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Daniele Prada, Marco Bellini, Ivica Stevanovic, Senior Member, IEEE, Laurent Lemaitre, ... D. Prada is with the Department of Mathematical Sciences, Indi-.
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On the Performance of Multiobjective Evolutionary Algorithms in Automatic Parameter Extraction of Power Diodes Daniele Prada, Marco Bellini, Ivica Stevanovi´c, Senior Member, IEEE, Laurent Lemaitre, James Victory, Senior Member, IEEE, Jan Vobeck´y, Senior Member, IEEE, Riccardo Sacco, and Peter O. Lauritzen, Life Senior Member, IEEE

Abstract—In this paper, a general, robust, and automatic parameter extraction of nonlinear compact models is presented. The parameter extraction is based on multiobjective optimization using evolutionary algorithms, which allow fitting of several highly nonlinear and highly conflicting characteristics simultaneously. Two multiobjective evolutionary algorithms which have been proved to be robust for a wide range of multiobjective problems [1]–[3], the nondominated sorting genetic algorithm II and the multiobjective covariance matrix adaptation evolution strategy, are used in the parameter extraction of a novel power diode compact model based on the lumped charge technique. The performance of the algorithms is assessed using a systematic statistical approach. Good agreement between the simulated and measured characteristics of the power diode shows the accuracy of the used compact model and the efficiency and effectiveness of the proposed multiobjective optimization scheme. Index Terms—Evolutionary algorithms (EAs), multiobjective optimization, parameter extraction, power semiconductor devices.

I. INTRODUCTION ELIABLE circuit simulations need accurate and computationally efficient compact models and device model parameter sets. In recent years, there have been considerable advances in compact modeling of power electronics semiconductor devices [4]–[6]. Unlike in the case of compact models in CMOS technology [7], [8], there are no standard compact models for power devices. However, the compact power device models based on the lumped charge technique [9]–[12] have emerged as a good candidate when it comes to simplicity,

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Manuscript received May 16, 2014; revised August 8, 2014; accepted September 14, 2014. Date of publication September 30, 2014; date of current version April 15, 2015. Recommended for publication by Associate Editor K. Ngo. D. Prada is with the Department of Mathematical Sciences, Indiana University–Purdue University, Indianapolis, IN 46202 USA (e-mail: [email protected]). M. Bellini is with ABB Schweiz AG, 5405 Baden-D¨attwil, Switzerland (e-mail: [email protected]). I. Stevanovi´c is with the Federal Office of Communications, 2501 Biel/Bienne, Switzerland (e-mail: [email protected]). L. Lemaitre is with Freelance EDA Consultant, 01630 P´eron, France (e-mail: [email protected]). J. Victory is with Fairchild Semiconductor, 85609 Aschheim, Germany (e-mail: [email protected]). J. Vobeck´y is with ABB Switzerland Ltd. Semiconductors, 5600 Lenzburg, Switzerland (e-mail: [email protected]). R. Sacco is with the Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy (e-mail: [email protected]). P. O. Lauritzen is with the Department of Electrical Engineering, University of Washington, Seattle, WA 98195 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TPEL.2014.2360864

accuracy, numerical stability, and computational efficiency (features that any compact model should possess [12], [13]). Additionally, these models allow a simpler parameter extraction procedure compared to more accurate but more complex compact models [14]–[16]. In any parameter extraction procedure, an optimal parameter set is extracted for a given device model by fitting the simulated electrical characteristics to the measured electrical characteristics of the device. The electrical characteristics of power device compact models are highly nonlinear functions of the model parameters. Moreover, the parameter set should ideally minimize the errors between all the measured and simulated characteristics, which are very often conflicting; good fitting of one characteristic can very often correspond to a poor fitting of another characteristic. Therefore, multiobjective optimization techniques, such as the ones described in the present study, are necessary to find an optimal parameter set that simultaneously minimizes all error estimators. This paper is organized as follows. In Section II, different parameter extraction approaches commonly found in the literature are discussed and several variants of two multiobjective evolutionary algorithms (MOEAs), which have been proved to be robust for a wide range of multiobjective problems [1]–[3], are introduced. Although the multiobjective optimization procedure proposed in this section can be applied to the parameter extraction of any compact model, in this paper, we focus on a novel power diode model based on the lumped charge approach. Details of the used model and the proposed extensions implemented are given in Section III. Finally, for the first time, Section IV presents the results of the evolutionary algorithms (EAs) used in the parameter extraction of this highly nonlinear power diode and discusses their performance evaluated using a systematic statistical procedure. Good agreement between the simulated and measured characteristics of the power diode is evidence of both: accuracy of the model and efficiency of the proposed multiobjective optimization. II. MULTIOBJECTIVE OPTIMIZATION A typical parameter extraction procedure can be performed following the steps shown in the flowchart in Fig. 1 [17], [18]. The starting point in the parameter extraction procedure is the initial model parameter estimation from measurements and/or technological process data [10], [18], [19]. Once the initial

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Fig. 2. Example of nondominated sorting. Minimization of both the objectives z1 and z2 is assumed. The sets F1 , F2 , and F3 are the nondominated fronts, and F1 is the Pareto front.

Fig. 1.

Flowchart of a typical parameter extraction procedure.

starting point is known, a sensitivity analysis may be performed around the initial parameter set, by varying one parameter at a time and observing the effect on the simulated electrical characteristics. This step helps to determine a subset of parameters that have strong influence on a given target. The steps 3, 4, and 5 in the flowchart are parts of the optimization procedure. The error is computed between the measured characteristics and simulated characteristics for the given parameter set. As error estimators, the sum of absolute errors (RSM) and the sum of squared errors (RSS) can be used [17], [18]. If the error is larger than a given threshold, the parameter set is perturbed and the optimization procedure continues. An optimal parameter set is found once the termination criteria are satisfied. Care is needed when employing conventional optimization techniques, since model equations are strongly coupled and very small changes in parameters lead to very different results. Moreover, the parameter set returned by the optimization procedure should ideally minimize the errors between all the measured and simulated characteristics. This can be very hard to achieve because a good fitting of one characteristic can correspond to a bad fitting of another characteristic. A problem where several error estimators have to be minimized simultaneously is called a multiobjective optimization problem (MOP) [2], [20]. Solving a MOP involves a tradeoff between the complexity of the optimization algorithm and the quality of the solution. In the following paragraphs, simpler techniques are introduced and evaluated. Then, the rationale for a more complex algorithm is presented. A first (and most simple) approach to optimization in the context of parameter extraction is a succession of optimizations of one characteristic at a time, until convergence is reached. Each optimization returns a value for a particular subset of the model parameters, which have been selected through the sensitivity analysis at step 3 of the above procedure. An advantage of such a method is that it can be carried out with a single objective optimization algorithm. The quality of the solution found by

