Modified rainflow algorithm for temperature-time-dependent counting in lifetime estimation of power devices

ABSTRACT

Power electronic systems include many fragile elements, with power devices being the most prone to failure. Cycle counting algorithms are essential to evaluate power devices degradation and system reliability. Among the available options, the rainflow algorithm is the most widely used. Since the rainflow algorithm was originally designed for fatigue analysis, it faces challenges when applied to power devices. Specifically, the conventional rainflow algorithm cannot compute the effective heating time and the time-dependent equivalent mean temperature of the thermal cycles. Additionally, it counts cooling half-cycles, which contradicts the basis of lifetime models, as these models rely on data associated with heating temperature gradients. To address these limitations, this work introduces a modified rainflow algorithm for cycle counting in the lifetime estimation of power devices. This methodology enhances the conventional rainflow by enabling the calculation of effective heating time and mean temperature while also filtering out cooling half-cycles. The results demonstrate that the modified rainflow algorithm significantly affects the lifetime predictions for all critical joints in an IGBT module, regardless of the mission profile. Across all case studies, the utilization of the modified rainflow algorithm resulted in a damage reduction exceeding 53%.

KEYWORDS
cycle counting algorithm; modified rainflow; lifetime estimation; power devices; reliability

I. INTRODUCTION

Since the electrification of modern society continues to advance rapidly, the electrical power system must expand continuously to meet the growing energy demands. A significant portion of this expansion is driven by power electronics technology, which enables the efficient control of power flow [¹], [²]. In this context, ensuring reliable operation is essential for any power electronics system, since even a single component failure can result in significant maintenance costs and operational downtime. However, power electronic systems are composed of several fragile elements, including power semiconductor devices, capacitors, magnetics, controllers, sensors, and auxiliary devices. Among these elements, the power devices are the most susceptible to failure [³].

Wear-out failure of a device can be predicted to some extent if the degradation mechanisms are well understood. Consequently, the lifetime of a power electronic system can be extended by employing design and/or control techniques aimed at reducing wear-out of power devices [⁴]. Since power devices consist of various materials, thermal cycling induces thermomechanical stress in their joints. This stress causes deformations that gradually lead to damage accumulation and, ultimately, result in the failure of critical joints, such as solders and bondwires [⁴]. Moreover, the shear stress-strain response of these joints to periodic thermal cycling forms stress-strain hysteresis loops. On this basis, the degradation of the critical joint material, as estimated by lifetime models, is directly related to the area enclosed by these hysteresis loops [⁴].

Lifetime estimation can be categorized into two types of thermal cycling: short-term and long-term [⁵]. Short-term analysis focuses on thermal stress caused by the grid fundamental frequency during the power converter's operation, while long-term analysis focuses on thermal stress caused by the mission profile variations [⁵]. The thermal cycling in short-term analyses is already well-defined due to the regularity of grid frequency. In contrast, the long-term thermal cycling exhibits irregular and unpredictable behavior driven by mission profile dynamics [⁶]-[⁸]. However, lifetime models are typically designed for constant conditions [⁴], [⁹]. Most models are based on power cycling tests, in which power devices are actively heated by losses in the semiconductor and cooled again with the aid of cooling equipment [¹⁰]. Moreover, the temperature cycle is generally defined by the mean temperature ($T_{[j,c]m}$), temperature swing ($\Delta T_{j,c}$) and heating time ($t_{on}$), as exemplified for a thermal cycle in Fig. 1. For this reason, a cycle counting algorithm is required to compute the hysteresis loops based on regular data series of $T_{[j,c]m}$, $\Delta T_{j,c}$ and $t_{on}$, enabling assessment of damage accumulation under variable thermal cycling conditions [⁶].

FIGURE 1
Example of thermal cycle.

The literature describes several cycle counting methods, such as half-cycle peak through counting, maximum edge peak through counting, rising edge peak through counting, and rainflow counting [¹¹], [¹²]. Among these, rainflow cycle counting is the most widely used for power devices, since it results in lower error [¹²]. This method is based on the principle of treating hysteresis loops of a variable amplitude loading as equivalent cycles of constant amplitude loading [¹³].

