A Novel DC-Power Control Method for Cascaded Converter under DC Voltage-Imbalance Condition
This paper proposes an interphase power control algorithm (IPCA) and submodule power control algorithm (SPCA) based on state transition PWM (ST-PWM) for cascaded H-bridge inverter with unbalanced DC sources. In this method, the correction value helps control the interphase power distribution and submodule power distribution by directly modifying the duty cycle of each switching state. In order to achieve linear modulation, an in-depth analysis of ST-PWM is conducted. Combining ST-PWM and IPCA, balanced AC currents and accurate power control on DC side can be achieved with an unbalanced DC source. The proposed strategies are simulated using MATLAB/Simulink and tested on a five-level laboratory prototype. Simulation and experimental results demonstrate the advantage and effectiveness of the proposed algorithm.
The cascaded H-bridge converter (CHB) has become an outstanding type of converter circuitry in the medium-voltage converter field due to its merits, such as low switching voltage stress, improved output waveform quality, and modular and extendible structure [1–5]. As a suitable topology for high power applications, the CHB has been widely researched and applied both in research and industries, such as renewable energy systems , high-voltage power AC transmission systems , and large-scale photovoltaic power plants .
The CHB suits large photovoltaic power plants by providing separate DC links to photovoltaic (PV) strings and has better scalability to reach high power ratings. Therefore, the cascaded H-bridge converter is particularly attractive for photovoltaic grid-connected applications [8–11]. However, when the CHB serves as the interfacing converter for a PV system, the submodules will generate different power and DC voltages. This imbalance is mainly caused by nonuniform solar irradiances, unequal temperature, or inconsistent module degradation of different solar panels . Unequal voltages and output power of DC sources in each submodule make the output voltage seriously distorted and lead to unbalanced AC current flowing to the grid, which could potentially make the whole system disconnected from the grid [10, 11]. Similar imbalance problems can be found in electric vehicles (EV) applications and energy storage systems . In the electric vehicle applications [13, 14], for instance, the CHBs with batteries powering the vehicles are not equal and will vary while in use . Because of the different characteristics of battery packs, unequal DC voltages become inevitable. This will result in unbalanced output power in electric vehicle . But, with effective measures when the fault battery pack is cut off, the entire system can still work well. It is therefore essential to design suitable control strategies to tackle power imbalance problems.
In , a power balancing method is introduced to regulate the actual power transmitted through the dual active bridge (DAB) parallel modules. With this power balance control method, the current and power transmitted through each DAB module can be balanced. In , a distributed maximum power point tracking (MPPT) control method is proposed to increase the efficiency of PV systems. Therein, a control scheme with modulation compensation is also proposed to acquire a balanced grid current under power imbalance conditions. In [10, 11], a fundamental frequency zero-sequence injection (FFZSI), double min-max (DMM), and double 1/6 third harmonic injection (DTHI) are proposed to balance the interphase power and grid current. There are mainly three drawbacks to the above methods: (1) the trigonometric calculations are complicated, and this might lead to overflow issues; (2) methods based on FFZSI might be ineffective when the power generation ratios are approximately equal; and (3) the relationship between the linear modulation index and power generation ratio limitation is unclear, which is important for PV systems under different modulation indexes. Moreover, these methods only analyze the adaptability in PV systems under equal DC sources but do not provide an efficient solution for CHB systems with unbalanced DC sources.
To solve the power imbalance problem for CHBs with unbalanced DC sources, in this paper, we propose an interphase power control algorithm (IPCA) and submodule power control algorithm (SPCA) based on state transition PWM (ST-PWM), to realize accurate power control for CHB with unbalanced DC sources. The power generation ratios per phase can be maintained to their expected value precisely by modifying the correction value derived from the two power control algorithms. Balanced AC currents at the AC side and accurate power control performance on the DC side are acquired with the proposed algorithms when the CHB operates under unbalanced DC sources, even under severe voltage-imbalance conditions.
2. Proposed Modulation Strategy with Unbalanced DC Sources
2.1. Introduction of the Simplified PWM Strategy under Unbalanced DC Sources
A typical CHB converter is shown in Figure 1, where Udc_Xn (n = 1, 2, 3, …) represents the DC-source voltage of each submodule. The terminal voltage of each submodule is Udc_Xn in the P state, –Udc_Xn in the N state, and 0 V in the O state. Udc_A, Udc_B, and Udc_C stand for the total DC-source voltage per phase.
Udc_ave, the average value of Udc_Xn, is expressed as (2). rX in (3) stands for the interphase voltage-imbalance coefficient.
