A Virtual Negative Resistor Based Common Mode Current Resonance Suppression Method for Three-Level Grid-Tied Inverter with Discontinuous PWM

: The output LC ﬁlter of a photovoltaic (PV) string three-level grid-tied inverter that connects the ﬁlter capacitor neutral point to dc-link capacitor neutral point can reduce the common-mode (CM) current injected to the grid by letting the CM current circulate within the inverter. However, the internal CM current may resonate because of the existence of the resonant frequency of the internal CM LC circuit. Compared with the traditional continuous pulse-width modulation (CPWM), the resonance can be worse if discontinuous pulse-width modulation (DPWM) is applied, for the zero sequence quantity of DPWM contains more harmonics than that of CPWM. In this paper, a virtual negative resistor based common mode current resonance suppression method for a three-level grid-tied inverter is proposed to overcome the CM current resonance problem in DPWM application. Di ﬀ erent positions of the virtual negative resistor in the equivalent CM circuit with di ﬀ erent feedback variables are analyzed theoretically. The virtual negative resistor connected in series with the inductor in the equivalent CM circuit is selected to damp the CM current resonance for simpliﬁcation and damping performance. Di ﬀ erent from the implementation in CPWM where a pair of small voltage vectors exist and are used to adjust the CM voltage directly, the proposed method for DPWM application is implemented indirectly by adding the CM adjustment quantity to di ﬀ erential-mode (DM) control quantity with appropriate coe ﬃ cients. Depending on the sector of DM control quantity in the αβ reference frame, the coe ﬃ cients are calculated using one of three speciﬁc voltage vectors. Experimental results are given to demonstrate the e ﬀ ectiveness of theoretical analyses and the proposed method.


Introduction
Among all renewable energy sources, photovoltaic (PV) systems have experienced rapid growth both in residential and commercial applications. The power quality of current generated by the PV grid-connected inverter is of importance for the grid. According to IEEE Std 519-2014, the recommended harmonic distortion limit of line-to-neutral voltages is that the voltage total harmonic distortion (VTHD) at the point of common coupling (PCC) is 8% for the bus voltage less than or equal to 1 kV, 5% for the bus voltage between 1 and 69 kV, 2.5% for the bus voltage between 69 and 161 kV, etc., [1]. Switching ripple filters (SRFs), like L, LC, LCL, and LLCL filter, is a significant part of PV grid-connected inverter interfacing with the grid. On one hand, SRFs can maintain a coupling connection and integration the topology can be found in [14]. Here, only the CM circuit model is given in Figure 2. Its transfer function Gp and bode diagram is given in Equation (3) and Figure 3, respectively. The CM resonance frequency in the bode diagram is given in Equation (4). The CM current resonance would be excited in the CM current path if no CM current resonance suppression technique is applied. Such resonance current would be more serious when DPWM is used as the modulation algorithm of PV string inverter, which is explained next.
1 2 where, iCM and vCM are the CM current and voltage, ia, ib, ic are three-phase inductor current, respectively. vAo, vBo, and vCo are three-phase bridge output voltage. Gp and fr are the model and the resonant frequency of the equivalent CM circuit, respectively. Lf and Cf are the inductance and capacitance of LC filter. S is a complex variable in S domain, and S=2 , f is the frequency of current and voltage. Energies 2020, 13, x FOR PEER REVIEW 3 of 16 the topology can be found in [14]. Here, only the CM circuit model is given in Figure 2. Its transfer function Gp and bode diagram is given in Equation (3) and Figure 3, respectively. The CM resonance frequency in the bode diagram is given in Equation (4). The CM current resonance would be excited in the CM current path if no CM current resonance suppression technique is applied. Such resonance current would be more serious when DPWM is used as the modulation algorithm of PV string inverter, which is explained next.
1 2 where, iCM and vCM are the CM current and voltage, ia, ib, ic are three-phase inductor current, respectively. vAo, vBo, and vCo are three-phase bridge output voltage. Gp and fr are the model and the resonant frequency of the equivalent CM circuit, respectively. Lf and Cf are the inductance and capacitance of LC filter. S is a complex variable in S domain, and S=2 , f is the frequency of current and voltage.

Harmonics Comparison of Zero Sequence Quantity of DPWM and CPWM
The symmetrical PWM [18,19] is used as CPWM in this paper. The zero sequence quantity of symmetrical PWM is given by Equation (5). A detailed description of DPWM used in this paper can be found in [16]. The zero sequence quantity of DPWM used in this paper is given by Equations (6) and (7) [20]. DPWM is widely used in the PV string inverter for its higher efficiency than that of CPWM.
where, i CM and v CM are the CM current and voltage, i a , i b , i c are three-phase inductor current, respectively. v Ao , v Bo , and v Co are three-phase bridge output voltage. G p and f r are the model and the resonant frequency of the equivalent CM circuit, respectively. L f and C f are the inductance and capacitance of LC filter. S is a complex variable in S domain, and S = 2π f i, f is the frequency of current and voltage.