The original rainflow algorithm was first proposed by M. Matsuishi and T. Endo in 1968 [¹⁴] to count full and half-cycles in strain-time signals for fatigue failure analysis in mechanical engineering. Due to this origin, rainflow counting presents challenges when implemented to real-time operating (online) damage assessment and/or when employed for lifetime estimation of power devices [⁴], [¹⁵]. Specifically, rainflow counting requires the complete time history (past and future) of the variable amplitude loading. This requirement poses a significant obstacle in estimating the remaining useful life of a power device during its operation. In practice, it would necessitate an indefinite amount of memory for online fatigue calculations. Some works proposed strategies to solve this issue, such as optimized algorithms, preset of the amplitude range and maximum time window [¹⁵]-[¹⁸]. Online rainflow application is beyond the scope of this work.

Several algorithms have been developed to implement rainflow counting [¹⁹]. One of the most widely referenced and utilized algorithms was created by Downing and Socie in 1982 [²⁰]. This algorithm, based on a three-point rainflow counting rule [²⁰], was later incorporated into ASTM E1049-85 as a standard practice for cycle counting in fatigue analysis [¹¹]. Sandia National Laboratories implement the rainflow procedure employing the three-point counting rule with equivalent data information for fatigue analysis in wind turbine components [²¹]. In 2009, a rainflow function was written by Nieslony [²²] to be used in the MATLAB software environment. This function was written according to the ASTM standard and made available for download in MATLAB Central [²²]. More recently, since MATLAB 2017b version, a rainflow function following ASTM standards is available directly on the software environment.

Conventional rainflow algorithms do not adequately consider the temperature-time-dependence crucial for the lifetime estimation of power devices [²³], [²⁴]. These algorithms often compute the mean temperatures independent of cycle time. Nevertheless, the stress induced by temperature is inherently time-dependent in power devices [²³], [²⁵].

The heating time significantly influences damage accumulation in power devices [²⁴], [²⁶], as exemplified in the lifetime curves of an ABB HiPak IGBT in Fig. 2 (a) and (b) for $t_{on}=1$ s and $t_{on}=60$ s. However, conventional rainflow computes the time only at the maximum and minimum temperature points ($T_{max}$ and $T_{min}$). While some approaches approximate heating time as half of the cycle time ($t_{cycle}$) derived from the rainflow function, this assumption relies on the premise of equal heating and cooling time ($t_{off}$), which may not always hold true [²⁷].

The rainflow algorithm counts all half-cycles of the data. Nevertheless, most lifetime models are based on power cycling tests, in which only the heating process is actively controlled [⁹], [¹⁰]. Consequently, these lifetime models consider only heating transients. As discussed in reference [²⁴], only half-cycles representing heating transients should be considered for lifetime estimation of power devices.

FIGURE 2
Power cycling lifetime model of chip solder of ABB HiPak IGBT as a function of $T_{jm}$, $\Delta T_j$ for: (a) $t_{on}=1s$ (b) $t_{on}=60s$ [⁹].

Reference [²⁶] proposed a methodology for cycle counting in the lifetime estimation of power devices: the modified rainflow algorithm. The methodology adapts the three-point rainflow algorithm (conventional rainflow) available in MATLAB environment to a four-point rainflow algorithm (modified rainflow) [²⁵]. The modified rainflow algorithm is capable of computing the effective heating time and the effective mean temperature while filtering out cooling half-cycles counted with the conventional rainflow. However, this algorithm also filters all heating half-cycles except for the last one.

This work extends the methodology presented in [²⁶]. The new version of the modified rainflow algorithm, while still adapting the conventional MATLAB rainflow to a four-point algorithm to compute the effective heating time and mean temperature, now also eliminates only cooling half-cycles, preserving all heating half-cycles. Thus, the modified rainflow aims to extract from the temperature profiles the parameters that characterize the thermal cycling and which are consistent with the parameters that should be used as input in the lifetime models. The impact of this refined methodology on lifetime estimation is evaluated using different mission profiles. Moreover, the modified rainflow function is available for download in [²⁸]. A general lifetime estimation procedure applicable to any power converter application is presented. Finally, the impact of the modified rainflow algorithm is compared to conventional rainflow in terms of damage accumulation, system-level reliability, and processing time.

This paper is organized as follows. Section II introduces the long-term lifetime estimation and reliability procedure for power devices. Section III introduces the proposed modified rainflow algorithm. The case studies are described in Section IV. In Section V, the obtained results are presents. Ultimately, Section VI presenters the conclusions of this work.

II. LIFETIME ESTIMATION OF POWER DEVICES

Fig. 3 illustrates the long-term lifetime estimation procedure. Real-world mission profiles, representative of the operational conditions of the power converter in the field, are used. The power device losses (P) are determined using a power loss model based on the operating conditions of the power converter. A thermal model is then used to estimate the junction ($T_j$) and case ($T_c$) temperature profiles of the power devices.