this first approach depends heavily on the number of waveforms being optimized and on the quality of the initial parameter set. As the number of characteristics to fit increases, or the quality of parameter initial guess decreases, the method will either fail to converge to a meaningful solution [19] or be trapped in whatever local minimum is nearest to the given starting guess [21], [22]. This first approach could even fail to converge if the output characteristics or the corresponding objective functions are conflicting. (two or more waveforms are conflicting if there is no parameter set achieving the minimum fitting error for all of them at the same time.) A second approach is to combine all the objective functions into a single composite function, see, for example, [23]. In this case, construction of a single objective is possible with methods such as the weighted sum method [20], [24], [25], but the problem lies in the proper selection of the weights to characterize the decision-maker’s preferences. Compounding this drawback is that scaling amongst objectives is needed and small perturbations in the weights can sometimes lead to quite different solutions. Moreover, the weighted sum approach can fully explore the solution space only under some convexity assumptions of the underlying MOP [20]. Such optimization approach would return a single solution rather than a set of solutions that can be examined for tradeoffs. In a preliminary study, we applied these first two approaches to the parameter extraction of the power diode presented in this paper. We used a scalar optimization algorithm, the covariance matrix adaptation evolution strategy (CMA-ES) [26]. The CMA-ES is a very effective algorithm for difficult nonlinear, nonconvex black-box optimization problems [27]. However, due to highly conflicting dc and reverse recovery characteristics of the diode model, these approaches failed in providing a good simultaneous fitting of all the target characteristics. A third (and most complex) approach is to determine an entire optimal solution set or a representative subset. An optimal set for a MOP is a set of solutions that are nondominated with respect to each other. While moving from one nondominated solution to another, there is always a certain amount of tradeoff in one objective(s) to achieve a certain amount of gain in the other(s) [20]. This concept is illustrated in Fig. 2, which also shows that the solution set for a MOP can contain more than one point. Potentially, there can be countably or uncountably

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infinitely many solutions. The space which objective values are taken from is called the objective space. Nondominated solutions are also called Pareto optimal solutions. The set of all the feasible nondominated solutions in the MOP search space is referred to as the Pareto optimal set, and for a given Pareto optimal set, the corresponding objective function values in the objective space are called the Pareto front [28]. Due to the analytical and numerical complexity of the parameter extraction process, it is extremely difficult to identify the exact Pareto optimal set. Therefore, an approximated description of this set is an appealing alternative. MOEAs [2] are a particular class of approximation techniques, which are not affected by MOP convexity or nonconvexity and return a whole set of approximated nondominated solutions. EAs are a generic term for identifying several stochastic search methods, which computationally simulate the natural evolutionary process. In the EA terminology, a solution vector x is called an individual or a chromosome. Chromosomes are made of discrete units called genes, which control chromosomes’ features. In parameter extraction problems, an individual is a particular realization of the parameter set of a device model. Each model parameter represents a specific gene. Finally, a given set of chromosomes is termed a population. Usually, the population is randomly initialized. As the algorithm proceeds, the population includes fitter and fitter solutions. Just as in nature, EAs use three major operators to generate solutions with higher and higher fitness: recombination, mutation, and selection. In recombination, few chromosomes, called parents, are combined together to form new chromosomes, called offspring. The parents are selected among existing chromosomes in the population with preference toward fitness so that offspring is expected to inherit good genes which make the parents fitter. By iteratively applying the recombination operator, genes of good chromosomes are expected to appear more frequently in the population. The mutation operator introduces random changes into chromosomes’ gene and so assists the search to escape from local optima. Above-average individuals in the population are finally selected to become members of the next generation (more often than below-average) individuals. MOEAs are variants of EAs for MOP. In this study, we used the nondominated sorting genetic algorithm II (NSGA-II) [29], which is one of the literature’s most cited and imitated MOEAs (see, for example, [1]), and the multiobjective covariance matrix adaptation evolution strategy (MO-CMA-ES), which has been shown to have some additional benefits in comparison to the NSGA-II [3]. The general procedure of both the NSGA-II and the MO-CMA-ES is the following: 1) set t = 1. Randomly generate N solutions to form the first population P1 ; 2) generate an offspring population Qt from Pt by using recombination and mutation operators; 3) set Rt = Pt ∪ Qt , evaluate and assign a fitness value to each solution x ∈ Rt based on its objective function values and infeasibility; 4) select N solutions from Rt based on their fitness and copy them to Pt+1 ;

5) if the stopping criterion is satisfied, terminate the current search and return the current population, else, set t = t + 1 and go to step 2. Both in the NSGA-II and the MO-CMA-ES, fitness assignment is based on nondominated sorting. The population is ranked according to the Pareto dominance rule and, then, each solution is assigned a fitness value based on its rank in the population. A lower rank corresponds to a better solution in this case. Fig. 2 illustrates an example of the nondominated ranking method. In order to maintain a good spread of nondominated solutions, a second fitness criterion is used in the NSGA-II and the MO-CMA-ES to distinguish between solutions in the same nondominated front. In particular, the NSGA-II uses a crowdedcomparison approach [29], while the MO-CMA-ES uses the contributing hypervolume [3]. In the NSGA-II, the simulated binary crossover operator [30] and the polynomial mutation operator [31] are used to generate an offspring population. In the MO-CMA-ES, new candidate solutions are sampled according to multivariate normal distributions associated with parent solutions. The MO-CMA-ES is a method to update the covariance matrices of these distributions efficiently. This makes the method feasible on badly conditioned and even noncontinuous problems, as well as on noisy problems. Moreover, linear transformations of the search space do not affect the performance of this algorithm. In this study, we use two variants of the MO-CMA-ES, which improve its performance. In the first variant, which we call MO-CMA-ES-P, an improved step size (i.e., mutation strength) adaptation scheme is used [32]. In the second variant, which we call MO-CMA-ES-P-REC, this step size adaptation scheme is combined with a new method for the covariance matrix adaptation [33]. Almost all the routines have been implemented in Python, an open source language with extensive computing libraries. We used the inspyred library, a framework for creating biologically inspired computational intelligence algorithms in Python, including multiobjective evolutionary computation [34]. In particular, inspyred provides an easy-to-use canonical version of the NSGA-II for users who do not need much customization. Unfortunately, the MO-CMA-ES is not currently available in Python, so we created an interface between inspyred and the Shark machine learning library, a C++ framework which implements a basic version of the MO-CMA-ES [35]. Some routines had to be added to Shark in order to implement the MO-CMAES-P and the MO-CMA-ES-REC. III. DIODE MODEL In the lumped-charge approach, local hole and electron densities are lumped to specific locations in the device, called charge-storage nodes, approximating a continuous distribution of charge with a discretized one. Connection nodes are also needed to link junction voltage to the charge-carrier concentration level and are located on the junction depletion edge. The charge-storage nodes are then linked to internal voltages and currents by physical equations. Using this systematic approach,

