**Forecasting the Reliability of Components Subjected to Harmonics Generated by Power Electronic Converters**

#### **Giovanni Mazzanti 1,\* , Bassel Diban <sup>1</sup> , Elio Chiodo <sup>2</sup> , Pasquale De Falco <sup>3</sup> and Luigi Pio Di Noia <sup>4</sup>**


Received: 18 May 2020; Accepted: 3 August 2020; Published: 7 August 2020

**Abstract:** This paper aims at refining an experimentally based reliability model for the insulation of power components subjected to the randomly varying harmonics generated by power electronic converters. Compared to previous papers of the same authors and to the existing literature, here the model is re-formulated from the theoretical viewpoint focusing on the foremost role played by low percentiles of time to failure—in particular by the 1st percentile—selected as the rated life in the framework of modern probabilistic design of components. This is not only more correct from the viewpoint of component design, but also on the safe side as for the reliability of devices. Moreover, the application of the model is broadened to treat the whole sequence of odd voltage harmonics from the 5th to the 25th, i.e., those taken as the most significant in power systems according to international standards. The limits to voltage distortion set in Standard EN50160 are the reference for establishing parametrically a series of typical distorted voltage waveshape analyzed in the applicative part, which account for the possible phase-shift angles between voltage harmonics. The effect of current harmonics is also considered, from both the theoretical and applicative viewpoint. As a last, but not least novelty, the reliability model is used here for life and reliability estimates not only of Medium Voltage (MV)/Low Voltage (LV) capacitors and cables—already studied in the previous stages of this investigation—but also of induction motors and transformers in the presence of harmonics from power converters.

**Keywords:** current harmonics; voltage harmonics; power electronic converters; reliability; cables; capacitors

#### **1. Introduction**

The worldwide diffusion of electric transportation systems and of smart grid technologies call for better performance of power electronic converters and components. However, in turn power electronic devices are well known to act as distorting loads, which inject voltage and current harmonics into the alternating current (AC) grid. These harmonics may hamper the reliability of power components—such as cables, capacitors, transformers, electrical motors—connected to the grid, because of the potential increase of thermal and electrical stress associated with current and voltage harmonics. Such a situation raises the reliability challenges to power components in electric transportation systems and smart grid installations. For this reason, in the last three decades some international standards have set distortion limits. IEEE 519 in 1993 [1] established limits to voltage and current harmonics, but later on IEC 61000-2-2 [2], IEC 61000-2-4 [3], EN 50160 [4] set limits to voltage distortion only, by fixing the maximum values of low-order voltage harmonics in LV/MV grids in unperturbed conditions.

Forecasting the reliability of the components in the presence of current and voltage harmonics is not an easy task. Traditional approaches rely on accelerated life testing (ALT) and on historical failure databases, but the fast technological development-leading to highly reliable devices-and the difficulty of performing sound testing campaigns under harmonic distortion make these data hardly available. The estimation of reliability in distorted conditions is made more difficult by the random nature of harmonic distortion brought about by power electronic devices. This requires proper statistical models and methods capable of correlating the electric and thermal stress, associated with voltage and current harmonics, to life and the reliability of components.

To overcome all these difficulties, a probabilistic electro-thermal life model—that can also be referred to as "electro-thermal reliability model" [5]—has been developed and proved to be capable of forecasting the life and reliability of components subjected to randomly time-varying harmonics generated by power electronic converters [6–9]. This model is based on a broad and innovative experimental campaign of testing insulating specimens (flat samples, mini-cables, twisted pairs) for power components (cables, capacitors, transformers, rotating machines) subjected to a big deal of combinations of voltage harmonics, as shown in [10–13]. Other tests of this kind are described e.g., in [14–18], but with particular emphasis on water tree growth in cross-linked polyethylene (XLPE) insulation for power cables in a wet environment. However, results of aging tests of insulation under distorted voltage are rare in the literature, due to the experimental difficulties in arranging test set-ups for aging of insulation in the presence of distorted voltage. For this reason, attention in the literature of reliability models under harmonic distortion is mainly concentrated on the thermal effect of current harmonics [19–23] although applications relevant to the electrical aging can also be found e.g., in [24–30].

Following the streamline of this investigation over the years, in the very first application of this electro-thermal reliability model for insulation under harmonic distortion the level of distortion was either set parametrically [6] or derived experimentally for a particular case [7]. Then, the model was used to estimate the reliability of MV/LV components affected by voltage harmonics matching exactly the limits established in [4] (which are numerically the same as those in [2,3]); the study was broadened from the 11th, 13th voltage harmonics treated in [8], to the combination of the 5th, 7th, 11th, 13th treated in [9], showing that—notwithstanding the compliance with [4]—the reliability of components decreased significantly with respect to rated sinusoidal conditions.

Here, as the ultimate stage of the investigation specifically conceived for this Special Issue, the reliability model is refined and better formulated from the theoretical viewpoint. Furthermore, its application is significantly broadened in this paper, since here many new case studies are examined and two more power system components (induction motors and transformers) are treated in addition to those (cables and capacitors) already studied in the previous stages of this investigation. The main novelties in this paper compared to previous papers devoted to the same topic [8,9] are as follows.

From the theoretical viewpoint, in our previous papers and in the existing literature the theory of life and reliability estimation of the insulation of components in distorted conditions relied on the 63.2th percentile of failure time and on the mean time to failure (MTTF). Here, in the theoretical treatment the 63.2th percentile is replaced with design life, *LD*, given at design failure probability, *PD*. As a consequence, the theory is focused on a conservatively-low percentile of times to failure (in particular on the 1st, see below): this is both more correct from the viewpoint of modern probabilistic design of power components and on the safe side as for power component reliability [5]. Furthermore, the whole theory is formalized more extensively and carefully, with a more detailed treatment of the

effect of current harmonics and of a possible increase of thermal and electric stress associated with the sinusoidal components of voltage and current.

From the applicative viewpoint, a first novelty compared to previous papers is the treatment of both the 5th, 7th, 11th, 13th voltage harmonics already studied in [9], and of the 17th, 19th, 23rd, 25th, tackled here for the first time. Such harmonics are characteristic of 6-pulse—some of them of 12-pulse—alternating/direct current (AC/DC) converters; they play a major role, since are not only the highest in the spectra of voltage and current harmonics typically measured at the bus-bars of MV/LV grids [31,32], but also those for which Standards IEC 61000 and EN50160 set limits [2–4].

Another applicative novelty is that in our previous papers and in the existing literature the calculations were concentrated on the MTTF (design life itself was given as the design value of the MTTF) and only a spot estimation of reliability was carried out at a service time equal to design life [8,9]; moreover, the amplitudes of voltage harmonics were selected so as to match the limits in [4] exactly, only. Here, on the contrary—consistently with what is said above—the calculation is focused on the 100 × *PD*th percentile of times to failure, and for failure probability a conservatively-low value *PD* = 0.01 = 1% is chosen to be on the safe side as for power component reliability. One more novelty is that, pragmatically, design life *LD* of power components is the typical service life of power systems affected by harmonics where the components are located. In addition, as for reliability estimation, reliability is evaluated throughout the service life of power components. Regarding the selected amplitude of voltage harmonics, beside the case where the amplitudes of voltage harmonics match the limits after EN 50160 exactly, two more cases are considered where voltage harmonics are 25% above and 25% below the limits in [4], so as to illustrate respectively the problems that may arise if these limits are overcome and the problems which may still remain even if these limits are matched with an apparently-broad safety margin—e.g., resorting to passive and/or active filters [31].

As a last, but not least applicative novelty-as hinted at above—two more components of power systems, i.e., induction motors and MV/LV transformers, are studied here in the presence of harmonics from power electronic converters in addition to cables and capacitors, already studied in the previous stages of this investigation.

On the whole, it is worth emphasizing that the calculations in this paper are completely new compared with those in previous ones [8,9], that have intentionally been cited here to allow a direct comparison. As highlighted hereafter at Section 3 in comprehensive tables for the cable and the capacitor, as well as for the motor and the transformer, such a comparison shows that all results and figures in this paper—although being intentionally homologous to those in previous papers for enabling a straightforward comparison—are not only different as for the values obtained and the curves plotted, but also on the safe side with respect to those in [8,9]. This highlights the need for the more accurate analysis performed for the first time in this paper vs. the simpler one carried out in previous papers. This overall comparison is also opportune in a Special Issue paper, which can take the chance for reviewing and completing previous investigations so as to outline the state of the art and the ultimate achievements in this field.

Last but not least, an aspect to mention as a closure of this Introduction is that—as hinted at above—similar models were also used focusing on the effect of either current harmonics on thermal aging as in [33,34], or voltage harmonics on electrical aging as in [35,36], with results which agree with those found in this investigation. However, the full application to electrical and thermal aging of insulation in under current and voltage harmonics has been carried out only in the streamline of the development and refinement of the electro-thermal reliability model used here, which finds in this paper its conclusive application.

#### **2. Theoretical Background of Insulation Aging in the Presence of Voltage and Current Harmonics**

The weakest part of a power system component is mostly its insulation [37]. In the presence of the harmonic distortion generated by a power electronic device, a reduction of insulation life at a given failure probability—or, conversely, of insulation reliability at a given service time—may be observed vs. rated sinusoidal life and/or reliability due to a possible rise of [6,7]:


Let us now treat cases 1 and 2 separately (Sections 2.1 and 2.2), then combine them (Section 2.3) and later on recast them into a probabilistic time-varying framework (Sections 2.4 and 2.5).

#### *2.1. The Role Played by Current Harmonics*

Treating case 1 of Section 2 first, let us assume that a power electronic device generates *M* current harmonics (of root mean square (rms) value *Ih*, *h* = 1, ... , *M*), and that such current harmonics lead to a non-negligible increase (current harmonics superimposed to fundamental current can only increase the losses in conducting parts, thus the temperature of the component. Of course, such an increase might be negligible, i.e., non-measurable.) Δ*Tarm* of the temperature (herein, all temperatures *T* are meant in K, while the corresponding temperatures in degrees Celsius are indicated as θ) of the insulation of a nearby power component. Let us further hypothesize that the temperature of the insulation of the affected component increases from the nominal–design–sinusoidal temperature *TS* to a "non-sinusoidal" temperature *TNS* equal to:

$$T\_{\rm NS} = T\_s + \Delta T\_{arm} \tag{1}$$

Then, the following thermal life model can be used for the estimation of the time to failure of the insulation of the power component in the presence of the *M* current harmonics generated by the power electronic device and affecting the component [6–9]:

$$L\_{NS,I} = L\_S \exp(-B\Delta T\_{harm}') \tag{2}$$

where:


$$
\Delta T\_{harm}' = \frac{1}{T\_S} - \frac{1}{T\_{NS}} = \frac{1}{T\_S} - \frac{1}{\left(T\_S + \Delta T\_{arm}\right)}\tag{3}
$$

From (3) it is readily seen that the greater the increase Δ*Tarm*—if any—of the temperature of the insulation, the shorter is life in the presence of current harmonics.

Of course, the use of model (2) requires that the relationship between the *M* current harmonics and the relevant variation Δ*Tarm* of the temperature of the insulation is known.

#### *2.2. The Role Played by Voltage Harmonics*

Coming now to case 2 of Section 2, let us assume that a power electronic device generates *N* voltage harmonics (of rms value *Vh*, *h* = 1, ... , *N*), and that such voltage harmonics distort non-negligibly the nominal–design–sinusoidal voltage waveshape at power frequency applied to the insulation of a nearby component connected to the AC power grid. The distortion is said to be "non-negligible" if at least one of the three following quantities associated with the distorted voltage waveshape change sensibly (i.e., in a measurable way) from their nominal sinusoidal value:

1. the rms value of voltage, which changes from the nominal sinusoidal rms value *VS* = *V*1,*<sup>n</sup>* (rms nominal sinusoidal voltage of harmonic order *h* = 1 or rms fundamental voltage at power frequency (For components connected directly to the AC power grid, the frequency of the fundamental harmonic *f* <sup>1</sup> coincides with the power frequency, *f* <sup>0</sup> = 50/60 Hz.)) to *VNS*, the rms value of distorted voltage. Then, the rms value of distorted voltage in p.u. of the rms value of rated sinusoidal voltage, *vNS*, can be written as follows:

$$v\_{\rm NS} = \frac{V\_{\rm NS}}{V\_{\rm S}} = \frac{V\_{\rm NS}}{V\_{1,n}}\tag{4}$$

2. the peak value of voltage, which changes from the peak value of the nominal sinusoidal voltage *VS,p* = *V*1*,n,p* to *VNS,p*, the rms value of distorted voltage. Then, the peak value of distorted voltage in p.u. of the peak value of rated sinusoidal voltage, *vNS,p*, can be written as follows:

$$v\_{NS,p} = \frac{V\_{NS,p}}{V\_{S,p}} = \frac{V\_{NS,p}}{V\_{1,n,p}}\tag{5}$$

3. the rms value of the derivative of voltage, which changes from the rms value of the derivative of nominal sinusoidal voltage *hV*1,*<sup>n</sup>* = *V*1,*<sup>n</sup>* to the rms value of the derivative of distorted voltage, where *N* different voltage harmonics of order *h* are now present. Then, by defining the ratio "rms value of the hth voltage harmonics" over "rms fundamental voltage at power frequency" α*<sup>h</sup>* = *Vh*/*V*1,*n*, the rms value of the derivative of each of these voltage harmonics in p.u. of the rms value of nominal sinusoidal voltage can be written as follows:

$$hV\_h/V\_{1,n} = h\alpha\_h\tag{6}$$

The distortion of the voltage waveshape, in turn, may result in an increase of the electric stress acting on the insulation of the power component. Such increase—if any—is due to the distorted electric field (of rms value *ENS*) within the insulation caused by the non-sinusoidal voltage (of rms value *VNS*). The increase of the electric stress in distorted regime is quantified by the so-called "voltage waveshape factors", which can be defined from Equations (4)–(6) as follows [6–10]: peak factor:

$$K\_p = \upsilon\_{NS,p} \tag{7}$$

waveshape factor (or slew rate) (If—differently from here—one wants to study the reliability of the insulation of a component not directly connected to the AC power grid—e.g., a PWM inverter controlling a three-phase motor whose working frequency *f* <sup>1</sup> is different from power frequency *f* 0, then the frequency of the fundamental harmonic is *f* <sup>1</sup> and the ratio *f* <sup>1</sup>/*f* <sup>0</sup> should appear as a multiplying factor of the right-hand side of Equation (8)):

$$\mathcal{K}\_w = \sqrt{\sum\_{h=1}^{N} h^2 \alpha\_h^2} \tag{8}$$

rms factor:

$$\mathbb{K}\_{\mathbb{Y}} = v \tag{9}$$

Equations (7)–(9) clearly demonstrate that, when nominal sinusoidal voltage is applied, the three voltage waveshape factors are equal to unity, namely *Kp* = *Kw* = *Kr* = 1. By contrast, when *N* voltage harmonics are superimposed onto nominal sinusoidal voltage, it is observed that:


$$THD\_v = \sqrt{\sum\_{h=2}^{N} h^2 \alpha\_h^2} = \sqrt{K\_w^2 - 1} \tag{10}$$

Then, the following electrical life model can be used for the estimation of the time to failure of the insulation of a power component in the presence of the *N* voltage harmonics generated by the power electronic device [6–10]:

$$L\_{NS,V} = L\_S K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \tag{11}$$

where:


From Equation (11) it is readily seen that the greater the values of *Kp*, *Kw*, *Kr* with respect to unity, the shorter is life in the presence of voltage harmonics. However, it must be highlighted that—as pointed out above and in [36]—it is not granted that a distorted voltage waveshape necessarily leads to a value of *Kp* > 1, as clearly shown in Section 3.

