Time-Variant Reliability Optimization for Stress Balance in Press-Pack Insulated Gate Bipolar Transistors

Stress imbalance significantly affects the performance of a press-pack insulated gate bipolar transistor (IGBT). Time-variant loads and conditions lead to the stress fluctuations, exacerbating the impacts. The conventional reliability optimization faces efficiency barriers due to the nested time-variant reliability analysis and design optimization. In this paper, a time-variant reliability optimization approach for press-pack IGBTs is proposed to address the efficiency issue of the IGBT reliability optimization. The performance functions of the maximum and typical stresses are formulated as the optimization objective and constraint. A time-variant reliability optimization model is formulated considering the stress balance reliability degradation within the service cycle. A decoupling algorithm is proposed to transform the nested optimization into a sequential iteration of static reliability optimization and time-variant reliability analysis. The reliability analysis utilizes the performance function continuity in the time domain to reduce the evaluations for the most likelihood points, thereby enhancing efficiency. Numerical and experimental results on an actual IGBT demonstrate the accuracy of the stress balance performance analysis. The time-variant reliability optimization based on the performance functions improves the stress balance performance by 16.3% and meets the reliability requirements within the service cycle. Compared with the conventional double-loop approach, the difference between the solution of the proposed approach with the reference solution is 0.4%, and the efficiency is 334 times that of the double-loop approach. The performance advantages in accuracy and efficiency exhibit the application potential of this approach.

Nomenclature

AbbreviationExpansion

$R $	Performance reliability
$f$	Objective function
$g$	Constraint function
$g_{j} $	$j$ -th Constraint function
$X$	Design vector
$X_{i} $	$i$ -th Design parameter
$P $	Uncertain vector
$P_{l} (t)$	$l$ -th stochastic process
$\rho_{l} \left({\tau }\right)$	auto-correlation of $P_{l} (t)$
$\mu $	Mean vector
$\beta $	Reliability index
$\Phi $	Cumulative distribution function
$E$	Expectation operator
$S_{U} $	Maximum chip stress
$S_{T} $	Typical chip stress
$L_{0} $	Chip side length
$N_{F} $	Simulation evaluation
$T_{F} $	Time for FEM simulation
$t_{0} $	Start of the service cycle
$t_{f} $	End of the service cycle
$t_{i} $	$i$ -th period unit
$m$	Number of period unit
$\rho_{j} $	Correlation coefficient matrix of $g_{j}$
$P^{M}$	Most likelihood point
$L_{j} $	Linear approximation to $j$ -th
$\sigma $	Standard deviation operation
$\varepsilon $	Convergence limit
$c $	Safety factor
$\rho $	Auto-correlation function
Superscripts
$L$	Lower bound
$R$	Upper bound
$thr$	Threshold value
$t$	Target value
$k$	Iteration step
Abbreviation
IGBT	Insulated gate bipolar transistor

SECTION I. Introduction

An insulated gate bipolar transistor (IGBT) is a fully controlled voltage-driven power semiconductor device. It is widely used in high-voltage and large-capacity power electronics fields, such as traffic traction, industrial frequency converters and flexible direct current transmission [1]. Compared with welded IGBTs, the press-pack IGBT has the advantages of easy series connection, double-sided heat dissipation and high reliability [2]. Therefore, it presents better prospects in high power-density and voltage applications. The clamping force applied to the press-pack IGBT ensures the mechanical and electrical connection between the components and establishes the conductive paths between the heat sink and sources. Excessive pressure may cause the IGBT chips to break. Too little pressure cannot ensure effective thermal contact between components, resulting in thermal failure of the self-heating chips. According to the specification provided by IGBT manufacturers (e.g., ABB, WESTCODE), the external clamping pressure should be controlled at about 12 MPa [3]. The chip self-heating and the component thermal expansion mismatch will lead to the stress imbalance of the IGBT [4]. The contact stress between components determines their electrical/thermal contact states. The performance degradation caused by inappropriate contact stress exacerbates the imbalance [5]. Thus, the stress balance is a crucial factor affecting the electrical/thermal performance of the press-pack IGBT.

Various uncertainties exist in actual engineering, such as manufacturing tolerances, material properties, conditions, and loads. Under the combined influence of the uncertainties, the IGBT performances may fluctuate significantly or even fail [6]. The uncertainties can be divided into two categories [7]. The first is static uncertainty, described as a random variable, such as manufacturing tolerances (e.g., sizes, flatness, roughness) and material properties (e.g., thermal conductivity, modulus of elasticity). The second is time-variant uncertainty, described as a stochastic process consisting of a sequence of time-dependent random variables. For example, in an offshore wind power system, the IGBT clamping force affected by the platform vibration should be considered a stochastic process [8]. In distribution network applications with flexible multi-state switches, the IGBT current load changes with the source/load states, which can be described as a stochastic process [9]. Such time-variant uncertainties lead to varying electrical/thermal stresses on the IGBT components. Excessive stress will cause cumulative damage to the components, significantly reducing the IGBT reliability during the service life [10]. Due to the crucial impact of stress imbalance on performance, it is necessary to explore a time-variant reliability optimization approach for the press-pack IGBT involving stochastic processes.