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TABLE I PARAMETERS OF THE EXTENDED MA MODEL [11] Parameter

Fig. 3. Location of charge nodes in the p+ n− n+ diode structure. Electron and hole carrier concentration variables at each node and voltage drop along the device are also shown.

any power semiconductor device can be represented as a combination of the following elements: 1) p+ n− structure (or n+ p− ); 2) n− n+ structure (or p− p+ ); 3) the MOS structure. Over the years, diode [9]–[11] and IGBT [12], [36] models have been developed based on this approach [37]. In this study, we present a novel extension of a well-known lumped-charge diode model [10], which includes impact ionization and dynamic avalanche during the reverse recovery phase [11] and allows better fit of the dc characteristics. In the lumped charge model of [10], three nodes indicated by 2, 3, and 4 in Fig. 3 are chosen for the lightly doped n− region. Nodes 2 and 4 are connection nodes, while node 3 is a charge-storage node. The heavily doped end regions are discretized with nodes 1 and 5. They are both connection and charge-storage nodes. This model makes the following assumptions: 1) during forward condition, the drift component of the current injected in the drift region is approximated on the basis of the lowest carrier concentration (in the middle of the drift region); 2) during reverse recovery, the current generated by the Shockley–Read–Hall generation-recombination in the depleted region dominates; 3) average carrier lifetime is used in the base region. The main features of the model are 1) it accounts for dc and switching characteristics in low- and high-level injection conditions (with and without emitter recombination) and 2) it includes voltage-dependent reverse recovery. Note, however, that this model does not include self-heating and it does not scale with temperature. Therefore, it will be valid only for a given operating temperature at which the model parameters have been extracted. The model of [10] has been extended in [11] to include impact ionization, the Chynoweth law [38], [39] is used to calculate avalanche ionization rates and a linear approximation is chosen for the electric field in the space charge region [11], [19]. In order to numerically control excessive impact ionization currents of this approximate model at very large electric fields, a field dependent average lifetime has been introduced. The average lifetime takes an “artificial” higher recombination rate τH at fields larger than a given threshold Eth to self-limit the impact ionization. In this paper, we also suggest the use of two

Cj0 m Rs vj Tn 0 τL τH QB QB p NN NE Er φ B , IB b Eth ap , bp

Description

Unit

Zero bias junction capacitance Grading coefficient Series resistance Built-in potential Electron transit time Carrier lifetime at low electric field values Carrier lifetime at high electric field values Thermal equilibrium electron charge in the base Thermal equilibrium hole charge in the base Low current ideality parameter Medium-high current ideality parameter Symmetric end-region recombination parameter Voltage-dependent reverse-recovery parameters Mobility ratio of electrons and holes Threshold value of electric field Impact ionization parameters

F Ω V s s s C C

sC V, A Vm −1 m −1 , Vm −1

ideality parameters NN and NE in the p-n junctions equations of [10], which allow a better fit of different regions in the dc characteristics:   v 12 qp2 = QBp exp V T NN   v   45 −1 . qp4 = QB exp V T NE Table I lists the 18 model parameters. The model has been implemented in the Verilog-A hardware description language, leading to fast and stable SPICE simulations. IV. RESULTS A. Experimental Goals The objective of this study is to find a good parameter set for the extended Ma model in order to fit four separate characteristics obtained from TCAD simulations reproducing the behavior of an ABB fast diode [40]: 1) junction capacitance versus reverse-bias voltage; 2) DC forward-bias i-v characteristic; 3) reverse recovery current and reverse recovery voltage at a load current of 6000 A and a supply voltage of 3200 V. The reference temperature is 140 ◦ C. In the following, the term measurements will refer to this dataset. The complete parameter list of the extended Ma model is reported in Table I. The threshold value of the electric field Eth = 1.3 × 105 Vm−1 is taken from [41], whereas the impact ionization parameter bp is manually set to 1 × 105 V m−1 [19]. Sensitivity analysis shows that the shape of the capacitance characteristic is mainly influenced by the zero bias junction capacitance Cj0 , the grading coefficient m and the built-in potential vj0 [19]. These three parameters are then extracted by minimizing the RSM between measured and simulated capacitance data with the nonelitist CMA-ES algorithm [26]. The computed optimum is depicted in Fig. 4. The optimal values of Cj0 , m, and vj0 are kept constant in subsequent optimizations. The remaining 13 diode parameters

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Fig. 5. Hypervolume indicator measures the size of the dominated region, bounded by some reference point. Here four different sets A, B, C, D are shown by increasing order of darkness of the shaded region. The increasing order of darkness denotes a decreasing order of preference. A is different to B mainly in the extent, B is better than C in proximity to the Pareto front, and C is better than D mainly in evenness. Thus, the hypervolume indicator is capable of detecting differences in any of these different aspects. Fig. 4. Comparison between measured and optimized simulated capacitance characteristic.

are optimized to fit dc and transient characteristics. Sensitivity analysis shows that optimizations of the dc and transient characteristics are clearly conflicting. On the other hand, since the current and the voltage are physically coupled in the phenomena of diode turn-off, a good fitting of reverse recovery current is expected to yield a good fitting of the reverse recovery voltage as well. Therefore, we optimize only the dc and the reverse recovery current characteristics. Mathematically, this is translated into a MOP with two objectives to be minimized: 1) RSM between measured and simulated dc data. This error estimator helps to redress the imbalance between errors at low voltage values, where current values are also low, and errors at high voltage values, where the current is very high; 2) RSS between measured and simulated reverse recovery current data. This error estimator focuses more on matching the switching losses rather than the on-state losses [17]. We select three MOEAs to solve the MOP: NSGA-II, MOCMA-ES-P, and MO-CMA-ES-P-REC. Our aim is to investigate whether there is a best overall method for parameter extraction problems. B. Measures of Performance In this section, the term approximation set will be used as an alias for an approximated Pareto front returned by an optimizer. MOEAs are stochastic. If a stochastic multiobjective optimizer is applied several times to the same problem, each time a different approximation set may be returned. By running a specific algorithm several times on the same problem instance, one obtains a sample of approximation sets. Now, comparing two stochastic optimizers means comparing the two corresponding approximation set samples. There exist two basic approaches in the literature to analyze several samples of Pareto approximation sets.

