FSC-MPC Switching Phase Control for Interleaved DC/DC Vehicle Charger

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an conference paper published 2025 IEEE International Conference on Industrial Technology (ICIT) available at: 10.1109/ICIT63637.2025.10965309 “© 2025 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” FSC-MPC Switching Phase Control for Interleaved DC/DC Vehicle Charger Mat´ ıas Aguirre Member, IEEE, Sergio Vazquez Fellow, IEEE, Jose I. Leon Fellow, IEEE, Leopoldo G. Franquelo Life Fellow, IEEE Abstract—Interleaved converter configurations are a common solution for the charging of electric vehicles. Carrier phase shift is a common and important design element for these converters. Recent research shows that a proper regulation of the phase shift can improve the system performance. This requires more complex controllers and actuation capabilities. Finite Control Set Model Predictive Control (FCS-MPC) has the capability to manage multiple control. Recent research has managed to achieve a satisfactory switching frequency control through a period control approach (PCA) for a single power converter. However, it does not control the switching phase required for an interleaved operation with parallel converters. This paper presents the design and implementation of a switching phase control for PCA-FCS-MPC, allowing the operation of an interleaved configuration. The performance is evaluated for different operating point through experimental validation. Index terms— Model Predictive Control, Finite Control Set, Fixed Switching Frequency, Interleaved Converter, Vehicle Charger I. INTRODUCTION The transition from fossil fuel vehicles to hybrid or fully electrical ones has increased in recent years and many countries aim for a full electrification in the near future. A major aspect in the execution of this transition is the infrastructure required to accommodate this technologies. Mainly in this category is the availability, capabilities and robustness of electric chargers [1]. Technological advancement in materials, software and manufacturing have made it easier over time to jump-start the creation of a functional network to support the integration of electric vehicles. Nonetheless, there is still much to do to achieve an ease of use comparable to fossil fuel vehicles in terms of charging speed, reliability or availability. In terms of charging speed, one common strategy to achieve the required power without incurring in costly components, is the use of interleaved converters [2]–[7]. These configurations allow multiple modules to distribute the power output while improving the overall control performance of the relevant electrical variables. This also provides flexibility and resilience when dealing with malfunctions or unconventional requirements. The most common scheme to control these converters is through PWM along linear controllers. Another alternative is through the use of Finite Control Set Model Predictive Control (FCS-MPC), which offers more flexibility in terms of actuation, while also providing a more flexible control scheme capable of multiple control objectives and of different nature. Nonetheless, due to the variable switching frequency achieved by the conventional implementation of FCS-MPC, its use for the control of interleaved converters has been limited. In recent publications, a solution to the switching frequency of FCS-MPC has been proposed, named Period Control Approach (PCA-FCS-MPC), that maintains the time discrete nature of the controller [8], [9]. The achieved switching pattern is close to a PWM, while performing like FCS-MPC in most relevant metrics. An important drawback of this solution is the lack of control over the switching phase, thus being unable to benefit at full from the interleaved configuration. This has been solved in [10], where a switching phase control is implemented into PCA-FCS-MPC. Using this approach, the phase switch required by the interleaved configuration can be realized and further optimized according to different operating points and objectives. The proposed strategy has been validated in steady state, but a real implementation for a electric vehicle charger would also require quick response times to changes in the load, to further optimize the charging process and quickly respond to potential disturbances. This paper aim to expand the analysis presented