The voltage-second balancing principle defined in  for a three-level NPC converter is still suitable for CHB. Assuming that the variables (Udc_Xn, TXn) remain constant within Ts, the volt-second product formula can be represented as follows:
where stands for the three-phase reference voltages, Ts is the switching period, uz is the equivalent zero-sequence voltage, TX is the duration time of each phase, and TXn is the duration time of each submodule. (SXn = SXn1 – SXn2) is the switching functions of each submodule, and SXni = 0,1 (i = 1,2).
To simplify the calculation process, duty cycle DX is adopted instead of the duration time. The voltage-second balance in (4) can be rewritten as follows:
where ∆DX represents the correction value that is added to . In order to achieve the frequency multiplier effect of cascade H-bridge, the carrier phase shift technology is adopted, so the duty cycle of each submodule is the same before submodule power control, as shown in (6).
It should be noted that the duty cycle of each submodule is equally distributed before the submodule power control, and it will be modified during the introduction of the SPCA to achieve the submodule power control.
2.2. Extending Linear Modulation Range
The maximum value of modulation index m under unequal DC sources is reduced due to the distorted hexagon. In fact, it cannot reach the maximum modulation index when the traditional SVPWM method is used, and voltage distortion is not avoidable. The essence of the problem is that the partial vector is assigned to the wrong sector because of the simplified PWM strategy. It can be effectively solved by the proposed method based on state transition , to achieve the linear modulation range extension and the waveform quality improvement.
Assuming that is defined as (7), where m is the modulation index. DX can be acquired by substituting (7) into (5).
In order to get the linear modulation, DX should be in the range of 0 to 1. Thereby, the existence condition of ∆D can be expressed by inequality as follows:
Thus, the maximum linear modulation index mmax derived in  can be expressed by
where mmax is decided by the minimum value of rX. In a comprehensive analysis in , the state transition includes the following two categories:
Accordingly, the range of DX is extended to [−1 1] or [0 2], and the existence condition of ∆D can be expressed by inequality as follows:
Through the state transition, the maximum value mmax_ST can be obtained as follows by solving inequality (14). Obviously, mmax_ST is determined by the medium and minimum voltage value.
The upper and lower limits of ∆D can be defined as follows:
To meet the target of extending the linear modulation, ∆D in the whole range should satisfy constraint ∆Din [∆Dmin – ST ∆Dmax-ST] in all six sectors.
3. Analysis of the Proposed Power Control Method
When the CHB has assumed the interfacing converter for a battery system, the submodules will generate unbalanced output power and DC voltages, which must be considered in practice. Aiming at the interphase power and submodule power distribution problem, a power control algorithm under unbalanced DC sources is proposed here, to achieve precise interphase and submodule power control and maintain three-phase output current balance.
3.1. Definition of Control-Period Power and Average Power in DC Links
Let ffund, fcarrier, and fsample represent the fundamental frequency of , the carrier-wave frequency, and sampling frequency, respectively. For convenience, both the control period Tcontrol and carrier-wave period are equal to the sampling period. Assuming that the variables (Udc_X, iX, DX, and ) keep constant within Tcontrol, the control period (CP) power pX is given in the following equation:
where iX represents the three-phase current.
If sgn(iX) = sgn(), then pX > 0, which means that the DC sources release the power in the controlling period; otherwise, the DC sources absorb power.
Within Tfund, the single-phase average DC-power (SPA) PX can be expressed as follows:
where N refers to the ratios of fcontrol/ffund.
The three-phase average DC-power, Pave, can thus be derived as follows:
From (20), we know that average DC-power Pave is determined by three-phase reference voltages and AC currents, not related to ΔD.
3.2. Computational Principle of SPA Power in DC Links
The concept of the sliding window is introduced in  to calculate the SPA power. The width of the sliding window is equal to N. The calculation of SPA power within the previous N control period has been derived (19). After the sliding window moves one step, the SPA power can be expressed by
Thus, the SPA power flowing though individual DC sources can be acquired in each controlling period. As we can see from (21), is decided by PX, pX-in, and pX-out, revealing that can be adjusted by ∆D in the current controlling period.
3.3. Proposed Unbalanced DC-Power Control Method
The adjustable DC-power ∆PX within one controlling period is defined by
Referring to (19), the adjustable DC-power ΔPX within Tfund and the one in the current controlling period is expressed as follows:
In the generalized unbalanced case, kX in the previous N controlling period can be rewritten as follows:
The following steps outline the proposed active unbalanced DC-power control method when kX ≠ kX. ∆PX-in in the current controlling period should be changed to regulate the ratios.