#### *2.3. The Combination of Current and Voltage Harmonics*

When a power system component is subjected to both current and voltage harmonics generated by a power electronic device, then a combination of the life models (2) and (11) is required for a thorough and complete evaluation of the effect of distorted current and voltage on the life of the component. By combining these models, the following electro-thermal life model for distorted conditions is obtained:

$$L\_{NS} = L\_S \exp(-B\Delta T\_{harm}') \mathcal{K}\_p^{-n\_p} \mathcal{K}\_w^{-n\_w} \mathcal{K}\_r^{-n\_r} \tag{12a}$$

where *LNS* is insulation life in the presence of non-sinusoidal voltage, *VNS*, and non-sinusoidal temperature *TNS*.

A careful analysis of Equations (4)–(11) emphasizes that voltage waveshape factors *Kp*, *Kw*, *Kr* are defined with respect to nominal sinusoidal voltage, hence life models (11) and (12) implicitly assume that the fundamental component of the distorted voltage waveshape (of rms value *V*1) is equal to the rated sinusoidal voltage (of rms value *VS* = *V*1,*n*). However, it might happen that the fundamental component of distorted voltage has rms value *VH* higher-than-rated sinusoidal voltage *VS*, for instance because a highly capacitive load—e.g., a capacitor bank—is also supplied by the voltage source (only an increase of sinusoidal voltage is considered, since a lower than rated sinusoidal voltage is not common in unperturbed conditions, even under full inductive load; moreover considering a decrease of sinusoidal voltage is not on the safe side as for the estimation of life and reliability of power components). This situation tends to reduce the life of the insulation of power components with respect to nominal sinusoidal life, too: not because of voltage harmonics, but because a higher-than-rated sinusoidal electric field, *EH*, is applied to the insulation. Such an effect has to be accounted for, resorting to the well-known Inverse Power Law electrical life model (IPM), that can be written as follows:

$$L\_H = L\_S (E\_H / E\_S)^{-n\_S} \tag{12b}$$

where:


Thereafter, by combining Equations (12a) and (12b), one obtains the following comprehensive life model:

$$L\_{NS} = L\_S (E\_H / E\_S)^{-n\_S} \exp(-B\Delta T\_{harm}') K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \tag{13}$$

In addition, when looking carefully at Equations (1)–(3) it can be argued that non-sinusoidal temperature *TNS* is defined with respect to nominal sinusoidal temperature *TS*, hence life models (2), (12) and (13) implicitly assume that the fundamental component of the distorted current waveshape (of rms value *I*1)—let us call it "sinusoidal current"—is equal to the rated sinusoidal current (of rms value *IS* = *I*1,*n*). However, it might happen that the fundamental component of distorted current has rms value *IHL* higher/lower than rated sinusoidal current *IS*, for instance due to a temporary overload/underload—e.g., because linear loads demand a higher/lower-than-rated current. This situation tends to reduce/increase the life of the insulation of the power component with respect to nominal sinusoidal life, too: not because of current harmonics, but because a higher/lower-than-rated "sinusoidal" temperature, *THL*, is applied to the insulation. If this is the case, let us write *THL* as follows:

$$T\_{HL} = T\_S + \Delta T\_{HL} \tag{14a}$$

where Δ*THL* is higher/lower than zero depending on whether linear loads demand a higher/lower-than-rated current. Moreover, let us introduce the overall temperature variation Δ*Ttot* with respect to nominal sinusoidal temperature *TS* resulting from:


Thus Δ*Ttot* can be written as follows:

$$
\Delta T\_{\text{tot}} = T\_{\text{S}} + \Delta T\_{\text{arm}} + \Delta T\_{\text{HL}} \tag{14b}
$$

and a quantity Δ*T*- *tot*—analogous to Δ*T*- *trm* defined in Equation (3)—can be introduced, as follows:

$$T\_{\rm tot}' = 1/T\_S - 1/\left(T\_S + \Delta T\_{\rm arm} + \Delta T\_{\rm HL}\right) = 1/T\_S - 1/\left(T\_S + \Delta T\_{\rm tot}\right) \tag{15}$$

Therefore, by combining Equation (13) with Equations (14)–(15) one obtains the following electro-thermal life model holding in the presence of voltage and current harmonics, as well as of higher-than-rated sinusoidal voltage and current [9]:

$$L\_{NS} = L\_S \left( E\_H / E\_S \right)^{-n\_\S} \exp(-B \Delta T\_{tot}') K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \tag{16}$$

Model (16) is a comprehensive relationship which accounts for:


#### *2.4. Reliability Model in the Presence of Current and Voltage Harmonics*

The electro-thermal breakdown of the insulation of a power component is an inherently random phenomenon. This is due first and foremost to the non-uniform composition of the dielectric material which constitutes the insulation, as well as to the randomly distributed defects within the insulation itself. A further reason for the random behavior of insulation breakdown is the uncertainty of the values of applied electrical and thermal stress [40,41]. For these reasons, all the above models—and in particular the general electro-thermal life model (16) holding in the presence of voltage and current harmonics—have to be recast into a probabilistic framework, whereby life (time to failure) of insulation is a random variable associated with a certain failure probability. This requires the introduction of a proper probability distribution of failure times. As is well known, failure statistics in polymeric insulation fit well the Weibull probability distribution [5,42], whereby insulation time to failure *tF* can be represented by means of a 2-parameter Weibull cumulative probability distribution function (cdf) of failure times, given in the following equation:

$$P(t\_F) = P = 1 - \exp[-\left(t\_F/\alpha\_l\right)]^{\beta\_l} \tag{17}$$

where *P* is cumulative failure probability corresponding to time to failure *tF*, α*<sup>t</sup>* is the scale parameter (the 63.2th percentile of the life) and β*<sup>t</sup>* is the shape parameter of the Weibull probability distribution function. This means that insulation time to failure *tF* is a random variable associated with a certain value of failure probability *P*, or—conversely—of reliability *R* = 1 − *P*. Equation (17) can be easily recast in terms of α*<sup>t</sup>* as follows:

$$\alpha\_t = t\_F / \left[ -\ln(1 - P) \right]^{1/\beta\_t} \tag{18}$$

As a consequence, the reliability of a power component subjected to current and voltage harmonics can be estimated from the general model (16) by assuming that component life is Weibull-distributed according to (17). In this respect, it is worth pointing out that here—contrary to previous papers [6–9] based on the 63.2th percentile of time to failure, α*NS*, and on the expected value of time to failure, μ*NS*—the theory and the calculations for distorted conditions are focused on noteworthy 100 × *P*th percentiles of the Weibull distribution of times to failure in distorted conditions, *tP*,*NS*, which are compared in the applicative section with design life in sinusoidal conditions, *LD*, given at a certain cumulative failure probability, *PD*: this is consistent with the modern probabilistic approach to the design of power component insulation. Therefore, in order to recast model (16) in terms of *tP*,*NS*, *LD* and *PD*, the following procedure is required.

Let us first write Equation (18) twice:

1. a first time for *tP*,*NS*, namely insulation life in non-sinusoidal conditions at a generic failure probability *P*, as follows:

$$P(t\_{P, \text{NS}}) = P = 1 - \exp[-(t\_{P, \text{NS}}/a\_{\text{NS}})]^{\beta\_t} \tag{19}$$

where α*NS* indicates the 63.2th percentile (or scale parameter) of non-sinusoidal life distribution; 2. a second time for *LD*, namely insulation life in nominal sinusoidal conditions at design failure probability *PD*—indeed life is now a random variable, hence also design life *LD* is associated with a certain fixed and known design failure probability, *PD*, see above—as follows:

$$P(L\_D) = P\_D = 1 - \exp[-\left(L\_D/a\_S\right)]^{\beta\_l} \tag{20}$$

where α*<sup>S</sup>* indicates the 63.2th percentile (or scale parameter) of sinusoidal life distribution.

Thereafter, using Equation (18) let us explain (19) and (20) in terms of the scale parameter, as follows:

$$
\sigma\_{\rm NS} = t\_{\rm P,NS} / \left[ -\ln(1 - P) \right]^{1/\beta\_t} \tag{21}
$$

$$
\alpha\_{\mathcal{S}} = \mathcal{L}\_{\mathcal{D}} / \left[ -\ln(\mathbf{1} - \mathbf{P}\_{\mathcal{D}}) \right]^{1/\beta\_t} \tag{22}
$$

Considering now model (16), it is readily seen that this model has to be written for a given failure probability as well, in such a way that life in the presence of current and voltage harmonics, *LNS* (at the left-hand side) and sinusoidal life *LS* at the right-hand side are relevant to the same failure probability: indeed, *LNS* at the left-hand side and *LS* at the right-hand side are so far the only random variables in model (16) (for the moment, *Kp*, *Kw* and *Kr* are assumed as deterministic and known, since the *N* voltage harmonics are assumed as deterministic and known so far. In the following, also the typical random variation of harmonics is accounted for). Therefore, model (16) can be written, e.g., for 63.2% failure probability, so that *LNS* = α*NS* and *LS* = α*S*. In this way, one obtains:

$$
\alpha\_{\rm NS} = \alpha\_{\rm S} (E\_H / E\_S)^{-n\_{\rm S}} \exp(-B\Delta T\_{tot}') K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \tag{23}
$$

Then, by expressing α*NS* via (21) and α*<sup>S</sup>* via (22), relationship (23) becomes:

$$\frac{t\_{P,NS}}{[-\ln(1-P)]^{1/\beta\_t}} = \frac{L\_D}{[-\ln(1-P\_D)]^{1/\beta\_t}} \left(E\_H / E\_S\right)^{-n\_S} \exp(-B\Delta T\_{tot}') K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \tag{24}$$

which can be eventually recast in the following form—reported here for the first time:

$$t\_{P,NS} = L\_D \frac{[-\ln(R)]^{1/\beta\_l}}{[-\ln(1-P\_D)]^{1/\beta\_l}} (E\_{H}/E\_S)^{-n\_S} \exp(-B\Delta T\_{tot}') K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \tag{25}$$

Relationship (25) is the so-called electro-thermal "probabilistic life model"—or "reliability model"—for the insulation of a power component affected by current and voltage harmonics. It is a quite powerful tool that, for given values of design life *LD* and design failure probability *PD*, relates the 100 × *P*th percentile of insulation time-to-failure (life) in non-sinusoidal regime, *tP,NS*, to reliability *R* = 1 − *P* and applied stresses, i.e., temperature associated with linear and no linear loads, sinusoidal voltage and voltage waveshape factors. Therefore, this relationship yields the main functions and parameters needed for a thorough reliability analysis of power system components, namely [8]:

Reliability function

$$R(t\_{P,NS}) = \exp\left\{-\left\{\frac{[-\ln(1-P\_D)]^{1/\beta\_t} \, t\_{P,NS}}{L\_D(E\_H/E\_S)^{-n\_S} \exp\{-B\Delta T\_{tot}^r\} \mathcal{K}\_p^{-n\_p} \mathcal{K}\_w^{-n\_w} \mathcal{K}\_r^{-n\_r}}\right\}^{\beta\_t}\right\}\tag{26}$$

*Electronics* **2020**, *9*, 1266

Failure probability

$$P(t\_{P,NS}) = 1 - R(t\_{P,NS}) \tag{27}$$

Hazard function

$$h(t\_{P,NS}) = \frac{[-\ln(1 - P\_D)]^{1/\beta\_t} \left\| t\_t(t\_{P,NS})^{\beta\_t - 1} \right\|}{\left\{ L\_D (E\_H / E\_S)^{-n\_S} \exp(-B\Delta T\_{tot}') K\_p^{-n\_p} K\_w^{-n\_w} K\_r^{-n\_r} \right\}^{\beta\_t}} \tag{28}$$

Mean Time To Failure (MTTF)

$$MTTF = \frac{L\_D}{[-\ln(1 - P\_D)]^{1/\beta\_l}} \left( E\_H / E\_S \right)^{-n\_S} \exp\left( -B\Lambda T\_{tot}^{\prime} \right) \mathbf{K}\_p^{-n\_p} \mathbf{K}\_w^{-n\_w} \mathbf{K}\_r^{-n\_r} \Gamma(1 + 1/\beta\_l) \tag{29}$$

where Γ is the Euler Gamma function.

#### *2.5. Reliability Model in the Case of Randomly Time-Varying Distortion*

Voltage and current harmonics have amplitude and phase-angle that are stochastically varying with time. Therefore, they have to be regarded as random variables (RVs), in line with [4] and as done hereafter from both the theoretical and the applicative viewpoint.

If the probability density functions (pdfs) of voltage and current harmonics at a certain node of the grid are known, life and reliability of power components at that node can be inferred by applying the cumulative damage law of Miner [43] to the reliability model (25). Then, the following probabilistic electrothermal life model for randomly time-varying distortion is obtained:

$$t\_{P, \text{NS}} = \frac{\left[ -\ln(\text{R}) \right]^{1/\beta\_I}}{\int\_0^{l\_{1, \text{max}}} \cdots \int\_0^{l\_{M, \text{max}}} \int\_0^{V\_{1, \text{max}}} \cdots \int\_0^{V\_{N, \text{max}}} \cdots \frac{\int\_{l\_1}^{V\_{1, \text{max}}} \dots \int\_{l\_{\text{max}}} \frac{f(l\_1, \dots, l\_{\text{N}}; V\_1, \dots, V\_{\text{N}}) \prod\_{k=1}^{N} dV\_k \prod\_{l=1}^{M} dl\_l}{\frac{l\_{\text{D}}}{[-\ln(1 - p\_{\text{D}})]^{1/\beta\_I}} \left( \frac{E\_{\text{H}}}{E\_{\text{S}}} \right)^{-n\_{\text{S}}} \exp(-R \Lambda T\_{\text{M}}^{\prime}) K\_{\text{p}}^{-n\_{\text{P}}} K\_{\text{w}}^{-n\_{\text{W}}} K\_{\text{r}}^{-n\_{\text{F}}}} } \tag{30}$$

where: *f*(*I*1, ... , *IM*; *V*1, ... , *VN*) is the multivariate pdf of rms current harmonics (*I*1,..,*IM*) and rms voltage harmonics (*V*1, ... , *VN*); *I*1,*max*,..*IM*,*max* and *V*1,*max*,..*VN*,*max* are, respectively, the maximum values reached with time by rms harmonic currents and voltages. It must be emphasized that in LV and MV systems Δ*Tarm*—thus Δ*T*- *tot*, see Equation (15)—depends directly on the amplitudes of current harmonics only (Dielectric losses are practically negligible in LV and MV systems.), while *Kp*, *Kw* and *Kr* depend directly on the amplitude and phase of voltage harmonics (the dependence on the phases of voltage harmonic is omitted in (30) for the sake of simplicity).