Reliability optimization can improve structural performance and ensure reliability without eliminating uncertainties [11]. The reliability optimization establishes the links between the uncertain parameters and design options through probability constraints, thereby achieving reliable design solutions. In recent decades, the reliability optimization has become an important research direction in academia and engineering. It has been applied to various fields, such as aerospace [12], automobile [13], electronics [14], [15], pharmacology [16] and civil engineering [17]. Conventional reliability optimization methods only apply to time-invariant systems [18], [19]. To address time-variant reliability problems, Jiang et al. proposed a general solution framework based on time-invariant equivalent strategies [20]. Yu et al. developed a time-variant reliability analysis approach combining the extreme value moment and improved maximum entropy methods for problems with multiple failure modes and temporal parameters [21]. Li et al. explored a direct probability integral method for a ten-story building with tuned mass damper under near-fault stochastic impulsive motions [22]. Wu et al. proposed a time-variant probabilistic feasible region approach using the equivalent inverse most probable point for enhancing efficiency [23]. Objectively speaking, the research on time-variant reliability optimization is still preliminary, and the primary technical bottleneck is efficiency. The solution involves a two-layer nested optimization process. The outer layer optimizes the design variables, and the inner layer performs the time-variant reliability analysis. The time-variant reliability analysis is a challenging issue in the uncertain design field, and its computational burden is much higher than that of static reliability analysis. The nested optimization calls heavily for time-variant reliability analysis involving the time-consuming simulation of IGBT performance. It results in extremely low efficiency.

The literature on the mechanical modeling of IGBT, the time-variant uncertainty measurement of IGBT, and the general time-variant reliability design methods are summarized in Table 1, forming the motivation for this study. A time-variant reliability optimization approach is proposed to address the reliability optimization modelling and solution for the press-pack IGBT involving stochastic processes. The maximum and typical stress of the chips are considered as performance functions, and time-variant reliability constraints are established to limit the stress balance reliability degradation caused by stochastic processes. The time-variant continuity of the stress response is utilized to reduce the evaluations of time-consuming simulation models. The rest of the paper is organized as follows. Section II explains the influence mechanism of time-variant uncertainties on IGBT stress balance. Section III creates an IGBT time-variant reliability optimization model. Section IV proposes a decoupling algorithm and flowchart. Section V demonstrates the effectiveness of the proposed approach through an actual IGBT application. Section VI concludes.

TABLE 1 Summary of Literature Related to This Study

SECTION II. Influence of Time-Variant Uncertainties on Stress Balance

A. The Stress Balance Issue in a Press-Pack IGBT

A typical press-pack IGBT device is shown in Fig. 1. It contains several sub-modules, divided into two categories: the IGBT sub-module and the fast recovery diode sub-module. In the sub-modules, the components of a silver sheet, bottom molybdenum sheet, chip and top molybdenum sheet are stacked in a plastic housing from bottom to top. The difference is that the IGBT sub-module has a spring pin to connect the chip gate and printed circuit board. The sub-modules are parallel between a collector copper block and emitter copper block. They are clamped into a ceramic package to form an IGBT device. In practice, a clamping force is applied to the upper and lower copper blocks to maintain electrical/thermal contact between components. The silver sheets are used to relieve the stress imbalances between the chips. Liquid cooling conditions are usually set on the top and bottom surfaces of the IGBT device to maintain heat dissipation.

FIGURE 1.

A cross-section view of the typical IGBT [25].

The IGBT stress balance analysis is a multi-physics problem coupled with electricity, thermal and mechanics [24]. The contact pressure between components determines the contact thermal resistance and electrical resistance, as shown in Fig. 2. Thermal resistance and electrical resistance are vital parameters for thermal analysis, affecting the temperature distribution response inside the IGBT. Under the action of non-uniform temperature distribution and mismatched thermal expansion coefficients, the stress imbalance between the chips emerges. Also, the chip heat dissipation is affected by temperature and stress. Therefore, stress balance optimization is the key to improving the IGBT electrical/thermal performance.

FIGURE 2.

The influence mechanism of contact pressure on IGB [25].

B. Stress Balance Reliability Degradation Caused By Time-Variant Uncertainty

For the press-pack IGBT, the function of $g$ characterizes its stress balance performance. $g\ge 0$ indicates that the stress balance meets the requirements. The IGBT contains $m$ -numbers of uncertain parameters, written as ${ \boldsymbol P}=\left ({{P_{1},\;P_{2},\;\ldots \;,P_{m}} }\right)\;$ . ${P}$ causes the performance response of $g$ to be uncertain. The performance reliability is the probability that the stress balance meets requirements under uncertainty, expressed as:\begin{equation*} R=\textrm {Pr}\left ({{g\left ({{ \boldsymbol X,{ \boldsymbol P}} }\right)\ge 0} }\right) \tag{1}\end{equation*} where ${ \boldsymbol X}=\left ({{X_{1},\;X_{2},\;\ldots \;,X_{n}} }\right)\;$ is an $n$ -dimensional design vector. Pr represents a probabilistic operation.

Due to time-variant conditions and loads, the IGBT performance response exhibits time-variant characteristics. For example, the chip current changes with the source and load states of the system, and it leads to the time-variant characteristics of the chip heat consumption. For distributed sources of a wind power system, the output power is determined by the wind speed, which is a typical stochastic process [26], [27]. The system load is a dynamic process with statistical laws and random noise, usually described as a stochastic process [28], [29]. Correspondingly, the heat consumption of the chips in the IGBT is a stochastic process.

The time-variant uncertain vector ${P}$ in Eq. (1) is described as a stochastic process vector: ${ \boldsymbol P}\left ({t }\right)=\left ({{P_{1} \left ({t }\right),\;P_{2} \left ({t }\right),\ldots \;,P_{n} \left ({t }\right)} }\right)\;$ . A stochastic process is a combination of random variables, which have auto-correlation characteristics in a time domain. $P_{l} \left ({t }\right)$ denotes the $l$ -th stochastic process of ${ \boldsymbol P}\left ({t }\right)$ . Its auto-correlation during the service life of $\left [{ {t_{0},\;t_{T}} }\right]$ is described as $\rho _{l} \left ({\tau }\right)$ , where $\tau $ is a time interval. $\rho _{l} \left ({\tau }\right)=0$ means that the random variables included in the stochastic process are independent during the service cycle. Thus, a random variable can be treated as a special case of stochastic process.