FIGURE 3
Long-term lifetime estimation procedure.

To employ lifetime models, the profile data must be prepared appropriately. This requires a cycle counting algorithm for thermal cycling classification [⁴]. The modified rainflow algorithm proposed in this work is used for this purpose. This algorithm flowchart is presented in Fig. 4 and detailed in Section III.

FIGURE 4
Flowchart of modified rainflow algorithm.

To compute hysteresis loops, the modified rainflow algorithm should be applied only to the extreme points (peaks and valleys) of the temperatures. Therefore, the temperature data was preprocessed using a peaks and valleys detection algorithm (PVDA). This results in a filtered temperature dataset containing only peak and valley information ($T_{j,pv}$ and $T_{c,pv}$) [²³]. Subsequently, the modified rainflow algorithm is applied to this filtered data. This yields a regular data series containing $T_{[j,c]m}$, $\Delta T_{j,c}$ and $t_{on}$. Thereafter, the regular temperature data is applied to the lifetime model.

The lifetime of a power module is limited by the mechanical fatigue of the package. This fatigue arises from thermally induced mechanical stress caused by disparities in thermal expansion coefficients among the constituent materials. Since power devices incorporate materials with varying thermal expansion coefficients, these materials are constrained from freely expanding. This constraint results in thermally induced mechanical stress within critical joints of the power device, such as base plate and conductor solders (BPCS), bondwires (BW), and chip solders (CS) [²⁹]. These joints are placed in different locations within the power module. Thus, each lifetime model considers the temperature of the location of its respective joint.

Finally, the power device life consumption (LC) can be accumulated using the Miner's rule [³⁰], as follows:

(1) $$LC=\sum _k\frac{n_k}{N_{f,k}},$$

where $n_k$ is the $k^{th}$ number of repeated cycles obtained from the modified rainflow algorithm, $N_{f,k}$ is the $k^{th}$ number of cycles to failure, and LC is the life consumption for each critical joints of each power device in the converter.

To approximate the results of practical applications, Monte Carlo simulations with a population of 10,000 samples are carried out for each power device and respective critical joint, using the respective lifetime model [³¹], [³²]. Subsequently, the unreliability function of each joint is obtained from its respective lifetime distribution and combined to compute the system-level unreliability [³³]. The power device critical joints unreliability curves are computed and combined following the methodology described in [³¹], [³²]. Moreover, the unreliability function indicates the proportion of the failure population over time. Therefore, this is an important figure of merit to evaluate the probability of failure over time of the entire converter (system-level).

III. MODIFIED RAINFLOW ALGORITHM

Conventional rainflow algorithms are unable to determine the effective heating time and the time-dependent effective mean temperature. In essence, these algorithms only identify the time instances at the temperature extremes: the initial cycle ($t_i$) and the final cycle ($t_f$). These points are exemplified in Fig. 5 (a) and (b) for descending and ascending cycles, respectively. Although some approaches, such as those that utilize the MATLAB rainflow function developed by A. Nieslony [²²], approximate the heating time as half of the cycle time, this assumption relies on the premise of equal cooling and heating times, which may not always hold true [²⁷], as illustrated in Fig. 5 (a). Even with the newest MATLAB rainflow function introduced in MATLAB version R2017b [³⁴], only $t_i$ and $t_f$ are explicitly computed.

FIGURE 5
Modified rainflow algorithm counting for: (a) descending cycle; (b) ascending cycle.

Furthermore, the stress-strain hysteresis loop is temperature-time-dependent [²³]. As verified in reference [³⁵], the stress that metals can withstand decreases as the temperature increases. For this reason, reference [²⁵] proposed a four-point algorithm for conservative damage estimation. This work demonstrated that for temperatures below the threshold temperature of metals (usually < 250 °C) a time-dependent equation should be employed to compute an equivalent temperature of the stress-strain hysteresis loop. Since most power devices operate with a maximum rated temperature lower than 175 °C, the time-dependent equivalent mean temperature should be considered in the lifetime estimation. As exemplified in Table 1 and Table 2, the temperature values computed considering the simple mean temperature calculation of conventional rainflow and the equivalent temperature calculation can differ. Consequently, since the lifetime of power devices is highly sensitive to temperature, even minor discrepancies in temperature calculations, especially when accumulated over numerous data points, can significantly affect the lifetime estimation results.