Let us now assume that the temperature variation caused by harmonic currents, Δ*Tarm*, and the temperature variation caused by a change of sinusoidal current with respect to rated sinusoidal current, Δ*THL*, can be regarded as constant with time, since temperature exhibits moderate fluctuations of a few K around a mean value *Ttot* which is essentially constant in the rms sense [2,3]. Then *Ttot* can be essentially regarded as a deterministic quantity, constant with time in the rms sense as well; this means that the thermal effects of currents can be also essentially regarded as deterministic. As a consequence, (30) can be made much simpler, as follows:

$$t\_{P, \text{NS}} = \frac{\frac{L\_D \left[ -\ln(R) \right]^{1/\beta\_I}}{\left[ -\ln(1 - P\_D) \right]^{1/\beta\_I}} \left( \frac{F\_{H\_2}}{E\_S} \right)^{-n\_S} \exp(-B\{\Delta T\_{tot}' \})}{\int\_0^{K\_{p, \text{max}}} \int\_0^{K\_{w, \text{max}}} \int\_0^{K\_{r, \text{max}}} \frac{f\left(K\_p, K\_w, K\_r\right) dK\_p dK\_w dK\_r}{\left(K\_p^{-n\_P} K\_w^{-n\_W} K\_r^{-n\_r}\right)}} \tag{31}$$

where:

• *f*(*Kp*, *Kw*, *Kr*) is the multivariate pdf of *Kp*, *Kw* and *Kr*, correlated in turn with *f*(*V*1,..,*VN*; φ1, ... , φ*N*), the multivariate pdf of rms values, *V*1,..,*VN*, and phase-shift angles φ1, ... , φ*<sup>N</sup>* of voltage harmonics;

• Δ*T*- *tot* is defined as follows (see Equation (15)):

$$
\langle \Delta T\_{\text{tot}}' \rangle = 1/T\_S - 1/\left(T\_S + \langle \Delta T\_{\text{arm}} + \Delta T\_{\text{HL}} \rangle \right) = 1/T\_S - 1/\left(T\_S + \langle \Delta T\_{\text{lat}} \rangle \right) \tag{32}
$$

Equation (31) can be rewritten in a more compact form as follows:

$$t\_{P, \text{NS}} = \frac{L\_D \left[ -\ln(\mathcal{R}) \right]^{1/\beta\_l} (E\_H / E\_S)^{-n\_S} \exp(-\mathcal{B} \langle \Delta T\_{\text{tot}}' \rangle)}{[-\ln(1 - P\_D)]^{1/\beta\_l} \, E \Big[ K\_p^{n\_p} K\_w^{n\_w} K\_r^{n\_r} \Big]} \tag{33}$$

where *E Knp <sup>p</sup> <sup>K</sup>nw <sup>w</sup> <sup>K</sup>nr r* indicates the expected value of *<sup>K</sup>np <sup>p</sup> <sup>K</sup>nw <sup>w</sup> <sup>K</sup>nr r* .

The above hypothesis that Δ*Tarm* and Δ*THL* can be regarded as constant with time is meaningful, particularly as far as Δ*Tarm* is concerned [2,3]. Indeed, load current does vary with time, but the associated variation of Joule losses is more or less compensated by the thermal inertia of the insulation of the component and of the surrounding environment, which have a much longer thermal time constant than the typical variation period of harmonic currents—and sometimes even of sinusoidal current. Therefore, the overall thermal effect of load current can be well approximated as a steady variation of the temperature of the component, proportional to rms load current (the use of rms current is due to the dependence of Joule losses in conducting parts on the rms value of current).

Then, from Equation (33) it follows that life and reliability of the insulation of the component affected by harmonic distortion can be estimated in two steps:

1. evaluation of the 100 × *P*th failure time percentile based on the effect of voltage harmonics only, *tP*,*NS*,*V*, resorting to the following equation (derived from Equation (33) by setting Δ*T*- *tot* = 0, which implies *exp*(−*B*Δ*T*- *tot*) = 1):

$$t\_{P,NS,V} = \frac{L\_D [-\ln(R)]^{1/\beta\_l} (E\_H / E\_S)^{-n\_S}}{[-\ln(1-P\_D)]^{1/\beta\_l} E \left[K\_p^{n\_p} K\_w^{n\_w} K\_r^{n\_r}\right]} \tag{34}$$

2. evaluation of the further possible variation of the 100 × *P*th failure time percentile due to the effect of the mean value of Δ*T*- *tot*, Δ*T*- *tot*—guessed in turn through rms load current—resorting to the following equation, derived comparing Equation (33) with Equation (34):

*tP*,*NS* = *tP*,*NS*,*<sup>V</sup> exp*(−*B*Δ*T*- *tot*) (35)

Focusing on the 1st step, a proper application of Equation (34) requires an appropriate description of the time variation of voltage harmonics. This variation is inherently stochastic, but two typical situations can occur at a certain node of a grid where a component of interest is located:


In both cases, once the multivariate pdf *f*(*Kp*,*Kw*,*Kr*) has been obtained, the life and reliability of the power component subjected to randomly time-varying voltage harmonics can be evaluated by means of Equation (34) by computing the expected value *E Knp <sup>p</sup> <sup>K</sup>nw <sup>w</sup> <sup>K</sup>nr r* .

*Electronics* **2020**, *9*, 1266

Coming to the 2nd step, the further effect of time-varying current harmonics—involving in turn a constant (in the rms sense) variation of insulation temperature Δ*T*- *tot*—can be assessed by means of Equation (35).

Since the focus in this paper is on the 1st percentile of times to failure in the presence of harmonic distortion, *t*1%,*NS*, its calculation requires rewriting Equation (33) for *R* = *RD* = 0.99 (i.e., *PD* = 0.01), namely:

$$t\_{1\%, \text{NS}} = \frac{L\_{D\exp}(-B\langle \Delta T\_{\text{tot}}' \rangle) \left(E\_H / E\_S\right)^{-n\_S}}{E \left[\mathcal{K}\_p^{n\_p} \mathcal{K}\_w^{n\_w} \mathcal{K}\_r^{n\_r}\right]} \tag{36}$$

Moreover, reliability is calculated as a function of time elapsed in service, *tE*, by explaining Equation (33) as a function of *R* with *RD* set to 0.99, as follows:

$$\mathcal{R}(t\_E) = \exp\left\{-\left\{\frac{[-\ln(0.99)]^{1/\beta\_t} \operatorname{t}\_E \operatorname{E}\left[\mathbf{K}\_p^{n\_p} \mathbf{K}\_w^{n\_w} \mathbf{K}\_r^{n\_r}\right]}{\operatorname{L}\_D(\operatorname{E}\_H/\operatorname{E}\_S)^{-n\_S} \exp\{-B(\Delta T\_{tot}')\right\}}\right\}^{\beta\_t} \tag{37}$$

By setting Δ*T*- *tot* = 0 in (36), (37), homologous relationships valid in the presence of voltage harmonics only are obtained, i.e.,:

$$\text{At}\_{1\%, \text{NS}} = L\_D \left( E\_H / E\_S \right)^{-n\_S} / E \left[ \mathbf{K}\_p^{n\_p} \mathbf{K}\_w^{n\_w} \mathbf{K}\_r^{n\_r} \right] \tag{38}$$

$$R(t\_E) = \exp\left\{-\left[\left[-\ln(0.99)\right]\right]^{1/\beta\_l} t\_E \operatorname{E}\left[\mathcal{K}\_p^{\text{u}\_p} \mathcal{K}\_w^{\text{u}\_w} \mathcal{K}\_r^{\text{u}\_r}\right] / \left[\operatorname{L}\_D(E\_H/E\_S)^{-\text{n}\_S}\right] \right\}^{\beta\_l} \tag{39}$$

The reliability model for time-varying distortion as newly formulated here relies on Equations (38) and (39) or (36) and (37)—depending on whether Δ*T*- *tot* = 0 or - 0, respectively—which differ from those used in [8,9], since these latter are based on the 63.2th percentile of failure time. For the sake of simplicity, hereafter the sinusoidal component of electric field at power frequency is taken equal to rated sinusoidal electric field, namely *EH* = *ES.*

#### **3. Application of the Reliability Model for Randomly Time-Varying Distortion**

#### *3.1. Selected Case Studies*

In previous papers [8,9] the reliability model for randomly time-varying distortion was applied to power system components affected by voltage harmonics whose amplitudes matched the limits after EN 50160 [4] exactly. The calculations were concentrated on the 63.2th percentile of time to failure, α*NS*, and on the mean time to failure (MTTF), μ*NS*; design life itself was given as the design value of the MTTF in sinusoidal conditions set to 40 and 30 years for the cable and the capacitor, the two treated components. Reliability was estimated at a service time equal to design life only.

Here, the calculations are focused on the 100 × *PD*th percentile of times to failure, which is consistently compared with design life in sinusoidal conditions, *LD*, given at design failure probability *PD* = 0.01 = 1%. This is in line with the modern probabilistic design of the insulation of power components, which requires high reliability [5]. Moreover, the design life of power components is set pragmatically to *LD* = 20 years, i.e., the typical order of magnitude of the service life (based on the duration of the mortgages and the amortization times) of industrial plants, traction systems for railways or subways and renewable power plants for photovoltaic (PV) and wind energy generation [44–46]. Grid-connected three-phase power electronic converters are found to connect these plants with the AC power grid and they generate voltage and current harmonics which affect nearby components. This has the first and foremost effect that all estimates obtained here for failure time percentiles and reliability are different from those in previous papers.

In addition, in place of the "spot" reliability estimate at design life in [8,9], reliability is evaluated here in terms of design life of power components—and even beyond it, although results are omitted for brevity. Indeed, as service time *tE* goes by, maintenance and/or repair and/or replacement actions might be required on some components of the plant/grid, if their residual reliability is unsatisfactory; conversely, other components may remain in service without any action even after a service time *tE* equal to—or even longer than—design life *LD*, if their residual reliability is unsatisfactory. For this reason, beside proper diagnostic techniques, a sound reliability model can be extremely useful to provide indications about the residual reliability of components all along their time on duty.

Furthermore, the case studies treated in this paper have been changed and broadened considerably compared to [8,9], as shown in the next section. First of all, the voltage harmonics treated in [8] were the 11th, 13th, to which the 5th, 7th were added in [9], while here for the first time the 17th, 19th, 23rd, 25th are included. It is worth emphasizing that the implementation of the new calculations including the whole set of odd harmonics from the 5th to 25th has required a non-negligible effort for restructuring and modifying the code: analyzing the whole sequence of odd voltage harmonics from the 5th to the 25th, as done here for the first time, is all but trivial and inexpensive from the viewpoint of theoretical and computational efforts. It is also worth outlining that the reliability model (33),(36),(37) works for all harmonic orders [10], but harmonics of order *h* > 25 are neglected here since in [4] no limits are given for voltage harmonics above the 25th, as they are said to be "*usually small, but largely unpredictable due to resonance e*ff*ects*".

Moreover, three cases differing as for the magnitude (In agreement with [4] let us consider for each harmonic—in place of the rms value—the "*10 min mean rms value*", called "magnitude" from now on, and indicate it simply as *Vh* like the rms value.) of voltage harmonics are treated here:


Three more applicative novelties compared to previous papers [8,9] of this investigation are:


As for the magnitude of voltage harmonics, it can be argued that, anyway, the reference case for setting the correct values of μ*<sup>h</sup>* and σ*<sup>h</sup>* in cases (i)–(iii) above is case (i). Since measurements of harmonic voltages that match the probabilistic limits set in [4] exactly are rare in the literature, the random magnitudes of harmonic voltages matching the limits set in [4] exactly are derived here via a heuristic parametric approach [8,9] based on a priori hypotheses about the Coefficient of Variation *CVh* = σ*h*/*Vh*

of harmonic voltages. Namely, two different values of *CVh* are hypothesized so as to reproduce a narrower and a broader spread of *Vh*:


These values of *CVh* were already chosen in [8,9] for the 5th, 7th, 11th, 13th voltage harmonics, being consistent with those measured in the power supply grid of a subway traction system, where *CVh* ranged from ≈10% to ≈27% [32]. They are kept here and extended to the 17th, 19th, 23rd, 25th voltage harmonics, for the sake of comparison.

As a result of this heuristic parametric approach, the pdfs (the pdfs describing the randomness of the parameters *Vh* in Figure 1, as well as the one of other parameters, e.g., *Kp* and *Kw*, may be also considered as the "a priori" pdfs in a Bayesian inference framework, which is outside the scope of the present paper. The details of this approach are discussed for the problem under study in [48]) of the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonic magnitudes are obtained, which are Gaussian with μ*<sup>h</sup>* and σ*<sup>h</sup>* such that their magnitudes match exactly the limits set in [4], namely: 6%, 5%, 3.5%, 3%, 2.0%, 1.5%, 1.5%, 1.5% of rms fundamental voltage *V*<sup>1</sup> for the 95th percentiles of *V*5, *V*7, *V*11, *V*13, *V*17, *V*19, *V*23, *V*25, respectively [48]. The cdfs of all voltage harmonics, *F*(*Vh*), are shown for the sake of illustration in Figure 1a,b for the cases *CVh,*<sup>1</sup> = 0.1 and *CVh,*<sup>2</sup> = 0.3, respectively: is readily seen that the limits after [4] are exactly matched for all voltage harmonics.

Now, life and reliability estimation according to model (36) and (37) requires knowledge of the expected value *E Knp <sup>p</sup> <sup>K</sup>nw <sup>w</sup> <sup>K</sup>nr r* . As it can be argued from Equations (30), (31), (33), this in turn requires the knowledge of the multivariate pdf of *Kp*, *Kw*, *Kr*, *f*(*Kp*, *Kw*, *Kr*), correlated with *f*(*V*5, *V*7, *V*11, *V*13, *V*17, *V*19, *V*23, *V*25; φ5, φ7, φ11, φ13, φ17, φ19, φ23, φ25), the multivariate pdf of magnitudes and phase-shift angles (with respect to the fundamental *V*1) of the treated voltage harmonics. The derivation of the pdfs/cdfs of the magnitude of voltage harmonics (Figure 1) has solved the problem of deriving *f*(*V*5, *V*7, *V*11, *V*13, *V*17, *V*19, *V*23, *V*25), but the phase-shift angles between voltage harmonics and the fundamental is also needed to estimate the part of multivariate pdf relevant to phase-shift angles. This problem is solved—as in [8,9] for the 5th, 7th, 11th, 13th voltage harmonics—by taking phase-shift angles φ5, φ7, φ11, φ13, φ17, φ19, φ23, φ<sup>25</sup> as known deterministic quantities, both to make the treatment easier and to carry out a parametric sensitivity analysis; thereafter, *f*(*Kp*, *Kw*, *Kr*) can be readily derived.

Such analysis focuses on three different cases (a), (b), (c) of phase-shift angles, which have quite different (or even extreme and opposite) effects on the distorted voltage waveshape. These cases are illustrated in the reference Figure 2, which shows the mean waveform of harmonic voltage pdfs for phase-shift cases (a), (b), (c) obtained using *CVh,*<sup>1</sup> = 0.1 and μ*<sup>h</sup>* as the magnitude of the *h*th harmonic in the case of exact matching of the limits after [4]. Cases (a), (b), (c) of phase-shift angles are as follows.


φ<sup>13</sup> = 0, φ<sup>17</sup> = −16.6◦, φ<sup>19</sup> = −17.4◦, φ<sup>23</sup> = −19◦, φ<sup>25</sup> = 11◦. The values of φ5, φ7, φ11, φ13, φ23, φ<sup>25</sup> are the mean of the measured values for these harmonics, while the values of φ17, φ<sup>19</sup> have been attained through a linear interpolation of the mean values of φ<sup>13</sup> and φ23—the 17th and 19th voltage harmonics are not found in [32]. The experimental case leads to a distorted voltage waveshape fairly similar to those observed in the field, as obvious when considering its experimental origin.

**Figure 1.** Gaussian cumulative distribution functions (cdfs) of the magnitudes of 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics, with μ*<sup>h</sup>* and σ*<sup>h</sup>* such as to match the limits in [4] exactly (6%, 5%, 3.5%, 3%, 2%, 1.5%, 1.5%, 1.5% of *V*<sup>1</sup> for the 95th percentiles): (**a**) *CVh* = 0.1; (**b**) *CVh* = 0.3.