The performance reliability evolves into the probability that the stress balance in the IGBT meets the requirements during the service cycle. The time-variant reliability is formulated as:\begin{equation*} R^{T}=\textrm {Pr}\left ({{g\left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right)\ge 0} }\right),\quad \forall t\in \left [{ {t_{0},\;t_{T}} }\right] \tag{2}\end{equation*}

Also, the time-variant reliability can be described by a reliability index $\beta ^{T}$ . The relationship between the two is written as:\begin{equation*} R^{T}=\Phi \left ({{\beta ^{T}} }\right),\quad \beta ^{T}=\Phi ^{-1}\left ({{R^{T}} }\right) \tag{3}\end{equation*} where $\Phi $ and $\Phi ^{-1}$ denote the standard normal cumulative distribution function and its inverse, respectively. Compared with the static uncertain condition, the stress balance of IGBT presents a greater probability of failure during the service cycle, i.e., reliability degradation.

SECTION III. Time-Variant Reliability Optimization Modelling for Stress Balance

An IGBT design problem with $n$ -number of sub-modules is considered. There are two constraints for the stress balance. ① The maximum chip stress $\left ({{S_{U}} }\right)$ should be less than a threshold $\left ({{S_{U}^{thr}} }\right)$ to avoid structural cracking. ② The typical chip stress $\left ({{S_{T}} }\right)$ should be greater than a threshold $\left ({{S_{T}^{thr}} }\right)$ to ensure adequate thermal contact. The typical stress means the stress at the chip centre. The objective is to optimize the stress balance, defined as $\textrm {E}\left ({{{S_{T}} \mathord {\left /{ {\vphantom {{S_{T}} {S_{U}}}} }\right. } {S_{U}}} }\right)$ . E stands for expectation operator.

In the IGBT, the silver sheets adjust the stress difference between the sub-modules [25]. A softer silver sheet (with lower elastic modulus) can compensate for thermal deformation and improve pressure balance [2]. The shape of a silver sheet is a square sheet, and different deformation resistance can be obtained by changing its side length, namely the equivalent elastic modulus. It is used as a design vector, written as ${ \boldsymbol X}=\left ({{X_{1},\;X_{2},\;\ldots \;,X_{n}} }\right)\;$ . For a silver sheet, the relationship between the equivalent elastic modulus $\left ({{X_{i}} }\right)$ , side length $\left ({{L_{i}} }\right)$ , and material elastic modulus $\left ({{E_{0}} }\right)$ is expressed as:\begin{equation*} L_{i} =L_{0} \cdot \sqrt {\frac {X_{i}}{E_{0}}} \tag{4}\end{equation*} where $L_{0}$ denotes the chip side length. When the side lengths of a silver sheet and a chip are equal, $X_{i} =E_{0}$ . When the stress simulation is invoked by the optimization process, the finite element model usually requires re-meshing. It may lead to a convergence barrier. The equivalent of Eq. (4) avoids the re-meshing in the finite element simulations, thereby solving the convergence issue.

During the service cycle of $\left [{ {t_{0},\;t_{T}} }\right]$ , the time-variant chip heat consumption is measured as ${ \boldsymbol P}\left ({t }\right)=\left ({P_{1} \left ({t }\right),\;P_{2} \left ({t }\right),}\right. \left.{\;\ldots \;,P_{l} \left ({t }\right),\ldots,P_{n} \left ({t }\right) }\right)\;$ . The auto-correlation of $P_{l} \left ({t }\right)$ can be formulated as a function of $\rho (-\tau _{\mathrm {}})$ . The stress balance performances are the functions associated with ${X}$ and ${P}$ , i.e., $S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol P}\left ({t }\right)} }\right)$ and $S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol P}\left ({t }\right)} }\right)$ , forming the function of $g$ in Eq. (2). The objective $\textrm {E}\left ({{{S_{T}} \mathord {\left /{ {\vphantom {{S_{T}} {S_{U}}}} }\right. } {S_{U}}} }\right)$ can be rewritten as $\frac {S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)}{S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)}$ . ${ \boldsymbol \mu }_{ \boldsymbol X},\;{ \boldsymbol \mu }_{ \boldsymbol P}$ represent the mean of ${X}$ and ${P}$ . Thus, the stress balance time-variant reliability optimization model of the press-pack IGBT is formulated as follows:\begin{align*} & \! \max \limits _{ \boldsymbol X} f={S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} \mathord {\left /{ {\vphantom {{S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} {S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)}}} }\right. } {S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} \\ & \textrm {s.t.} ~\beta _{j}^{T} \ge \beta _{j}^{t},\quad j=1,2 \\ &\hphantom { \textrm {s.t.} }\beta _{j}^{T} =\Phi ^{-1}\left ({{\textrm {Pr}\left ({{g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right)\ge 0} }\right)} }\right) \\ &\hphantom { \textrm {s.t.} }g_{1} =S_{U}^{thr} -S_{U} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right) \\ &\hphantom { \textrm {s.t.} }g_{2} =S_{T} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right)-S_{T}^{thr} \\ &\hphantom { \textrm {s.t.} }\forall t\in \left [{ {t_{0},\;t_{T}} }\right],\quad X_{i}^{L} \le X_{i} \le X_{i}^{R}, i=1,2,\cdots,n \tag{5}\end{align*} where $X_{i}^{R} $ and $X_{i}^{L}$ are the upper and lower bounds of the design variables, respectively. $\beta _{j}^{t},\;j=1,2$ represent the target reliability indexes.

The time-variant reliability optimization is essentially nested, as shown in Fig. 3. The outer layer optimizes the design points, and the inner layer analyzes the time-variant reliability at each design point. Even if high-efficiency algorithms (e.g., the quasi-Newton algorithm and sequential quadratic programming) are adopted [30], dozens or hundreds of reliability analyses (Iteration $k \times $ Dimensions $n$ ) are required. The static reliability analysis is an optimization process that repeatedly invokes the time-consuming simulation of IGBT. Further considering the time-variant reliability analysis of the service cycle, the simulation evaluations $\left ({{N_{F}} }\right)$ are dozens of times that of the static analysis. Time for a single finite element simulation $\left ({{T_{F}} }\right)$ may take several minutes or hours. The computational cost of IGBT time-variant reliability optimization is calculated as $k\times n\times N_{F} \times T_{F}$ . According to a rough estimate, the solution time is 10⁵ hours $\left ({{10\times 10\times 10^{3}\times 1} }\right)$ , i.e., 11.4 years. The solution efficiency cannot meet the needs of engineering applications. Therefore, it is necessary to develop an efficient algorithm according to the features of IGBT time-variant reliability optimization.