3.3.1. Priority Control Phase Selection
The power generation ratio deviation is defined as ∆kX = |kX-kX|. Since the phase with the largest value of ∆kX deviates worst, that phase will be set as the priority control phase. And the direction of ∆D will be decided by this one based on
3.3.2. Power Generation Ratio Comparison
By comparing kX with kX, whether is increased or not can be determined. Furthermore, ΔPX-in will be decided based on (24) as shown in the following equation:
3.3.3. Identify the Direction of ∆D in Current Controlling Period
Since the direction of ΔPX-in has been decided by step 2, the variation direction of ∆D in the current controlling period is expressed by (28) based on (22). When substituting (19) and (18) into (24) gives equation (29), the relationship between kX and ∆D can be acquired, that is, the proper value of ∆D can be chosen to keep the three-phase power generation ratios under control accurately.
3.3.4. Value Identification of ∆D in Current Controlling Period
According to the analysis mentioned in sectors A and B, the range of ∆D is [∆Dmin ∆Dmax]. So ∆D is identified as follows:where the expressions of ∆Dmax and ∆Dmin are different in different cases in Section 3.
DX, as derived from the steps above, can be used to generate the switching signals directly, if it satisfies constraint DX_in [0 1]. Otherwise, the process of state transition in (13) is required. So far, the combination of both proposed algorithms (ST-PWM and IPCA) has been completed.
3.4. Control Flow of the Proposed Submodule Power Control Algorithm
Due to the series of structures of submodules in each phase, currents at the AC sides are identical. Conventional methods realize power control through superposing voltage components on the module reference voltage. In fact, submodule power control redistributes the output power of each module, which is related to the current direction, the switch state, and the duration time of the submodule. After the interphase power control, the duty cycle of each submodule defined as DXn in the following equation is a determined value:
Following the SPCA, the DC-power pXn generated by the submodules within TS is expressed as (32). The compensating value ∆DXn is introduced to redistribute the duty cycle of the submodules, and is the value after compensating.
In order to realize accurate individual submodule power control, the power generation ratios per module lXn are defined as (33), where PX stands for the DC power per phase. lXn must satisfy the equality as follows:
where PXn represents the DC-power generated by each module.
Based on (21), PXn can be updated as follows:
Submitting (35) into (32) gives equation (36), ∆DXn can be acquired to control the distribution of submodule power:
The correction ∆DXn should satisfy constraint (37), when (DX – ∆DXn(k))in [−1 1].
When it is stable in interphase power control, the power of each phase is constant, that is, = . Therefore, the sum of ∆DXn of each phase is equal to zero. In other words, the duty cycle correction injected in each module can achieve submodule power control without influencing the interphase power control.
Once ∆DXn reaches its limit, (38) does not work. To avoid an impact on interphase power control, the correction of the last module must be obtained as follows:
Finally, in SVPWM applications, the duty cycle needs to be converted to duration time as follows:
3.5. Loss Calculation and Blocking Voltage Stress
The loss is mainly determined by the IGBT conduction loss and switching loss and the diode conduction loss and reverse recovery loss.
As to the IGBT, the total loss can be written as follows:
where is the initial saturation voltage drop of IGBT and is the conduction resistance of IGBT.where is the switching frequency and is the switching energy of IGBT.
As to the diode, the total loss can be written as follows:where is the initial saturation voltage drop of the diode and is the conduction resistance of the diode.where is the switching frequency and is the reverse recovery energy of the diode.
Blocking stress varies with the DC link voltages across the power switches. As to the high power converter design, we should obtain parasitic inductances in each loop in order to minimize the turn-off voltage spike of the IGBT.
3.6. Stability Analysis
The DC-power control strategy is an open-loop control. The DC-power references are used as the command to modify the correction value ∆D. So there is no stability problem. The converter is controlled as a grid-connect converter, and a voltage and current double closed-loop control are adopted. This may raise stability problems. The voltage and current regulators used are all PI controllers. To acquire the stable system, the current inner loop is set into a typical type I system; then the bandwidth of the outer loop is tuned lower then inner loop; and it is set into a typical type II system.
4. Simulation Verification
The simulation model of the inverter is set up to analyze the relationship between voltage-imbalance coefficients and modulation index and the relationship between power generation ratios and modulation index, which can provide a reference for the grid-connected systems. The proposed ST-PWM strategy is simulated in MATLAB/Simulink to control a three-phase five-level CHBI. The simulated parameters are listed in Table 1. And the battery cells are set as in Table 2.
