**4. System-Level Reliability Benchmarking**

With the static damage obtained in the previous section, the lifetime of the power devices can be obtained as certain fixed values. This is far from reality, since the power device lifetime could present variations due to the uncertainties in device parameters and experienced stresses. Therefore, the lifetime prediction should consider these uncertainties and, thus, provide statistical lifetime values. In this section, a statistical approach that is based on the Monte-Carlo analysis is applied [31], in which the variations of the model parameters in (2) and the thermal stresses are introduced with 5% variations to represent the uncertainties. Notably, to assist the analysis, the dynamic stress parameters (i.e., *<sup>T</sup>*j(min), <sup>Δ</sup>*T*j, and *t*on) are normally converted into equivalent static ones (i.e., *<sup>T</sup>* j(min), <sup>Δ</sup>*<sup>T</sup>* j , and *t* on), which can produce the same one-year *LC* when applying them to the *LC* calculation process [32]. By doing so, the system-level reliability can be predicted.

The Monte-Carlo simulations are conducted when considering a population of 10,000 samples, following which the obtained lifetime data for a certain device are fitted with the Weibull distribution as [33]

$$f(\mathbf{x}) = \frac{\beta}{\eta} \left(\frac{\mathbf{x}}{\eta}\right)^{\beta - 1} e^{-\left(\frac{\mathbf{x}}{\eta}\right)^{\beta}}, \quad F(\mathbf{x}) = 1 - e^{-\left(\frac{\mathbf{x}}{\eta}\right)^{\beta}} \tag{3}$$

where *f*(*x*) and *<sup>F</sup>*(*x*), respectively, represent the probability density function (PDF) of the Weibull distribution and the cumulative density function (CDF, also referred to as the unreliability function) with *x*, *η*, and *β* being the operation time, the scale parameter, and the shape parameter, respectively.

Subsequently, the reliability assessment of the PV-battery system follows the steps from the component level, the converter level, to the system level, and the corresponding lifetime values are obtained in terms of *B*10 lifetime, which represents the total operation time when 10% of the populations will fail. For the component-level reliability analysis, the *B*10 lifetime of each power device can be obtained from the corresponding *<sup>F</sup>*(*x*) curve. While investigating the converter-level and system-level reliability, it can be performed by using the Reliability Block Diagram (RBD), which describes the reliability interaction between each device and subsystem in the entire system. Figure 11a shows the RBD of the considered PV-battery systems. For the converter-level RBD, if any of the IGBTs or diodes fails, it is considered that the converter cannot function. Thus, the series-connected RBD is considered for these power converters, as shown in Figure 11a. Subsequently, the unreliability function for the converters *<sup>F</sup>*con(*x*) can be calculated as

$$F\_{\rm con}(\mathbf{x}) = 1 - \prod\_{i=1}^{n} \left( 1 - F\_{\rm comp}(i) \right) \tag{4}$$

in which *<sup>F</sup>*comp(*i*)(*x*) represents the unreliability function of the *ith* device in the converters. Regarding the system-level RBD, as shown in Figure 11b, the series connection of the converter-level RBDs is considered for both the DC- and AC-coupled system configuration. Thus, the system unreliability is calculated as

$$F\_{\rm sys}(\mathbf{x}) = 1 - \left(1 - F\_{\rm PV}(\mathbf{x})\right) \left(1 - F\_{\rm but}(\mathbf{x})\right) \tag{5}$$

where *<sup>F</sup>*pv(*x*) and *<sup>F</sup>*bat(*x*) represent the unreliability functions of the PV converter and the battery converter, respectively.

**Figure 11.** Series connection of the reliability block diagram: (**a**) converter level and (**b**) system level, where *<sup>F</sup>*comp*<sup>i</sup>*(*x*) represents the unreliability function of the *ith* device in the converter, and *<sup>F</sup>*pv(*x*) and *<sup>F</sup>*bat(*x*) represent the unreliability functions of the PV and battery converters, respectively.

The converter-level unreliability functions of the power converters within the two considered PV-battery systems are shown in Figure 12, along with the corresponding component-level functions of their power devices. The *B*10 lifetime results are in accordance with the *LC* comparison in the previous section. For the PV inverters, as shown in Figure 12a,b, the reliability of the two inverters is dominated by the reliability of the clamping diodes (e.g., D5 referring to Figure 4) and, consequently, the *B*10 lifetime of the PV inverter with the DC-coupled BESS is 101.6 years, which is even higher than the twice of the *B*10 lifetime of the PV inverter with the AC-coupled BESS. Notably, the results also imply that the two inverters have excessive design margins for the considered Denmark mission profile. Regarding the BESS converters, for the battery inverter, as shown in Figure 12c, instead of the clamping diodes, the outer diodes (e.g., D1 referring to Figure 7b) will have the lowest *B*10 lifetime due to its bidirectional operation, and the inverter *B*10 lifetime is 77.1 years. As observed in Figure 12d, the reliability of the battery converter for the DC-coupled BESS is mainly affected by the lower IGBTs (e.g., T2 referring to Figure 7a) and its *B*10 lifetime is 68.8 years. Notably, both the battery inverter and the battery converter are less reliable than the corresponding PV inverters, which will seriously affect the overall reliability of the PV-battery systems.

**Figure 12.** Unreliability functions of the different power converters: (**a**) PV inverter with the AC-coupled BESS, (**b**) PV inverter with the DC-coupled BESS, (**c**) battery inverter for the AC-coupled BESS, and (**d**) battery converter for the DC-coupled BESS.

Figure 13 (the solid lines) shows the system-level unreliability functions of the considered PV-battery systems, where the corresponding converter-level unreliability functions are also given (the dashed lines). The *B*10 lifetime of the DC-coupled configuration is 68.8 years, which is six years shorter than that of the AC-coupled configuration. As expected, both the reliability of the DC-and AC-coupled configuration are dominated by the reliability of their battery converters, especially for the DC-coupled case. This is expected, as the battery is used to balance and smooth the fluctuating power from the PV array. Although the integrated DC-coupled BESS can enhance the reliability of the PV inverter to a large extent, the much less reliable battery converter will limit the overall reliability performance. Hence, for the case study in this paper, the AC-coupled configuration is a better option, featuring a more balanced and higher reliability performance. It should be mentioned that the redundancy of the interleaved battery converter is not considered in the above comparison for comparing the two configurations under the full power smoothing capability. Obviously, it is expected that the reliability of the DC-coupled configuration can be improved with a proper redundancy design of the battery converter. For instance, adding one more paralleled bidirectional converter stage to the battery converter (see Figure 7a), which can achieve a three-out-of-four redundancy [34]. By doing so, the reliability of the battery converter will be improved considerably, as shown in Figure 13b, achieving higher system-level reliability than the AC-coupled configuration in Figure 13a, while its hardware cost might probably still less than the three-level battery inverter for the AC-coupled configuration.

**Figure 13.** Unreliability function of the DC- and AC-coupled PV-battery system: (**a**) AC-coupled configuration and (**b**) DC-coupled configuration.