FIGURE 3.

The nested optimization in the time-variant reliability optimization.

SECTION IV. Formulation of Decoupling Algorithm

To address the efficiency issue, a decoupling algorithm is proposed. It decouples the time-variant reliability optimization into a sequential iteration with time-variant reliability analysis and static reliability optimization. The time-variant continuity of stress in the IGBT is utilized to reduce the times of searches for the most likelihood point in the reliability analysis.

A. Time-Variant Reliability Analysis

The improved time-variant progress discretization method is employed to solve the time-variant reliability constraints in Eq. (5) [31]. Firstly, the time-variant reliability analysis is transformed into a system reliability problem by discretizing the stochastic processes and performance functions during the service cycle. Secondly, the static reliability analysis is performed on the period unit, and the correlation coefficient matrices of the performance functions are calculated. Finally, the IGBT time-variant reliability is calculated using the unit reliability analysis results and correlation coefficient matrices.

The service cycle $\left [{ {t_{0},\;t_{T}} }\right]$ is discretized into $m$ -number of equal period units $\left ({{t_{1},t_{2},\ldots,t_{i},\ldots,t_{m}} }\right)$ . Correspondingly, each time-variant performance function is discretized into $m$ -number of static performance functions. $m$ is the discrete number. $t_{i}$ represents the $i$ -th unit, i.e., $t_{i} ={\left ({{i-0.5} }\right)\cdot \left ({{t_{T} -t_{0}} }\right)} \mathord {\left /{ {\vphantom {{\left ({{i-0.5} }\right)\cdot \left ({{t_{T} -t_{0}} }\right)} m}} }\right. } m$ . The time-variant reliability analysis is converted to the reliability analysis of the series system composed of the units. Thus, $\textrm {Pr}\left ({{g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right)\ge 0} }\right)$ in Eq. (5) is equivalent to \begin{equation*} R_{j}^{T} =\textrm {Pr}\left ({{\mathop \cap \limits _{i=1}^{m} \left ({{g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{i}} }\right),\;t_{i}} }\right)\ge 0} }\right)} }\right) \tag{6}\end{equation*} where ${ \boldsymbol P}\left ({{t_{i}} }\right)=\left ({{P_{1} \left ({{t_{i}} }\right),P_{2} \left ({{t_{i}} }\right),\ldots,P_{n} \left ({{t_{i}} }\right)} }\right) $ is the $n$ -dimensional random vector in the unit of $t_{i}$ after ${ \boldsymbol P}\left ({t }\right)$ is discretized. According to the reliability analysis principle of a series system [32], Eq. (6) can be solved by \begin{equation*} R_{j}^{T} =\Phi _{m} \left ({{\beta _{j} \left ({{t_{1}} }\right),\beta _{j} \left ({{t_{2}} }\right),\ldots,\beta _{j} \left ({{t_{m}} }\right),{ \boldsymbol \rho }_{j}} }\right) \tag{7}\end{equation*} where $\Phi _{m}$ is the $m$ -dimensional standard normal distribution function, and multi-dimensional normal distribution calculation refers to the literature [33]. $\beta _{j} \left ({{t_{i}} }\right)$ represents the reliability index of the performance function of $g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{i}} }\right),\;t_{i}} }\right)$ . ${ \boldsymbol \rho }_{j}$ denotes the correlation coefficient matrix of $g_{j}$ .

The first-order second-moment method is adopted to calculate $\beta _{j} \left ({{t_{i}} }\right)$ and $g_{j}$ [11]. ${ \boldsymbol P}^{M}\left ({{t_{i}} }\right)$ denotes the most likelihood point, with the maximum joint probability density. A linear approximation to $g_{j}$ at the point can be used to calculate the correlation coefficient matrix [20]. The linear approximation is derived as \begin{align*} L_{j}& =g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}^{M}\left ({{t_{i}} }\right),\;t_{i}} }\right)+\left ({{ \boldsymbol P\left ({{t_{i}} }\right)-{ \boldsymbol P}^{M}\left ({{t_{i}} }\right)} }\right)^{\textrm {T}} \\ &\quad \cdot \nabla g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}^{M}\left ({{t_{i}} }\right),\;t_{i}} }\right) \tag{8}\end{align*} where $\nabla g_{j}$ represents the gradient vector at the point of ${ \boldsymbol P}^{M}\left ({{t_{i}} }\right)$ . A component of $\rho _{j} \left ({{t_{i},\;t_{i} +\tau } }\right)$ in ${ \boldsymbol \rho }_{j}$ represents the correlation coefficient between $L_{j} \left ({{t_{i}} }\right)=L_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{i}} }\right),\;t_{i}} }\right)$ and $L_{j} \left ({{t_{i} +\tau } }\right)=L_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{i} +\tau } }\right),\;t_{i} +\tau } }\right)$ . $\tau $ means a time interval. $\rho _{j} \left ({{t_{i},\;t_{i} +\tau } }\right)$ is derived as \begin{align*} &\rho _{j} \left ({{t_{i},\;t_{i} +\tau } }\right) \\ &=\frac {\textrm {COV}\left ({{L_{j} \left ({{t_{i}} }\right),L_{j} \left ({{t_{i} +\tau } }\right)} }\right)}{\sigma \left ({{L_{j} \left ({{t_{i}} }\right)} }\right)\cdot \sigma \left ({{L_{j} \left ({{t_{i} +\tau } }\right)} }\right)} \\ &=\frac {\sum \limits _{l=1}^{n} {\nabla g_{j,\;l} \left ({{t_{i}} }\right)\cdot \nabla g_{j,\;l} \left ({{t_{i} +\tau } }\right)\cdot \rho \left ({{P_{l} \left ({{t_{i}} }\right),P_{l} \left ({{t_{i} +\tau } }\right)} }\right)} }{\sigma \left ({{L_{j} \left ({{t_{i}} }\right)} }\right)\cdot \sigma \left ({{L_{j} \left ({{t_{i} +\tau } }\right)} }\right)} \\{}\tag{9}\end{align*} where COV indicates a covariance operation. $\sigma $ denotes a standard deviation operation. $\nabla g_{j,\;l} \left ({{t_{i}} }\right)$ represents the $l$ -th component of $\nabla g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}^{M}\left ({{t_{i}} }\right),\;t_{i} } }\right)$ . $\rho \left ({{P_{l} \left ({{t_{i}} }\right),P_{l} \left ({{t_{i} +\tau } }\right)} }\right)$ represents the auto-correlation function of the stochastic process $P_{l}$ .