The more popular approach first transforms the approximation set samples into samples of real values using quality indicators, the resulting samples of indicator values are compared based on standard statistical testing procedures. A quality indicator I is a function that assigns to each approximation set a real number. Thereby, we can compare two approximation sets A and B by comparing their corresponding indicator values; A is preferable to B, if and only if I(A) > I(B), assuming that the indicator values are to be maximized. Whenever an approximation set A dominates another approximation set B, the indicator value for A should be at least as good as the indicator value for B. Such indicators are called Pareto compliant. In this study, we use two Pareto compliant indicators: the hypervolume − 1 and the additive epsilon indicator Iε+ . indicator IH The hypervolume indicator IH measures the hypervolume of that portion of the objective space that is dominated by an approximation set A and has to be maximized (see Fig. 5) [2], [42]. In order to measure this quantity, the objective space must be bounded by some reference point. − (A) between We will consider the hypervolume difference IH − (A) = a reference set R and an approximation set A, i.e., IH IH (R) − IH (A). For this indicator, smaller values correspond to higher quality, in contrast to the original hypervolume IH . In − . the following, the term hypervolume indicator will refer to IH 1 The additive epsilon indicator Iε+ (A) has been introduced in [43]. It gives the minimum factor ε, which can be added to each point in any reference set R such that the resulting transformed approximation set is dominated by A. This indicator has to be minimized (see Fig. 6). Different quality indicators normally reflect different decision-maker preferences. A combination of Pareto compliant indicators can yield interpretations that are more powerful than what can be said by using a single indicator. In some cases, the hypervolume and the epsilon indicators may return opposite preference orderings for a pair of approximation sets A and B. − 1 and Iε+ are Pareto compliant, A and B In that case, since IH are incomparable, i.e., neither A dominates B nor B dominates A (see Fig. 6).

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Fig. 6. Two incomparable approximation sets A and B. Under the hypervolume indicator, B is the best set, but under the additive epsilon indicator A is better with respect to the reference set (the two X points connected by the dashed line), since Iε1+ (A) = 2 and Iε1+ (B) = 3. This discrepancy indicates that the two sets must be incomparable.

Regarding the choice of the reference set, the Pareto front is the ideal reference because it can yield more statistical power to the indicators. However, in parameter extraction problems the Pareto front is unknown. Therefore, in this study, the reference set is formed by the nondominated vectors among the combined approximation sets generated by all the optimizers. The advantage of this method is that the reference set dominates all the approximation sets; however, whenever additional approximation sets are included in the comparison, the reference set needs to be re-computed. The alternative approach for comparing the approximation sets of two stochastic optimizers, the attainment function method, summarizes a sample of approximation sets in terms of a so-called empirical attainment function (EAF) [2], [42], [44], [45]. Suppose that a certain stochastic multiobjective optimizer is run once on a specific problem. For each objective vector z in the objective space, there is a certain probability p that the resulting approximation set contains an objective vector that dominates z. We say that p is the probability that z is attained in one run of the considered algorithm. The true attainment function is usually unknown, but it can be estimated on the basis of the approximation set samples, one counts the number of approximation sets by which each objective vector is attained and divides the resulting number by the total number of approximation sets. EAFs can be used for visualizing the outcomes of multiple runs of an optimizer. For instance, one may be interested in plotting all the objective vectors that have been attained (independently) in k% of the runs. This is defined in terms of a k%-attainment surface. The k%-attainment surface divides the objective space into two parts: the objective vectors that have been attained with a frequency of at least k% (EAF ≥ k/100) and the objective vectors that have not been attained with a frequency of at least k% (EAF < k/100). These concepts are depicted in Fig. 7, which shows some results obtained by the MO-CMA-ES-P on the extended Ma model. For two-objective problems, plotting significant differences in the EAFs of two optimizers can be done by color-coding the levels of difference in the sample probability, as shown in Fig. 9. Using EAFs is complementary to the use of quality indicators, looking at the approximation set shape can provide an insight into the strengths and weaknesses of an optimizer, or provide information about

Fig. 7. Top: Approximation sets returned by eight runs of the MO-CMA-ES-P on the fitting of the dc and reverse recovery characteristics with the extended Ma model. Bottom: Corresponding EAF. The 25%, median(50%), 75% and worst (100%) attainment surfaces represent the objective vectors attained in at least 2, 4, 6 and 8 runs, respectively.

how it is working. For example, an optimizer may converge well in the center of the Pareto front only, or more at the extremes. C. Statistical Testing The statistical inference we would like to make, if it is true, is that one optimizer’s underlying approximation set distribution is better than another one’s. Boxplots are a first useful tool supporting statistical inferences made from quality indicator samples. In this study, the null hypothesis H0 always states that two samples A and B of indicator values are drawn from the same distribution or from distribution with the same mean value. The alternative hypothesis, HA states that sample A and sample B are from different distributions. We rely on nonparametric statistical tests, which make no assumptions about the distributions of the variables [46]. The initial population and the random seeds used by the MOEAs are matched in corresponding runs, so that the runs (and hence the final quality indicator values) are taken together. In this case, the statistical testing reveals whether there is a difference in the indicator value distributions given the same initial population, and the inference relates to each optimizer’s ability to improve the initial population. − 1 and the Iε+ We use the Friedman test to compare the IH distributions of the NSGA-II, the MO-CMA-ES-P and the

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MO-CMA-ES-P-REC [47]. If there is sufficient statistical evidence to reject the null hypothesis H0 , we can proceed with a posthoc test in order to find the concrete pairwise comparisons which produce differences. If we used the same group of approximation sets for all the comparisons, statistical tests would give misleading results. This problem is known in statistics as the multiple testing issue. Imagine that the same set of data are used to make five different inferences, and each has a significance level α of 0.05. The chance that at least one of the inferences will be a type-1 error (i.e., the null hypothesis is wrongly rejected) is 1 − (0.955 )  23%, when assuming that the null hypothesis is true in every case. In order to minimize the issue of multiple testing, we consider the two approaches. 1) Independent data are generated to investigate differences − 1 and Iε+ separately. in the distributions of IH 2) For each indicator, pairwise comparisons of all the MOEAs are performed on the same data, but the Bergmann–Hommel’s procedure is used to correct the p-values for the reduction in confidence associated with data reuse [48]. Such modified p-values can be compared directly with any chosen significance level α. D. Design of the Experiment For each MOEA, we choose a reasonable set of values and make no effort in finding the best strategy parameter setting. For the NSGA-II, we use a population size equal to 100 and a generation number of 250. These values are chosen according to the guidelines in [49], and represent a tradeoff between exploration of the objective space and computational effort requested by the optimizer. We set the mutation probability to pm = 0.08 and the crossover probability to pc = 0.9 [29]. The distribution indices of the mutation and crossover operators are set to ηm = 20 and ηc = 10, respectively. The same values are used in [50] to solve a pool of MOPs with a large number of objectives. In the two variants of the MO-CMA-ES, we set the population size μ to 100, the number of offspring per parent λ to 1, and the generation number to 250 to allow for a better comparison with the NSGA-II. For the strategy parameters of each individual, we use the same setting as in Igel et al. [3]. In particular, since we rescale all the diode model parameters between 0 and 1, the initial step size σ (0) is set to 0.3 [26]. For each indicator-MOEA pair, 30 MOEA runs are performed. Therefore, the size of the approximation set samples used for the statistical testing procedure is equal to 30, the total number of initial populations to be generated is np op = number of indicators · sample size = 2 · 30 = 60, and the total number of runs is nruns = number of indicators · sample size · number of MOEAs = 2 · 30 · 3 = 180. The whole randomized procedure to set up the MOEA comparison is described in [19]. E. Analysis of the Results In this section, the NSGA-II, the MO-CMA-ES-P, and the MO-CMA-ES-P-REC are compared using the techniques described in Section IV-B and IV-C. The whole MOEA performance assessment flow is shown in Fig. 8. The starting point

Fig. 8.