Coming to the role played by current harmonics, as outlined in Section 2.5 it is assumed that Δ*Ttot* can be replaced by its mean value (in the rms sense) <Δ*Ttot*>, constant with time and deterministic value. Two values of <Δ*Ttot*> are considered here, namely:


in Section 2.5; this implies Δ*T*- *tot*-0, thus reliability model (36) and (37) shall be used, including the role played by current harmonics and sinusoidal current. Setting a value of <Δ*Ttot*> < 0 implies acknowledging that the component—notwithstanding the presence of current harmonics—works typically at a lower- than-design temperature. This holds in particular for MV distribution cables—whose design temperature oscillates between 75 ◦C for "vintage" paper-insulated lead-covered cables to 90 ◦C for extruded cables—which have usually an overall load well below the rated one [50].

**Figure 2.** Mean voltage waveshapes for case studies (**a**–**c**) obtained using *CVh,*<sup>1</sup> = 0.1 and μ*<sup>h</sup>* as the magnitude of the *h*th harmonic in the case of exact matching of the limits after [4]. Left: positive halfwaves; right: zoomed-in view showing the phase-shift between harmonics. *vdist*(*t*) is total distorted voltage with all harmonics, while *vdist*(*t*) [5,7,32,38] with the 5th, 7th, 11th, 13th only as in [9]. THD*v* = 8.3%.

By combining above points (i)–(iii), (1)–(2), (a)–(c), A.–B., the case studies of distorted voltage waveshapes treated here are 36, as summarized in Table 1, where the relevant value of *THDv* is also reported; for making the table more compact, the cases corresponding to points (i)–(iii) above (i.e., pdfs of voltage harmonic magnitude *Vh* that either match the limits in [4] exactly, or overcome them by 25%, or are below them by 25%) are quoted on one single row: this implies that any one of case studies (a)–(l) degenerates in turn into three more case studies depending on the pdfs of *Vh*. For the sake of illustration, the mean voltage waveshapes of case studies (a), (b), (c) for the exact matching of the limits in [4] (obtained by using μ*<sup>h</sup>* as the magnitude of the *h*th harmonics) are plotted in Figure 2a–c. Other distorted voltage waveshapes than these may exits, but they tend to be in between those considered here, thus they tend to have intermediate effects on insulation life and reliability.

**Table 1.** Summary of cases of distorted voltage waveshapes studied here (interestingly, with the whole set of odd harmonics non-multiple of 3 from the 5th to the 25th, the 8% limit for *THDv* after [4] is slightly overcome when *CVh* = 0.1 even when the amplitudes of each single harmonic voltage strictly match the limits in [4]. This does not happen when 5th, 7th, 11th, 13th harmonics only are treated [9]. This is another new outcome of this study, which would deserve a deeper investigation).


The reliability model for distorted current and voltage (36)–(39) is applied to these 36 case studies for estimating the life and reliability of:


The values of the parameters of Equations (36)–(39) used in the application are quoted in Table 2. As far as the cable and the capacitor are concerned, the values of *np*, *nw*, *nr*—derived by processing the results of ALTs performed on insulating samples subjected to various combinations of voltage harmonics [11,12]—and of *B* in Table 2 are the same as in [8,9] for the sake of comparison, but here the analysis is broader and different with respect to that carried out in [8,9]. Indeed here, as partly explained above:



**Table 2.** Values of the parameters of model (36)–(39) used in the application.

As far as the induction motors and MV/LV transformers are concerned, the values of *np*, *nw*, *nr* and *B* in Table 2 have been derived by reprocessing the results in [10,51,52]. In fact, experimental results relevant to twisted pairs (i.e., twin varnished copper conductors wrapped on each other which reproduce on a smaller scale the full-size insulation of induction motors) indicate for *np* a value of 8–10, that can be used also for induction motor insulation. In the case of transformers, the dielectric performances of both epoxy resin and oil-paper insulation for MV/LV transformers suggest a slightly higher value, namely *np* = 10 − 12. For parameter *B*, a value of the order of those reported in Table 2 for the cable and the capacitor seems reasonable for both the induction motor and the transformer (from [52] a value of *B* = 12,600 can be derived for a bisphenolic epoxy resin), even if the thermal endurance of the epoxy insulation is usually larger than that of XLPE and PP.

A first observation drawn from Table 2 is that for all components the value of *np* overwhelms that of *nw* and *nr*; hence, *Kp* should have a foremost effect on the life and reliability of components subjected to voltage harmonics, if *Kp*, *Kw* and *Kr* have the same order of magnitude.

#### *3.2. Results*

For the XLPE cable and the all-film capacitor working in the distorted conditions of case studies (a)–(f) of Table 1, i.e., those relevant to Δθ*tot* = 0, as well as of case studies (g)−(l) of Table 1, i.e., those relevant to Δθ*tot* = −15 ◦C, Table 3 reports the estimates of the following quantities:


Table 3 also reports the 1st percentile of failure time, *t*1%,*NS,OLD*, in p.u. of design life *LD*, computed via Equation (38) for Δθ*tot* = 0 with the voltage harmonics treated in [9] only, i.e., the 5th, 7th, 11th, 13th.

It is worth recalling that the cases 25% above and 25% below the limits after EN 50160 are totally new and treated here for the first time. Moreover, also the values of *t*1%,*NS,OLD* in Table 3 differ from the values of the 1st percentile of failure time in [9]; indeed, the design hypothesis in [9] was to set the MTTF in sinusoidal conditions to 40 and 30 years for the cable and the capacitor, while here design life in sinusoidal conditions is set for both components to *LD* = 20 years at design failure probability *PD* = 0.01 = 1%.

Analogously to Table 3, the homologous Table 4 reports for the induction motor and MV/LV transformer, working in the distorted conditions of case studies (a)–(f) (Δθ*tot* = 0) and (g)–(l) (Δθ*tot* = −15 ◦C) of Table 1, the estimates of the following quantities:


Table 4 also reports the 1st percentile of failure time, *t*1%,*NS,OLD*, in p.u. of design life *LD*, computed via Equation (38) for Δθ*tot* = 0 with the voltage harmonics treated in [9] only, i.e., the 5th, 7th, 11th, 13th.

It is worth recalling that all cases in Table 4 are totally new and treated here for the first time, since they refer to the induction motor and the MV/LV transformer, tackled here for the first time in the streamline of this investigation.

**Table 3.** Values of 1st percentile of time to failure, *t*1%,*NS*, and reliability at rated life *LD*, *R*(*LD*), estimated with reliability models (36)–(39) for the XLPE cable and the all-film capacitor subjected to 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics in case studies (a)–(f) (Δθ*tot* = 0), and (g)–(l) (Δθ*tot* = −15 ◦C), for the various matching of the limits in [4], see Table 1. The values of 1st percentile of time to failure, *t*1%,*NS*,*OLD*, calculated with the 5th, 7th, 11th, 13th voltage harmonics only as in [9], are also reported.


**Table 4.** Values of 1st percentile of time to failure, *t*1%,*NS*, and reliability at rated life *LD*, *R*(*LD*), estimated with reliability models (36)–(39) for the induction motor and MV/LV transformer subjected to 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics in case studies (a)–(f) (Δθ*tot* = 0), and (g)–(l) (Δθ*tot* = −15 ◦C), for the various matching of the limits in [4], see Table 1. The values of 1st percentile of time to failure, *t*1%,*NS*,*OLD*, calculated with the 5th, 7th, 11th, 13th voltage harmonics only as in [9], are also reported.



**Table 4.** *Cont.*

Figure 3 illustrates the reliability vs. service time (in p.u. of design life) for the XLPE cable with Δθ*tot* = 0 and voltage harmonics that: (a) exactly match, (b) are 25% above, (c) are 25% below the limits in [4]. In the legend, *RS*(*t*) indicates reliability vs. time in the rated sinusoidal case, while *Ra*(*t*), *Rb*(*t*), *Rc*(*t*), *Rd*(*t*), *Re*(*t*), *Rf*(*t*), denotes reliability vs. time in distorted case studies (a)–(f).

**Figure 3.** Reliability vs. service time (in p.u. of design life) for the XLPE cable with voltage harmonics that: (**a**) exactly match, (**b**) are 25% above, (**c**) are 25% below the limits in [4]. Δθ*tot* = 0.

Figure 4 shows the same quantities in the same conditions as Figure 3, but for the capacitor. Similar figures could have been drawn for the induction motor and MV/LV transformer, but they are omitted here for brevity.

**Figure 4.** Reliability vs. service time (in p.u. of design life) for the all-film capacitor with voltage harmonics that: (**a**) exactly match, (**b**) are 25% above, (**c**) are 25% below the limits in [4]. Δθ*tot* = 0.

Figure 5a, reports the pdfs of *Kp* for case studies (a), (b), (c) with the strict match of the limits in [4] in the presence of the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics, as well as of the 5th, 7th, 11th, 13th only as in [9]. Figure 5b is the same as Figure 5a, but for *Kr* instead of *Kw*. Figure 6 displays the pdfs of *Kr* for case studies (a), (b), (c), (d) with the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics, as well as for case studies (a), (b) with 5th, 7th, 11th, 13th only.

**Figure 5.** Probability density functions of: (**a**) *Kp* for case studies (a), (b), (c) with the strict match of the limits in [4] in the presence of the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics, as well as of the 5th, 7th, 11th, 13th only; (**b**) *Kw* for case studies (a), (b), (c), (d) with the strict match of the limits in [4] in the presence of the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics, as well as for case studies (a), (b) with 5th, 7th, 11th, 13th only.

**Figure 6.** Probability density functions of *Kr* for case studies (a), (b), (c), (d) with the strict match of the limits in [4] in the presence of the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics, as well as for case studies (a), (b) in the presence of the 5th, 7th, 11th, 13th only.

Figure 7 illustrates the reliability vs. service time (in p.u. of design life) for the XLPE cable with Δθ*tot* = −15 ◦C and voltage harmonics that: (a) exactly match, (b) are 25% above, (c) are 25% below the limits in [4].

**Figure 7.** Reliability vs. service time (in p.u. of design life) for the XLPE cable with voltage harmonics that: (**a**) exactly match, (**b**) are 25% above, (**c**) are 25% below the limits in [4]. Δθ*tot* = −15 ◦C.

#### *3.3. Discussion*

#### 3.3.1. The Effect of Voltage Harmonics Only

Let us discuss the results of case studies (a)–(f) in Table 3 first, so as to deal with the effect of voltage harmonics only. The 1st percentile of failure time is lower—in most cases far lower—than design life for all case studies (a)–(f) and for both the cable and the capacitor; correspondingly, the reliability at design life *LD* is always lower than design reliability *RD* = 1 − *PD* = 0.99 = 99%. Hence, for such devices a combination of the 5th, 7th, 11th, 13th, 17th, 19th, 23rd, 25th voltage harmonics seems to be detrimental irrespective of their phase-shift angle.

This is also confirmed by the plots of reliability vs. service time for the cable and the capacitor reported in Figures 3 and 4, respectively, which show that the reliability in distorted cases (a)–(f) is always below that in rated sinusoidal conditions (solid red curve). Surprisingly, this holds not only when the harmonics comply with the limits in [4] exactly—as already found in [8] for the 11th, 13th harmonics and in [9] for the 5th, 7th, 11th, 13th—but also when they are 25% below these limits: hence, this analysis suggests that the distorted voltage waveshapes of case studies (a)–(f) can reduce the life and reliability of the insulation of the cable and the capacitor with respect to design life and reliability even if harmonics have a 25% safety margin compared to the limits in standards [2–4].

Obviously, the lowest values of *t*1%,*NS* are found when harmonics are 25% above the limits: thus the limits in [4] are still a key reference for designing effective filters to limit voltage harmonics [31].

The effect on insulation life and reliability is different among cases (a)–(f): indeed, Table 3 and Figures 3 and 4 indicate that the reduction of component life and reliability is:


From Table 3, case studies (a)–(f), it is also clear that the 1st percentile of failure time and reliability at design life increase as *CVh* rises from 0.1 for cases (a), (b), (c) to 0.3 for cases (d), (e), (f)—i.e., with the scatter of the magnitude of voltage harmonics. This is because the 95th percentiles of the magnitude of voltage harmonics are fixed—i.e., either equal to, or 25% above, or 25% below the limits in [4]—and greater dispersion means that the cdfs *F*(*Vh*)/pdfs *f*(*Vh*) encompass lower values of *Vh*, as deduced by comparing Figure 1b with Figure 1a.

Overall, it is worth emphasizing again that the calculations in this paper for the cable and the capacitor—summarized in Table 3—are completely new compared to those in previous ones [8,9], that have intentionally been cited here to allow a direct comparison. Such comparison shows that all results and figures in this paper—although being intentionally homologous to those in previous papers for enabling a straightforward comparison—are different as for the values obtained and the curves plotted from those reported in the previous papers. However, although being different, the results reported in this paper for the cable and the capacitor are in line with those previously obtained, but is also interesting to observe that—for both the cable and the capacitor—the whole sequence of odd voltage harmonics from the 5th to the 25th analysed here is more challenging than the simple combination of the 5th, 7th, 11th, 13th voltage harmonics treated in [9]. Indeed, in Table 3 the values of *t*1%,*NS,* are all lower than the homologous values of *t*1%,*NS,O*: thus, it is worth performing the much more cumbersome analysis carried out here, since it is not only more accurate, but also on the safe side from the viewpoint of life and reliability estimation of power components.

Another essential comment stemming from the results in Table 3 and Figures 3 and 4 for case studies (a)–(f) is that the main outcomes hold for both the cable and the capacitor. Therefore, these outcomes seem to be general and independent of the values of *np*, *nw* and *nr* and of the uncertainty in their evaluation: indeed, in Table 2 the values of exponents *np*, *nw* and *nr* for the cable are far apart from those for the capacitor, thus their effect on a same set of values of *Kp*, *Kw*, *Kr* will be different as well.

This comment gives a chance of a deeper understanding of cases (a)–(f), which requires dealing precisely with the values of *Kp*, *Kw*,*Kr*. In fact, it should be understood that the values of *Kp*, *Kw*, *Kr* are not the same, either in a given case study, or among different case studies, as outlined in Figures 5 and 6.

Starting from Figure 5a, it is readily seen that the pdfs of *Kp* in case studies (a)–(c) fully agree with what previously observed at points (1)–(3) above for the same case studies, i.e., the pdf of *Kp* is located in correspondence of:


Furthermore, Figure 5a shows that the pdfs of cases (a), (b), (c) are located differently when the 5th, 7th, 11th, 13th voltage harmonics only are dealt with: the consideration of all harmonics moves the pdf of *Kp* towards higher values in case (a), slightly lower values in cases (b), (c). This agrees well with Figure 2a–c, since Figure 2a exhibits a higher peak of distorted voltage, while Figure 2b,c a slightly lower peak of distorted voltage when all harmonics are considered rather than the 5th, 7th, 11th, 13th only. This is due to the more favourable combination of phase-shift angles of cases (b),(c) when all harmonics than when the 5th, 7th, 11th, 13th only are included, since a better compensation of the peaks of the various harmonics is achieved in the former than in the latter case.

Analogous comments hold for the pdfs of *Kp* in the worst case (d), best case (e) and experimental case (f)—omitted in Figure 5a for brevity. However, in these case studies the pdfs of *Kp* are displaced towards lower values, for the same reasons explained above for the pdfs of *Vh*: namely cases (d), (e), (f) have a higher *CVh*—0.3 vs. 0.1—than cases (a), (b), (c).

