Bayesian Remaining Useful Lifetime Prediction of Thermally Aged Power MOSFETs

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1 Bayesian Remaining Useful Lifetime Prediction of Thermally Aged Power MOSFETs Mehrdad Heydarzadeh, Serkan Dusmez, Mehrdad Nourani, Bilal Akin 1 Department of Electrical Engineering, The University of Texas at Dallas, {mehhey, serkan.dusmez, nourani, bilal.akin}@utdallas.edu Abstract The demand for more reliable power conversion is ever increasing. This necessitates smart gate drivers or smart system controllers that monitor the components that are susceptible to failure. Together with the electrolytic capacitors, power semiconductor devices are among the weakest components in a power converter. In an effort to predict power MOSFET aging, this paper proposes an remaining useful lifetime (RUL estimation algorithm for degraded power MOSFETs, which are exposed to high amplitude thermal cycles. The relative change in on-state resistance is identified as the fault signature. A data-driven RUL estimation algorithm based on a linear model approximation is proposed. The outliers present particularly in the beginning part of the data decrease the accuracy of the estimation with classical least-squares method. In this paper, a Bayesian Interference estimator is proposed to improve the accuracy through incorporating prior knowledge to estimation. The accuracy of the proposed RUL estimation tool is verified on the collected experimental data of thermally aged discrete power MOSFETs. Keywords Bayesian inference, Fault diagnosis, Failure prognostics, Power MOSFET, On-state resistance, Gate threshold voltage, Health monitoring, Remaining useful lifetime, I. INTRODUCTION Power electronic converters play an important role in determining the reliability of a system. The field experience and data suggest that failure rate of Si based power semiconductors namely, insulated gate bipolar transistors (IGBTs and metaloxide field-effect transistors (MOSFETs, and electrolytic capacitors is much higher than the remaining components of a power converter [1], [2]. Many research efforts have been devoted to open-circuit and short-circuit fault detection in a system, but only a few has attempted to detect the non-invasive incipient failures [3 9]. Different than detecting instantaneous failures where the current/voltage sharply rises and drops, incipient fault diagnosis requires tracking the changes in aging-variant I- V characteristics so called failure precursors. This feature is desirable and essential to avoid strenuous periodic check-ups and costly interruptions, and can be integrated into the main controller or gate driver of a power converter. Some of the studies have investigated the potential failure precursors. In [3], collector-emitter voltage was monitored and shown as a degradation indicator for IGBT. Another study focused on the maximum peak voltage of the collector-emitter ringing at turn off [4]. In [5], the turn off time of the switch was studied and recognized as failure precursor. On the other hand, only a few has focused on the failure precursors for power MOSFETs. The failure related to die solder attachments of discrete power MOSFETs was related with increased on resistance, and gate threshold voltage variation is recognized as failure indicator for gate oxide degradation [6 8]. Although the precursors have been well identified, there are a few studies that put effort on estimating the remaining useful lifetime (RUL of power switches. In model based approaches, typically junction temperature information is obtained from the power losses and thermal impedance model of the switch while temperature cycles are counted using algorithms such as rainflow counting [1]. In the data-driven methods, experimental data is processed to derive an empirical degradation model. In [11], relevance vector machine is used to train the degradation data of power MOSFETs to obtain representative vectors, which are fitted by a degradation model. A threshold value is defined and remaining useful lifetime is estimated using the proposed degradation model. In [9], a linear model is used for prognostics of power MOSFETs. The outliers to this linear model are automatically detected and removed by RANSAC algorithm. Even though RANSAC improves the overall estimation accuracy, the non-linearity in the initial part of the data cannot be effectively reduced without incorporating prior knowledge. In this study, a Bayesian inference estimator, which introduces prior probability to limit the choices of estimators to the region of parameter space, is proposed to improve the estimation accuracy particularly at the early stage of aging. II. EXPERIMENTAL RESULTS OF ON-STATE RESISTANCE 4V/11A power MOSFETs are thermally aged on a custom built test-bed, shown in Fig. 1. The switches are degraded under three different thermal conditions, which are a = 16 C T j,max = 24 C, b = 14 C T j,max = 22 C, c = 13 C T j,max = 21 C, at drain current of 5.2 A. The setup details and analysis of the failure mechanisms can be found in [12, 13]. The results of the online R ds,on measurements are given in Fig. 2(a. It is clearly seen that even though the same thermal swing is applied to the same type of switch, the degradation curves are slightly different. The zoom-in profiles of the results plotted on a logarithmic x-axis, showing the variation between 7mΩ, are given in Fig. 2(b. From this figure, it is clear that all of the switches degrade exponentially up to a certain value. After this value, at least one or more switches exhibit very high and/or unstable on-state resistances till they completely fail. The devices aged to this level can be classified as faulty, as the characteristics deviate from normal operating conditions, such as dropped breakdown voltage. From the results given in /17/$ IEEE 2718