in [10], particularly in regards to the time response of the overall control strategy, thus validating its feasibility for its use in the application of electric vehicle charging stations. The structure of this paper is as follows. Section II starts with a description of the discrete model of the selected converter for its use in PCA-FCS-MPC Also, this section includes a brief summary of the classic control operation with equal phase shift between modules’ carriers, and the possibility to optimize this phase shift for different operating conditions. After this, in Section III, the control strategy of PCA-FCSMPC is described in detail to facilitate its implementation. In Section IV, the proposed strategy for phase control is described, both for phase measurement and control. In Section V, experimental results are presented to illustrate and evaluate the performance of the proposed strategy. Section VI finishes this paper with conclusions and closing remarks. II. CONVERTER MODEL AND OPERATION The selected system is an interleaved dc/dc buck power converter. It is composed of three modules in parallel, as shown in Fig. 1. The high voltage capacitor steady state voltage, vi, is fixed by an external voltage source. The bandwidth of this voltage source is limited, which means it is expected to reach a stable voltage, vi, and current, ii, in steady state, but the capacitor is still susceptible to high frequency dynamics or distortions due to the converter’s operation. The low voltage capacitor voltage, vo, and inductors’ currents i1,i2and i3, Fig. 1. Circuit scheme of the dc/dc interleaved power converter. are the dc/dc buck controller objective variables. The load battery is represented in simulation by a resistor, without loss or generality. In order to implement an FCS-MPC method, a discrete model of the system is required. A forward Euler approximation with sampling period τsis selected [11]. Therefore, the equations that describe the behavior of the proposed system are I= [i1i2i3]T,S= [S1S2S3]T Ik+1 =Ik+ (vi,kSk9vo,k)τs/L (1a) vi,k+1 =vi,k +ii9ST kIkτs/Ci(1b) vo,k+1 =vo,k +1T 3Ik9io,kτs/Co(1c) where 1n= [1 · · · 1]Tof length ‘n’. A. Interleaved Operation A common operation strategy for an interleaved power converter is to distribute the power between modules, through the use of PWM [12]–[14] and a phase shift θxfor the carrier of each module. θx,r =x2π/M , (2) where Mis the number of modules in parallel [15]–[20] and x∈ {1,2,· · · , M}. One advantage of this approach is the reduction of low frequency harmonic content of the current at the high voltage side of the converter. This extends the expected lifetime of the capacitors, as shown in [21]. Another advantage is the effective multiplication of the switching frequency observed at the total output current, reaching a factor of Mover the switching frequency implemented at each module, thus achieving a better performance without overexerting the semiconductors. Examples of the effects achieved with a distributed phase shift operation is shown in Fig. 2 and the balanced results of Fig. 3. Fig. 2 shows the current contribution of each module and the capacitor current, ici, in steady state. Fig. 2(b) illustrates the reduced ripple in the output current, compared to the inductor’s current ripple. Fig. 3, illustrates the effect on the current spectra for the high voltage capacitor. The results clearly illustrates the absence of low frequency harmonics for the balanced system. These frequencies components are of major importance, since they are among the main factors in the reduction of the capacitor lifetime expectancy [21]. 4 5 6 0 0.5 1 1.5 2 2.5 3 -2 0 2 4 (a) (b) Fig. 2. Inductors and capacitors currents for a (left) balanced, (center) unbalanced and (right) optimized operation in time, at 2kHz switching frequency. 