B. The Flowchart of Decoupling Algorithm

The time-variant reliability analysis method for the IGBT stress balance is given in the previous section. It is an optimization process embedded with time-consuming simulations. The time-variant reliability optimization requires repeated calls to the reliability analysis, thus involving a nested optimization. To improve efficiency, a decoupling algorithm with the sequential iteration of static reliability optimization and time-variant reliability analysis is proposed.

In the $k$ -th iteration, the time-variant reliability analysis is performed at the previous solution ${ \boldsymbol X}^{\left ({{k-1} }\right)}$ . According to the research in Part A, two issues need to be solved: the efficiency of searching for the most likelihood point, and how to determine the discrete number (m). ① The time-variant continuity of the stress response is exploited to reduce the times of searches for the most likelihood points. The stress response varies with time, but its functional form is constant for each period unit. Thus, the most likelihood points are close in the standard normal space [9]. The most likelihood point is searched in the first unit $t_{1}$ and extended to other units $t_{i} =t_{2},t_{3},\ldots,t_{m}$ . $m$ -number of searches are converted into a single search, so the efficiency is significantly improved. ② A convergence mechanism is developed to determine the discrete number $m$ . A trial-by-trial identification is used to search for $m$ , i.e., starting from $m=1$ to calculate $R_{j}^{T} \left ({m }\right)$ based on Eq. (7) until two-step continuous convergence. The convergence criterion is established as \begin{align*} \left |{ {\frac {R_{j}^{T} \left ({m }\right)-R_{j}^{T} \left ({{m-1} }\right)}{R_{j}^{T} \left ({m }\right)}} }\right |\le \varepsilon \;\& \;\left |{ {\frac {R_{j}^{T} \left ({m }\right)-R_{j}^{T} \left ({{m-2} }\right)}{R_{j}^{T} \left ({m }\right)}} }\right |\le \varepsilon \\{}\tag{10}\end{align*} where $\varepsilon \;$ indicates the convergence limit. The value of $m$ obtained in the first iteration applies all iterations.

After solving the above issues, an iterative mechanism for the time-variant reliability optimization is created. A nonlinear equation is formulated according to Eq. (5) and Eq. (7), expressed as:\begin{align*} \Phi ^{-1}\left ({{\Phi _{m} \left ({{\begin{array}{l} c_{j}^{\left ({{k+1} }\right)} \cdot \beta _{j}^{\left ({k }\right)} \left ({{t_{1}} }\right),c_{j}^{\left ({{k+1} }\right)} \cdot \beta _{j}^{\left ({k }\right)} \left ({{t_{2}} }\right) \\[2pt],\ldots,c_{j}^{\left ({{k+1} }\right)} \cdot \beta _{j}^{\left ({k }\right)} \left ({{t_{m}} }\right),{ \boldsymbol \rho }_{j} \\ \end{array}} }\right)} }\right)=\beta _{j}^{t} \\{}\tag{11}\end{align*} where $c_{j}^{\left ({{k+1} }\right)}$ represents the safety factor for the ${j}$ -th constraint at the $k+1$ -th iteration. $\beta _{j}^{\left ({k }\right)} \left ({{t_{i}} }\right)$ denotes the static reliability index on the $t_{i}$ unit at the $k$ -th iteration. Eq. (11) is a root-finding problem for the nonlinear equation, which can be solved by the Newton iterative algorithm [34].

The time-variant reliability optimization as Eq. (5) is converged into a static reliability optimization, formulated as:\begin{align*} & \! \max \limits _{ \boldsymbol X} {f=S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} \mathord {\left /{ {\vphantom {{f=S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} {S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)}}} }\right. } {S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} \\ & \textrm {s.t.}~ \beta _{j}^{\left ({k }\right)} \left ({{t_{1}} }\right)\ge c_{j}^{\left ({k }\right)} \cdot \beta _{j}^{t},\quad j=1,2 \\ &\hphantom { \textrm {s.t.} }\beta _{j}^{\left ({k }\right)} =\Phi ^{-1}\left ({{\textrm {Pr}\left ({{g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{1}} }\right),t_{1}} }\right)\ge 0} }\right)} }\right) \\ &\hphantom { \textrm {s.t.} }g_{1} =S_{U}^{thr} -S_{U} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{1}} }\right),t_{1}} }\right) \\ &\hphantom { \textrm {s.t.} }g_{2} =S_{T} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({{t_{1}} }\right),t_{1}} }\right)-S_{T}^{thr} \\ &\hphantom { \textrm {s.t.} }X_{i}^{L} \le X_{i} \le X_{i}^{R}, \quad i=1,2,\cdots,n \tag{12}\end{align*} The above formula can be solved by existing methods, such as the incremental shifting vector method [35], thereby obtaining the solution ${ \boldsymbol X}^{\left ({k }\right)}$ of the current iteration. The time-variant reliability analysis and reliability optimization are performed alternately until meeting the convergence criteria as follows:\begin{align*} \begin{cases} \displaystyle \beta _{j}^{T} \ge \beta _{j}^{t},\;j=1,2 \\ \displaystyle \left |{ {\frac {f^{\left ({k }\right)}-f^{\left ({{k-1} }\right)}}{f^{\left ({k }\right)}}} }\right |\le \varepsilon \end{cases} \tag{13}\end{align*} The algorithm flowchart is summarized in Fig. 4. The computational cost of the solution depends on the evaluations of the finite element model. The ${X}$ and ${P}$ with the higher dimensions result in more evaluations of the finite element model for design optimization and reliability analysis in each iteration. Moreover, the combination of high-dimensional ${X}$ and ${P}$ leads to more iterations. Therefore, the dimension of the optimization model and the complexity of the finite element model are the key to the proposed approach efficiency.

