a more thorough description, we can evaluate mechanical quantities that are likely to explain degradation such as stress or strain. As these quantities are hard to evaluate experimentally, we proceed by estimating them numerically using the ANSYS Workbench platform [16].

#### 3.5.1. Geometric modelling

A geometric model that replicates the modules used in experiments is first needed. From Fig. 6, we can see that this architecture is complex, which makes simulations hard to converge in a timely manner. To circumvent this issue, we simplify drastically the geometry to only consider the contact geometry, which is the area of interest. The simplified geometry is presented in Fig. 7.

Modelling a crack in the top side of the module would require

<figure><img src="image_7.png" /><figcaption>Fig. 5. Synthetic runs to failure obtained using Gaussian Process Regression showing damage evolution with respect to the number of cycles.</figcaption></figure>

changing the initial geometry. We define a contact area between the metallization and the wire, which consists of a truncated cone where the top basis is connected to the wire and the bottom basis is connected to the metallization. Both bases are elliptical with an equal eccentricity to match the initial contact form. The top ellipse remains identical to the initial contact, whereas the bottom ellipse's semi-major axis is reduced by $l_c^i$ to model an even crack from both sides of a total length $l_c$. Fig. 8 highlights the bottom basis, whereas Fig. 9 highlights the top basis.

Material assignments for each component as well as their properties have been chosen to match those of the tested power modules and are identical to the work in [3]. More information is available in Appendix B.

A fixed support boundary condition was applied to the bottom area of the lower copper layer, alongside thermal conditions consisting of an ambient temperature value of $T_{min} = 55$ °C, and an identical value was used for cooling in the bottom area of the lower copper layer, with a film coefficient of $h = 4.10^{-3} \text{ W} \cdot \text{mm}^{-2} \cdot \text{C}^{-1}$. Lateral areas of each component are perfectly insulated, and a 0 Volt boundary condition is applied to the sectioned area of the wire due to our simplified model.

The loading is applied by injecting a current $I_{load}$ to the bottom area of the upper copper layer, calibrated to induce a temperature swing equal to $\Delta T$, which is the temperature variation of the loading profile at the cycle of interest. Degradation is due to thermal stresses, which induce physical variations in the power module. We opt to use the average stress, strain, and displacement in the contact area to best characterize the state of the module subject to a loading cycle.

#### 3.5.2. Accelerated simulations using machine learning

Despite the simplifications made, numerical simulations remain slow (~14 min per simulation), which limits their use. To tackle this issue, we only simulate a number of scenarios of degradations $l_c$ subject to loading $\Delta T$ prior to launching the iterative pipeline. We opt to use this approach instead of running the simulations inside the prediction loop to drastically reduce computational time while keeping accuracy losses limited. 365 different values of $l_c$ were chosen, ranging from 0.01 mm to 0.664 mm, alongside 6 different values of $\Delta T$ varying from 115 °C to 165 °C. We note that the original contact length is equal to 0.723 mm to give an order of magnitude to the reader. A total of 1062 combinations were obtained and ran through the finite element simulations. A database is thus created where the input is the couple $(l_c, \Delta T)$ and the output is the triplet consisting of the average contact stress, strain, and displacement $(\sigma, \epsilon, u)$. This is a classical regression task for machine learning, and we opted to train different algorithms and compare their scores. Results are presented in Table 1, where the best-performing metrics are highlighted in bold. The first metric used is the mean squared error (MSE), which calculates the distance between the predictions and the true values, and the second metric is the R-squared $R^2$, which calculates the correlation between the predictions and the true values. Ideally, the chosen algorithm has a high $R^2$ (close to 1) and a low MSE (close to 0).

Results show that support vector regression is the best performing algorithm for this task, with other algorithms having comparable scores. To visualize its performance, we present a plot showing the predicted average stress in the contact compared to the true simulated values in Fig. 10, alongside the residual error plot in Fig. 11 which is useful to spot biases or outliers in the prediction.