4.1. Control Performance of the Proposed ST-PWM Strategy with IPCA
To evaluate the performance of a proposed ST-PWM strategy with IPCA, the simulation is carried out with the battery cells settings in Table 2. Therefore, the interphase voltage-imbalance coefficients are rA = 2/3, rB = 1, and rC = 4/3. In this case, the maximum modulation index mmax_ST is 0.96 based on (15). To simplify the relationship between m and kAmax/min, phase A is set as the priority control phase at all times. This operation means that ∆D is decided by the power generation ratio kA. Figure 2(a) is drawn to describe the trends when rA = 2/3, rB = 1, rC = 4/3, RX = 10 Ω, and LX = 4 mH. kAmax decreases as the modulation index m increases. Meanwhile, kB and kC increase as m increases when kA = kAmax. The trends of kB and kC, when kA = kAmin, are opposite to when kA = kAmax. Noting that the value of kX can be negative, which means the corresponding phase will absorb power from the system. The ranges of kX become smaller when m ∈ [0.7 0.96] as shown in Figure 2(a).
Figure 2(b) shows the trend of three-phase variance S2 defined in (44), where PX_real and PX_theoretical represent the real and theoretical DC-power generated by DC sources per phase, respectively. This figure is obtained under conditions of (kA = kB = kC = 1), (rA = 2/3, rB = 1, rC = 4/3), and RX = 10 Ω, and LX = 4 mH. Obviously, most values of S2 with the proposed method are smaller than 0.25 W2 in simulation, which suggests that the power control performance of the proposed ST-PWM strategy is accurate. The simulation results show that the proposed method is (1) capable of providing more accurate results for DC-power control and (2) suitable for DC-power control under unbalanced DC sources, even under the severely unbalanced voltage condition.
4.2. Control Performance of Proposed Power Control Methods
To evaluate the proposed power control methods on grid-tied conditions, the simulation combined with IPCA and SPCA is implemented with the parameters in Table 3. Figure 3 shows the control block diagram of the grid-connected inverter with proposed methods. The control is divided into two subcontrols: (1) conventional decoupled control based on the d-q rotational frame and (2) power-distribution control. Moreover, the power distribution control is divided into the extension of modulation index, interphase power control, and submodule power control. First, three-phase voltage references acquired by conventional decoupled control are used for calculating the duration time. Then the correction value ∆D derived from the IPCA is added to the duty cycle of each phase to control the power distribution among the three phases. Once corrected DX falls outside of [0 1], the state transition algorithm is needed to extend the linear modulation range. At last, the compensating value ∆DXn is added to the duty cycle of each submodule to realize the submodule power control.
Figure 4 shows interphase power control and submodule power control, where the three-phase power generation ratios keep constant as kA = 0.6, kB = 1.1, and kC = 1.3 in the whole period and the submodule power generation ratios are set as lA1 = 0.6, lB1 = 0.5, lC1 = 0.4 after 0.3 s. Figure 5(a) shows that interphase power and submodule power can be accurately controlled after introducing IPCA and SPCA, and the submodule power control has little negative impact on interphase power control. It should be noted that the submodule power is allocated with the submodule voltage as lA1 = 2/3, lB1 = 4/9, and lC1 = 1/3 when the submodule power control algorithm is absent. From Figure 4(b), it is obvious that balanced AC currents can be achieved, even when submodule power generation ratios are changed.
5. Experimental Verification
A 750 VA three-phase five-level CHB inverter system has been developed in our laboratory to verify the performance of the proposed method as depicted in Figure 5. Two groups of experiments are implemented under voltage-imbalance conditions. A three-phase downscaled grid-connected converter is constructed. The corresponding experimental environment and configuration are shown in Figure 1. The parameters are the same as the simulation in Table 3, and the switching frequency is 8 kHz. It should be noted that the H-bridge unit is coupled with a lead-acid battery to provide power to the grid rather than a solar power unit, due to our limited experimental conditions. A DC-DC step-up converter, which is usually required for MPPT of solar PV systems, is omitted in the experiments due to the use of the battery. The proposed algorithms are implemented in TMS320F28335 with Xilinx FPGA boards.