Fig. 1. R DS_ON (Ω R DS_ON (Ω 1..9.8.7.6.5.4.3 Custom built accelerated aging setup. R4a-R9a =16 C =24 C =14 C =22 C R1a R2a R4a R5a =13 C =21 C R6a R7a R8a R9a.2.1 5 1 15 2 25 3 35 Cycles.7.6.5.4.3.2.1 End of Life (a 1 1 Cycles (b Fig.

2 Fig. 1. R DS_ON (Ω R DS_ON (Ω Custom built accelerated aging setup. R4a-R9a =16 C =24 C =14 C =22 C R1a R2a R4a R5a =13 C =21 C R6a R7a R8a R9a Cycles End of Life (a 1 1 Cycles (b Fig. 2. Online R ds,on monitoring during thermal cycling under three different conditions; (a full scale, (b zoomed into 7 mω region. Fig. 2, it is clear that a safe threshold value can be determined after exhaustive experiments. The results also suggest that an exponential empirical model can be established representing the data points till this chosen threshold value. Considering the results of the other samples, the safe threshold R ds,on value is determined as 5 mω for the given switch type. III. RUL ESTIMATION FOR DEGRADED POWER MOSFETS The experimental results show a an exponential trend in R ds,on which can be used for RUL estimation. The measured R ds,on can be expressed in linear form by taking the natural log ( R ds,on R4a R5a R6a Number of Cycles R7a R8a R9a Fig. 3. R ds,on of experiments in log scale. A linear model can be fitted to data in log scale and used for RUL estimation. logarithm as y = ln(r ds,on = β 1 t + β + ɛ (1 where, β 1 and β are the model parameters and ɛ is the modeling error, which is considered as a zero mean Gaussian noise with σɛ 2. Fig. 3 shows the results in log scale. This model can be used for estimating RUL by defining a failure threshold for the resistance, i.e. R τ. It is possible to predict the failure and estimate RUL as T F = However, in order to use this model for prediction, the vector of parameters β = [β, β 1 ] t, needs to be estimated using past samples of the system. Suppose y = [y 1, y 2,, y N ] is the vector of observed values of the resistance up to current time (t N. Then, β can be estimated using the ordinary least square (OLS : β OLS = (X t X 1 X t y (2 where X = Rτ β β 1. [ 1 1 t 1 t N ] t (3 Although, the OLS approximation is an unbiased estimator, it is not robust to outliers, which are defined as extreme cases of noise. Since OLS is an averaging technique, outliers can change the direction of fitting line. In addition, when the assumption of linearity is not valid the OLS causes a big error. This becomes more critical, when the number of samples in the estimation is low. Fig. 4 shows the estimated line for one of the experiments at various stages of estimation. As seen in Fig. 4, although the existing trend in the data is almost linear, the initial points do not conform this trend. These points are considered as outliers and affect the accuracy of the estimated line, and hence the RUL estimation. This issue becomes more serious in the beginning of the part of the data, since the number of data points is limited (Fig. 4(a. As the more data is processed, the error becomes smaller (Fig. 4(b. If all data points are available, the estimated line has an acceptable accuracy as seen from Fig. 4(c. However, in an online estimation tool, only data up to current time is available. 2719