0 2 4 6 8 10 12 14 16 0 0.5 1 1.5 2 Fig. 3. Comparison between the current spectrum of Cifor a balanced, unbalanced and optimized operation, at 2kHz switching frequency. B. Optimal Phase Shift A phase shift of 360◦ Mbetween consecutive modules achieves a good performance when the modules are balanced, i.e. all modules operate under the same conditions of voltage and current. Nonetheless, this is no longer the case when the power managed by each module differs. This can be the case of multiple charging stations connected to vehicles with different charging requirements, or a required imbalance to manage the remaining life time of the components. It can also be the case that one of the modules is defective, which may require an uneven distribution of the current, or disconnection of a module, thus being unfit for the same phase shift. This is illustrated in Fig. 2(center) and Fig. 3, where an unbalanced configuration leads to an increase in the low frequency harmonics of the current at the high voltage side of the converter. The optimal phase shift to minimize low frequency harmonics can be computed as a function of the operating point and the different parameters of the system [21]. The effect of using the optimized phase shift is illustrated at Fig. 2(right), and further emphasized by the spectra in Fig. 3. III. PREDICTIVE CONTROL The use of Period Control Approach (PCA) has shown to be effective to fix the switching frequency and shape the spectrum of the FCS-MPC [8], [9]. Therefore, it is possible to implement this algorithm in the present system, achieving similar performances as with PWM. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 0 10 20 Fig. 4. Example of period computation. A. Period Control Approach Period measurement is achieved by counting the time in between similar commutations for each module of the converter. Every time a commutation takes places, the respective counter, Quor Qd, is reset to 1. Fig. 4 illustrates how this process unfold. The period values Tuand Tdare only updated when the respective commutation takes place, as Q△= [Q1uQ2uQ3u]T,T△= [T1uT2uT3u]T, Q▽= [Q1dQ2dQ3d]T,T▽= [T1dT2dT3d]T, E△ kE▽ k=E△▽ k= [(Sk>Sk91) (Sk<Sk91)] (3a) Q△ kQ▽ k=Q△▽ k=Q△▽ k91◦E△▽ k+ 1 (3b) T△ kT▽ k=T△▽ k=T△▽ k91◦E△▽ k+τsQ△▽ k91◦E△▽ k(3c) where ‘◦’ corresponds to the elementwise product. When a very high or very low output is required, the control signal would not switch, but the internal count is still updated as if a zero width commutation were implemented. This virtual commutation is achieved as Ev k=Q△ k>Qr∧Q▽ k<Qr1T 2(4a) T△▽ k=T△▽ k◦Ev k+τsQrEv k,(4b) Q△▽ k=Q△▽ k9QrEv k.(4c) The harmonic mean is used to compute the effective period per module, T, and the system’s average period, T, as Tk= 2 T△▽ k 911291 ,(5a) Tk= 3 1T 3T91 k91.(5b) where the ‘91’ exponent operates elementwise. The prediction stage, signaled by the subindex ‘p′follows a different algorithm based on the predicted actuation, as E△ p= (Sp>Sk),E▽ p= (Sp<Sk)(6a) Q△▽ p=Q△▽ k+E△▽ p.(6b) B. Cost Function The count and current predicted values, Q△▽ pand Ip, are included in the cost function, J, through quadratic error as Qr= [Q1rQ2rQ3r]T,Ir= [I1rI2rI3r]T Jq= Q△ p9Qr  2+ Q▽ p9Qr  2(7a) Ji=∥Ip9Ir∥2(7b) J=λiJi+λQJQ(7c) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 Fig. 5. Phase measurement example. where λiand λQare the weight factors of current and period respectively, and ∥X∥2=XTX, for single column vectors. C. Frequency Reference Correction As it is, PCA-FCS-MPC achieves a steady switching frequency with a steady state error [8]–[10]. This problem is addressed in [9], where an offset, Fo, is included to the switching frequency reference, Fr, to compensate for the error, as Fo,k =Fo,k91+Fr9T91λo,(8) where λois equivalent to an integrator gain in a PI controller. Due to the integrating nature of this offset, a saturation limit of ±Fr1 2is applied. Thus, the base reference used in the cost function is Qr,k =τs(Fr+Fo,k)91.(9) This reference is the same for all modules of the converter. IV. PREDICTIVE PHASE MEASUREMENT AND CONTROL PCA-FCS-MPC achieves control of the switching frequency with a slight deviation from the reference. This deviation is not critical in most applications, thus being free to drift as required by the controller. This natural drifting can be used to address the switching phase control. A. Switching Phase Measurement The critical variable for performance improvement is the phase difference, or relative phase, between modules, Φxy, which, as illustrated in Fig. 5, is computed based on the count measurements, Q, used for period control as Φ△= [Φ12uΦ23uΦ31u],⇃X= [XnX1· · · Xn91]T, Φ▽= [Φ12dΦ23dΦ31d],↿X= [X2· · · XnX1]T, Φ△ kΦ▽ k=Φ△▽ k=2π QrQ△▽ k9 ↿Q△▽ k(10) where ⇃Xand ↿Xindicates a vertical shift of the elements of the respective vector or matrix. Each phase reference, θx,r, can be obtained through an optimization algorithm, and the difference between consecutive modules is used as the