### 3.6. Markov chain predictions

As mentioned in Section 3.3, we would like to estimate the damage variation $\delta l_c^i$ at a cycle $i$, given the previous states of the power module. We first need to determine the length of the historic observations $h_0$ (length of the previous states that we want to work with) and the length of predictions $f$. Historic observations consist of previous crack lengths from cycle $i-p.h_0$ to cycle $i$ (we subtract $p.h_0$ instead of $h_0$ because we make predictions over a window of cycles, as mentioned in Section
<figure><img src="image_8.png" /><figcaption>Fig. 6. Geometric model of a power module.</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>Fig. 7. Simplified geometry of a power module.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Fig. 8. Dimensions of the bottom basis of the truncated conic contact.</figcaption></figure>

3.4.1), as well as their corresponding mechanical descriptions due to loading. These historic observations are obtained by transforming the health indicator values available using Gaussian Process Regression and we infer the mechanical state using support vector regression, where the loading is extracted from the loading profile, as mentioned in Section 3.5.2.

We are then interested in finding the set of possible damage variations for the next f.p cycles: $\delta l_c^{i,f.p}$, $\delta l_c^{i,f.p}$, $\delta l_c^{i+2p}$, ..., $\delta l_c^{i+f.p}$, given the set of observations $(l_c^i, \sigma^i, \epsilon^i, u^i)$, $(l_c^{i+P}, \sigma^{i+P}, \epsilon^{i+P}, u^{i+P})$, ..., $(l_c^{i.p.ho}, \sigma^{i.p.ho}, \epsilon^{i.p.ho}, u^{i.p.ho})$. This translates mathematically to determining $(\delta l_c^{i+f.p}, ..., \delta l_c^i|O^i...O^{i-p.ho})$, where $O^i = (l_c^i, \sigma^i, \epsilon^i, u^i)$.

<figure><img src="image_11.png" /><figcaption>Fig. 9. Dimensions of the top basis of the truncated conic contact.</figcaption></figure>

<table><caption>Table 1
Machine learning algorithms' scores for regression on numerical simulation results.</caption><thead><tr><td>Algorithm</td><td>MSE</td><td>R<sup>2</sup></td></tr></thead><tbody><tr><td>Bayesian neural networks</td><td>0.261</td><td>0.733</td></tr><tr><td>Linear regression</td><td>0.274</td><td>0.719</td></tr><tr><td>Support vector regression</td><td><b>0.247</b></td><td><b>0.746</b></td></tr><tr><td>Order 4 polynomial regression</td><td>0.257</td><td>0.736</td></tr></tbody></table>

The problem boils down to finding the probability distribution

$$P(\delta l_c^{i+f.p}, \dots, \delta l_c^i|O^i \dots O^{i-p.ho})$$

For this purpose, we use Kernel Density Estimation [17], which is a non-parametric method to estimate a probability distribution from observations. To best visualize this effect, we will consider the case where $h_o = f = 1$ and $O^i = \sigma^i$.

Kernel density estimation consists of scattering the observation in an n-dimensional space, where n is the dimension of the observations, equal to 2 in the aforementioned illustration case. The scatter plot is shown in Fig. 12.

We then apply a kernel to each scattered observation with a kernel being a window function whose purpose is to add weight around the area of the observation. By applying a kernel, we increase the
<figure><img src="image_12.png" /><figcaption>Fig. 10. Predicted average contact stress using support vector regression compared to numerical simulation results.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. Residual errors plot for average contact stress predictions using support vector regression.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Fig. 12. Scatter plot of the normalized average stress w.r.t. the normalized crack length variation.</figcaption></figure>

probability in order to have a higher density where observations are concentrated, and we make the distribution continuous. A Gaussian kernel was applied with a bandwidth parameter $σ_b$, resulting in the density shown in Figs. 13 and 14, alongside a colormap indicating the different density values. From this density, we can sample the candidate $\delta l_c$ value to complete the loop of the pipeline. Monte-Carlo sampling was used, from which we constructed our confidence intervals, highlighting the variance in the predictions.

One may notice that we are using numerical simulations that can present biases and inaccuracies due to simplifications and approximations. However, these quantities were used as descriptors of the runs to failure that result from real-life experiments for a kernel-based method, where we use the distance between the descriptors rather than their values so as to eliminate any unwanted bias in the numerical simulations.

Using mechanical quantities to describe these runs to failure can be seen as a form of feature extraction, which is a technique used in machine learning to obtain relevant descriptors that help algorithms converge and improve their performance.