MOEA performance assessment flow.

is the set of runs for all combinations of initial populations and MOEAs. Three operations are performed on the approximation sets found by the optimizers. 1) Bound: This procedure calculates lower and upper bounds of all the objective vectors. 2) Normalize: Based on the bounds determined above, this procedure transforms all the objective vectors such that all values lie in the interval [1], [2]. Normalization is necessary in order to allow different objectives to contribute − 1 and Iε+ . approximately equally to IH 3) Filter: This procedure collects all normalized objective vectors from all populations and extracts the objective vectors which are nondominated. In this manner, a refer− 1 and Iε+ is generated. ence set for computing IH In Fig. 9, differences in the EAFs of the three optimizers are plotted. We can observe that 1) the NSGA-II converges better than the two variants of the MO-CMA-ES in the center of the Pareto front [see Fig. 9(a), left and Fig. 9(b), left], whereas the latter optimizers can better capture the whole spectrum of the Pareto front [see Fig. 9(a), right and Fig. 9(b), right]; 2) the MO-CMA-ES-P converges better than the MO-CMAES-P-REC in the center of the Pareto front [see Fig. 9(c)]. − and the additive As a next step, the hypervolume indicator IH 1 epsilon indicator Iε+ are computed for the collection of normalized approximation sets and the reference set. The distributions − 1 and Iε+ are shown in Fig. 10(a) and (b), respectively. of IH These boxplots suggest that there are performance differences between the two variants of the MO-CMA-ES and the NSGA-II − 1 and, independently, under Iε+ . This is confirmed by under IH the results given by the Friedman test. The p-values computed by − 1 and Iε+ are 2 × 10−10 Friedman test for the distributions of IH −8 and 4 × 10 , respectively. Since these p-values are extremely small, we can conclude that the optimizers are statistically sig− 1 and, independently, under Iε+ . nificantly different under IH The adjusted p-values computed by the Bergmann–Hommel’s dynamic test for pairwise comparisons between the MOEAs are shown in Table II. Under the hypervolume indicator, there is a strong statistical evidence against the following hypotheses (see Table II, column 2): 1) NSGA-II equivalent to MO-CMA-ES-P-REC; 2) NSGA-II equivalent to MO-CMA-ES-P.

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Fig. 9. Differences between the probabilities of attaining different goals with the NSGA-II and the two variants of the MO-CMA-ES on the problem of fitting the dc and reverse recovery characteristics with the extended Ma model. In each subfigure, the left side shows the values of the left EAF minus the right EAF and the right side shows the differences in the other direction. The values in the legends represent the levels of EAF differences that are plotted. The 0% and 100%-attainment surfaces over all data of each MOEA comparison are shown. In addition, the 50%-attainment surface of each dataset is plotted.

TABLE II ADJUSTED p-VALUES Hypothesis NSGA-II ≡ MO-CMA-ES-P-REC NSGA-II ≡ MO-CMA-ES-P MO-CMA-ES-P ≡ MO-CMA-ES-P-REC

Fig. 10. (a) Comparison between hypervolume indicator distributions. (b) Comparison between epsilon indicator distributions.

Under the epsilon indicator, only the NSGA-II and the MOCMA-ES-P are statistically significantly different (see Table II, column 3). Using an overall significance level α = 5%, there is enough statistical evidence to reject the following additional hypotheses: 1) NSGA-II equivalent to MO-CMA-ES-P-REC; 2) MO-CMA-ES-P-REC equivalent to MO-CMA-ES-P.

Adjusted p (hyp) Adjusted p (eps) 2 × 10 −8 2 × 10 −8 1

0.02 2 × 10 −8 5 × 10 −4

Thus, the hypervolume and the epsilon indicators do not agree on the statistical significance of performance differences between the optimizers. In conclusion, there is enough statistical evidence to argue that the MO-CMA-ES-P performs better than the NSGA-II under the hypervolume indicator and, independently, under the epsilon indicator. This can be due to the efficient adaptation of the search distribution in the MO-CMA-ES-P. Additionally, the hypervolume indicator and the epsilon indicator may contradict one another on the performance ordering of the NSGA-II and the MO-CMA-ES-P-REC. Therefore, it is likely for the approximation sets generated by these two algorithms to be incomparable. Finally, we are interested in measuring the convergence of solutions to the Pareto optimal front. The aim is to understand whether the generation number can be decreased, while maintaining good quality solutions or it has to be increased. − indicator. In order to measure convergence, we use the IH For each MOEA run, we compute the difference in hypervolume between the reference set and the current approximation set at each generation. Second, for each MOEA, we collect hypervolume values of corresponding generations among all the runs and we plot them using boxplots. In these figures, smaller values correspond to approximation sets that are closer to the reference set. Fig. 11 shows that, for each optimizer, the hypervolume distribution does not noticeably improve after 160 generations. A final choice has to be made among nondominated solutions computed by the MOEAs. An automated selection of a single preferred Pareto optimal solution can be performed through the reference point method by Deb and others [50].

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− Convergence study for the NSGA-II, the MO-CMA-ES-P, and the MO-CMA-ES-P-REC based on the IH indicator.