Anyway, as pointed out above when commenting on Table 3, the life and reliability of components is lower when all harmonics are considered. This is mainly because of the effect of the shape factor *Kw*, as straightforwardly understood from Figure 5b. Figure 5b—when compared to Figure 5a—demonstrates that, while *Kp* depends strongly on phase-shift angles φ*h*, *Kw* does not, since it is affected only by the order *h* and rms value *Vh* of voltage harmonics, see Equations (7)–(9). Thus the pdf of *Kw* is the same for worst case (a) as for best case (b) and experimental case (c) (this latter omitted in Figure 5b for the sake of graphical clearness)—since these cases differ only as for the values of φ*h*. Similarly, the pdf of *Kw* is the same for worst case (d) as for best case (e) and experimental case (f) (this latter omitted, too). However, when all voltage harmonics are considered the pdf of *Kw* moves towards much higher values than with the 5th, 7th, 11th, 13th harmonics only, and since exponent *nw* is significantly >1—although being much lower than *np*—this involves that life and reliability of the component with all voltage harmonics decrease with respect to the 5th, 7th, 11th, 13th harmonics only. On the other hand, such a decrease is much more remarkable for the cable than for the capacitor, as *nw* is much higher for the former than for the latter.

One could guess the rms factor *Kr* plays a more significant role than *Kw* for the capacitor. Indeed, as shown in Table 2, the capacitor features *nr* = 1.8 > *nw* = 0.56, while the cable features *nr* = 1.2 << *nw* = 4.9. This is not true, as seen in Figure 6—the same as Figure 5b, but for *Kr* instead of *Kw*. In fact, Figure 6 shows that *Kr* ≈ 1 in all case studies treated here (Experimental cases (c) and (f) are omitted in Figure 6 like in Figure 5b for the sake of graphical clearness, since they yield the same pdfs as cases (a),(b) and (d),(e), respectively.), hence its effect on life and reliability of power components is negligible in these cases. This conclusion can be essentially generalized to the most applicable cases, unless the total harmonic distortion of voltage *THDV* is really severe, say, above 20% (see Equation (10) for the relationship between *THDV* and *Kr*): here *THDV* ≤ 10%, see Table 1.

Figures 5 and 6 also confirm that—as stated in Section 2.1 when commenting on Equations (7)–(9)—*Kp* can be either <1 or >1, while *Kw* and *Kr* are always ≥1. However, *Kw* can attain much higher values than:


Like the pdf of *Kp*, also the pdfs of *Kw* and *Kr* are displaced towards lower values of *Kw* and *Kr* for a higher dispersion: the reason is same as for the pdfs of *Vh* and *Kp*, see above.

As for the differences between the cable and the capacitor, all quantities in Table 3 are much greater for the all-film capacitor than for the XLPE cable in all case studies. This is because the cable features much greater values of the exponents *np* and *nw* (see Table 2), which weigh the role played by *Kp* and *Kw*, the most dominant ageing factors not only in the case studies analysed here, but under distorted voltage in general [35,36]. On the other hand, unforeseen parallel resonances between the capacitor and the equivalent impedance of the grid—skipped here for the sake of brevity—might increase a few voltage harmonics strongly, thereby causing a premature failure of the capacitor.

Another difference between capacitor and cable is that the former is truly placed at a certain node in the network, where voltage distortion changes only with time, while cable lines actually stretch from one node to another, experiencing a voltage distortion that typically changes both with time and along the line. Hence, voltage distortion at a given time is more or less deterministically distributed along cable lines. Here cables are implicitly regarded as a discrete number of series-connected cable lengths with a constant—in space—voltage distortion level within each length.

Coming now to the induction motor and the transformer, as done in the case of Table 3 for the cable and the capacitor, let us discuss the results of case studies (a)–(f) in Table 4 first, so as to deal with the effect of voltage harmonics only. Table 4 confirms the following observations already made when commenting on the homologous case studies (a)–(f) in Table 3 for the cable and the capacitor.

	- (i) the greatest for worst cases (a), (d) leading to the highest peak of distorted voltage, see Figure 2a;
	- (ii) the smallest for best cases (b), (e) leading to the lowest peak of distorted voltage, see Figure 2b;
	- (iii) intermediate for experimental cases (c), (f) leading to an intermediate peak of distorted voltage, see Figure 2c.

Other comments stemming from a comparison between Table 3 for the cable and the capacitor and Table 4 for the transformer and induction motor—as well as from the inspection of Figure 5—is that the role played by *Kw* vs. *Kp* is more noteworthy for the cable, less significant for the other components, essentially because the capacitor, motor and transformer have all a lower value of *nw*, as readily seen in Table 2. In this respect it can be said that the motor and transformer have a closer behavior to the capacitor than to the cable; this is not surprising, when considering that typically the composite/lapped structure of the insulation of motors and MV/LV transformers is more similar to the lapped insulation of the PP capacitor than to the quasi-homogeneous insulation of the XLPE cable.

On the other hand, it is also interesting to point out that the motor and the transformer exhibit lower values of time to failure and reliability than the capacitor in the same case studies. This is essentially due to the higher value of *np* featured by the motor (*np* = 9) and the transformer (*np* = 11) compared to the capacitor (*np* = 6.2). For the same reason, in turn, the transformer exhibits lower values of time to failure and reliability than the motor in the same case studies.

In summary, the hierarchy of the values of failure times and reliability is as follows: capacitor < motor < transformer < cable. However, it should be pointed out that this hierarchy has to be taken with great care, since it has been obtained from the values of reliability model parameters reported in Table 2, which are subjected to all uncertainties related to the relevant experimental campaigns and failure of data processing. Moreover, these values hold for the particular insulating specimens considered in those campaigns, which of course cannot be considered as fully representative of all insulations of existing capacitors, motors, transformers and cables found in MV and LV systems.

#### 3.3.2. The Effect of Current Harmonics and Sinusoidal Current

Coming now to the effect of current harmonics, one could guess that the relevant Joule losses might cause a further reduction of lifetime and reliability of power components compared to those quoted for case studies (a)–(f) in Table 3 and in Table 4, as outlined in [6,7]. However, load demand and common service practices imply that distribution grids and the relevant power components mostly work at a load substantially lower than the design "sinusoidal" load. In this way, the thermal effect of harmonic currents is insufficient to let the maximum temperature of components reach the rated sinusoidal temperature. This is also because some design practices recommended to oversize conductors to be on the safe side as for possible harmonic currents. This situation, rather common especially for MV distribution cables [50], might hide the decrease of life and reliability under distorted voltage found for case studies (a)–(f) in Table 3. To check this effect, Table 3 includes case studies (g)–(l), identical to case studies (a)–(f) apart that <Δθ*tot*>=<Δθ*arm*>+<Δθ*HL*> = −15 ◦C (see Table 1).

Case studies (g)–(l) in Tables 3 and 4 show that this effect can really occur. Indeed, for the treated components all values of failure times and reliability increase significantly in case studies (g)–(l) with respect to their homologous values in case studies (a)–(f). In particular, the 1st percentile of time to failure rises by a factor *exp*(−*B*Δ*T*- *tot*)—compare Equation (36) with Equation (38)—which is ≈4.4 for the cable and the capacitor, ≈4.5 for the motor and the transformer: thus this factor is quite similar for all treated components, since the relevant values of *B* are quite close (see Table 2). However, being times to failure at <Δθ*tot*> = 0 ◦C the highest for the capacitor, the least for the cable, intermediate for the motor and the transformer, the increase involved by <Δθ*tot*> = −15 ◦C leads to:

	- worst case (g) with all matching types of the limits in [4];
	- worst case (j) with voltage harmonics matching exactly or 25% above the limits in [4];
	- worst cases (g) and (j) with all matching types of the limits in [4];
	- experimental cases (i) and (l) with voltage harmonics matching exactly or 25% above the limits in [4];

Focusing on the cable, these potentially harmful cases are also confirmed by the plots of reliability vs. service time for the cable with <Δθ*tot*> = −15 ◦C reported in Figure 7, which show the many previously listed cases where cable reliability in distorted conditions falls below that in rated sinusoidal conditions (solid red curve). Therefore, as such "pessimistic" cases cannot be fully ruled out for a cable subjected to voltage and current harmonics generated by power electronic devices, such a cable faces a risk of premature failure although its average temperature is 15 ◦C below design temperature. This holds also for potentially harmful cases for the motor and the transformer emphasized with bold characters in Table 4.

#### 3.3.3. Further Remarks

As a closure of this discussion, it must be emphasized that the analysis carried out here in the streamline of the present investigation does not have a particular type of converter as a reference—e.g., the 6-pulse or the 12-pulse, although the 6-pulse and 12-pulse SCR-based converters are quite often encountered in LV and MV grids. Rather, this analysis aims at reproducing the typical situation of harmonic distortion level which can be typically found in LV and MV grids, based on the typical distorting loads found in such grids and on the typical filtering strategies to limit the effects of such distorting loads. This is the reason why international standards EN50160, IEC 61000-2-2, IEC 61000-2-4 are taken as a reference here; based on the limitations established by these authoritative international standards to voltage harmonics—and in particular by the probabilistically-established limits after EN50160—we have chosen as case-study examples three typical sets of possible voltage distortion levels, i.e.,:


However, each of these three criteria is only one among infinite possible ways to set the voltage distortion level. Once the distortion level is set parametrically or known from measurements in a particular point of common coupling (PCC) in the grid or internal point of coupling (IPC) in a plant, the method proposed here—namely Equations (31)–(39)—can be applied anyway.

#### **4. Conclusions**

This paper has refined the theory and broadened the applications of an electro-thermal probabilistic life model—also referred to as electro-thermal reliability model—developed in previous studies for the insulation of power components affected by current and voltage harmonics. In line with previous papers, the outcomes of this article confirm that the limits to voltage harmonics set in international standards may be not totally conservative as for the reliability of components subjected to the harmonics generated by power electronic converters. This holds even if a fairly broad safety margin of 25% is kept from such limits. This effect is due not only to the peak of the distorted voltage, accounted for by the peak factor *Kp*, but also to the shape of the distorted voltage—thus to its time derivative—accounted for by the shape factor *Kw*. The effect of voltage harmonics may be compensated for by a lower-than-rated load current of components. However, only a strong reduction of the load can hide the effect of distorted voltage totally.

Another important conclusion of this study is that for all treated components (cables, capacitors, induction motors, transformers) and for all case studies examined, the whole sequence of odd voltage harmonics from the 5th to the 25th analysed here is more challenging that the simple combination of

the 5th, 7th, 11th, 13th voltage harmonics treated previously. Thus, it is worth performing the much more cumbersome analysis carried out here, since it is not only more accurate, but also on the safe side from the viewpoint of life and reliability estimation of power components.

The results also point out that a wide variety of distorted regimes can be encountered in practice, each having different effects on component reliability. Therefore, a reliability model like that presented here is a quite useful tool to be used in each case of interest beside other methods, like careful experimental measurements and sound diagnostic techniques. All these techniques employed together may provide a fairly exhaustive picture of the effect of voltage and current harmonics generated by power electronic devices on the life and reliability of power components.

It is also worth emphasizing that the reliability model is used here for life and reliability estimates of the MV/LV capacitor and cable, but it can be extended to other insulations, e.g., those of induction motors and transformers. Indeed, the treated components feature quite different values of the parameters of the reliability model: since the main results obtained here for all case studies hold for both the cable and the capacitor, these results appear to be general and independent of the values of the parameters, as well as of the uncertainty in their evaluation.

Furthermore, the investigations presented in this paper for the international standards EN50160, IEC 61000-2-2, and IEC 61000-2-4 could be conducted in future works considering also other standards from the IEC 61000 series that are related to specific equipment and converters.

As a final remark, it has to be pointed out that—due to the simplifying assumptions made and the "quasi-parametric" approach followed—the results obtained should be always generalized with care, being indicative but not exhaustive. Indeed, all results reported here have been obtained from particular values of reliability model parameters, which are subjected to all uncertainties related to the relevant experimental campaigns and failure of data processing. Moreover, these values hold for the particular insulating specimens considered in those campaigns, which of course cannot be considered as fully representative of all insulations of existing capacitors, motors, transformers and cables found in MV an LV system.

**Author Contributions:** Conceptualization, G.M.; methodology G.M.; validation, G.M. and B.D.; formal analysis, G.M., B.D., E.C., P.D.F., L.P.D.N.; investigation, G.M.; writing—original draft preparation, G.M.; writing—review and editing, G.M. and B.D. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **The Study of the Single Event Effect in AlGaN/GaN HEMT Based on a Cascode Structure**

**Yanan Liang 1, Rui Chen 1,2,\*, Jianwei Han 1,2, Xuan Wang 1, Qian Chen <sup>1</sup> and Han Yang <sup>1</sup>**


**Abstract:** An attractive candidate for space and aeronautic applications is the high-power and miniaturizing electric propulsion technology device, the gallium nitride high electron mobility transistor (GaN HEMT), which is representative of wide bandgap power electronic devices. The cascode AlGaN/GaN HEMT is a common structure typically composed of a high-voltage depletionmode AlGaN/GaN HEMT and low-voltage enhancement-mode silicon (Si) MOSFET connected by a cascode structure to realize its enhancement mode. It is well known that low-voltage Si MOSFET is insensitive to single event burnout (SEB). Therefore, this paper mainly focuses on the single event effects of the cascode AlGaN/GaN HEMT using technical computer-aided design (TCAD) simulation and heavy-ion experiments. The influences of heavy-ion energy, track length, and track position on the single event effects for the depletion-mode AlGaN/GaN HEMT were studied using TCAD simulation. The results showed that a leakage channel between the gate electrode and drain electrode in depletion-mode AlGaN/GaN HEMT was formed after heavy-ion striking. The enhancement of the ionization mechanism at the edge of the gate might be an important factor for the leakage channel. To further study the SEB effect in AlGaN/GaN HEMT, the heavy-ion test of a cascode AlGaN/GaN HEMT was carried out. SEB was observed in the heavy-ion irradiation experiment and the leakage channel was found between the gate and drain region in the depletion-mode AlGaN/GaN HEMT. The heavy-ion irradiation experimental results proved reasonable for the SEB simulation for AlGaN/GaN HEMT with a cascode structure.

**Keywords:** AlGaN/GaN HEMT; cascode structure; single event effects; technology computer-aided design simulation; heavy-ion irradiation experiment

#### **1. Introduction**

As high-power and miniaturizing electric propulsion technology devises continue to develop, high-performance and high-reliability wide bandgap power electronic devices begin to play an important role in air and space vehicles. Gallium nitride high electron mobility transistors (GaN HEMT) are representative of wide bandgap power electronic devices and could be an attractive candidate for space and aeronautic applications due to their excellent electrical characteristics, such as high electron mobility, high breakdown voltage, and high thermal conductivity. To realize space and aeronautic applications, the radiation effect should be considered, including the total ionizing dose (TID) effect, single event effect (SEE), and the displacement damage dose (DDD) effect. Due to the existence of two-dimensional electron gas (2DEG), AlGaN/GaN HEMT, fabricated on AlGaN/GaN heterojunction, is in depletion mode. To ensure the reliability of the device in space applications, enhancement-mode devices can be adopted. The enhanced mode is mainly realized in AlGaN/GaN HEMT by the following: p-GaN [1,2], F ion implantation [3,4], MIS HEMT [5,6], and cascode structure [7,8]. The total ionizing dose and displacement damage effect on GaN HEMT have been studied by many researchers [9–14]. Because there

**Citation:** Liang, Y.; Chen, R.; Han, J.; Wang, X.; Chen, Q.; Yang, H. The Study of the Single Event Effect in AlGaN/GaN HEMT Based on a Cascode Structure. *Electronics* **2021**, *10*, 440. https://doi.org/10.3390/ electronics10040440

Academic Editor: Elio Chiodo Received: 23 December 2020 Accepted: 20 January 2021 Published: 10 February 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

is no gate dielectric layer in AlGaN/GaN HEMT, GaN HEMT is less sensitive to the TID effect. Additionally, some references [12,13] have shown that the DDD effect impacts the direct-current (DC) characteristics of the AlGaN/GaN HEMTs and the failure mechanisms have been basically studied. However, as one of the most important effects of space environment radiation, many studies focused on the enhancement-mode AlGaN/GaN HEMT based on the p-GaN structure [15–17]. The SEE characteristics and AlGaN/GaN HEMT failure mechanisms based on the cascode structure are not clear. Therefore, in this paper, we focused on studying the SEE of the AlGaN/GaN HEMTs and analyzing the failure mechanism of the single event burnout (SEB) effect.