5.1. Control Performance of Interphase Power Generation
Experiments with rA = 2/3, rB = 1, and rC = 4/3 were conducted to evaluate the dynamic performances of the proposed modulation strategy when interphase power generation ratios changed. In Figure 2(a), the maximum value of kA is 1.057 when m = 0.89. Hence, the reference power generation ratios can be set as follows: kA = 2/3, kB = 1, and kC = 4/3 in the first semiperiod and kA = kB = kC = 1 in the second semiperiod. Figure 6(a) gives the overall waveforms of power generation ratios kX, which reach their expected values precisely in the whole period. Besides, the detailed view shows that the transient process lasts about 20 ms. It requires one fundamental period to get a complete update of the average power. Figure 6(b) shows the line voltages uAB and three-phase currents. It is obvious that balanced AC currents can be achieved when power generation ratios change, and this change has little effect on the AC currents. Figure 6(c) shows grid phase voltage eX and inverter current iA. Figure 6(d) shows the waveforms of the submodule’s DC-side current and phase A voltage. From the voltage and current waveforms of each module, we can see that module 2 is always in the state of maximum power output, while module 1 is in the supplementary state. The DC side current has changed significantly before and after the interphase power switching, Thus, the effectiveness of the proposed power control is verified.
5.2. Control Performance of Submodule Power Generation
Experiments with submodule power control with rA = 2/3, rB = 1, and rC = 4/3 are conducted to evaluate the performance of the proposed modulation strategy when power generation ratios changed. Figure 7(a) provides the overall waveforms of power generation ratios kX, which remains constant during the entire period. Figure 7(b) shows the submodule power generation ratios. The submodule power is allocated with the submodule voltage when there is no submodule power control. As can be seen from Figure 7(b), the submodule power generation ratios reach the expected value over a short period of time. Grid phase voltage eX and inverter current iA are shown in Figure 7(c). Figure 7(d) shows line voltage uAB and three-phase currents in the switching process, which indicates that the submodule power control has little or no effect on the AC currents.
This paper has proposed a novel power control method for the CHB with unbalanced DC sources. First, the limitation of the modulation index was deeply analyzed to acquire the maximum linear modulation index. Then, the range of correction values is derived in detail, and the modulation index is extended by the maximum value with the ST-PWM strategy. Meanwhile, interphase and submodule power control algorithms have been proposed to maintain power generation ratios at the expected values. For simulation and experimental verification, the proposed control algorithm has been applied to a five-level CHB, which has achieved balanced grid-side currents and accurate power control. In order to acquire precise DC-power control, the switching frequency will be higher than traditional control strategies such as carrier phase-shifting control. So the efficiency of the system will be reduced slightly.
The simulation model and experimental setup are established and tested by our research group in the laboratory. We would like to share all our simulation and experiment data with others through e-mail [email protected]. Xia, the first author, can provide data details and necessary explanations to querists.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
This work was supported by the youth fund of the Foundation Research Project of Jiangsu Province (BK20160219).
S. Kouro, M. Malinowski, K. Gopakumar et al., “Recent advances and industrial applications of multilevel converters,” IEEE Transactions on Industrial Electronics, vol. 57, no. 8, pp. 2553–2580, 2010.View at: Publisher Site | Google Scholar
A. Marzoughi, R. Burgos, D. Boroyevich, and Y. Xue, “Investigation and Comparison of Cascaded H-Bridge and Modular Multilevel Converter Topologies for Medium-Voltage Drive Application,” in Proceedings of the Industrial Electronics Society, IECON 2014 - 40th Annual Conference of the IEEE, pp. 1562–1568, Dallas, TX, USA, November, 2014.View at: Publisher Site | Google Scholar
N. Salehi, H. Martínez-García, and G. Velasco-Quesada, “Modified cascaded Z-source high step-up boost converter,” Electronics, vol. 9, no. 11, p. 1932, 2020.View at: Publisher Site | Google Scholar