1) Provide a reference point z (in the objective space), representing target values of the fitting error between data and model. 2) Select the nondominated solution which minimizes the weighted Euclidean distance from the reference point  M   fi (x) − z i 2 wi d= fim ax − fim in i=1 where M is the number of objective functions, fi (x) is the ith objective value of the nondominated solution x, fim ax and fim in are the population maximum and minimum function values of the ith objective, and wi , i = 1, . . . , M , are weights that can be chosen in order to emphasize some objectives more than others. Fig. 12 shows some nondominated points found by the MOCMA-ES-P selected through manual inspection of the computed Pareto front. Reverse recovery voltage is plotted to check whether a good fitting of this characteristic is achieved even if it was not included in the optimization. The fitting error is calculated up to 12 μs. A decision maker could be interested in selecting the nondominated solution depicted in Fig. 12(b), since it represents a good tradeoff among all the objectives. This is in accordance with the strong coupling between current and voltage deriving from the variation during turn-off of the carrier densities within the diode. V. CONCLUSION In this paper, a novel lumped-charge diode is presented. The model has two extensions compared to the well-known power diode model of Ma et al. [10]: 1) modeling of impact ionization; 2) ideality factors that allow for better fit of different regions in the dc characteristics. Additionally, this model exhibits highly conflicting dc and reverse recovery (objective) characteristics, parameter set that fits well one characteristics fails when applied to the other. That is the reason why standard optimization techniques, such as choosing a random starting point, launching a scalar optimization algorithm and repeating until the fitting between data and model is good enough, fail to converge. In order to overcome this difficulty, a multiobjective optimization procedure has been

proposed. This procedure also offers other advantages with respect to the single objective optimization approaches. 1) It computes nondominated solutions which illustrate the capability of a model to represent the semiconductor device. 2) It identifies tradeoffs between fitting different aspects of the device (e.g. switching losses, voltage or current slopes, snappiness, specific “signatures” of physical phenomena such as avalanche generation and so on). These features cannot easily be identified or discriminated in a single objective approach. 3) It provides the decision maker with a family of models, rather than a single generic model, which can be examined to select the most accurate one for each specific simulation task. Two MOEAs, which have been proved to be robust for a wide range of multiobjective problems [1]–[3], the NSGA-II and the MO-CMA-ES, have been applied, for the first time, to the parameter extraction of the lumped-charge power diode model. Good agreement between the simulated and measured characteristics of the power diode shows that the model is accurate, and the proposed multiobjective optimization is as follows. 1) Robust: The user does not need either to compute initial guesses of diode model parameters or to tune MOEA parameters. 2) Effective: It can effectively solve parameter extraction problems with conflicting objectives, and it is not affected by objective space convexity or nonconvexity. It also returns several solutions in a single run, thus providing the decision maker a true picture of tradeoffs. 3) Simple and High Impact: A detailed understanding of neither the device compact model nor the EAs is requested. Once the physical ranges of the diode model parameters are defined, the user does not have to monitor the progress of the MOEAs. − and Two quality indicators, the hypervolume indicator IH 1 the additive epsilon indicator Iε+ , have been used to statistically compare the solutions generated by the MO-CMA-ES-P, the MO-CMA-ES-P-REC, and the NSGA-II. Using an overall significance level α = 5%, there is enough statistical evidence to assert that the two variants of the MO-CMA-ES algorithm − 1 and, independently, Iε+ . outperform the NSGA-II under IH

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Fig. 12. Some Pareto points found by the MO-CMA-ES-P on the extended Ma diode model. (a) Pareto point with the best fitting of the dc forward-bias i–v characteristic. (b) Best tradeoff between the objectives. (c) Pareto point with the best fitting of the reverse recovery current.

Surprisingly, the MO-CMA-ES-P-REC does not appear to be superior to the MO-CMA-ES-P, implying that combining step size adaptation with recombination of strategy parameters as proposed in [33] does not improve the performance of the MOCMA-ES. This could be due to a very rugged search landscape which prevents effective transfer of information between neighboring individuals. The performed statistical tests suggest that the MO-CMAES-P would be a preferred algorithm of choice, when applied to multiobjective parameter extraction of power diodes. In this algorithm, multiobjective selection is based on the contributing hypervolume, whose computation is exponential in the number of objectives [51]. We do not regard this scaling behavior as a severe drawback, because in real-world applications the costs for generating offspring and selection can often be neglected compared to the time needed for objective function evaluation. Calculation of complex nonlinear objective functions can be completed faster by decomposing the computational load across two or more processors. Evaluating more solutions in the same or reduced time may result in a larger or higher fidelity representation of possible outcomes. When dealing with the parameter extraction of other compact models, we suggest to start with the first two standard approaches described in Section II, which can be carried out by means of a scalar optimization algorithm [23]–[25]. If these procedures exhibit poor convergence characteristics without a good initial guess, or difficulty with simultaneous objective optimizations, the multiobjective optimization procedure proposed in this paper should be used, instead. Good fitting obtained by the NSGA-II and the MO-CMA-ES-P on the highly conflicting dc and reverse recovery characteristics of the power diode model we presented suggests that MOEAs are a very promising tool for the parameter extraction of any compact model. The NSGA-II and the MO-CMA-ES should also be “stress tested” by fitting as many electrical characteristics of a device as possible. One could also try to optimize several different devices at the same time. However, in this case, the optimization process could allow deficient fitting of one device, or indeed the circuit, to be compensated by incorrect parameter variation elsewhere. Hence, the models become more behavioral and less physical.

This may be acceptable for a single operating condition, but as the conditions will generally vary in actual circuit operation this will lead to inaccurate simulation results [17].