3 (a (b (c Fig. 4. Fitted line using OLS with different number of observed data points; (a 3 (15 th cycle, (b 6 (3 th cycle, (c 9 (45 th cycles. The accuracy of the fitted line is sensitive to number of observed points. The associated prediction error for a low number of points is large. Polynomial (2 Polynomial (3 Polynomial (2 Polynomial (3-1 Polynomial (2 Polynomial ( (a (b (c Fig. 5. Fitted second and third order polynomial using OLS with different number of observed data points; (a 3 (15 th cycle, (b 6 (3 th cycle, (c 9 (45 th cycles. Using a higher order model (polynomial is not helpful for reducing the prediction error even it is associated with a larger errror. Since the data points show non-linear variation, one might think that choosing a nonlinear model can help in improving accuracy. If a second or third order polynomial function is fitted using all data points up to time, it provides quite satisfactory results, as shown in Fig. 5(a. But, as a matter of fact, it makes the RUL estimation worse in the initial part as seen from Fig. 5(b-5(c. This is because they are associated with higher extrapolation error; OLS is a maximum likelihood ( estimation with Gaussian assumption [14]. In framework, the parameter is determined in the way the likelihood of data is maximized: β = arg max β p(y β (4 For this problem, this likelihood can be written as p(y β = N i=1 p(y i β, where p(y i β is Gaussian since ɛ is Gaussian. This type of estimator just evaluates the available data (y and chooses β in order to maximize its likelihood. So, the extrapolation error becomes large since there is no data available for future observations. But, based on the prior knowledge, we know that a linear trend exists in the data and we have an initial guess for the parameters of the line by evaluating a number of test data. Thus, the estimator can be modified in such a way that it assigns a high cost for choosing β values, which are not close to the defined prior knowledge. In Bayesian Inference (BI, it is assumed that parameter β is a random unknown. Therefore, instead of finding a specific value for β, it looks for a conditional probability distribution function (pdf for β given the observed data, i.e. p(β y which is called the posterior pdf and related to likelihood by the Bayes rule: p(β y = p(y βp(β (5 p(y where p(β is called prior probability and models our prior knowledge of β. One way of inferring a specific value for β is the maximum a posteriori method (, which chooses β to maximize p(β y: β = arg max β p(y βp(β (6 where p(y is omitted from denominator since it is not a function of β.the difference of β and β is in the prior probability function (compare Eqn. 4 and Eqn. 6. The likelihood pdf models our knowledge obtained by samples 272

4 (a (b (c Fig. 6. Fitted line using the proposed Bayesian Inference estimator with different number of observed data points; (a 3 (15 th cycle, (b 6 (3 th cycle, (c 9 (45 th cycles. Using a biased estimator increase the accuracy of estimation when the number of samples is low RUL Trajectory for MOSFETA RUL Trajectory for MOSFETA RUL Trajectory for MOSFET-11A RUL Trajectory for MOSFET-12A Fig. 7. The comparison of the estimated RULs for four different devices using and. from the system. The prior pdf is another source of information which is obtained by doing experiment and it biases the estimator. Intuitively, this bias forces the estimator to choose a value for β close to values observed in prior experiments. To derive the estimator, we assume that the prior probability, p(β, is also Gaussian with mean µ β of and covariance matrix of C β. In such constraints, it can be shown that the posterior pdf is also a Gaussian [15]. Also, a Gaussian curve is maximized at its mean. So, the mean of the posterior pdf needs to be calculated. It can be shown that the mean of posterior [14] is equal to β = µ β + C β X t (XC β X t + σ 2 ɛ I 1 (y Xµ β (7 This estimator simplifies to OLS if prior knowledge is not considered, i.e. µ β =, C β = I and σɛ 2 =. In order to use this estimator, µ β, C β and σɛ 2 have to be determined. These parameters are estimated by method for each experiment in the training set using all points in dataset. Next, the estimated values can be used in estimator in Eq. 7 for a device under test. Fig. 6 shows the result of estimation for R5a data. As seen, the overall accuracy of the estimation is improved. This is because the prior probability acts a cost function, and limits the choices of model parameters to the region of defined parameter space. Fig. 7 shows the estimation of RUL for four different experiments in cycle units. The black lines in the middle show the actual RUL, which decrease linearly in time. The red curves show the estimated RUL using the proposed Bayesian estimator. As it can be seen, the estimation is much more accurate in comparison to that obtained with for all samples. IV. CONCLUSION Incipient fault prognosis for power semiconductors is very critical to avoid periodic system check-ups or complete system failures. Accelerated aging tests help to identify the fault precursors in shorter time. In this study, power MOSFETs are exposed to thermal cycling and field data is approximated by a linear function, even though there is non-linearity particularly in the initial part of the data. When model parameters are found by ordinary least squares method or approximated with high order non-linear functions, the estimation error becomes significantly high. To deal with the outlier effect and improve the estimation accuracy, Bayesian Inference estimator is proposed. It takes advantage of the prior knowledge and limits the choices of model parameters accordingly. The findings on degradation and experimental degradation data suggest that the proposed real-time data-driven remaining useful lifetime estimation method can be utilized for failure prognostics. REFERENCES [1] Y. Song and B. Wang, Survey on reliability of power electronic systems, IEEE Transactions on Power Electronics, vol. 28, no. 1, pp , 213. [2] U.-M. Choi, F. Blaabjerg, and K.-B. Lee, Study and handling methods of power igbt module failures in power electronic converter systems, 2721