reference for the relative phase Θr= [θ1,r θ2,r θ3,r]T Φr=Θr9 ↿Θr(11) The relative phase computation of (10) gives pairs of measurements, Φ△ kand Φ▽ k, that should reach the same value 0 10 20 30 0 0.5 1 1.5 2 2.5 3 3.5 4 0 10 20 30 (a) (b) Fig. 6. Phase adjustment example, with reference Φxy,r =20 25 πrad, when the relative phase starts (a) further apart and (b) closer than the reference. in the steady state operation. Therefore, each pair is given the same reference. Thus, the phase error is computed as Φ△▽ e=Φ△▽ k9Φr1T 2(12) When a commutation takes place, the value of the respective count is reset to 1and the respective phase measurement would experience a large step. This would cause a large step in the phase error, which could destabilize the phase control. Since the phase error is an angle, measured in radians, the step problem is solved by changing the error such that it remains in the range [9π π], by adding or subtracting multiples of 2π as required. B. Phase Control If the relative phase errors, Φxyu,e and Φxyd,e, are positive, it means that the phase of module xis further than needed from the phase of module y. Therefore, to fix the relative phase, module xshould drift forward in time or module yshould drift backwards, as illustrated in Fig. 6(a). If the relative phase errors are negative, the opposite correction is due, as illustrated in Fig. 6(b). To achieve this correction, the count reference given to module xis offset from the base reference, Qr, as function of the errors related to it. For the 3 module converter, the offset references for each module are Qr=Qr+Φ△▽ e9 ⇃Φ△▽ e12λΦQ(13) where λΦQis a conversion factor, to change units from radians to step counts. A final consideration to take into account, is the implementation of a limit in the offset. A high offset in the count reference might lead to unstable or unpredictable behavior since it would force large deviations from the actual reference. This limit is evaluated in simulations, to verify stability. Thus, the values in parenthesis in equation (13), are limited to be between ±π/6. V. EXPERIMENTAL RESULTS The proposed strategy for phase shift control is implemented in a 1007 dSPACE control platform, and an interleaved converter prototype was constructed, as shown in Fig. 7. The Fig. 7. Laboratory prototype of the interleaved power converter with 3 parallel modules. TABLE I EXPERIMENTAL PARAMETERS. Symbol Name Value RLoad Resistor 7.5 Ω CoOutput Capacitor 470 µF CiInput Capacitor 940 µF LxLine Inductor 15 mH FrSwitching Frequency Reference 1.5kHz FsSampling Frequency 50 kHz τsSampling Period 20 µs viInput Voltage 140 V voOutput Voltage 70 V parameters used for the experimental test are presented in table I An initial test is done to compare the performance achieved by PCA-FCS-MPC, without special consideration for the phase between modules, and without and with the proposed algorithm for phase control. For this test, all modules have the same current reference, thus being a balanced converter. Fig. 8 shows the resulting current through each module, in yellow, green and magenta, and the current at the high voltage side of the converter in cyan. Two different experiments are plotted. The right side shows the performance when there is no control over the switching phase of each module, and the left, when the switching phase control is active. As illustrated, the switching frequency is properly controlled, and a stable steady state is achieved. This base test shows in rough terms the correct operation of the phase control in the interleaved converter. Figure Fig. 9 shows the transient when the phase control is activated, at 5ms. It can be observed that the both current control and frequency control are maintained successfully during the transient, while reaching the desired phase. To properly evaluate the phase tracking performance, one needs the phase values as measured by the controller, as well as the spectral distribution of the currents. With a phase reference of 0rad, 2 3πrad and 4 3πrad (relative phase between modules of 2 3πrad), the following results are obtained. As show in Fig. 10(a), without phase control (left), the current through each module has the same mean value, as expected from predictive control, but the resulting capacitor current does not have a consistent shape. Fig. 8. Inductors current (yellow, green and magenta), and Cicurrent (cyan), without phase control (left) and with phase control (right). 