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[15] A. Bryant, P. Palmer, E. Santi, and J. Hudgins, “A compact diode model for the simulation of fast power diodes including the effects of avalanche and carrier lifetime zoning,” in Proc. IEEE 36th Power Electron. Spec. Conf., Recife, Brazil, Jun. 2005, pp. 2042–2048. [16] N. Jankovic, T. Pesic, and P. Igic, “All injection level power PiN diode model including temperature dependence,” Solid-State Electron., vol. 51, no. 5, pp. 719–725, 2007. [17] A. T. Bryant, X. Kang, E. Santi, P. R. Palmer, and J. L. Hudgins, “The use of a formal optimization procedure in automatic parameter extraction of power semiconductor devices,” in Proc. Power Electron. Spec. Conf., Acapulco, Mexico, Jun. 2003, vol. 2, pp. 822–827. [18] A. T. Bryant, X. Kang, E. Santi, P. R. Palmer, and J. L. Hudgins, “Twostep parameter extraction procedure with formal optimization for physicsbased circuit simulator IGBT and p-i-n diode models,” IEEE Trans. Power Electron., vol. 21, no. 2, pp. 295–309, Mar. 2006. [19] D. Prada, “Multiobjective optimization for parameter extraction of power electronics devices,” M.S. thesis, Dept. Math., Polytechnic of Milan, Milano, Italy, 2012. [20] M. Ehrgott, Multicriteria Optimization, 2nd ed. New York, NY, USA: Springer, 2005. [21] J. Nocedal, and S. Wright, Numerical Optimization (Springer Series in Operations Research and Financial Engineering), 2nd ed. New York, NY, USA: Springer, 2006. [22] M. Keser, and K. Joardar, “Genetic algorithm based MOSFET model parameter extraction,” in Proc. Tech. Int. Conf. Model. Simul. Microsyst., Mar. 2000, pp. 341–344. [23] M. J. Neath, A. K. Swain, U. K. Madawala, and D. J. Thrimawithana, “An optimal PID controller for a bidirectional inductive power transfer system using multiobjective genetic algorithm,” IEEE Trans. Power Electron., vol. 29, no. 3, pp. 1523–1531, Mar. 2014. [24] R. Yu, B. Pong, B. W. Ling, and J. Lam, “Two-stage optimization method for efficient power converter design including light load operation,” IEEE Trans. Power Electron., vol. 27, no. 3, pp. 1327–1337, Mar. 2012. [25] S. H. Kang, D. Maksimovi´c, and I. Cohen, “Efficiency optimization in digitally controlled flyback DC-DC converters over wide ranges of operating conditions,” IEEE Trans. Power Electron., vol. 27, no. 8, pp. 3734–3748, Aug. 2012. [26] N. Hansen, The CMA Evolution Strategy: A Tutorial, Cedex, Orsay, France, Jun. 2011. [27] N. Hansen, A. Auger, R. Ros, S. Finck, and P. Poˇs´ık, “Comparing results of 31 algorithms from the black-box optimization benchmarking BBOB2009,” in Proc. 12th Annu. Conf. Companion Genetic Evol. Comput., Jul. 2010, pp. 1689–1696. [28] R. Bosshard, J. W. Kolar, J. Muehlethaler, I. Stevanovic, B. Wunsch, and F. Canales, “Modeling and η-α-Pareto optimization of inductive power transfer coils for electric vehicles,” IEEE J. Emerg. Sel. Topics Power Electron., vol. PP, no. 99, Mar. 11, 2014. DOI: 10.1109/JESTPE.2014.2311302 [29] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast elitist multiobjective genetic algorithm: NSGA-II,” IEEE Trans. Evol. Comput., vol. 6, no. 2, pp. 182–197, Aug. 2000. [30] K. Deb and R. B. Agrawal, “Simulated Binary Crossover for Continuous Search Space,” Dept. Mech. Eng., Indian Inst. Technol. Kanpur, Kanpur, India, Rep. IITK/ME/SMD-94027, 1994. [31] K. Deb and M. Goyal, “A combined genetic adaptive search (GeneAS) for engineering design,” Comput. Sci. Informat., vol. 26, pp. 30–45, 1996. [32] T. Voß, N. Hansen, and C. Igel, “Improved step size adaptation for the MOCMA-ES,” in Proc. 12th Annu. Conf. Genetic Evol. Comput., Portland, OR, USA, Jul. 2010, pp. 487–494. [33] T. Voß, N. Hansen, and C. Igel, “Recombination for learning strategy parameters in the MO-CMA-ES,” in Proc. 5th Int. Conf. Evol. MultiCriterion Optim., 2009, pp. 155–168. [34] A. Garrett. (2012). A framework for creating bio-inspired computational intelligence algorithms in Python. [Online]. Available: https://pypi.python.org/pypi/inspyred [35] C. Igel, V. Heidrich-Meisner, and T. Glasmachers. (2012). Shark: A C++ machine learning library. [Online]. Available: http://sharkproject.sourceforge.net/index.html [36] F. Chimento, N. Mora, M. Bellini, I. Stevanovi´c, and S. Tomarchio, “A simplified spice based IGBT model for power electronics modules evaluation,” in Proc. IEEE 37th Annu. Conf. Ind. Electron. Soc., Melbourne, VIC, Australia, Nov. 2011, pp. 1155–1160. [37] P. O. Lauritzen. (2014). Compact Models for Power Semiconductor Devices. [Online]. Available: http://www.ee.washington.edu/research/ pemodels [38] A. G. Chynoweth, “Ionization rates for electrons and holes in silicon,” Phys. Rev., vol. 109, pp. 1537–1540, 1958.

[39] J. Lutz, H. Schlangenotto, U. Scheuermann, and R. D. Doncker, Semiconductor Power Devices. New York, NY, USA: Springer, 2011. [40] Fast diode die 5SLY 12N4500, ABB Switzerland Ltd, Semiconductors, Lenzburg, Switzerland, Jan. 2008. [41] R. V. Overstraeten and H. D. Man, “Measurement of the ionization rates in diffused silicon p-n junctions,” Solid-State Electron., vol. 13, pp. 583–608, 1970. [42] J. D. Knowles, L. Thiele, and E. Zitzler, “A tutorial on the performance assessment of stochastic multiobjective optimizers,” Comput. Eng. Netw. Lab., ETH Z¨urich, Z¨urich, Switzerland, TIK- 214, Feb. 2006. [43] E. Zitzler, L. Thiele, M. Laumanns, C. M. Fonseca, and V. G. da Fonseca, “Performance assessment of multiobjective optimizers: An analysis and review,” IEEE Trans. Evol. Comput., vol. 7, no. 2, pp. 117–132, May 2003. [44] V. G. da Fonseca and C. M. Fonseca, “The attainment-function approach to stochastic multiobjective optimizer assessment and comparison,” in Experimental Methods for the Analysis of Optimization Algorithms. Berlin, Germany: Springer, 2010, pp. 103–130. [45] M. L´opez-Ib´an˜ ez, L. Paquete, and T. St¨utzle, “Exploratory analysis of stochastic local search algorithms in biobjective optimization,” in Experimental Methods for the Analysis of Optimization Algorithms. Berlin, Germany: Springer, 2010, pp. 209–222. [46] J. W. Conover, Practical Nonparametric Statistics, 3rd ed. New York, NY, USA: Wiley, 1999. [47] J. Demˇsar, “Statistical comparisons of classifiers over multiple data sets,” J. Mach. Learn. Res., vol. 7, pp. 1–30, 2006. [48] S. Garc´ıa and F. Herrera, “An extension on ‘Statistical comparisons of classifiers over multiple data sets’ for all pairwise comparisons,” J. Mach. Learn. Res., vol. 9, pp. 2677–2694, 2008. [49] K. Deb. (2011, July. 8). Multi-objective NSGA-II code in C. Kanpur Genetic Algorithms Laboratory. [Online]. Available: http://www.iitk.ac.in/kangal/codes.shtml [50] K. Deb, J. Sundar, U. B. Rao, and S. Chaudhuri, “Reference point based multi-objective optimization using evolutionary algorithms,” Int. J. Comput. Intell. Res., vol. 2, no. 3, pp. 273–286, 2006. [51] L. While, “A new analysis of the LebMeasure algorithm for calculating hypervolume,” in Proc. 3rd Int. Conf. Evol. Multi-Criterion Optim., 2005, vol. 3410, pp. 326–340.