Harmonics Comparison of Zero Sequence Quantity of DPWM and CPWM
The symmetrical PWM [18,19] is used as CPWM in this paper. The zero sequence quantity of symmetrical PWM is given by Equation (5). A detailed description of DPWM used in this paper can be found in [16]. The zero sequence quantity of DPWM used in this paper is given by Equations (6) and (7) [20]. DPWM is widely used in the PV string inverter for its higher efficiency than that of CPWM.
where V z is zero sequence quantity, V x is three-phase voltage, V x is the intermediate variable, x = a, b, c. V max is the voltage among V x which have maximum absolute value. (a mod b) delivers the remainder of the division a/b. The zero sequence quantity of CPWM and DPWM according to different modulation index and time are shown in Figure 4a,c, respectively. Using discrete Fourier transform, the harmonics comparison of zero sequence quantity between CPWM and DPWM are shown in Figure 4b,d respectively, which shows that the zero sequence quantity of DPWM contains more harmonics than that of CPWM. Hence, the zero sequence quantity of DPWM can lead to more serious CM current resonance than that of CPWM. As is shown in Figure 5, when the modulation method is changed from DPWM to CPWM, the resonance of CM current in i o is alleviated.

VNRBAD for CM Current Resonance
In order to suppress the CM current resonance, VNRBAD is proposed in this subsection. Using the same method as in [10], a general block diagram of the equivalent CM current power stage is proposed, as is shown in Figure 6. Although the same method as in [10] is used, it should be noticed that the analysis equivalent circuit are the major difference: model in [10] is for and only for DM circuit and DM current resonance damping, model in this paper focus on CM circuit and CM current resonance damping. Gad is a VNRBAD controller and is given by Equation (8). Gt is the transfer function from iCM to inductor current iL, capacitor current ic, or capacitor voltage vc, and is given by Equation (9). The virtual negative resistor has three positions in the modified equivalent CM circuit, and the feedback variable can be iL, ic or vc. The modified power stage and its corresponding Gad is given in Table 1 without considering the delay effects.

VNRBAD for CM Current Resonance
In order to suppress the CM current resonance, VNRBAD is proposed in this subsection. Using the same method as in [10], a general block diagram of the equivalent CM current power stage is proposed, as is shown in Figure 6. Although the same method as in [10] is used, it should be noticed that the analysis equivalent circuit are the major difference: model in [10] is for and only for DM circuit and DM current resonance damping, model in this paper focus on CM circuit and CM current resonance damping. Gad is a VNRBAD controller and is given by Equation (8). Gt is the transfer function from iCM to inductor current iL, capacitor current ic, or capacitor voltage vc, and is given by Equation (9). The virtual negative resistor has three positions in the modified equivalent CM circuit, and the feedback variable can be iL, ic or vc. The modified power stage and its corresponding Gad is given in Table 1 without considering the delay effects.