## 4. Results

The proposed pipeline was tested to measure its capability of standard prediction, interpolation, and extrapolation. We use the same metrics that were defined in [9] that describe the quality of the confidence interval. The first metric is the Relative Bounding Time (RBT), which determines the quality of the interval across the entirety of the evolution, whereas the Relative Minimal Strict Bounding Time (RMSBT) evaluates the time it takes, in terms of loading cycles, for our model to accurately predict the health indicator's evolution. Both of these metrics vary from 0 to 1, and a higher value would indicate better predictions. These metrics are linked, and RBT is guaranteed to be greater or equal to RMSBT.

The history size for all prediction tasks was set to $h_o = 2$, and the prediction size is $f = 1$.

### 4.1. Standard prediction

We call “standard predictions” the task of predicting the outcome of a test that belongs to the same distribution as the training data. For this application, it is predicting a run-to-failure that shares the same loading profile as a run-to-failure that was used to create our pipeline.

<figure><img src="image_15.png" /><figcaption>Fig. 13. Probability map resulting from applying a Gaussian Kernel to numerical observations.</figcaption></figure>
<figure><img src="image_16.png" /><figcaption>Fig. 14. Probability distribution giving the probability of occurrence of every possible value of ($l_o, \sigma$).</figcaption></figure>

In Section 3.4.1, we mention that 18 runs to failure were provided, which can be divided into 3 categories according to the temperature swing value $\Delta T$. For the standard prediction task, 17 runs were used for training, and the remaining run was used for testing. We varied the test run each time to cover all 3 cases of $\Delta T$, and we used 100 Monte-Carlo iterations with a bandwidth parameter $\sigma_b$ equal to 0.01 for the Kernel Density Estimation.

### 4.1.1. Case 1: $\Delta T = 70 \,^{\circ}C$

The training set for this case consisted of 5 runs to failure where the loading cycle presents a temperature variation of $70 \,^{\circ}C$ and 6 runs to failure each where the temperature variation is equal to $90 \,^{\circ}C$ and $110 \,^{\circ}C$. Results are shown in Fig. 15 and metrics are given in Table 2.

### 4.1.2. Case 2: $\Delta T = 90 \,^{\circ}C$

The training set for this case consisted of 5 runs to failure where the loading cycle presents a temperature variation of $90 \,^{\circ}C$ and 6 runs to failure each where the temperature variation is equal to $70 \,^{\circ}C$ and $110 \,^{\circ}C$. Results are shown in Fig. 16 and metrics are given in Table 3.

<figure><img src="image_17.png" /><figcaption>Fig. 15. In-distribution predictions with a 90 % confidence interval on a test to failure run with a $70 \,^{\circ}C$ loading temperature variation.</figcaption></figure>

<caption>Table 2</caption>

Metrics for in-distribution predictions on a run-to-failure with a $70 \,^{\circ}C$ loading temperature variation.

<table><thead><tr><td>ΔT</td><td>RBT</td><td>RMSBT</td></tr></thead><tbody><tr><td>70 °C</td><td>0.702</td><td>0.637</td></tr></tbody></table>

<figure><img src="image_18.png" /><figcaption>Fig. 16. In-distribution predictions with a 90 % confidence interval on a test to failure run with a $90 \,^{\circ}C$ loading temperature variation.</figcaption></figure>

<caption>Table 3</caption>

Metrics for in-distribution predictions on a run-to-failure with a $90 \,^{\circ}C$ loading temperature variation.

<table><thead><tr><td>ΔT</td><td>RBT</td><td>RMSBT</td></tr></thead><tbody><tr><td>90 °C</td><td>0.802</td><td>0.802</td></tr></tbody></table>

### 4.1.3. Case 3: $\Delta T = 110 \,^{\circ}C$

The training set for this case consisted of 5 runs to failure where the loading cycle presents a temperature variation of $110 \,^{\circ}C$ and 6 runs to failure each where the temperature variation is equal to $70 \,^{\circ}C$ and $90 \,^{\circ}C$. Results are shown in Fig. 17 and metrics are given in Table 4.