To study the radiation response and failure mechanism of the SEB effect for GaN HEMTs, TCAD simulation and heavy-ion irradiation experiments were carried out. The remainder of the paper is organized as follows.

Firstly, according to the circuit structure shown in Figure 1, the cascode structure of AlGaN/GaN HEMT, including the depletion-mode AlGaN/GaN HEMT and Si MOS-FET, were modeled. Because low-voltage Si MOSFET is not sensitive to the SEB effect, we focused on studying the SEE in the depletion-mode AlGaN/GaN HEMT. The simulation results showed that (1) the SEB effect of the AlGaN/GaN HEMT was influenced by heavy-ion energy and track position. Further, (2) a leakage channel between the gate electrode and drain electrode was observed after heavy-ion striking in depletion-mode AlGaN/GaN HEMT.

**Figure 1.** The schematic of aluminium gallium nitride /gallium nitride high electron mobility transistor (AlGaN/GaN HEMT), based on the cascode structure.

Heavy-ion irradiation experiments of the commercial cascode AlGaN/GaN HEMT were conducted in an off-mode condition. The experimental results showed that (1) the depletion-mode AlGaN/GaN HEMT was burnt out under Vds = 50 V for Ge ions at LET = 28.5 MeV·cm2/mg (about 0.31 pC/μm). Moreover, it showed (2) a leakage channel between the gate electrode and the drain electrode in the depletion-mode, where AlGaN/GaN HEMT was formed after the heavy-ion striking.

Finally, the possible SEE failure mechanism was proposed. We proved that the enhancement of the impact ionization mechanism at the edge of the gate might be an important factor in the formation of the leakage channel.

#### **2. Simulation Results and Discussion**

As shown in Figure 1, a high voltage depletion-mode AlGaN/GaN HEMT and a low voltage enhancement-mode silicon (Si) MOSFET were connected to form a high voltage enhancement-mode AlGaN/GaN HEMT in the cascode structure. For the cascode AlGaN/GaN HEMT circuit, the drain electrode of GaN HEMT served as the drain port and the gate electrode of Si MOSFET acted as the gate port. The source port of the device was formed by connecting the source electrode of Si MOSFET and the gate electrode of GaN HEMT. It is well known that the low voltage power MOSFETs are robust to the SEB effect [18–20]. Therefore, to investigate the mechanism of the SEB induced by heavy ion irradiation on cascode AlGaN/GaN HEMT, we studied the SEB effect of the depletion-mode AlGaN/GaN HEMT. TCAD simulation was carried out with the Sentaurus TCAD simulator.

#### *2.1. Modeling*

In the simulation, the generic model of the depletion-mode AlGaN/GaN HEMT supported by the foundry was adopted. According to previous reports [21–23], the architecture considered in the simulation is shown in Figure 2. The structure consisted of a Si substrate, a GaN buffer layer, an unintentional doped GaN channel layer, an unintentional doped AlGaN barrier layer, and a SiN passivation layer. The breakdown voltage of the simulated device was higher than 900 V. The breakdown, transfer, and output characteristics of the simulated structure are shown in Figure 3. The threshold voltage (Vth) of the AlGaN/GaN HEMT was approximately −2.5 V.

**Figure 2.** The schematic of the depletion-mode AlGaN/GaN HEMT.

**Figure 3.** The characteristics of the depletion-mode AlGaN/GaN HEMT: (**a**) the breakdown characteristic, (**b**) the transfer characteristic, and (**c**) the output characteristics.

In the simulation, we used a heavy-ion model in Sentaurus software. The impact ionization model and drift-diffusion model were adopted. To simplify the simulation process, thermal equations (lattice heating) were not considered in this paper. The heavyion was vertical incidence on the device from the front side. The Gaussian track radius was 20 nm spatially.

#### *2.2. Simulation Results and Discussion*

#### 2.2.1. SEB Characteristics of HMET

To trigger the device's SEE effect, we adopted a track length of 10 μm and linear energy transfers (LET) of 1 pC/μm. The drain, source, and gate port of the depletionmode AlGaN/GaN HEMT were biased at 400 V, 10 V, and 0 V, respectively (Vgs = −10 V, Vds = 390 V), which ensured that the device was in the off state. The transient drain, source, and gate currents are presented in Figure 4. The currents increased immediately after a heavy-ion struck the device. The drain current increased to about 2.5 mA, the gate current increased to about 1.5 mA, and the source current increased to about 1 mA. Approximately 1 ns after striking, the source current returned to 0.1 μA and the drain current and gate current reached 0.2 mA. The drain current and gate current were about 0.1 mA at 50 ns

after striking. This indicated that a leakage channel between the gate electrode and the drain electrode in depletion-mode AlGaN/GaN HEMT was formed.

**Figure 4.** The transient drain, source, and gate currents after heavy-ion striking.

2.2.2. Influence Factors of SEB

• Heavy-ion energy

Heavy-ion energy, heavy-ion track length, and heavy-ion track position are important parameters that affect the single event effect of the depletion-mode AlGaN/GaN HEMT. Heavy-ion energy is related to the electron-hole pairs produced in the device. To study the influence of heavy-ion energy on the SEE of the depletion-mode AlGaN/GaN HEMT, simulation experiments with different LETs (10, 5, 1, 0.1, 0.01, 0.005, 0.001 pC/μm) were carried out. The track length (10 μm) and track position (close to the drain electrode) were adopted in the simulation. The transient drain currents after heavy ion impacts for different LETs were shown in Figure 5. The transient drain current increased with the LET increasing and reached a saturation value until the LET was higher than 0.1 pC/μm. It indicated that the electron-hole pairs generated in the AlGaN and GaN layers increased with the LET grew and reached the maximum at 0.1 pC/μm [16].

• Heavy-ion track length

In addition to heavy-ion energy, the heavy-ion track length is also an important parameter. The impacts of heavy-ion with different track lengths were also studied. In the simulation, the heavy-ion energy (1 pC/μm) and track position (close to the drain electrode) were adopted and the track lengths were 2, 5, 10, and 15 μm. The transient drain currents after heavy ion impacts for different track lengths were shown in Figure 6. The amplitude of transient drain current almost kept constant under these simulation conditions. In these simulation conditions, the heavy-ion had passed through the GaN channel layer and reached the GaN buffer layer or the Si substrate layer. In other words, the heavy-ion had passed through the active region of the depletion-mode AlGaN/GaN HEMT. Therefore, the

transient drain currents did not change with heavy-ion track length under these simulation conditions. These results indicated when the heavy-ion passed through the GaN channel layer and reached the GaN buffer layer, the heavy-ion track length had little effect on the SEE of the depletion-mode AlGaN/GaN HEMT.

**Figure 5.** The transient drain currents after heavy-ion striking with different linear energy transfers (LETs).

**Figure 6.** The transient drain currents after heavy-ion striking with different track lengths.

• Sensitive region of SEB

To localize the sensitive region of the depletion-mode AlGaN/GaN HEMT, different track positions of heavy-ion were studied. The heavy-ion energy (1 pC/μm) and track length (10 μm) were adopted in the simulation. The schematic of heavy-ion tracks on the depletion-mode AlGaN/GaN HEMT is revealed in Figure 7. The transient drain currents after heavy-ion impact on different track positions are shown in Figure 8. The drain currents were about 0.1 mA at 50 ns after the heavy-ions striking at P7, P8, and P9, while the drain currents were lower than 10−<sup>8</sup> A at 50 ns after the heavy-ions striking at P1 to P6. These indicated that a SEB current was triggered when the heavy-ion impact on the location of P7, P8, and P9. Moreover, the closer the track position is to the drain region, the more sensitive it is to the SEB device.

**Figure 7.** The cross-section of AlGaN/GaN HEMT with different heavy-ion track positions.

**Figure 8.** The transient drain currents after heavy-ion striking with different positions.

#### 2.2.3. The Failure Mechanism of SEB

To understand the SEB mechanism, the cross-sections of the total current density after the heavy-ion impact on location P5 and P8 are shown in Figures 9 and 10, respectively. Figures 9a and 10a show the initial states of the total current density before the heavy-ions striking. Figures 9b and 10b show the total current density after the heavy-ions striking at P5 and P8 for 0.05 ns. After the heavy-ions striking, the total current density in the GaN buffer layer increased (from green to orange), indicating that the GaN buffer layer forming a leakage channel. Figures 9c and 10c show the total current density after the heavy-ions strike at P5 and P8 for 50 ns. These showed that the leakage channel disappeared at 50 ns after the heavy-ions striking at P5, while it was still a leakage channel at 50 ns after the heavy-ions striking at P8.

**Figure 9.** The cross-sections of the total current density with time after the heavy-ion striking at P5: (**a**) before heavy-ion striking, (**b**) 0.05 ns after heavy-ion striking, and (**c**) 50 ns after heavy-ion striking. The white line represented the depletion region.

**Figure 10.** The cross-sections of the total current density with time after heavy-ion striking at P8: (**a**) before heavy-ion striking, (**b**) 0.05 ns after heavy-ion striking, and (**c**) 50 ns after heavy-ion striking. The white line represented the depletion region.

After the heavy-ions strike the device, the electron-hole pairs were generated. Electrons flowed toward the drain under the high drain voltage and collected by the drain electrode. Holes were left in the GaN buffer layer, leading to the electrons injection of the source and gate electrode [16,24]. Thus, the large transient current was generated in the source, gate, and drain regions. The leakage channel of the GaN buffer layer was formed. The electron-hole pairs generated by heavy-ions decreased with time, leading to the leakage current reduced.

In order to further understand the SEB mechanism, the cross-sections of the impact ionization after the heavy-ion striking at P5 and P8 were studied. Figures 11a and 12a were the initial value of the impact ionization before the heavy-ions struck. Figure 11b,c and Figure 12b,c show the impact ionization at 0.05 ns and 50 ns after the heavy-ions striking at P5 and P8. As shown in Figure 11, after the heavy-ions striking at P5, the impact ionization at the edge of the gate region decreased at 0.05 ns and the value was still low at 50 ns. As shown in Figure 12, after the heavy-ions striking at P8, the impact ionization at the edge of the gate region and the drain region increased at 0.05 ns and the value was still

high at 50 ns. The electron-hole pairs were generated after the heavy-ion injected to the AlGaN/GaN HEMT. Then the generated electron-hole pairs drifted under the applied voltage. Electrons flowed toward the drain under the high drain voltage and collected by the drain electrode. Holes were left in the GaN buffer layer, leading to the electrons injection of the source, and gate electrode. The movement of the electron-hole pairs led to the changing of the potential distribution. When the heavy-ions struck at P5, which was close to the gate electrode, the electrons were injected from the source and gate electrode quickly. This process decreased the potential near the drain, which decreased the impact ionization at the edge of the gate region. When the heavy-ions struck at P8, which was close to the drain electrode, electrons were quickly collected by the drain and holes left in the GaN buffer layer accumulated. This process increased the potential near the drain, which enhanced the impact ionization at the edge of the gate region. The enhancing impact ionization increased the number of generated carriers, which increased the probability of electron tunneling of the gate region. This might be attributed to the burnout of the gate electron of the AlGaN/GaN HEMT after heavy-ion striking at P8.

**Figure 11.** The cross-sections of the impact ionization after the heavy-ion striking at P5: (**a**) before heavy-ion striking, (**b**) 0.05 ns after heavy-ion striking, and (**c**) 50 ns after heavy-ion striking.

**Figure 12.** The cross-sections of the impact ionization after the heavy-ion striking at P8: (**a**) before heavy-ion striking, (**b**) 0.05 ns after heavy-ion striking, and (**c**) 50 ns after heavy-ion striking.

#### **3. Heavy-Ion Experimental Results and Discussion**

*3.1. Experiment Samples and Setup*

To further study the phenomenon of the drain current increase obtained in the simulation section, a commercial GaN HEMT with a breakdown voltage of 900 V was chosen. The devices under test (DUTS) were commercial devices employing a cascode structure from transform (TP90H180PS). The radiation experiments were carried out on the heavyion accelerator of China Atomic Energy Research Institute, shown in Figure 13. In the heavy-ion radiation experiments, the device was biased at the off state. The source port and gate port were grounded (Vgs = 0 V). The drain port was biased at 50 V (Vds = 50 V). The heavy-ion parameters used in the irradiation experiment are shown in Table 1. For the heavy-ion irradiation with the flux of about 10<sup>4</sup> particle/cm2/s, the devices were irradiated to the fluence of 5 × 106 particle/cm2. The drain currents were monitored by Keithley 2470 during the heavy-ion irradiation. The drain current and gate current were measured by Keithley 2470 and 2450 after heavy-ion irradiation.

**Figure 13.** The heavy-ion accelerator of China Atomic Energy Research Institute.

**Table 1.** Heavy ions used in the experiment.


#### *3.2. Experiment results and discussion*

The transient drain currents during the heavy-ion irradiation are shown in Figure 14. When the drain port was biased at 50 V, the drain current quickly increased to 10 mA (current limitation), which took about 45 s at LET = 28.5 MeV·cm2/mg (about 0.31 pC/μm).

**Figure 14.** The drain currents of the device at Vds = 50 V for Ge ions at LET = 28.5 MeV·cm2/mg (about 0.31 pC/μm)**.**

After heavy-ion irradiation, the drain current and gate current were measured (Vgs =0V), as shown in Figure 15. The drain leakage current increased to approximately 5 mA and the gate leakage current was less than 5 nA at Vds = 20 V, Vgs = 0 V. It indicated that a leakage channel was formed between the source port and drain port in the Cascode AlGaN/GaN HEMTs. Analyzing from Figure 1, the source port and drain port of the Cascode AlGaN/GaN HEMTs corresponded to the gate electrode and the drain electrode of the depletion-mode AlGaN/GaN HEMT. Therefore, a leakage channel was formed between the gate electrode and the drain electrode in the depletion-mode AlGaN/GaN HEMT.

**Figure 15.** The gate and drain currents of the device after heavy-ion irradiation.

The picture of the DUT before and after heavy-ion irradiation is shown in Figure 16. Figure 16a shows the structure of the cascode AlGaN/GaN HEMT under test and Figure 16b shows a picture of the device after heavy-ion irradiation. The left device was the low voltage enhancement-mode Si MOSFET and the right device was the high voltage depletion-mode AlGaN/GaN HEMT. As shown in Figure 16b, the depletion-mode AlGaN/GaN HEMT was burnt out under Vds = 50 V for Ge ions at LET = 28.5 MeV·cm2/mg (about 0.31 pC/μm). The Si MOSFET was not sensitive to SEB. Moreover, it was found that the burned area of the depletion-mode AlGaN/GaN HEMT was located at the gate electrode of the device, using layout photographing technology. A burnout area between the drain and gate was found on the surface of depletion-mode AlGaN/GaN HEMT from the heavy-ion experimental results, as shown in Figure 16b, which indicated the sensitive region was between the drain electrode and gate electrode.

Analyzing from the simulation results, the sensitive region of the depletion-mode AlGaN/GaN HEMT was located near the drain. As shown in Figure 16b, the sensitive region was between the drain electrode and gate electrode. The results of the heavy-ion experiment proved reasonable of the SEB simulation for AlGaN/GaN HEMT with the cascode structure. In a word, the simulation and heavy-ion experimental results both indicated that a leakage channel between the gate electrode and the drain electrode in depletion-mode AlGaN/GaN HEMT was formed after the heavy-ion striking. This may be the main failure mechanism leading to the SEB effect.