To solve these issues, the modified rainflow algorithm is proposed. The flowchart of the modified rainflow algorithm is exhibited in Fig. 4. The modified rainflow algorithm adapts the three-point rainflow algorithm (conventional rainflow) available in the MATLAB environment to a four-point rainflow counting capable of computing the effective heating time and effective mean temperature while simultaneously filtering out cooling half-cycles.

As observed in Fig. 4, $t_i$ and $t_f$, resulting from the MATLAB rainflow function, are used to identify the corresponding thermal cycle within the original data and to determine its direction. Based on the respective temperatures at $t_i$ ($T_i$) and $t_f$ ($T_f$), the cycle is classified. If $T_i>T_f$, the cycle counted by MATLAB rainflow function is descending (Fig. 5 (a)). Otherwise, if $T_i<T_f$, the cycle that was counted is ascending (Fig. 5 (b)).

Furthermore, $t_e$ represents the time at which the thermal cycle is fully completed, defined in [²³]:

(2) $$t_e=\frac{T_i-T_{be}}{T_{af}-T_{be}}(t_{af}-t_{be})+t_{be},$$

where $t_{be}$ and $T_{be}$ are the time and temperature, respectively, in the original data at the point immediately preceding the end of the thermal cycle. Similarly, $t_{af}$ denotes the time in the original data at the point immediately following the end of the thermal cycle.

This verification defines $t_L$ as the time of the lowest temperature, in which the heating starts in the thermal cycle, and $t_H$ as the time of the highest temperature, in which the heating ends. If $T_i>T_f$, $t_L=t_f$ and $t_H=t_e$. Otherwise, if $T_i<T_f$, $t_L=t_i$ and $t_H=t_f$.

Subsequently, the heating time intervals ($\Delta t_{o,j}$) between $t_L$ and $t_H$, are calculated and summed to determine the effective heating time ($t_{on}=\sum _j\Delta t_{o,j}$).

The equivalent mean temperature is calculated by considering each time interval within the original data (with sample time $T_s$) that falls within the full thermal cycle (from $t_i$ to $t_e$). For each interval, the mean temperature is calculated and then weighted by the proportion of the duration of the interval to the total cycle time ($t_e-t_i$), as described in [²³]:

(3) $$\Delta T_{mean,j}=\frac{T_2-T_1}{2}\frac{t_2-t_1}{t_e-t_i},$$

where $t_1$ and $T_1$ represent the time and temperature, respectively, in the original data at the beginning of the interval. $t_2$ and $T_2$ represent the time and temperature, respectively, in the original data at the end of the interval. Subsequently, the equivalent mean temperature values within each interval ($\Delta T_{mean,j}$) between $t_i$ and $t_e$ are computed and summed to determine the effective mean temperature ($T_{mean}=\sum _j\Delta T_{mean,j}$).

To demonstrate the proposed algorithm, conventional and modified rainflows are applied to the temperature profile of Fig. 6. The conventional rainflow output and modified rainflow output are presented in Table 1 and Table 2, respectively, for this temperature profile.

FIGURE 6
Example for a temperature profile.

For the temperature profile of Fig. 6, two half-cycles (k = 4 and k = 6) are filtered/eliminated by modified rainflow, as indicated in Table 2 as non-counted (n.c.). These half-cycles present cooling transients that should not be considered in the lifetime estimation of power devices. The half-cycle (k = 5) presenting heating transient is counted by modified rainflow since should be considered in the lifetime estimation. Regarding the $\Delta T_j$, the values computed by conventional rainflow are maintained after modified rainflow implementation, since the modified rainflow does not affect this parameter. On the other hand, $t_{on}$ and $T_{j,m}$ obtained from modified rainflow differ from conventional rainflow, since the effective $t_{on}$ and $T_{j,m}$ are computed by the modified rainflow algorithm.

TABLE 1

Conventional rainflow output and respective $t_{on}$ computed.

k	$n_k$	$T_{j,m}$ [°C]	$\Delta T_j$ [°C]	$t_i$ [s]	$t_f$ [s]	$t_{on}$ [s]
1	1	55.0	6	900	1200	300
2	1	56.0	10	600	1500	900
3	1	57.5	13	2100	2400	300
4	0.5	58.5	15	0	3300	3300
5	0.5	56.0	10	3300	3600	300
6	0.5	60.5	1	3600	3900	300

TABLE 2

Modified rainflow output.

k	$n_k$	$T_{j,m}$ [°C]	$\Delta T_j$ [°C]	$t_{on}$ [s]
1	1	55.0	6	200
2	1	56.4	10	400
3	1	60.2	13	300
4	0.5	n.c.	n.c.	n.c.
5	0.5	56.0	10	300
6	0.5	n.c.	n.c.	n.c.