012345678910 2 2.5 3 3.5 4 Fig. 9. Transient results for the activation of phase control. It can be seen in Fig. 10(a), that without a phase control, they can drift randomly, which explains why ici has a different shapes at different times, when comparing Fig. 8 and Fig. 10(a). When the phase control is activated (right), the relative phase between all modules reaches 2 3πrad, and dramatically improves the performances observed in ici. This is more clearly appreciated in Fig. 10(c), where the current spectra for low frequency harmonics is much higher without the phase control. With the balanced operation verified, the next step is to check the performance when an unbalanced current is required for the converter. For this, a defective module is simulated through a lower current reference. Fig. 11 shows the response time of the proposed strategy for changes in the input voltage, from 150[V] to 250[V] (left) and output load from 7.5 [Ω] to 3.75 [Ω] (right). These results illustrate the effectiveness of the phase and frequency control. Regardless of the current at each module, as long as a set of phases is available to optimize the power converter operation, then it is possible to apply a phase control to PCA-FCS-MPC and reach a desirable performance. VI. CONCLUSION The presence of low frequency harmonics is a major contributor in the decay of the capacitor lifetime, and is in general an undesired element. Interleaved converters can address this issue when PWM control techniques are used in balanced systems, by shifting the carrier phase of each module. However, it is an unsolved problem when FCS-MPC strategy is used to operate the converter. In this paper, PCA-FCS-MPC is successfully implemented to achieve an accurate phase control without major changes in -4 -2 0 2 4 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 3.6 4 -120 -60 0 60 120 (a) (b) 0123456789 0 0.1 0.2 0.3 0.4 (c) Fig. 10. Experimental results for a balanced system. 2 4 6 8 10 12 50 100 150 200 -10 0 10 20 30 1.4 1.6 1.8 -10 0 10 20 30 (a) (b) (c) (d) (e) (f) Fig. 11. Unbalanced system transient response to (a-c) an input voltage step and (d-f) a load step. the core algorithm. This phase control achieves an improvement over the current spectral distribution, as expected with a proper phase shift for an unbalanced system. The core idea of the strategy is that, once a a steady switching frequency is achieved, the period reference for each module can be adjusted through an offset, to briefly nudge the frequency, such that a phase adjustment is achieved. The phase control is shown to be effective and unharmful to other control objectives, thus maintaining a good control performance over voltages and currents, and a fast transient when activated. REFERENCES [1] S. Rivera, S. Kouro, S. Vazquez, S. M. Goetz, R. Lizana, and E. Romero-Cadaval, “Electric vehicle charging infrastructure: From grid to battery,” IEEE Industrial Electronics Magazine, vol. 15, DOI 10.1109/MIE.2020.3039039, no. 2, pp. 37–51, 2021. [2] H. J. Raherimihaja, S. Lu, G. Sun, X. Xu, L. Tang, and B. He, “A combined single-phase boost rectifier and dcdc buck converter into threephase open-winding machine drive for ev battery charger,” in 2024 International Conference on Electrical, Computer and Energy Technologies (ICECET, DOI 10.1109/ICECET61485.2024.10698232, pp. 1–5, 2024. [3] B. N. Alajmi, I. Abdelsalam, M. I. Marei, and N. A. Ahmed, “Two stage single-phase ev on-board charger based on interleaved cascaded non-inverting buck-boost converter,” in 2023 IEEE Conference on Power Electronics and Renewable Energy (CPERE), DOI 10.1109/CPERE56564.2023.10119584, pp. 1–6, 2023. [4] B. Kim, H. Kim, and S. Choi, “Three-phase on-board charger with three modules of single-stage interleaved soft-switching ac-dc converter,” in 2018 IEEE Applied Power Electronics Conference and Exposition (APEC), DOI 10.1109/APEC.2018.8341592, pp. 3405–3410, 2018. [5] K. Kumar, S. Garlapati, M. S. Devi, and T. R. Choudhury, “Estimation of power losses in interleaved buck-boost converter for ev charging applications,” in 2023 IEEE 2nd International Conference on Industrial Electronics: Developments & Applications (ICIDeA), DOI 10.1109/ICIDeA59866.2023.10295242, pp. 107–111, 2023. [6] G. Balen, A. R. Reis, H. Pinheiro, and L. Schuch, “Modeling and control of