FIGURE 4.

The flowchart of the decoupling algorithm.

SECTION V. Engineering Application

A press-pack IGBT application, as shown in Fig. 5, was used to verify the performance of the proposed time-variant reliability optimization approach. The IGBT device contained twelve sub-modules of IGBT and four sub-modules of fast recovery diode. Clamping forces $F=72\;\textrm {KN}$ with equal magnitude and opposite directions were applied to the top and bottom surfaces of the IGBT. To dissipate the IGBT heat consumption, the upper and lower surfaces were given liquid cooling conditions. The stress balance performance was defined as an objective function of $f={S_{U} } \mathord {\left /{ {\vphantom {{S_{U}} {S_{T}}}} }\right. } {S_{T} }$ . The reliability indexes of the constraints $g_{1} =S_{U}^{thr} -S_{U} \ge 0$ and $g_{2} =S_{T} -S_{T}^{thr} \ge 0$ should be greater than $\beta _{j}^{t} =2.5,\;j=1,2$ , where $S_{U}^{thr} =\textrm {78}\;\textrm {MPa}$ and $S_{U}^{thr} =\textrm {5}\;\textrm {MPa}$ . The service cycle of IGBT was 24 hours, i.e., $\forall t\in \left [{ {0,\;24} }\right]$ . The design vector ${E}$ consisted of the equivalent elastic modulus of the silver sheet in the sub-modules. The time-variant uncertain parameters were the chip heat consumption, described as the stochastic processes in Table 2.

TABLE 2 Distribution Characteristics of the Heat Dissipation Stochastic Proces

FIGURE 5.

The structural diagram of the press-pack IGBT.

A. Stress Simulation and Experiment

The first step in the time-variant reliability optimization was to create a finite element model of the IGBT to analyze the stress balance performance, i.e., the maximum stress $\left ({{S_{U}} }\right)$ and typical stresses $\left ({{S_{T}} }\right)$ of the chips. The commercial finite element software of ABAQUS/CAE (Version 6.13) was employed. Considering the symmetry of the structure, boundary conditions and loads, a 1/4-type finite element model was established, as shown in Fig. 6. The model ignored the contact pins because their diameter is much smaller than the sub-module side length and had little influence on the stress distribution. The IGBT sub-modules are marked as white numbers ①~③, and the sub-module of the fast recovery diode is marked as ④. The internal structure of the sub-modules is shown in Fig. 1. Table 3 lists the mechanical and thermal properties of the IGBT components.

TABLE 3 The Material Properties of the Component

FIGURE 6.

The finite element model of the IGBT.

According to the IGBT operating conditions, the loads and boundary conditions in the finite element model were set as follows: A fixed constraint is placed on the emitter lower surface. A uniform force $F=12$ kN is applied on the collector upper surface. Heat exchange conditions were established on the surfaces to simulate the liquid cooling effect, i.e., the sink temperature of 35 °C and the heat transfer coefficient of 0.01 W/mm². The contact behaviour between the components was characterized by pressure-dependent thermal resistance and normal hard contact [36]. The initial values of the design variables are $X_{i} =E_{0},\;i=1,2,3,4$ , where $E_{0} =10.5\;\textrm {GPa}$ represents the elastic modulus of silver. It means that the side lengths of the silver sheets and chips are equal, i.e., $L_{i} =L_{0} =9\;\textrm {mm},\;i=1,2,3,4$ . The chip heat consumption was defined as the mean values of the random processes in Table 2, i.e., $\mu _{P_{i}} =120\;\textrm {W},\;i=1,2,3$ and $\mu _{P_{4}} =72\;\textrm {W}$ . The finite element model contains 18 elements and 28,624 eight-node thermally coupled hexahedral elements. Using a conventional computer (e.g., CPU_i7, RAM_8g) to solve it, a single simulation took about ten minutes. The stress simulation results are shown in Fig. 7, marking the Von-Mises stress at the centre of each chip. The stress distribution presents a significant imbalance. The upper bound (15.5 MPa) and the lower bound (5.8 MPa) appear on Chip_4 and Chip_3, and the former is 2.6 times larger than the latter.

FIGURE 7.

The stress distribution on the chips.

To verify the accuracy of the finite element simulation, a stress distribution experiment was carried out under a coupled mechanical/thermal condition, as shown in Fig. 8. The experimental bench was composed of the material testing machine (Model: CMT5504 by SASTEST) and the studied IGBT. A temperature-controlled heating membrane was pasted on the emitter lower surface to simulate the IGBT heat consumption. The clamping force was provided by the material testing machine. The mechanical and thermal loads were applied in three stages: contacting, heating and clamping. Firstly, a slight clamping force was loaded onto the IGBT to bring the components into contact. Secondly, the heating film was turned on and held for twenty minutes, with the temperature set at 45 °C until thermal equilibrium. Thirdly, the clamping force was increased to 72 kN at the rate of 12 kN/min, and maintained for five minutes.

FIGURE 8.