• ^*
  n.c. indicates the non-counted cycles.

IV. CASE STUDY

The performance of the modified rainflow algorithm is evaluated within the context of a modular multilevel converter-based static synchronous compensator (MMC-STATCOM) application. The MMC-STATCOM topology is illustrated in Fig. 7. This topology comprises N submodules (SMs) per arm. Each SM incorporates a capacitor bank represented by the equivalent capacitance (C) and four power devices (S ₁, S ₂, D ₁ and D ₂). In this work, the MMC-STATCOM design and control strategy follows the same approach of [³²]. MMC-STATCOM main parameters are presented in Table 3. In addition, ABB HiPak IGBT modules, part number 5SND0800M170100, rated at 1.7 kV and 800A, are considered.

FIGURE 7
MMC-STATCOM topology with half-bridge SMs.

TABLE 3
Parameters of the MMC-STATCOM.

Parameter	Value
SMs per arm (N)	21
dc-link voltage ($v_{dc}$)	21.61 kV
Rated power ($S_n$)	17 MVA
Line-to-line rms grid voltage ($V_g$)	13.8 kV
Grid frequency ($f_g$)	60 Hz
Switching frequency ($f_{sw}$)	200 Hz
SM capacitance (C)	14 mF
Capacitors in parallel per SM	7
Transformer inductance ($L_f$)	1.5 mH
Arm inductance ($L_a$)	2.97 mH

Wear-out analyses are performed using simulations implemented within the MATLAB/Simulink and PLECS software environments. The simulations were performed on a Dell Inspiron i15-3583-AS100P computer equipped with an Intel Core i7-8565U CPU (1.80 GHz), an AMD Radeon 520 dedicated graphics card, and 8 GB of RAM.

The mission profiles were derived from measurements collected over a year in locations within Southeastern Brazil, with a sampling rate of $T_s$ = 5 min. Fig. 8 (a) depicts the ambient temperature ($T_a$) mission profile. Furthermore, two distinct reactive power (q) mission profiles are utilized, as shown in Fig. 8 (b) and (c) for case studies 1 and 2, respectively. In Case 1 (Fig. 8 (b)), a seasonal variation in the reactive power is observed, characterized by a decrease in reactive power injection during the middle of the year. Besides, q exhibits both positive and negative values, indicating both lagging and leading reactive power operation, respectively. In Case 2 (Fig. 8 (c)), q demonstrates an oscillatory pattern throughout the year and exclusively exhibits positive values, signifying solely leading reactive power operation.

FIGURE 8
Mission profiles: (a) ambient temperature; (b) reactive power - Case 1; (c) reactive power - Case 2.

The power losses model is performed by look-up tables derived from the curves provided in the ABB HiPak IGBT module datasheet, considering a defined set of operational conditions. Moreover, the Foster thermal model [³⁶] with a single heatsink per SM is employed. The heatsink thermal impedance is designed to ensure that the case temperature remains below 80°C, adhering to the upper limit of temperature specified in the lifetime models [⁹], [³⁷].

Lifetime models, also based on look-up tables provided by ABB for HiPak IGBT modules [⁹], are utilized. All critical joints within the power module, including base plate and conductor solders (BPCS), chip solder (CS), and bondwires (BW), are considered. Each lifetime model incorporates the temperature to its respective joint location. Notably, the BW lifetime model exhibits negligible dependence on heating time. This observation is consistent with advancements in ABB HiPak technology, which have resulted in bond wire performance being minimally impacted by heating time [⁹]. Consequently, the inputs of each lifetime model are: $T_{c,m}$, $\Delta T_c$ and $t_{on}$ for BPCS; $T_{j,m}$, $\Delta T_j$ and $t_{on}$ for CS; and $T_{j,m}$ and $\Delta T_{j,c}$ for BW.

The lifetime models provided by ABB are based on power cycling experiments with a maximum heating time of 16 h. Therefore, it is assumed that viscoplastic deformation saturates for heating time exceeding 16 h [³⁸].

V. RESULTS

The results for Case 1 and Case 2 are presented in the following subsections. These results are based on the methodology described in Section II (detailed in [³²]) and Section III.