<figure><img src="image_19.png" /><figcaption>Fig. 17. In-distribution predictions with a 90 % confidence interval on a test to failure run with a $110 \,^{\circ}C$ loading temperature variation.</figcaption></figure>
<table><caption>**Table 4**

Metrics for in-distribution predictions on a run-to-failure with a 110 °C loading temperature variation.</caption><thead><tr><td>ΔT</td><td>SBT</td><td>RMSBT</td></tr></thead><tbody><tr><td>110 °C</td><td>0.509</td><td>0.509</td></tr></tbody></table>

## 4.2. Interpolation

The interpolation task for this pipeline is to predict the outcome of a run-to-failure that does not belong to the same distribution as the training data but lies between two distributions seen in the training. A concrete case is to estimate a run-to-failure where ΔT is equal to 90 °C, having only observed runs to failure where the temperature swing is 70 °C and runs to failure where the temperature swing is 110 °C.

For this experiment, we used all 12 runs to failure having ΔT = 70 °C or ΔT = 110 °C as training data, and the testing data consisted of one run-to-failure having ΔT = 90 °C. We used a bandwidth parameter σb equal to 0.00775, which results in the predictions shown in Fig. 18 and metrics in Table 5.

## 4.3. Extrapolation

Extrapolation consists of predicting the outcome of an observation that lies outside of the previous observation range. In our case, it is predicting a test run having a loading temperature variation of ΔT = 70 °C where the training data consists of runs where the loading temperature variation is ΔT = 90 °C and ΔT = 110 °C, or predicting a test run having ΔT = 110 °C where the training data consists of runs where ΔT = 70 °C and ΔT = 90 °C. The first case corresponds to extrapolating on lower test conditions, which is the more relevant case, as experimental loads are usually amplified compared to real life values.

We used all 12 runs to failure having ΔT = 90 °C or ΔT = 110 °C as training data, and the testing data consists of one run-to-failure having ΔT = 70 °C. We used a bandwidth parameter σb equal to 0.1974, which results in the predictions shown in Fig. 19 and metrics in Table 6.

## 4.4. Interpretation

Prediction results show the capability of our model for predicting different scenarios with varying degrees of precision. The task of standard predictions shows that the model can predict outcomes of runs to

<figure><img src="image_20.png" /><figcaption>**Fig. 18.** Interpolated predictions with a 90 % confidence interval on a test to failure run with a 90 °C loading temperature variation.</figcaption></figure>

<table><caption>**Table 5**

Metrics for interpolated predictions on a run-to-failure with a 90 °C loading temperature variation.</caption><thead><tr><td>ΔT</td><td>SBT</td><td>RMSBT</td></tr></thead><tbody><tr><td>90 °C</td><td>0.574</td><td>0.574</td></tr></tbody></table>

<figure><img src="image_21.png" /><figcaption>**Fig. 19.** Out-of-distribution predictions with a 90 % confidence interval on a test to failure run with a 70 °C loading temperature variation.</figcaption></figure>

<table><caption>**Table 6**

Metrics for out-of-distribution predictions on a run-to-failure with a 70 °C loading temperature variation.</caption><thead><tr><td>ΔT</td><td>SBT</td><td>RMSBT</td></tr></thead><tbody><tr><td>70 °C</td><td>0.615</td><td>0.563</td></tr></tbody></table>

failure where a test subject to the same loading has been observed beforehand. This is illustrated by the metrics as well as the narrow confidence interval, widening only when the failure is near to occur.

As for the more difficult tasks of interpolation and extrapolation, the model sees a loss in accuracy, presenting a small deviation from the target and having a larger confidence interval. Nevertheless, the predictions remain close, and the increased uncertainty is intuitive, as in-distribution training data would improve the model's performance, which is the case for the standard predictions.

## 5. Limitations

Despite this pipeline's performance, it still presents some limitations due to the complex nature of the problem. We will cite below the most relevant limitations that we could identify, along with some potential solutions.

The pipeline does not take into consideration the effect of t<sub>on</sub>. This is mainly due to the lack of diversity in the training data, alongside the ambiguity in the physical origin of the loading time dependency. This can be tackled by conducting tests to failure with variable loading durations, as well as identifying physical properties susceptible to explain the effect of loading duration, such as viscoplasticity.

This model is yet to be tested on data with a variable loading profile, due to the lack of experiments with this setup. Synthetic data can be created by combining health indicator evolutions from different runs to failure with a constant temperature variation.