interleaved buck converter for electric vehicle fast chargers,” in 2017 Brazilian Power Electronics Conference (COBEP), DOI 10.1109/COBEP.2017.8257412, pp. 1–6, 2017. [7] M. Singh, A. P. Mittal, and A. K. Singh, “An efficient regulated battery charger with minimum ripple current for electric vehicle,” in 2023 7th International Conference on Computer Applications in Electrical Engineering-Recent Advances (CERA), DOI 10.1109/CERA59325.2023.10455707, pp. 1–6, 2023. [8] M. Aguirre, S. Kouro, C. A. Rojas, J. Rodriguez, and J. I. Leon, “Switching frequency regulation for fcs-mpc based on a period control approach,” IEEE Transactions on Industrial Electronics, vol. 65, DOI 10.1109/TIE.2017.2777385, no. 7, pp. 5764–5773, 2018. [9] M. Aguirre, S. Kouro, C. A. Rojas, and S. Vazquez, “Enhanced switching frequency control in fcs-mpc for power converters,” IEEE Transactions on Industrial Electronics, vol. 68, DOI 10.1109/TIE.2020.2973907, no. 3, pp. 2470–2479, 2021. [10] M. Aguirre, S. Vazquez, A. M. Alcaide, R. Portillo, S. Kouro, J. I. Leon, and L. G. Franquelo, “Period control approach finite control set model predictive control switching phase control for interleaved dc/dc converters,” IEEE Transactions on Industrial Electronics, vol. 71, DOI 10.1109/TIE.2023.3325548, no. 8, pp. 8304–8312, 2024. [11] C. A. Silva and J. I. Yuz, “On sampled-data models for model predictive control,” in IECON 2010 - 36th Annual Conference on IEEE Industrial Electronics Society, DOI 10.1109/IECON.2010.5674939, pp. 2966– 2971, 2010. [12] D. Garth, W. Muldoon, G. Benson, and E. Costague, “Multi-phase, 2kilowatt, high-voltage, regulated power supply,” in 1971 IEEE Power Electronics Specialists Conference, DOI 10.1109/PESC.1971.7069141, pp. 110–116, 1971. [13] B. Miwa, D. Otten, and M. Schlecht, “High efficiency power factor correction using interleaving techniques,” in [Proceedings] APEC ’92 Seventh Annual Applied Power Electronics Conference and Exposition, DOI 10.1109/APEC.1992.228361, pp. 557–568, 1992. [14] C. Chang and M. Knights, “Interleaving technique in distributed power conversion systems,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 42, DOI 10.1109/81.386158, no. 5, pp. 245–251, 1995. [15] V. G. Monopoli, A. Marquez, J. I. Leon, M. Liserre, G. Buticchi, L. G. Franquelo, and S. Vazquez, “Applications and modulation methods for modular converters enabling unequal cell power sharing: Carrier variable-angle phase-displacement modulation methods,” IEEE Industrial Electronics Magazine, vol. 16, DOI 10.1109/MIE.2021.3080232, no. 1, pp. 19–30, 2022. [16] M. Esteki, B. Poorali, E. Adib, and H. Farzanehfard, “Interleaved buck converter with continuous input current, extremely low output current ripple, low switching losses, and improved step-down conversion ratio,” IEEE Transactions on Industrial Electronics, vol. 62, DOI 10.1109/TIE.2015.2397881, no. 8, pp. 4769–4776, 2015. [17] L. Bede, G. Gohil, T. Kerekes, and R. Teodorescu, “Optimal interleaving angle determination in multi paralleled converters considering the dc current ripple and grid current thd,” in 2015 9th International Conference on Power Electronics and ECCE Asia (ICPE-ECCE Asia), DOI 10.1109/ICPE.2015.7167932, pp. 1195–1202, 2015. [18] J. S. Siva Prasad and G. Narayanan, “Minimization of grid current distortion in parallel-connected converters through carrier interleaving,” IEEE Transactions on Industrial Electronics, vol. 61, DOI 10.1109/TIE.2013.2245620, no. 1, pp. 76–91, 2014. [19] S. Kavitha, S. E. Rajan, and R. P. Vengatesh, “Performance analysis of interleaved dc-dc boost converter for photo-voltaic power generation systems,” in 2014 International Conference on Green Computing Communication and Electrical Engineering (ICGCCEE), DOI 10.1109/ICGCCEE.2014.6922352, pp. 1–6, 2014. [20] J. W. Zapata, S. Kouro, M. Aguirre, and T. Meynard, “Model predictive control of interleaved dc-dc stage for photovoltaic microconverters,” in IECON 2015 - 41st Annual Conference of the IEEE Industrial Electronics Society, DOI 10.1109/IECON.2015.7392771, pp. 004 311– 004 316, 2015. [21] A. M. Alcaide, Y. Ko, M. Andresen, J. I. Leon, S. Vazquez, V. G. Monopoli, G. Buticchi, M. Liserre, and L. G. Franquelo, “Capacitor lifetime extension of interleaved dc-dc converters for multistring pv systems,” IEEE Transactions on Industrial Electronics, vol. 70, DOI 10.1109/TIE.2022.3187579, no. 5, pp. 4854–4864, 2023.