A. Case 1

Fig. 9 shows the case and junction temperature profile of the diode D ₁. In this case study, D ₁ emerges as the most critically stressed power device within the submodule, exhibiting the highest average temperatures throughout the year ($T_{c,avg}=25.79$ °C and $T_{j,avg}=26.08$ °C). Furthermore, this device also experiences the most significant maximum temperature variations throughout the year ($\Delta T_{c,max}=70.56$ °C and $\Delta T_{j,max}=74.68$ °C). As observed in Fig. 9, the temperature profiles closely mirror the shape of the reactive power mission profile, characterized by lower temperatures during the middle of the year.

FIGURE 9
Case 1 profiles of: (a) case temperature; (b) junction temperature.

Fig. 10 shows a comparison of the modified and conventional rainflow of the first 10 thermal cycles counted. As presented in Fig. 10 (a), the mean temperature counted by conventional and modified rainflow can exhibit discrepancies. For these 10 cycles, the percentage mean temperature difference achieves $\Delta T_{diff,max}=0.4\%$. This discrepancy highlights the significance of considering the time-dependence of temperature within the cycle counting process. Fig. 10 (b) demonstrates that the effective heating time computed using modified rainflow differs significantly from the approximation obtained by simply assuming $t_{on}=t_f-t_i$ in conventional rainflow. For these 10 cycles, the percentage heating time difference achieves $\Delta t_{diff,max}=82.0\%$. This result underscores the importance of calculating the effective heating time.

FIGURE 10
Modified and conventional rainflows data of the 10 first thermal cycles: (a) mean temperature; (b) heating time.

Fig. 11 shows the junction temperature histograms for modified and conventional rainflows. A similar distribution is observed for both rainflows. At the data point with the highest cycle count, the mean temperature is identical for both counting methods. However, $t_{on}$ decreased by approximately 73.7% when using the modified rainflow at this extreme point. This finding demonstrates the significant influence of the modified rainflow algorithm on heating time estimation for this mission profile. Moreover, conventional rainflow and modified rainflow computed 26,150 and 26,143 thermal cycles, respectively. This difference in the number of cycles indicated that the modified rainflow algorithm filtered 7 cooling half-cycles which should not be considered in the life computation.

FIGURE 11
Case 1 histogram of: (a) conventional rainflow; (b) modified rainflow.

Fig. 12 shows the static damage LC for all critical joints of the power devices, calculated using both modified and conventional rainflow methods. For D ₁, the modified rainflow resulted in a substantial reduction in LC for BPCS, CS, and BW by approximately 52%, 72%, and 53%, respectively. These results demonstrate the significant impact of the modified rainflow algorithm on lifetime estimations of all critical joints within the power devices. Notably, even the bondwire lifetime, which is unaffected by the heating time, is influenced by the modified rainflow due to the computation of the effective mean temperature and filtering of cooling half-cycles.

FIGURE 12
Case 1 static damages of: (a) base plate and conductor solders - BPCS; (b) chip solder - CS; (c) bondwire - BW.

Fig. 13 presents the processing time required from the PVDA output until the damage computation with Miner's Rule with conventional and modified rainflow. The error bars represent one standard deviation of uncertainty considering 10 executions. As observed in Fig. 13, the modified rainflow algorithm increased, on average, the processing time of the damage computation by 14.7 s. Nevertheless, part of the error bars of conventional and modified rainflows intercept each other. This indicates a small probability that the processing time of the modified rainflow is the same or lower than the conventional rainflow. This will depend on the computer processing speed during the procedure, which can vary due to other applications running in parallel.

FIGURE 13
Processing time of the damage computation with each counting algorithm of Case 1.

Fig. 14 exhibits the MMC-STATCOM unreliability at system-level, considering all power device critical joints. The time at which 10% of the samples fail ($B_{10}$ lifetime approach) is highlighted. As observed, the unreliability is higher when computed using conventional rainflow. Moreover, $B_{10}$ is 56.2% lower with modified rainflow. This is a consequence of the higher damage accumulation computed with the conventional rainflow algorithm.

FIGURE 14
Unreliability function at system-level for Case 1.