This pipeline does not present the physical explanation of the
phenomena at hand. However, statistical methods exist to infer re-
relationships between input and output variables, such as variable
importance analysis.

6. Conclusion

This paper presented a RUL estimation pipeline for power electronic
modules, a complex task due to lacks of physical knowledge that was
tackled by incorporating data-driven models. Different algorithms were
used depending on the task in the pipeline, with probabilistic models
being preferred for tasks where data is scarce and spread out, which is
typical in this setup.

The pipeline was conceived to provide cycle-by-cycle predictions,
making it robust to loading profile variations. We tested the approach on
various tasks to highlight its applicability in real-life scenarios. Further
improvements can be made by tackling some of the limitations identified
in this work; they will be the topic of future research.

The code used to run this pipeline is available at https://github.

com/MehdiGhrabli/PIMC4RUL

CRediT authorship contribution statement

**Mehdi Ghrabli:** Writing, Methodology, Software, Editing. **Mounira Bouarroudj:** Data acquisition, Supervision, Reviewing. **Ludovic Chaimoin:** Supervision, Reviewing. **Emanuel Aldea:** Supervision, Reviewing.

Grant information

CNRS (PhD funding).

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

**Appendix A. Graphs of runs to failure**

<figure><img src="image_22.png" /><figcaption>Fig. 20. Relative V<sub>ce</sub> variation curves for 6 modules with temperature variation ΔT = 70 °C.</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>Fig. 21. Relative V<sub>ce</sub> variation curves for 6 modules with temperature variation ΔT = 90 °C.</figcaption></figure>
<figure><img src="image_24.png" /><figcaption>Fig. 22. Relative $V_{ce}$ variation curves for 6 modules with temperature variation $\Delta T = 110^{\circ}C$.</figcaption></figure>

## Appendix B. Material assignments, properties, and geometry

<caption>**Table 7**</caption>

Material assignments for the power module's finite elements model.

<table><caption>**Table 8**</caption><thead><tr><td>Component</td><td>Material</td></tr></thead><tbody><tr><td>Wire</td><td>Aluminum (Al)</td></tr><tr><td>Metallization</td><td>Aluminum (Al)</td></tr><tr><td>Chip</td><td>Silicon (Si)</td></tr><tr><td>Solder</td><td>Silver (Ag)</td></tr><tr><td>Ceramic layer</td><td>Ceramic (Al<sub>2</sub>O<sub>3</sub>)</td></tr><tr><td>Upper/Lower copper layer</td><td>Copper (Cu)</td></tr></tbody></table>

Material properties of aluminum<sup>a</sup>.

<table><thead><tr><td>Property</td><td>Value</td></tr></thead><tbody><tr><td>Density (Kg·m<sup>-3</sup>)</td><td>2700</td></tr><tr><td>Coefficient of thermal expansion (ppm·K<sup>-1</sup>)</td><td>23.5</td></tr><tr><td>Young's modulus (GPa)</td><td>76</td></tr><tr><td>Poisson's ratio</td><td>0.33</td></tr><tr><td>Thermal conductivity (W·m<sup>-1</sup>·K<sup>-1</sup>)</td><td>238</td></tr><tr><td>Heat capacity (J·Kg<sup>-1</sup>·K<sup>-1</sup>)</td><td>897</td></tr><tr><td>Electrical resistivity<sup>b</sup> (Ω·m)</td><td>27.10<sup>-9</sup></td></tr></tbody></table>

<caption>**Table 9**</caption>

Material properties of silicon.

<table><thead><tr><td>Property</td><td>Value</td></tr></thead><tbody><tr><td>Density (Kg·m<sup>-3</sup>)</td><td>2330</td></tr><tr><td>Coefficient of thermal expansion (ppm·K<sup>-1</sup>)</td><td>4.1</td></tr><tr><td>Young's modulus (GPa)</td><td>131</td></tr><tr><td>Poisson's ratio</td><td>0.3</td></tr><tr><td>Thermal conductivity (W·m<sup>-1</sup>·K<sup>-1</sup>)</td><td>124</td></tr><tr><td>Heat capacity (J·Kg<sup>-1</sup>·K<sup>-1</sup>)</td><td>750</td></tr><tr><td>Electrical resistivity<sup>a</sup> (Ω·m)</td><td>10<sup>6</sup></td></tr></tbody></table>

<footnote><sup>a</sup> Non-linear behaviour was modelled on ANSYS workbench using data from [18].</footnote>

<footnote><sup>b</sup> Electrical resistivity is doubled in the metallization to simulate the effect of the reconstruction, following [3].</footnote>
a Anisotropic electrical resistivity was used to impose a vertical flux of the current. Resistivity in the vertical direction was modelled identically to [3].