The pressure distribution test bench.

Due to engineering limitations, the experimental bench was slightly different from the actual IGBT operating conditions. The limitations were in load application and stress measurement. The components are small and in contact, leaving no room for stress sensors and heating membranes. Thus, we replaced the chip self-heating effect with the temperature load. And the FUJI pressure-sensitive film was adopted to measure the pressure distribution, which is an intuitive and commonly used method for pressure distribution measurements [37]. Red spots appear on the film where pressure is loaded, and the colour density changes with the pressure level. The films are divided into several pressure levels according to the measurement ranges, and we choose the film with the pressure range of [2.5 MPa, 10 MPa].

The experimental results are shown in Fig. 9. It can be seen that the pressure distribution in the IGBT was not uniform under the coupled mechanical/thermal condition. The colour density at the border was higher than that at the middle, i.e., the sub-modules at the border were subject to the higher stresses. The conditions of the simulation model were adjusted to be consistent with the experiment, i.e., using the temperature load to simulate the chip power consumption. The experimental results were generally consistent with the simulation results. Besides, the simulation modeling in this study referred to the literature [36], which revealed the influence of temperature on the pressure distribution of the IGBT with 44 sub-modules. The accuracy of the simulation model was verified by the experimental results. Therefore, the experiments in this study and existing literature support that the simulation model can predict the IGBT stress distribution, providing effective performance functions for the time-variant reliability optimization.

FIGURE 9.

The numerical and experimental results of the pressure distribution.

B. Optimization Modelling and Solution

For the IGBT device with sixteen sub-modules, a time-variant reliability optimization model was established as:\begin{align*} & \! \min \limits _{ \boldsymbol X} f={S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} \mathord {\left /{ {\vphantom {{S_{U} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} {S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)}}} }\right. } {S_{T} \left ({{ \boldsymbol X,\;{ \boldsymbol \mu }_{ \boldsymbol P}} }\right)} \\ & \textrm {s.t.}~ \beta _{j}^{T} \ge \beta _{j}^{t} =2.5,\quad j=1,2 \\ &\hphantom { \textrm {s.t.} }\beta _{j}^{T} =\Phi ^{-1}\left ({{\textrm {Pr}\left ({{g_{j} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right)\ge 0} }\right)} }\right) \\ &\hphantom { \textrm {s.t.} }g_{1} =S_{U}^{thr} -S_{U} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right),\;S_{U}^{thr} =\;78\;\textrm {MPa} \\ &\hphantom { \textrm {s.t.} }g_{2} =S_{T} \left ({{ \boldsymbol X,{ \boldsymbol P}\left ({t }\right),t} }\right)-S_{T}^{thr},\;S_{T}^{thr} =\;5\;\textrm {MPa} \\ &\hphantom { \textrm {s.t.} }\forall t\in \left [{ {0,\;24} }\right],\quad X_{i}^{L} \le X_{i} \le X_{i}^{R}, i=1,2,3,4 \tag{14}\end{align*}

The numerical analysis software of MATLAB (Version R2013b) was adopted in the engineering application. $X_{i}^{0} =E_{0},\;i=1,2,3,4$ was selected as the initial design point. At this point, the maximum stress was $S_{U}^{\left ({0 }\right)} =89.45\;\textrm {MPa}$ , the typical stress was $S_{T}^{\left ({0 }\right)} =5.44\;\textrm {MPa}$ , and the objective value was $f^{\left ({0 }\right)}=\textrm {0.0608}$ . The time-variant reliability indexes of the two constraints were $\left ({{\beta _{1}^{T\left ({0 }\right)},\;\beta _{2}^{T\left ({0 }\right)}} }\right)=\left ({{ < -5,\;>5} }\right)$ . The first constraint was not satisfied, while the second was over-satisfied. The decoupling algorithm was applied to solve Eq. (14), and the optimal solution was obtained after four iterations, i.e., $X_{i}^{\ast } =\left ({{6744,\;2538,\;7469,\;2656} }\right)\;\textrm {MPa}$ . As shown in Fig. 10, the maximum stress was $S_{U}^{\ast } =73.15\;\textrm {MPa}$ , the typical stress was $S_{T}^{\ast } =5.17\;\textrm {MPa}$ , and the objective value is $f^{\ast }=\textrm {0.0707}$ at the optimal solution. The static reliability indexes of the constraints (i.e., the reliability indexes in the $t_{1}$ unit) were $\beta _{1}^{\ast } \left ({{t_{1}} }\right)=\textrm {2.89}$ . Considering the service cycle, the time-variant reliability indexes of two constraints were $\beta _{1}^{T\ast } =\textrm {3.22}$ and $\beta _{2}^{T\ast } =\textrm {2.54}$ , satisfying $\beta _{j}^{T\ast } \ge \beta _{j}^{t} =2.5,\;j=1,2$ . The iterative process of the objective value $\left ({{f^{\left ({k }\right)}} }\right)$ and reliability indexes $\left ({{\beta _{1}^{T\left ({k }\right)},\;\beta _{2}^{T\left ({k }\right)}} }\right)$ is shown in Fig. 11. The solution called the finite element simulation 316 times and took 53 hours.

FIGURE 10.

The stress distribution at optimal design.

FIGURE 11.

The iterations of the objective value and reliability indexes.

The optimal solution $\left ({{ \boldsymbol X^{\ast }} }\right)$ of the equivalent elastic modulus was converted into the side length ${ \boldsymbol L}^{\ast } =\left ({{7.2,\;4.4,\;7.6,\;4.5} }\right)\;\textrm {mm}$ of the silver sheet by Eq. (4). The finite element model described above is modified to obtain the maximum and typical stresses of $\left ({{73.41,\;5.14} }\right)\;\textrm {MPa}$ at the optimal side length. The results were close to $\left ({{S_{U}^{\ast },\;S_{T}^{\ast }} }\right)=\left ({{73.15,\;5.17} }\right)\;\textrm {MPa}$ , with errors of 0.36% and 0.39%. It verified the validity of the conversion between the equivalent elastic modulus and the structural size.