Daniele Prada received the M.Sc. degree in mathematical engineering from Politecnico di Milano, Milano, Italy, in 2012. He is currently working toward the graduate studies at the Indiana University–Purdue University (IUPUI), Indianapolis, IN, USA. In 2011, he was a Student Intern with ABB Corporate Research, Baden-D¨attwil, Switzerland, where he worked on the subject of multiobjective optimization for parameter extraction of power electronics devices. Mr. Prada received the School of Science Fellowship Award for graduate study toward the Ph.D. degree in mathematics from IUPUI in 2013. He also received an Outstanding Beginning Graduate Student Award in Mathematics presented by the Department of Mathematical Sciences, IUPUI in 2014.

Marco Bellini received the M.Sc. degree in electrical engineering from Politecnico di Milano, Milano, Italy, in 2002, and the Ph.D. degree from the Georgia Institute of Technology, Atlanta, GA, USA, in 2009, as a Fulbright Grantee. Since 2009, he has been with ABB Corporate Research Centre, Baden-D¨attwil, Switzerland, as a Principal Scientist, working on the power semiconductor related projects. His main research interests include semiconductor design, simulation, and compact modeling. He has authored and coauthored more than 40 scientific publications and a book chapter.

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Ivica Stevanovi´c (S’03–M’05–SM’12) received the Dipl.Ing. degree from the University of Belgrade, Belgrade, Serbia, in 2000, and the Ph.D. degree from the Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland, in 2005, both in electrical engineering. He held research and teaching positions in academia, including the California Institute of Technology, Pasadena, CA, USA, in 2000, and EPFL from 2000 to 2006, before joining the industrial R&D with Freescale Semiconductor, Geneva, Switzerland, from 2006 to 2008, as a Senior R&D Engineer, and with ABB Corporate Research, Baden-D¨attwil, Switzerland, from 2008 to 2013, as a Principal Scientist. Since 2013, he has been with the Federal Office of Communications, Biel/Bienne, Switzerland, where he is currently a Radio Spectrum Specialist. His research interests include numerical methods applied to electromagnetic modeling and to statistical and compact modeling of integrated circuits, with more than 70 scientific publications. Dr. Stevanovi´c received the National Science Foundation-Research Experiences for Undergraduates Award presented by Caltech in 2000.

Laurent Lemaitre received the M.S.E.E. degree in electrical engineering from the Institut National Polytechnique de Grenoble, Grenoble, France, and the ´ Ph.D. degree in electrical engineering from the Ecole Polytechnique F´ed´erale de Lausanne (EPFL), Lausanne, Switzerland, in 1987 and 1994, respectively. He is currently working as a Freelance EDA Consultant. His main interests include development and implementation of EDA solutions for RF design applications. He contributed to the standardization effort of the Verilog-AMS language. He built software Verilog-A interfaces for different spice simulators.

James Victory (M’90–SM’01) received the B.S.E.E., M.S.E.E., and Ph.D. degrees in electrical engineering from Arizona State University, Phoenix, AZ, USA, in 1990, 1992, and 1994, respectively. He is currently a Distinguished Member of the Technical Staff with Fairchild Semiconductor, Aschheim, Germany, working on research and development of physical models and design enablement platforms for discrete technologies. In June 2008, he cofounded Sentinel IC Technologies Inc., to develop and deploy innovative design solutions for RF and power technologies. He was the Executive Director of IC Design Enablement at Jazz Semiconductor from 2003 to 2008. From 2001 to 2003, he was the Director of Semiconductor Technology at XtremeSpectrum. He was with Motorola from 1992 to 2001, where he specialized in semiconductor device modeling for circuit simulation of RF analog and power technologies. He has more than 35 publications, including invited papers and workshop tutorials on semiconductor device modeling and design enablement for RF, power, analog, mixed-signal, and nanoscale technologies. He holds two U.S. patents.

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Jan Vobeck´y (M’92–SM’01) received the Graduation degree in electrotechnology, the Ph.D. degree in microelectronics, and the Dr. Sc. degree in electronics and vacuum technique from the Czech Technical University (CTU), Prague, Czechoslovakia, in 1981, 1988, and 1999, respectively. He was an Associate Professor and Full Professor at CTU in 1992 and 2000, respectively. His research interests include technology, characterization, and simulation of power devices with emphasis on lifetime and defect engineering. He was a Manager of several tens of projects with industrial partners in the field of power devices, device simulation, and testing of microcontrollers. Since 2007, he has been a Principal Engineer in ABB Switzerland Ltd. Semiconductors, he is currently focusing on bipolar technologies. He is developing next-generation power devices and related technologies. He is the author of about 20 patents and several hundred technical papers in the field of power devices. Dr. Vobeck´y was the President from 2002 to 2003 and the Vicepresident from 2004 to 2006 of Region 8 CS Section I EE. Since 2006, he has been the Member of Electron Devices Society AdCom (SRC Vicechair). Since 2010, he has also been the IEEE EDS Newsletter Regional Editor.

Riccardo Sacco received the M.Sc. degree in electronic engineering from Politecnico di Milano, Milano, Italy, in 1989, and the Ph.D. degree in applied mathematics from the University of Milan, Milano, in 1993. Since 2001, he has been an Associate Professor in numerical analysis at Politecnico di Milano. He has coauthored more than 70 papers in international peer-reviewed journals and conference proceedings, and a book on Numerical Mathematics (New York, NY, USA: Springer-Verlag) in the TAM Series, that has been translated in four languages. He is also the coauthor of a chapter on the Mathematical Modeling and Numerical Simulation of Semiconductor Devices in the Handbook of Numerical Analysis (New York, NY, USA: Elsevier) and of a chapter on Multiphysics Computational Modeling in Cartilage Tissue Engineering published by in the Series Studies in Mechanobiology, Tissue Engineering, and Biomaterials (New York, NY, USA: Springer-Verlag, 2009). His research interests include multiscale/multiphysics modeling and simulation of problems in applied sciences using the finite-element method. Covered applications include: Semiconductor devices in nano and optoelectronics, nonlinear fluid-mechanics, computational biology, and bioengineering.

Peter O. Lauritzen (S’58–M’62–SM’88–LSM’00) was born in Valparaiso, IN, USA, on February 14, 1935. He received the B.S. degree from the California Institute of Technology, Pasadena, CA, USA, and the M.S. and Ph.D. degrees from Stanford University, Stanford, CA, USA, all in electrical engineering, in 1956, 1958, and 1961, respectively. From 1961 to 1965, he worked on semiconductor device research and development with Fairchild Semiconductor Division, Palo Alto, CA. From 1965 to 1998, he was with the Electrical Engineering Faculty at the University of Washington, Seattle, where his research interests include semiconductor device noise, radiation effects, and power electronic device protection and modeling. He was the conference Chair for the IEEE Power Electronics Specialist Conference in 1993. He was a Fulbright Senior Lecturer at IIT Madras, Madras, India, in 1997, and a Danfoss Visiting Professor at Aalborg University, Denmark, in 1999. He is currently a Professor Emeritus at the University of Washington, Seattle, WA, USA.