M. Malinowski, K. Gopakumar, J. Rodriguez, and M. A. Pérez, “A survey on cascaded multilevel inverters,” IEEE Transactions on Industrial Electronics, vol. 57, no. 7, pp. 2197–2206, Jul.2010.View at: Publisher Site | Google Scholar
J. Aguayo-Alquicira and I. Vásquez-Libreros, “Reconfiguration strategy for fault tolerance in a cascaded multilevel inverter using a Z-source converter,” Electronics, vol. 10, no. 5, p. 574, 2021.View at: Publisher Site | Google Scholar
Z. Zheng, K. Wang, L. Xu, and Y. Li, “A hybrid cascaded multilevel converter for battery energy management applied in electric vehicles,” IEEE Transactions on Power Electronics, vol. 29, no. 7, pp. 3537–3546, 2014.View at: Publisher Site | Google Scholar
L. Liu, H. Li, S.-H. Hwang, and J.-M. Kim, “An energy-efficient motor drive with autonomous power regenerative control system based on cascaded multilevel inverters and segmented energy storage,” IEEE Transactions on Industry Applications, vol. 49, no. 1, pp. 178–188, 2013.View at: Publisher Site | Google Scholar
G. Farivar, B. Hredzak, and V. G. Agelidis, “A DC-side sensorless cascaded H-bridge multilevel converter-based photovoltaic system,” IEEE Transactions on Industrial Electronics, vol. 63, no. 7, pp. 4233–4241, 2016.View at: Publisher Site | Google Scholar
B. Xiao, L. Hang, J. Mei, C. Riley, L. M. Tolbert, and B. Ozpineci, “Modular cascaded H-bridge multilevel PV inverter with distributed MPPT for grid-connected applications,” IEEE Transactions on Industry Applications, vol. 51, no. 2, pp. 1722–1731, 2015.View at: Publisher Site | Google Scholar
Y. Yu, G. Konstantinou, B. Hredzak, and V. Agelidis, “Optimal Zero Sequence Injection in Multilevel Cascaded H-Bridge Converter under Unbalanced Photovoltaic Power Generation,” in Proceedings of the International Power Electronics Conference, pp. 1458–1465, Hiroshima, Japan, May, 2014.View at: Publisher Site | Google Scholar
Y. Yu, G. Konstantinou, B. Hredzak, and V. G. Agelidis, “Power balance of cascaded H-bridge multilevel converters for large-scale photovoltaic integration,” IEEE Transactions on Power Electronics, vol. 31, no. 1, pp. 292–303, 2016.View at: Publisher Site | Google Scholar
L. Maharjan, S. Inoue, H. Akagi, and J. Asakura, “State-of-Charge (SOC)-Balancing control of a battery energy storage system based on a cascade PWM converter,” IEEE Transactions on Power Electronics, vol. 24, no. 6, pp. 1628–1636, 2009.View at: Publisher Site | Google Scholar
L. J. Kere, M. L. Doumbia, S. Kelouwani, and K. Agbossou, “Cascaded H-bridge multilevel converter for electric vehicle speed control,” in Proceedings of the Vehicle Power and Propulsion Conference (VPPC), pp. 1–6, Montreal, QC, Canada, Octobar, 2015.View at: Publisher Site | Google Scholar
K. Kandasamy, M. Vilathgamuwa, and K. J. Tseng, “Inter‐module state‐of‐charge balancing and fault‐tolerant operation of cascaded H‐bridge converter using multi‐dimensional modulation for electric vehicle application,” IET Power Electronics, vol. 8, no. 10, pp. 1912–1919, 2015.View at: Publisher Site | Google Scholar
L. M. Tolbert, J. N. Chiasson, K. J. McKenzie, and D. Zhong, “Control of cascaded multilevel converters with unequal voltage sources for HEVs,” in Proceedings of the Electric Machines and Drives Conference, 2003. IEMDC'03, pp. 663–669, IEEE International, Madison, WI, USA, June, 2003.View at: Publisher Site | Google Scholar
T. Zhao, G. Wang, S. Bhattacharya, and A. Q. Huang, “Voltage and power balance control for a cascaded H-bridge converter-based solid-state transformer,” IEEE Transactions on Power Electronics, vol. 28, no. 4, pp. 1523–1532, 2013.View at: Publisher Site | Google Scholar
Z. Ye, T. Wang, S. Mao et al., “A PWM strategy based on state transition for cascaded H-bridge inverter under unbalanced DC sources,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 8, no. 2, pp. 1686–1700, 2020.View at: Publisher Site | Google Scholar
Y. Cho, T. LaBella, J.-S. Lai, and M. K. Senesky, “A carrier-based neutral voltage modulation strategy for multilevel cascaded inverters under unbalanced DC sources,” IEEE Transactions on Industrial Electronics, vol. 61, no. 2, pp. 625–636, 2014.View at: Publisher Site | Google Scholar
Z. Ye, Y. Xu, X. Wu, G. Tan, X. Deng, and Z. Wang, “A simplified PWM strategy for a neutral-point-clamped (NPC) three-level converter with unbalanced DC links,” IEEE Transactions on Power Electronics, vol. 31, no. 4, pp. 3227–3238, 2016.View at: Publisher Site | Google Scholar