VNRBAD for CM Current Resonance
In order to suppress the CM current resonance, VNRBAD is proposed in this subsection. Using the same method as in [10], a general block diagram of the equivalent CM current power stage is proposed, as is shown in Figure 6. Although the same method as in [10] is used, it should be noticed that the analysis equivalent circuit are the major difference: model in [10] is for and only for DM circuit and DM current resonance damping, model in this paper focus on CM circuit and CM current resonance damping. G ad is a VNRBAD controller and is given by Equation (8). G t is the transfer function from i CM to inductor current i L , capacitor current i c , or capacitor voltage v c , and is given by Equation (9). The virtual negative resistor has three positions in the modified equivalent CM circuit, and the feedback variable can be i L , i c or v c . The modified power stage and its corresponding G ad is given in Table 1 without considering the delay effects.
where, G p is the power stage circuit transfer function, G pm is the modified power stage circuit transfer function, G t is the transfer function from i CM to i L , i c or v c .
Energies 2020, 13, x FOR PEER REVIEW 6 of 16 where, Gp is the power stage circuit transfer function, Gpm is the modified power stage circuit transfer function, Gt is the transfer function from iCM to iL, ic or vc.
Using the parameters in Table 2, the bode diagram comparison between the original equivalent power stage and the three modified equivalent power stages are given in Figures 7-9, respectively. From these bode diagrams, it can be seen that the virtual negative resistor connected in series with the inductor has the best AD performance. Once the connection type of virtual negative resistor is selected, Gad = Rv is selected from Table 1 for simple implementation purposes.
where, Gp is the power stage circuit transfer function, Gpm is the modified power stage circuit transfer function, Gt is the transfer function from iCM to iL, ic or vc.
Using the parameters in Table 2, the bode diagram comparison between the original equivalent power stage and the three modified equivalent power stages are given in Figures 7-9, respectively. From these bode diagrams, it can be seen that the virtual negative resistor connected in series with the inductor has the best AD performance. Once the connection type of virtual negative resistor is selected, Gad = Rv is selected from Table 1 for simple implementation purposes.
where, Gp is the power stage circuit transfer function, Gpm is the modified power stage circuit transfer function, Gt is the transfer function from iCM to iL, ic or vc.
Using the parameters in Table 2, the bode diagram comparison between the original equivalent power stage and the three modified equivalent power stages are given in Figures 7-9, respectively. From these bode diagrams, it can be seen that the virtual negative resistor connected in series with the inductor has the best AD performance. Once the connection type of virtual negative resistor is selected, Gad = Rv is selected from Table 1 for simple implementation purposes.
Energies 2020, 13, x FOR PEER REVIEW 6 of 16 where, Gp is the power stage circuit transfer function, Gpm is the modified power stage circuit transfer function, Gt is the transfer function from iCM to iL, ic or vc.
Using the parameters in Table 2, the bode diagram comparison between the original equivalent power stage and the three modified equivalent power stages are given in Figures 7-9, respectively. From these bode diagrams, it can be seen that the virtual negative resistor connected in series with the inductor has the best AD performance. Once the connection type of virtual negative resistor is selected, Gad = Rv is selected from Table 1 for simple implementation purposes.
Using the parameters in Table 2, the bode diagram comparison between the original equivalent power stage and the three modified equivalent power stages are given in Figures 7-9, respectively. From these bode diagrams, it can be seen that the virtual negative resistor connected in series with the inductor has the best AD performance. Once the connection type of virtual negative resistor is selected, G ad = R v is selected from Table 1 for simple implementation purposes.    The virtual negative resistor connected in series with the inductor and Gad=Rv is applied in this paper. The 1.5 times sample period Ts delay is considered, and is given in Equation (10) which composed of computational and PWM delays in the digital control [10]. Since only the harmonics at CM resonance frequency fr is the harmonics needed to damp and usually it is high-order harmonics, the high pass filter (HPF) is applied to cut off the low frequency harmonics in iCM, and is given in Equation (11). The cutoff frequency fc can be chosen as a frequency which lower than fr. In this paper, fr is higher than 2000 Hz, and 1000 Hz is used as fc. The CM current resonance AD control loop with   The virtual negative resistor connected in series with the inductor and Gad=Rv is applied in this paper. The 1.5 times sample period Ts delay is considered, and is given in Equation (10) which composed of computational and PWM delays in the digital control [10]. Since only the harmonics at CM resonance frequency fr is the harmonics needed to damp and usually it is high-order harmonics, the high pass filter (HPF) is applied to cut off the low frequency harmonics in iCM, and is given in Equation (11). The cutoff frequency fc can be chosen as a frequency which lower than fr. In this paper, fr is higher than 2000 Hz, and 1000 Hz is used as fc. The CM current resonance AD control loop with   The virtual negative resistor connected in series with the inductor and Gad=Rv is applied in this paper. The 1.5 times sample period Ts delay is considered, and is given in Equation (10) which composed of computational and PWM delays in the digital control [10]. Since only the harmonics at CM resonance frequency fr is the harmonics needed to damp and usually it is high-order harmonics, the high pass filter (HPF) is applied to cut off the low frequency harmonics in iCM, and is given in Equation (11). The cutoff frequency fc can be chosen as a frequency which lower than fr. In this paper, fr is higher than 2000 Hz, and 1000 Hz is used as fc. The CM current resonance AD control loop with delay and HPF considered is shown in Figure 10. Its bode diagram with different virtual negative The virtual negative resistor connected in series with the inductor and G ad = R v is applied in this paper. The 1.5 times sample period T s delay is considered, and is given in Equation (10) which composed of computational and PWM delays in the digital control [10]. Since only the harmonics at CM resonance frequency f r is the harmonics needed to damp and usually it is high-order harmonics, the high pass filter (HPF) is applied to cut off the low frequency harmonics in i CM , and is given in Equation (11). The cutoff frequency f c can be chosen as a frequency which lower than f r . In this paper, f r is higher than 2000 Hz, and 1000 Hz is used as f c . The CM current resonance AD control loop with delay and HPF considered is shown in Figure 10. Its bode diagram with different virtual negative resistor values is shown in Figure 11, which shows that the admittance at resonance frequency can be effectively damped with a proper R v .
where, T s is the sample period, f c is the cutoff frequency of the first order high pass filter.
Energies 2020, 13, x FOR PEER REVIEW 8 of 16 where, Ts is the sample period, fc is the cutoff frequency of the first order high pass filter.