C. Discussion of Results

The stress balance performance comparison before and after the IGBT time-variant reliability optimization is listed in Table 4. After optimization, the stress balance performance is improved from $f^{\left ({0 }\right)}=\textrm {0.0608}$ to $f^{\ast }=\textrm {0.0707}$ , increasing by 16.3%. At the optimal solution, the static and time-variant reliability indexes of the first constraint are 3.45 and 3.22, and those of the second constraint are 2.89 and 2.54. As analyzed before, the time-variant uncertainty leads to the degradation of IGBT stress balance reliability during the service cycle. Therefore, to obtain a reliable solution, the IGBT reliability degradation during the service period must be considered. The solution of the proposed approach satisfies two time-variant reliability constraints and is feasible. The first constraint changes from unsatisfied to satisfied, and the second changes from over-satisfied to just-satisfied. It suggests that the proposed approach can solve the time-variant reliability design problem of the press-pack IGBT.

TABLE 4 The Results of Time-Variant Reliability Optimization

To investigate the efficiency and accuracy of the proposed approach, it was compared with a conventional double-loop approach [38]. The Double-loop approach means that the outer layer optimizes the design point by the sequential quadratic programming [30], and the inner layer performs the time-variant reliability analysis. The double-loop approach has high precision and is usually employed to obtain a reference solution, but its computational cost is high [39]. Unfortunately, the double loop approach with invoking the finite element model was not feasible in efficiency. To make the double-loop approach viable in this comparative study, second-order polynomial response surfaces were established for the finite element model. With one hundred random samples, the approximate analytical expressions of the performance functions were formulated as:\begin{align*} S_{U} &=58.6+10^{-3}\cdot (-1.36\cdot X_{1} +3.76\cdot X_{2} -0.02\cdot X_{3} \\ &\quad +\,2.53\cdot X_{4}) \\ &\quad +\, 10^{-7}\cdot (0.58\cdot X_{1}^{2} -1.76\cdot X_{2}^{2} -3.09\cdot X_{4}^{2})+10^{-2} \\ &\quad \cdot (10.5\cdot P_{2} -6.01\cdot P_{4})+10^{-7}\times (4.86\cdot X_{1} \cdot P_{1} \\ &\quad -\,12.21\cdot X_{2} \cdot P_{2}) \\ &\quad +\, 10^{-7}\cdot (-1.75\cdot X_{3} \cdot P_{3} +10.46\cdot X_{4} \cdot P_{4}) \\ S_{T} &=3.48+10^{-4}\cdot (1.18\cdot X_{1} -0.28\cdot X_{2} +0.16 \\ &\quad \cdot X_{3} -3.85\cdot X_{4}) \\ &\quad +\,10^{-8}\cdot (-0.44\cdot X_{1}^{2} +5.55\cdot X_{4}^{2})+10^{-3} \\ &\quad \cdot (8.46\cdot P_{2} +1.19\cdot P_{4})+10^{-7}\cdot (-2.68\cdot X_{1} \\ &\quad \cdot P_{1} +2.75\cdot X_{2} \cdot P_{2}) \\ &\quad +\, 10^{-7}\cdot (9.86\cdot X_{3} \cdot P_{3} +5.78\cdot X_{4} \cdot P_{4}) \tag{15}\end{align*}

For objective comparison, the proposed approach was applied again for the time-variant reliability optimization using Eq. (15) as the performance functions. Although the response surface was used in the comparative study, we suggest that the reliability optimization in practice should directly invoke the finite element model to avoid the errors introduced by the response surfaces. Thus, the time consumption of the finite element analysis was included in the computational cost.

The numerical results of the two approaches are listed in Table 5. Regarding efficiency, the proposed approach called the performance function 304 times, and the computational time was 50.7 hours. The functional evaluations were 101,632 times, and the computational time was 16,939 hours (1.93 years). The efficiency advantage of the proposed approach is significant, satisfying engineering requirements. In accuracy, the objective value of the proposed approach is close to the reference solution, with a difference of 0.4%. It demonstrates the approach’s accuracy.

TABLE 5 The Comparison of the Proposed and Double-Loop Approaches

SECTION VI. Conclusion

The stress imbalance in the press-pack IGBT seriously affects the thermal and electrical contacts between the components, resulting in its performance degradation. The effects of stress imbalances are exacerbated by time-variant uncertain loads and conditions. In this paper, a time-variant reliability optimization approach is proposed to address the reliability optimization modelling and solution for the press-pack IGBT involving stochastic processes. The contributions of this study are summarized below. Firstly, the performance functions of the maximum and typical stresses in the IGBT chips are established for the objective and constraints. And a time-variant reliability optimization model of IGBT is formulated considering the reliability degradation of stress balance. Secondly, a decoupling algorithm is proposed, separating the time-variant reliability analysis from the design optimization. The nested optimization is decoupled into a sequential iteration with static reliability optimization and time-variant reliability analysis, significantly reducing the reliability analysis evaluations. Thirdly, an approximation strategy is given to reduce the searches for the most likelihood point by utilizing the time-variant continuity of the stress response. It significantly improves the efficiency of time-variant reliability analysis. The numerical and experimental results on the actual IGBT demonstrate the accuracy of the stress simulation. On this basis, the time-variant reliability optimization improves the stress balance performance by 16.3%, and the reliability indexes meet the constraints. Taking the double-loop approach as a benchmark, the difference between the solution of the proposed approach and the reference solution is 0.4%, and the efficiency is 334 times that of the double-loop approach. The performance advantages in accuracy and efficiency support the excellent potential of the approach in engineering applications. In the future, we will explore experimental strategies to validate the proposed approach in an offshore wind power system and advanced time-variant reliability optimization approaches for power semiconductor devices involving high-dimensional stochastic processes.