B. Case 2

Fig. 15 shows the case and temperature profile of the diode D ₁. In this case study, D ₁ emerges as the most critically stressed power device within the submodule, exhibiting the highest average temperatures throughout the year ($T_{c,avg}=45.12$ °C and $T_{j,avg}=47.11$ °C). Furthermore, D ₁ experiences the most significant maximum temperature variations throughout the year ($\Delta T_{c,max}=53.64$ °C and $\Delta T_{j,max}=56.33$ °C). As observed in Fig. 15, the temperature profiles closely mirror the oscillatory pattern of the reactive power mission profile throughout the year. In particular, this case study demonstrates a more severe operating environment compared to Case 1, characterized by higher average temperatures.

FIGURE 15
Case 2 profile of: (a) case temperature; (b) junction temperature.

Fig. 16 shows a comparison of the modified and conventional rainflow data for the first 10 thermal cycles counted. As presented in Fig. 16 (a), the mean temperature calculated by modified and conventional rainflow methods can exhibit discrepancies. For these 10 cycles, the percentage mean temperature difference achieves approximately $\Delta T_{diff,max}=1\%$. For Fig. 16 (b), the heating time computed using modified and conventional rainflow were found to be identical ($\Delta t_{diff,max}=0\%$). This indicates that the 10 first cycles are ascending cycles.

FIGURE 16
Conventional and modified rainflows information of the 10 first thermal cycles: (a) mean temperature; (b) heating time.

Fig. 17 shows the junction temperature histograms for modified and conventional rainflows. A similar distribution is observed for both rainflows. At the data point with the highest cycle count, the mean temperature exhibits a minor difference of 0.01 °C between the rainflows. However, $t_{on}$ decreases by approximately 61.5% when using the modified rainflow approach at this extreme point. This finding demonstrates the significant influence of the modified rainflow algorithm on heating time estimation for this mission profile. Moreover, conventional rainflow and modified rainflow computed 25,008 and 24,997 thermal cycles, respectively. This difference in the number of cycles indicated that the modified rainflow algorithm filtered 11 cooling half-cycles, which should not be considered in the life computation.

FIGURE 17
Histogram of Case 2 of: (a) conventional rainflow; (b) modified rainflow.

Fig. 18 shows the static damage LC for all critical joints of the power devices, computed using modified and conventional rainflows. For D ₁ The modified rainflow approach resulted in a substantial reduction in LC for BPCS (52%), CS (69%), and BW (53%), respectively. These results demonstrate the significant impact of the modified rainflow algorithm on lifetime estimation for all critical joints within power devices. Notably, even the bondwire lifetime, which is unaffected by heating time, was influenced by the modified rainflow method due to the computation of the effective mean temperature and the filtering of cooling half-cycles.

FIGURE 18
Case 2 static damages of: (a) base plate and conductor solders - BPCS; (b) chip solder - CS; (c) bondwire - BW.

Fig. 19 presents the processing time required from the PVDA output until the damage computation with Miner's Rule with conventional and modified rainflow. The error bars represent one standard deviation of uncertainty considering 10 executions. As observed in Fig. 19, the modified rainflow algorithm increased, on average, the processing time of the damage computation by 69.4 s. Unlike in Case 1, part of the error bars of conventional and modified rainflows do not intercept each other.

FIGURE 19
Processing time of the damage computation with conventional and modified rainflows of Case 2.

Fig. 20 exhibits the MMC-STATCOM unreliability at system-level, considering all power devices critical joints. The time at which 10% of the samples fail ($B_{10}$ lifetime approach) is highlighted. As observed, the unreliability is higher when computed using conventional rainflow. Moreover, $B_{10}$ is 56.6% lower with modified rainflow. This is a consequence of the higher damage accumulation computed with the conventional rainflow algorithm.

FIGURE 20
Unreliability function at system-level for Case 2.

C. Modified rainflow effect on different mission profiles

Table 4 summarizes the main results of the case studies to evaluate the effect of modified rainflow applied to cases with distinct mission profiles. The reactive power mission profile exhibits seasonality, with reduced injection around mid-year, whereas Case 2 displays an oscillatory pattern throughout the year. Consequently, Case 1 and Case 2 exhibit the highest maximum temperature variation and average temperature over the year, respectively. According to [³³], maximum temperature variation and the average temperature of the profile correlate with life consumption. These parameters are directly linked to $T_{[j,c]m}$ and $\Delta T_{j,c}$, respectively, which serve as inputs for power device lifetime models.

TABLE 4
Summary of the main results of Case 1 and Case 2.