5 IEEE Transactions on Power Electronics, vol. 3, no. 5, pp , 215. [3] N. Patil, J. Celaya, D. Das, K. Goebel, and M. Pecht, Precursor parameter identification for insulated gate bipolar transistor (igbt prognostics, IEEE Transactions on Reliability, vol. 58, no. 2, pp , 29. [4] G. Sonnenfeld, K. Goebel, and J. R. Celaya, An agile accelerated aging, characterization and scenario simulation system for gate controlled power transistors, in 28 IEEE AUTOTESTCON. IEEE, 28, pp [5] D. W. Brown, M. Abbas, A. Ginart, I. N. Ali, P. W. Kalgren, and G. J. Vachtsevanos, Turn-off time as an early indicator of insulated gate bipolar transistor latch-up, IEEE Transactions on Power Electronics, vol. 27, no. 2, pp , 212. [6] J. R. Celaya, A. Saxena, P. Wysocki, S. Saha, and K. Goebel, Towards prognostics of power mosfets: Accelerated aging and precursors of failure, DTIC Document, Tech. Rep., 21. [7] J. Celaya, A. Saxena, S. Saha, and K. F. Goebel, Prognostics of power mosfets under thermal stress accelerated aging using data-driven and model-based methodologies, 211. [8] S. Dusmez, M. Heydarzadeh, M. Nourani, and B. Akin, Remaining useful lifetime estimation for thermally aged power mosfets with ransac denoising algorithm, in Energy Conversion Congress & Exposition (ECCE. IEEE, 216. [9] S. Dusmez, S. Ali, M. Heydarzadeh, A. Kamath, H. Duran, and B. Akin, Aging precursor identification and lifetime estimation for thermally aged discrete package silicon power switches, IEEE Transactions on Industry Applications in press. [1] H. Huang and P. A. Mawby, A lifetime estimation technique for voltage source inverters, IEEE Transactions on Power Electronics, vol. 28, no. 8, pp , 213. [11] Y. Zheng, L. Wu, X. Li, and C. Yin, A relevance vector machinebased approach for remaining useful life prediction of power mosfets, in 214 Prognostics and System Health Management Conference (PHM- 214 Hunan. IEEE, 214, pp [12] S. Dusmez and B. Akin, An accelerated thermal aging platform to monitor fault precursor on-state resistance, in 215 IEEE International Electric Machines & Drives Conference (IEMDC. IEEE, 215, pp [13] S. Dusmez and B. Akin, Comprehensive parametric analyses of thermally aged power mosfets for failure precursor identification and lifetime estimation based on gate threshold voltage, in 216 IEEE Applied Power Electronics Conference and Exposition (APEC. IEEE, 216, pp [14] S. Kay, Fundamentals of statistical signal processing. Englewood Cliffs, N.J: Prentice-Hall PTR, [15] C. Rasmussen, Gaussian processes for machine learning. Cambridge, Mass: MIT Press,

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