<caption>**Table 10**</caption>

Material properties of silver.

<table><caption>**Table 11**</caption><thead><tr><td>Property</td><td>Value</td></tr></thead><tbody><tr><td>Density (Kg·m<sup>-3</sup>)</td><td>7350</td></tr><tr><td>Coefficient of thermal expansion (ppm·K<sup>-1</sup>)</td><td>19.6</td></tr><tr><td>Young's modulus (GPa)</td><td>6.28</td></tr><tr><td>Poisson's ratio</td><td>0.37</td></tr><tr><td>Thermal conductivity (W·m<sup>-1</sup>·K<sup>-1</sup>)</td><td>250</td></tr><tr><td>Heat capacity (J·Kg<sup>-1</sup>·K<sup>-1</sup>)</td><td>230</td></tr><tr><td>Electrical resistivity (Ω·m)</td><td>4.10<sup>-8</sup></td></tr></tbody></table>

Material properties of ceramic.

<table><caption>**Table 12**</caption><thead><tr><td>Property</td><td>Value</td></tr></thead><tbody><tr><td>Density (Kg·m<sup>-3</sup>)</td><td>3780</td></tr><tr><td>Coefficient of thermal expansion (ppm·K<sup>-1</sup>)</td><td>8</td></tr><tr><td>Young's modulus (GPa)</td><td>370</td></tr><tr><td>Poisson's ratio</td><td>0.2</td></tr><tr><td>Thermal conductivity (W·m<sup>-1</sup>·K<sup>-1</sup>)</td><td>24</td></tr><tr><td>Heat capacity (J·Kg<sup>-1</sup>·K<sup>-1</sup>)</td><td>830</td></tr><tr><td>Electrical resistivity (Ω·m)</td><td>1.10<sup>12</sup></td></tr></tbody></table>

Material properties of copper<sup>a</sup>.

<table><thead><tr><td>Property</td><td>Value</td></tr></thead><tbody><tr><td>Density (Kg·m<sup>-3</sup>)</td><td>8690</td></tr><tr><td>Coefficient of thermal expansion (ppm·K<sup>-1</sup>)</td><td>17</td></tr><tr><td>Young's modulus (GPa)</td><td>97</td></tr><tr><td>Poisson's ratio</td><td>0.34</td></tr><tr><td>Thermal conductivity (W·m<sup>-1</sup>·K<sup>-1</sup>)</td><td>390</td></tr><tr><td>Heat capacity (J·Kg<sup>-1</sup>·K<sup>-1</sup>)</td><td>390</td></tr><tr><td>Electrical resistivity (Ω·m)</td><td>17.10<sup>-9</sup></td></tr></tbody></table>

a Non-linear behaviour of copper was modelled using the bilinear isotropic hardening of ANSYS.

The geometric model's components are cuboids with a length dimension of 1.5 mm and a width of 1 mm, aside from the wire, which is a circle of diameter 0.38 mm swept along a curved line that mimics the wire's path. The wire's contact area with the metallization is an ellipse of semi-major axis 0.3615 mm and a semi-minor axis of 0.146 mm. The center of the contact area coincides with the center of the topside area of the metallization. Components' depths are given in Table 13.

<caption>**Table 13**</caption>

Material assignments for the power module's finite elements model.

<table><thead><tr><td>Component</td><td>Depth (μm)</td></tr></thead><tbody><tr><td>Metallization</td><td>10</td></tr><tr><td>Chip</td><td>140</td></tr><tr><td>Solder</td><td>20</td></tr><tr><td>Upper copper layer</td><td>300</td></tr><tr><td>Ceramic layer</td><td>380</td></tr><tr><td>Lower copper layer</td><td>300</td></tr></tbody></table>

**Data availability**

The code and data used are available on GitHub. The link for the GitHub repository is available in the manuscript (Section 6).

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