#### **4. Conclusions**

Single event effects on cascode AlGaN/GaN HEMT were studied in this paper. To figure out the mechanism of single event effects, the simulation and experiment were carried out. The simulation results showed that the heavy-ion energy and track position influenced the SEB effect of the depletion-mode AlGaN/GaN HEMT. When the heavy-ion passed through the GaN channel layer and reached the GaN buffer layer, the heavy-ion track length had little effect on the SEE of the depletion-mode AlGaN/GaN HEMT. The sensitive region of the depletion-mode AlGaN/GaN HEMT was located near the drain. When the depletion-mode AlGaN/GaN HEMT was biased at Vgs = −10 V and Vds = 390 V, heavy-ion (LET = 0.1 pC/μm) struck near the drain and caused a leakage current between the gate electrode and drain electrode in the depletion-mode AlGaN/GaN HEMT. Moreover, the heavy-ion irradiation experiments of a commercial cascode AlGaN/GaN HEMT (TP90H180PS) were carried out. Experimental results showed that the drain current of the device increased to 10 mA (limiting by the instrument), which was about 45 s at LET = 28.5 MeV·cm2/mg (about 0.31 pC/μm) with Vds = 50 V. Analyzing from the cascode structure, the leakage channel was formed between the gate electrode and the

drain electrode in the depletion-mode AlGaN/GaN HEMT. The picture of the DUT after heavy-ion irradiation indicated that the burning area was between the gate electrode and the drain electrode of the depletion-mode AlGaN/GaN HEMT. The results of the heavy-ion experiment proved reasonable for the SEB simulation of AlGaN/GaN HEMT with cascode structure. The enhancement of the impact ionization at the edge of the gate was revealed using simulation and it might be an important factor for SEB effect. It was very useful for the radiation-hardened technique for GaN HEMT design.

**Author Contributions:** Conceptualization, Y.L.; data curation, Y.L.; formal analysis, X.W.; investigation, Q.C. and H.Y.; methodology, R.C.; project administration, R.C. and J.H.; writing—original draft, Y.L.; writing—review and editing, Y.L. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported in part by the Fast Support Project (Grant No. E11Z360101) and the Beijing Municipal Commission of Science and Technology (Grant No. E039360101).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


## *Article* **On the Lifetime Estimation of SiC Power MOSFETs for Motor Drive Applications**

**Carmelo Barbagallo , Santi Agatino Rizzo \* , Giacomo Scelba, Giuseppe Scarcella and Mario Cacciato**

Department of Electrical, Electronics and Computer Engineering, University of Catania, 95129 Catania, Italy; carmelo.barbagallo@unict.it (C.B.); giacomo.scelba@unict.it (G.S.); giuseppe.scarcella@unict.it (G.S.); mario.cacciato@unict.it (M.C.)

**\*** Correspondence: santi.rizzo@unict.it; Tel.: +39-095-738-2308

**Abstract:** This work presents a step-by-step procedure to estimate the lifetime of discrete SiC power MOSFETs equipping three-phase inverters of electric drives. The stress of each power device when it is subjected to thermal jumps from a few degrees up to about 80 ◦C was analyzed, starting from the computation of the average power losses and the commitment of the electric drive. A customizable mission profile was considered where, by accounting the working conditions of the drive, the corresponding average power losses and junction temperatures of the SiC MOSFETs composing the inverter can be computed. The tool exploits the Coffin–Manson theory, rainflow counting, and Miner's rule for the lifetime estimation of the semiconductor power devices. Different operating scenarios were investigated, underlying their impact on the lifetime of SiC MOSFETs devices. The lifetime estimation procedure was realized with the main goal of keeping limited computational efforts, while providing an effective evaluation of the thermal effects. The method enables us to set up any generic mission profile from the electric drive model. This gives us the possibility to compare several operating scenario of the drive and predict the worse operating conditions for power devices. Finally, although the lifetime estimation tool was applied to SiC power MOSFET devices for a general-purpose application, it can be extended to any type of power switch technology.

**Keywords:** AC motor drive; junction temperature; lifetime prediction; power MOSFET; loss modeling; reliability; SiC MOSFET

#### **1. Introduction**

The remarkable properties of silicon carbide (SiC) have made it a perfect candidate for replacing silicon-based power electronic devices in high power, high-temperature applications. In fact, SiC MOSFET technology provides excellent performance to MOSFET power devices in terms of low on-state resistance, high switching frequency, high breakdown voltage, high current capability also at very high temperature. Therefore, this technology represents a valid alternative to typical Si MOSFET and IGBT power devices for many applications [1]. The prospects for strong growth in SiC devices are high and are even stimulated by the increasing sales of plug-in hybrid and electric vehicles. Compared to the well mature Si technology, the field reliability of SiC devices must be demonstrated for various applications, and a voltage-derating design guideline needs to be established. This is especially important for applications in which reliability is extremely critical, such as the automotive and aerospace applications. Hence, it becomes imperative to apply in-depth studies on the SiC devices performance and operating limits, especially when they are used in very critical applications, for instance when they are integrated into three-phase traction inverters powering electric motors. In fact, because power conversion systems equipped with a SiC device potentially provide higher current density than one with Si devices, which leads to a larger thermal ripple, they need more stringent requirements for the package materials.

**Citation:** Barbagallo, C.; Rizzo, S.A.; Scelba, G.; Scarcella, G.; Cacciato, M. On the Lifetime Estimation of SiC Power MOSFETs for Motor Drive Applications. *Electronics* **2021**, *10*, 324. https://doi.org/10.3390/electronics 10030324

Academic Editor: Elio Chiodo Received: 31 December 2020 Accepted: 25 January 2021 Published: 30 January 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

Studies on the state-of-the-art SiC MOSFET's reliability evaluation and failure mode analysis were carried out in [2–4]; these studies pointed out the evolution and improvements as well as the future challenges of this promising device technology. The electrothermal co-simulation approach based on a PSpice-based model, including temperature dependency and a Simulink-based thermal network, was even proposed in [5] for SiC MOSFETs. Some model quantities have been obtained by FEM simulation for more accurate results, and a MATLAB script has been used to manage and interface the data from the different simulation tools. An accelerated power cycling test platform using a current source converter for SiC-MOSFET power modules was also presented in [6], where the junction temperature variations of the devices were monitored without the removal of silicone gel. Moreover, the analysis was used to examine some failure precursors and then to estimate the useful lifetime of SiC MOSFET modules. A comparison in the area of device reliability accounting for condition monitoring and active thermal control as well as the lifetime was recently carried out [7]. A method to obtain the thermal impedance of a SiC module by combining optical measurement and multi-physics simulations was proposed in [8], where the measurement of the junction temperature was performed by using fiber optic instead of temperature-sensitive electrical parameters. Several major achievements and novel architectures in SiC modules packaging were analyzed in [9], where the authors reported an accurate survey of the materials by considering their coefficient of thermal expansion and their proper combination to reduce the thermal stress in the material interfaces. The impact of different pulse width modulation control techniques on the power losses and thermal stress on SiC power modules used in a three-phase inverter was investigated in [10]. The advantages and problems due to the use of SiC MOSFET in a traction inverter were discussed in [11] to provide the guidelines for a viable solution. Electrical and thermal issues, safety and reliability problems, and challenges due to device paralleling and layout were analyzed by exploiting on-field experience in the industry sector, i.e., experimental tests, finite element analysis, and circuit simulations.

To reduce the total design and maintenance cost and to guarantee the service continuity as well as the human safety, it is also required to have an accurate prediction of the remaining lifetime prediction of power converters, which can help to prevent unwanted failures and generate better maintenance plans. While the reliability and lifetime prediction of silicon (Si) semiconductor device based power converters have been widely investigated in the literature [12–14], SiC MOSFETs are facing new reliability challenges. Hence, the design of more reliable SiC power converters requires an accurate lifetime prediction as well as online monitoring strategies for real-time lifetime prediction [15]. Different approaches can be applied to estimate the lifetime of the SiC power devices and power converters [16].

In the context discussed so far, this work presents a lifetime prediction method of SiC power MOSFETs integrated into three-phase inverters supplying a three-phase induction machine. The analysis exploits a suitable developed simulation tool realized in MATLAB and Simulink to keep the computational burden low, and it also provides a modular structure of the proposed procedure. The electric drive, the Coffin–Manson relation, the rainflow counting method, and Miner's rule are suitable for the analysis and are combined to calculate the lifetime prediction [17–28]. As it is simple and reliable, Miner's rule is the most widely used fatigue life prediction technique in this field.

In the proposed approach, the off-line prediction of the lifetime is performed, starting from the data obtained, by exploiting suitable modeling of the electric drive, operating at the conditions provided by the mission profile (MP) of the application. In the investigated case, several tests are carried out using 650 V, 45 A SiC power MOSFETs by customizing the MPs with suitable weights based on the operating conditions. Hence, the proposed procedure allows for the prediction of the behavior of the thermal stresses affecting the power switches in a wide operating range. Moreover, the option of creating several customized MPs allows us to quickly carry out many analyses and then making a comparison among them. In the

following, a step-by-step description of the developed activity is provided, along with the underlying advantages and limits of the approach.

#### **2. Framework of the MATLAB-Simulink Tool for SiC MOSFET Lifetime Estimation**

The tool provides a lifetime estimation by performing the main steps summarized in Figure 1 and listed below:


**Figure 1.** Workflow of the lifetime prediction starting from the electric drive system mission profile.

#### *2.1. Generation of the Mission Profile*

#### 2.1.1. Motor Drive System

The electric drive was modelled on Simulink, which is a MATLAB-based graphical programming environment and was exploited to model the entire electric drive. The model includes a three-phase asynchronous machine whose characteristics are shown in Table 1. The motor is supplied by an ideal three-phase inverter and controlled according to an indirect field oriented control (IFOC) [16]. The mission profile was carried out through the execution of a series of simulations by setting a matrix of reference speeds and reference torques values. The results of the simulations represent the dataset establishing the operating conditions of the inverter and thus power devices, according to the considered mission profile. The Simulink model of the electric drive is shown in Figure 2. It consists of different grouped blocks: vector control IFOC, modulation block SVPWM (space vector pulse width modulation), a three-phase inverter realized with ideal power switches, and the asynchronous motor. The block "Power Factor" performs the calculation of the quantity *cos*(*ϕ*(*t*)), while the block "To Workspace" allows the average values of the modulation index *ma*(*t*), the power factor *cos*(*ϕ*)(*t*), electric frequency *fe*(*t*) and the maximum current of the phase a, *ias*(*t*) to be available, thus allowing us to extract the following quantities:


**Table 1.** Technical specification of the induction motor.


**Figure 2.** Simulink model of the electric drive system.

The simulations were carried out by using a sampling time *Ts* = <sup>3</sup>×10−<sup>6</sup> s, an input dc voltage of the inverter *Vdc* = 560 V, and a switching frequency of the power switches equal to *fsw* = 10 kHz.

Figure 3 shows some of the results carried out from these simulations. In particular, it displays the waveforms of the above-listed electrical quantities when the reference speed and reference torque of the drive are 1400 rpm and 80 Nm, respectively. The blue traces represent the instantaneous values, while the orange traces represent the steady-state values. A campaign of simulations in which the speed is varied from 0 to 1400 rpm and the torque from 0 to 80 Nm was executed, allowing us to identify the variation ranges of the electrical quantities, which are summarized in the surfaces shown in Figure 4. These surfaces represent the input dataset required in the next step.

**Figure 3.** Results of the electric drive model operated at 1400 rpm and 80 Nm: (**a**) motor phase current a; (**b**) amplitude modulation index; (**c**) power factor; (**d**) output frequency.


**Figure 4.** Results of the electric drive model as a function of the reference speed and electromagnetic torque: (**a**) maximum current; (**b**) amplitude modulation index; (**c**) power factor; (**d**) frequency.

#### 2.1.2. Weights Assignment

For each operating condition, expressed in terms of speed and torque, a unique set of current, modulation index, power factor, and frequency values were obtained. A customizable MP, i.e., related to a specific use of the electric drive, can be obtained by assigning a time interval to each operating condition, according to a preset probability of occurrence. In this study, the sum of gaussian distributions, given by Equation (1), was used to achieve a customized time interval weights distribution, and was normalized so that their sum is equal to 100%.

$$f(\mathbf{x}, y) = \sum\_{i=1}^{n} A\_i \cdot \exp\left[ -\left( \frac{(\mathbf{x} - \mathbf{x}\_{0,i})^2}{2 \cdot \sigma\_{\mathbf{x},i}^2} + \frac{(y - y\_{0,i})^2}{2 \cdot \sigma\_{\mathbf{x},i}^2} \right) \right] \tag{1}$$

where:


The above expression was applied to the dataset carried out in the previous step, by substituting the values of speed and torque to the quantities *x* and *y* respectively, while *x*0,*<sup>i</sup>* and *y*0,*<sup>i</sup>* are the central values of each Gaussian distribution. In practice, Equation (1) provides the sum of Gaussian surfaces, which establishes the weight that has to be assigned to each operating condition of the drive; in the proposed procedure, these surfaces were generated by using a MATLAB algorithm. Figure 5 shows two examples of distribution when the index *i* = 1 (single Gaussian distribution) and *i* > 1 (multiple Gaussian distributions).

**Figure 5.** Examples of Gaussian surfaces: case *i* = 1 (**a**) a single Gaussian distribution "3D visualization" and (**b**) "2D visualization"; case *i* > 1 (**c**) multiple Gaussian distributions "3D visualization" and (**d**) "2D visualization".

#### 2.1.3. Grouping of the Motor Drive Operating Conditions

After having assigned the weights to each operating condition of the drive according to the MP, a grouping process is required to group similar operating conditions, thus reducing the computational burden required to the entire lifetime estimation procedure. The grouping is performed by initially splitting in a certain number of intervals the values of *I*0, *maavg*, *cos*(*ϕ*)*avg*, *fe*\_*avg*, defining the minimum, maximum, step, and central value of each interval, according to Table 2. Then, the values of the electrical quantities extracted from the simulations of the electric drive are inserted in the corresponding intervals. A suitable number of intervals have to be chosen to guarantee a good compromise between the representative conditions of the drive and the computational efforts needed to compute the lifetime of the SiC power device [22,23].

**Table 2.** Grouping of the electric drive operating conditions.


A MATLAB function was implemented to perform the grouping procedure by assigning each working condition of the drive to a specific set data, as seen in Figure 6. All empty intervals were discarded, and they are not included in the calculation of power losses. Obviously, such an approach can be adopted even when there are data from a real MP. Even in case of handling a huge number of operating condition datasets, this mechanism enables a significant reduction of operating conditions for which the power losses have to be calculated, thus reducing the lifetime prediction time.