Parameter	Case 1	Case 2
Seasonality over the year	yes	no
$T_{c,avg}$ [°C]	25.79	45.12
$T_{j,avg}$ [°C]	26.08	47.11
$\Delta T_{c,max}$ [°C]	70.56	53.64
$\Delta T_{j,max}$ [°C]	74.68	56.33
$\Delta T_{diff,max}$ for the 1st 10 cycles	0.4%	1%
$\Delta t_{diff,max}$ for the 1st 10 cycles	82.0%	0%
Cooling half-cycles filtered	7	11
Increased processing time [s]	14.7	69.4
$L{C}_{BPCS}$ reduction with MR¹	52%	54%
$L{C}_{CS}$ reduction with MR¹	72%	69%
$L{C}_{BW}$ reduction with MR¹	53%	53%
$B_{10}$ reduction with MR¹	56.2%	56.6%

• ^¹
  MR means modified rainflow.

For the first 10 thermal cycles counted, the maximum difference between the mean temperatures ($\Delta T_{diff,max}$) remains comparable for both cases. However, the maximum difference between the heating times ($\Delta t_{diff,max}$) for Case 1 is significantly higher than for Case 2. These findings indicate that the modified rainflow method can exhibit varying impacts depending on the specific characteristics of the mission profile. The histograms of Fig. 11 and Fig. 17 reveal that the application of modified rainflow has a less pronounced effect on the mean temperature distribution for both case studies, while exerts higher influence on the heating time computation. As shown for the extreme point highlighted in the histograms, $t_{on}$ can decrease 73.7% and 61.5% for Case 1 and Case 2, respectively. As presented in Table 4, both cases exhibit a comparable number of filtered cooling half-cycles and increased processing time when employing modified rainflow.

As observed in Table 4, the modified rainflow demonstrates a similar percentage effect on the estimation of the life consumption for the critical joints and the system-level unreliability (highlighted for $B_{10}$ lifetime). Moreover, the 53% reduction in $L{C}_{BW}$ underscores that the calculation of the effective mean temperature and the cooling half-cycles filtering of modified rainflow significantly influence lifetime estimation.

VI. CONCLUSIONS

This work introduces a modified rainflow algorithm for the lifetime estimation of power devices. This methodology adapts the conventional rainflow by incorporating the features of filtering cooling half-cycles and computing the effective heating time and effective mean temperature. According to the literature, these modifications allow to extract from the temperature profiles the parameters that characterize the thermal cycling and which are consistent with the parameters that should be used as input in the lifetime models. The impact of the modified rainflow algorithm on life consumption was assessed using an MMC-STATCOM and two case studies with distinct reactive power mission profiles.

The results demonstrate a more pronounced effect of the modified rainflow on the heating time computation, as evidenced by the temperature distributions, where $t_{on}$ decreases 73.7% and 61.5% for Case 1 and Case 2, respectively. Although the amount of cooling half-cycle filtering is inherently linked to the dynamics of the mission profile, both cases exhibited a comparable level of filtering. Furthermore, the impact of modified rainflow on processing time was minimal, remaining below 6% for both case studies.

Furthermore, this methodology exerts a substantial influence on the lifetime estimation of all critical joints within the power devices for both case studies. A reduction exceeding 53% in static damage is observed for all critical joints across both case studies when employing modified rainflow. Notably, even the lifetime estimation of bondwires, which are unaffected by the heating time, was influenced by modified rainflow due to the computation of effective mean temperature and the filtering of cooling half-cycles.

The modified rainflow function is readily available for download in [²⁸]. For future developments of this work, the investigation of the modified rainflow algorithm considering different converter applications and power devices (i.e., blocking voltages and manufacturers) is an interesting topic. Experimental verification using accelerated testing will be considered for future work.

ACKNOWLEDGMENT

The authors are grateful for the financial support provided by the P&D project ANEEL/CEMIG D0727, CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) - projects 311056/2020-2, 408058/2021-8 and 307172/2022-8 and FAPEMIG (Fundação de Amparo à Pesquisa do Estado de Minas Gerais) - project RED-00216-23. In addition, this work was carried out with the support of the CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) - Financing Code 001.

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Edited by

• Editor-in-Chief
  Telles B. Lazzarin https://orcid.org/0000-0001-8598-8641

• Associate Editor
  Roberto F. Coelho https://orcid.org/0000-0002-4672-0885

Publication Dates

• Publication in this collection
  11 Aug 2025
• Date of issue
  2025

History

• Received
  09 Sept 2024
• Accepted
  04 Apr 2025
• Published
  15 Apr 2025