**Figure 6.** Grouping procedure where each motor drive operation is included in a specific subset.

#### *2.2. SiC MOSFET Power Losses Computation*

The power losses of each SiC device composing the inverter are carried out from the dataset obtained in the grouping procedure. In this analysis, the lifetime estimation is based on solder degradation, strictly correlated to the Δ*Tj* variations of junction temperatures in the fundamental period of the stator voltages; therefore, the calculation of the power losses was implemented by considering the average value of the power, over each switching period *Tsw*. The following equations show the method used for the calculation [22–25]:

$$DC^{\mathbb{R}} = \left\{ \begin{array}{l} DC\_{MOS}^{\mathbb{R}} = 0.5 \cdot \left[1 + m\_{d} \cdot \sin(2\pi f\_{\varepsilon} \cdot nT\_{sw})\right] \text{ s\!e.0} \leq nT\_{sw} < \frac{T\_{\varepsilon}}{2} \\\ DC\_{SBD}^{\mathbb{R}} = 0.5 \cdot \left[1 + m\_{d} \cdot \sin(2\pi f\_{\varepsilon} \cdot nT\_{sw})\right] \text{ s\!e. } \frac{T\_{\varepsilon}}{2} \leq nT\_{sw} < T\_{\varepsilon} \end{array} \right\} \tag{2}$$

$$DC^{\eta} = \left\{ \begin{array}{l} DC\_{MOS}^{\eta} = 0.5 \cdot \left[ 1 + m\_{d} \cdot \sin(2\pi f\_{\varepsilon} \cdot nT\_{\rm sw}) \right] \text{ se } 0 \le nT\_{\rm sw} < \frac{T\_{\varepsilon}}{2} \\\ DC\_{SBD}^{\eta} = 0.5 \cdot \left[ 1 + m\_{d} \cdot \sin(2\pi f\_{\varepsilon} \cdot nT\_{\rm sw}) \right] \text{ se } \frac{T\_{\varepsilon}}{2} \le nT\_{\rm sw} < T\_{\varepsilon} \end{array} \right\} \tag{3}$$

$$I\_o^n = I\_o \cdot \sin(2\pi f\_c \cdot nT\_{sw} - \varphi) \tag{4}$$

where:


The voltage drops were carried out from the output characteristics of the SiC devices contained in the datasheets, as well as the switching energy loss and the reverse conduction characteristic. The characteristics of the SiC device under test are shown in Figure 7.

**Figure 7.** Technical specifications of the device under test: (**a**) output characteristic; (**b**) reverse conduction characteristic; (**c**) switching energy losses; (**d**) thermal impedance.

Figure 8 shows the power losses curve of a SiC device in a fundamental period, given for different operating conditions.

**Figure 8.** Power losses estimated for different operating conditions: (**a**) different currents; (**b**) different modulation indexes; (**c**) different power factors; (**d**) different fundamental frequencies.

#### *2.3. Junction Temperature Estimation*

The junction temperatures of the SiC Mosfets under test were estimated by exploiting a linear thermal model, whose generalized mathematical representation is given by [26]:

$$
\begin{bmatrix} T\_j^1(t) \\ T\_j^2(t) \\ T\_j^3(t) \\ \vdots \\ T\_j^n(t) \end{bmatrix} = \begin{bmatrix} Z\_{th}^{11}(t) & Z\_{th}^{21}(t) & \cdots & Z\_{th}^{n1}(t) \\ Z\_{th}^{12}(t) & Z\_{th}^{22}(t) & \cdots & Z\_{th}^{n2}(t) \\ Z\_{th}^{13}(t) & Z\_{th}^{23}(t) & \cdots & Z\_{th}^{n3}(t) \\ \cdots & \cdots & \cdots & \cdots \\ Z\_{th}^{1n}(t) & Z\_{th}^{2n}(t) & \cdots & Z\_{th}^{nn}(t) \end{bmatrix} \begin{bmatrix} P\_1(t) \\ P\_2(t) \\ P\_3(t) \\ \vdots \\ P\_n(t) \end{bmatrix} + \begin{bmatrix} T\_a(t) \end{bmatrix} \tag{5}
$$

where:


Given that in this analysis we are considering discrete SiC power devices, the thermal coupling with other devices can be assumed to be negligible, and thus the terms of mutual coupling can be nullified. Hence, Equation (5) can be rewritten as follows:

$$T\_{\dot{I}}\left(t\right) = \left[Z\_{th}(t) \cdot P(t)\right] + T\_a(t) \tag{6}$$

According to Equation (6), the computation of the junction temperature *Tj*(*t*) requires the knowledge of the thermal impedances *Zth*(*t*). The last is graphically provided in the datasheet of the power switch, but a circuital representation of *Zth*(*t*) is required to easily simulate variable power losses profiles. This goal is reached by combining a curve-fitting procedure followed by the implementation of an equivalent Foster thermal network, as intently described in the following step.

#### 2.3.1. Curve Fitting

Starting from the thermal impedance profile *Zth*(*t*), it is possible to implement a curve-fitting procedure on it by using Equation (7), where it is assumed to achieve a profile of *Zth*(*t*) with a Foster thermal network, whose parameters *Rth*,*<sup>n</sup>* and *Cth*,*<sup>n</sup>* constitute the *n*-th pole of the network [26,27].

$$\begin{cases} \ Z\_{th}(t) = \sum\_{n=1}^{N\_p} R\_{th,n} \cdot \left(1 - e^{-\frac{t}{\Pi}}\right) \\ \qquad \tau\_n = R\_{th,n} \cdot \mathbb{C}\_{th,n} \end{cases} \tag{7}$$

where:


It is important to underline the choice of the number of poles. A low number of poles does not guarantee the necessary accuracy, while a high number of poles will certainly be more accurate but results in an increase of the computational efforts. By applying the curve fitting algorithm to the SiC devices under tests, the best compromise has been obtained with seven poles, as shown in Figure 9.

**Figure 9.** Comparison of curve fitting using a different number of poles.

#### 2.3.2. Foster Network

The Foster network can be realized starting from the *Rth*,*<sup>n</sup>* and *Cth*,*<sup>n</sup>* parameters obtained from the previous step, reproducing the thermal behavior of the device under test. Figure 10 displays the Foster network realized to emulate the *Zth*(*t*) of Figure 7d.

**Figure 10.** Simulink model of the Foster network.

2.3.3. Examples of Junction Temperature Profiles

Figure 11 displays the temperature trends of the die junction temperature for different motor drive operating conditions. It should be noted that a significant temperature variation can be experienced during the fundamental period of the phase motor current, leading to high thermal stresses. Moreover, different power losses distributions can be appreciated at varying of the load conditions, which is consistent with the theoretical analysis.

**Figure 11.** Temperature trends carried out for: (**a**) different currents; (**b**) different modulation indexes; (**c**) different power factors; (**d**) different fundamental frequencies.

#### *2.4. Rainflow Counting*

The rainflow-counting (or cycle counting) algorithm is used to count the number of cycles present in a generic waveform. It is a statistical method for the analysis of the random load process, and its counting principle is carried out based on the stress-deformation behavior of the material, as shown in Figure 12. It combines load reversals by defining hysteresis cycles. Each of them has a range of deformation and average stress that can be compared with the constant amplitude.

**Figure 12.** Graphical representation of the rainflow method: (**a**) rainflow on a load profile; (**b**) stress–strain chart; (**c**) main parameters of the rainflow.

The rules for the rainflow counting method are the following: Let X be the range taken into consideration; let Y be the previous range, adjacent to X; let S be the starting point in the history of the load profile [17,18,28]:

	- a. If X <Y, go to step 1.
	- b. If X ≥ Y, go to step 4.

Figure 12 summarizes the rainflow counting procedure.

The load history of Figure 12a is used to illustrate the process. Details of the cycle counting are as follows [28]:



**Table 3.** Example of rainflow counting referred to Figure 12a [28].

After having satisfied the criterion for counting cycles *Nc*, the amplitude and the corresponding mean values of stress or strain for each cycle were calculated. A matrix with information on the cycle, mean value, and amplitude of stress or strain forming a cycle is thus created.

#### Example of Rainflow Counting

Figure 13 show an example of implementation of rainflow procedure on a temperature profile carried out in a specific drive operating condition, by applying the previous steps to the SiC MOSFETs under test; the temperature profile is first linearized to form peaks and valleys. The cycle count *Nc* is carried out by considering the jumps of the junction temperature Δ*Tj* and the average temperature *Tm* in each cycle.

**Figure 13.** Example of rainflow: (**a**) linearized temperature profile in peaks and valleys; (**b**) cycle counting neglecting the average temperature; (**c**) cycle counting considering the average temperature.

#### *2.5. Coffin–Manson's Equation*

The main factors influencing the life of the power devices under test are the jumps in the junction temperature Δ*Tj* and the average junction temperature *Tm*; in fact, the estimate of the number of thermal cycles to failure *Nf* is based on these two parameters.

Equation (8) shows the correlation between the number of cycles to failure with thermal jumps and the average temperature of the cycles [17,19,20]

$$N\_f = a \cdot \left(\Delta T\_{\bar{f}}\right)^{-n} \cdot e^{\frac{E\_d}{k\_B \cdot \bar{T}\_m}}\tag{8}$$


where:


The parameters *a* and *n* of Equation (8) are determined according to the power cycling tests. Figure 14a shows a type of power cycling (*PCsec*) in which an input stress is applied for a period *ton* < 15 *s*. In this case, the tests exert a stress in the interconnection areas close to the chips (bond wire, die-attach). On the contrary, Figure 14b shows a type of power cycling *PCmin* in which a stress is applied for a period *ton* > 15 *s*, stressing the solder layer. The resulting data of these tests give us information on the reliability of the devices under examination under specific operating conditions, and it is normally provided by the manufacturer [17,20,21].

**Figure 14.** Types of power cycling: (**a**) fast power cycling *PCsec*; (**b**) slow power cycling *PCmin*.

An example of Coffin–Manson curves that were calculated by considering the following parameters [20] are displayed in Figure 15.


$$\bullet \qquad E\_a = 0.814 \,\text{eV}$$

**Figure 15.** Coffin–Manson equation representation: (**a**) 2D curves (each one at a fixed *Tm*); (**b**) 3D Scheme.

#### *2.6. Miner's Rule*

The lifetime estimation is determined according to the Miner's rule, or the theory of the damage accumulation. In practice, the damage is associated with each stress condition by making the ratio between the number of thermal cycles *Nc* and the relative *Nf* associated with the same pair of Δ*Tj* and *Tm*. Extending this reasoning to all operating conditions, it is possible to assign damage to each of them. Palmgreen–Miner's theory of linear cumulative damage, called Miner's rule, is given by [17,19,20]:

$$D = \sum\_{i=1}^{k} \frac{N\_{c,i}}{N\_{f,i}} \tag{9}$$

$$T = \frac{1}{D} \tag{10}$$

where:


By computing the *Nc*,*i*/*Nf* ,*<sup>i</sup>* ratios for all the *i*-th working points of the electric drive, the overall damage of SiC power devices can be predicted. If the damage is less than 1, then the power device is capable of handling the thermal stresses to which it was subjected in the entire mission profile; otherwise (if *Nc*,*i*/*Nf* ,*<sup>i</sup>* is greater than or equal to 1), the device is considered faulty. Parameter *T*, which would be the inverse of *D*, provides an important indication of how many cycles the device still has before failure. For "cycle", we must consider the stress input to the rainflow; in our case, one cycle corresponds to one second of junction temperature at steady-state. To obtain the lifetime, Equation (11) was used:

$$Lifetime = \sum\_{i=1}^{k} \frac{1}{3600 \cdot h \cdot 365 \cdot w\_i \cdot D\_i} \tag{11}$$

where:


Therefore, the lifetime is estimated considering a predetermined number of hours of work per day.

#### **3. Case Studies**

In this section, the aforementioned life prediction procedure was applied to different operating scenarios for the drive so far considered: low speed–high torque, medium/high speed–low torque, and mixed operation. The goal of the following activity is to analyze the differences in lifetime estimation when different operating scenarios of the drive are considered. Figures 17, 19, and 21 show the lifetimes of each scenario as a function of daily working hours. For each scenario, daily use of the drive of 2 h is considered, although the working hours of the drive can be highly variable depending on the application.

#### *3.1. Case Study 1: Low Speed–High Torque*

This scenario is characterized by low speeds and a wide range of torques. Figure 16 shows the weight distribution considered for this mission profile, where the frequency of rotor speed is higher for values included in the range of 0–700 rpm. The conditions for which the speed is 0 rpm have a considerable weight; this could represent the numerous standing starts that normally occur in electric traction in the case of an urban cycle.

**Figure 16.** Weight distribution of case study 1: (**a**) 3D surface; (**b**) 2D surface.

Figure 17 shows the lifetime for this scenario, obtained by applying the step-by-step procedure defined in the previous section. Assuming an average use of the motor drive equal to 2 h per day, the lifetime of the SiC MOSFET composing the inverter of the electric drive is estimated to be about 12 years.

**Figure 17.** Lifetime estimation of case study 1.

#### *3.2. Case Study 2: Medium/High Speed–Low Torque*

In this scenario, the drive is mainly operated at medium-high speeds and low torques. A higher lifetime is expected because of the low torques, and thus low currents, which should lead to a lower degradation of the SiC power device under examination. Figure 18 shows the weight distribution associated to this scenario, which is projected towards medium-high speeds; in particular, a higher percentage of electric drive operation is centered around 700–800 rpm, even though the motor operation is observed up to 1400 rpm. The electromagnetic torque is normally kept quite low in this working profile, i.e., mainly around 20–30 Nm.

Figure 19 shows the lifetime estimation of case study 2. As expected, for the same hours of use, the lifetime of the SiC power MOSFET installed in the three-phase inverter is higher than in scenario 1. For instance, when an average daily use of the drive equal to 2 h is considered, the lifetime of the SiC devices is about 28 years.

**Figure 18.** Weight distribution of case study 2: (**a**) 3D surface; (**b**) 2D surface.

*3.3. Case Study 3: Mixed Operation*

In the case of mixed-use, an intermediate situation between the two previous scenarios is considered here. The speed and torque ranges are much wider than the previous cases. Figure 20 shows the weight distribution, and it can be noted how wide the speed range is, i.e., from 200 to 1000 rpm, while the torque is mostly between 15 and 45.

**Figure 20.** Weight distribution of the mixed-use: (**a**) 3D surface; (**b**) 2D surface.

Figure 21 shows the lifetime estimation of case study 3. In this case, an intermediate lifetime of previous use was found, as expected.

**Figure 21.** Lifetime estimation of mixed operation.

#### *3.4. Results Assessment*

According to the previous analysis, it is possible to note a variable lifetime that is consistent with the considered working scenarios. In fact, by assuming a daily work of the SiC-based inverter equal to 2 h, the proposed procedure provided the following lifetimes:


On the other hand, the same procedure allows for evaluating the number of working hours per day associated with each considered scenarios, when a lifetime equal to 15 years is imposed as a constraint for the SiC devices. The results are as follows:


Figure 22 shows the comparison between the lifetime results.

**Figure 22.** Comparison of the lifetime in case study 1, case study 2, and case study 3.

#### **4. Conclusions**

A detailed analysis of a lifetime estimation procedure devoted to discrete MOSFETs was presented in this paper. Such a technique was set with the aim of keeping the computational efforts and simulation times limited and low while providing an effective evaluation of thermal stresses and their effects on the power devices composing the electronic converter. The method is of a general application, and it can be used for any mission profile generated by the electric drive model. This allowed us to compare several operational states of the drive and to predict the worse operating conditions for the power devices. The

proposed approach can be extended to different operating scenarios and power devices technologies, even though the main focus of this study is the SiC technology. In the case of multi-chip modules, it is possible to easily extend the analysis also, while considering the effects of mutual thermal coupling.

**Author Contributions:** Conceptualization and methodology, C.B., G.S. (Giacomo Scelba) and G.S. (Giuseppe Scarcella); validation and data curation, C.B.; software and visualization C.B.; writing original draft preparation, C.B., G.S. (Giacomo Scelba) and S.A.R.; writing—review and editing, C.B., G.S. (Giacomo Scelba), S.A.R. and M.C.; supervision, G.S. (Giuseppe Scarcella) and M.C.; project administration, G.S. (Giacomo Scelba) All authors have read and agreed to the published version of the manuscript.

**Funding:** This work has been partially supported by the Italian Ministry for Economic Development (MISE), under the project "M9"—C32F18000100008 and by the University of Catania under the interdepartmental project PIA.CE.RI. 2020-2022 Line 2-TMESPEMES.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**