The Implementation of VNRBAD for DPWM Application
In [14], the output of CM current controller dCM is added into three final modulation quantities in CPWM application, as is shown in Figure 12 and Equation (12). However, it cannot be used in DPWM application. For example, if phase A is clamped to positive bus voltage, ma is clamped to 1, and dz can be arrived in Equation (13). Substituting Equation (13) into Equation (12), the final three modulation quantity ma, mb, mc can be calculated in Equation (14). From Equation (14), it can be seen that dCM have no effects on ma, mb, and mc, which means the dCM cannot be added directly into three final modulation quantities in DPWM application.
where, ma, mb, mc are three final modulation quantities, respectively; dDMa, dDMb, dDMc are three-phase components of DM current controller outputs; dCM is the CM Rv outputs, dz is the zero sequence quantity of DPWM.
where, Ts is the sample period, fc is the cutoff frequency of the first order high pass filter.

The Implementation of VNRBAD for DPWM Application
In [14], the output of CM current controller dCM is added into three final modulation quantities in CPWM application, as is shown in Figure 12 and Equation (12). However, it cannot be used in DPWM application. For example, if phase A is clamped to positive bus voltage, ma is clamped to 1, and dz can be arrived in Equation (13). Substituting Equation (13) into Equation (12), the final three modulation quantity ma, mb, mc can be calculated in Equation (14). From Equation (14), it can be seen that dCM have no effects on ma, mb, and mc, which means the dCM cannot be added directly into three final modulation quantities in DPWM application.
where, ma, mb, mc are three final modulation quantities, respectively; dDMa, dDMb, dDMc are three-phase components of DM current controller outputs; dCM is the CM Rv outputs, dz is the zero sequence quantity of DPWM.

The Implementation of VNRBAD for DPWM Application
In [14], the output of CM current controller d CM is added into three final modulation quantities in CPWM application, as is shown in Figure 12 and Equation (12). However, it cannot be used in DPWM application. For example, if phase A is clamped to positive bus voltage, m a is clamped to 1, and d z can be arrived in Equation (13). Substituting Equation (13) into Equation (12), the final three modulation quantity m a , m b , m c can be calculated in Equation (14). From Equation (14), it can be seen that d CM have no effects on m a , m b , and m c , which means the d CM cannot be added directly into three final modulation quantities in DPWM application.
where, m a , m b , m c are three final modulation quantities, respectively; d DMa , d DMb , d DMc are three-phase components of DM current controller outputs; d CM is the CM R v outputs, d z is the zero sequence quantity of DPWM.
Energies 2020, 13, 1595 Energies 2020, 13, x FOR PEER REVIEW 9 of 16 Zero sequence calculation using (5)  The implementation of the proposed method for DPWM application is indirect. Figure 13 show the voltage space vectors in the αβ reference frame for a three-level inverter. Based on voltage vector synthesis principle, once a reference DM voltage vector Vref in sector I in the αβ reference frame is given as a constant voltage vector, the dwell times t1, t11 and t01 will be known constant quantities, as is shown in Equation (15) when using CPWM and in Equation (16) when using DPWM. Similarly, the CM voltage VCM_cpwm using CPWM and VCM_dpwm using DPWM is acquired in Equations (17) and (18). From Equation (17), it can be seen that VCM_cpwm can be controlled by adjusting the distribution factor k without needing to adjust dwell times t1, t11 and t01, which means that there is no coupling relationship between VCM_cpwm and Vref. In CPWM application, DM duty cycle which controls inductor DM current and CM duty cycle which controls inductor CM current are decoupled by a pair of small voltage vectors. In this example, the small voltage vectors are V01_POO and V01_ONN. (1 )  Figure 12. d CM implementation in CPWM application [14].
The implementation of the proposed method for DPWM application is indirect. Figure 13 show the voltage space vectors in the αβ reference frame for a three-level inverter. Based on voltage vector synthesis principle, once a reference DM voltage vector V ref in sector I in the αβ reference frame is given as a constant voltage vector, the dwell times t 1 , t 11 and t 01 will be known constant quantities, as is shown in Equation (15) when using CPWM and in Equation (16) when using DPWM. Similarly, the CM voltage V CM_cpwm using CPWM and V CM_dpwm using DPWM is acquired in Equations (17) and (18). From Equation (17), it can be seen that V CM_cpwm can be controlled by adjusting the distribution factor k without needing to adjust dwell times t 1 , t 11 and t 01 , which means that there is no coupling relationship between V CM_cpwm and V ref . In CPWM application, DM duty cycle which controls inductor DM current and CM duty cycle which controls inductor CM current are decoupled by a pair of small voltage vectors. In this example, the small voltage vectors are V 01_POO and V 01_ONN .
The implementation of the proposed method for DPWM application is indirect. Figure 13 show the voltage space vectors in the αβ reference frame for a three-level inverter. Based on voltage vector synthesis principle, once a reference DM voltage vector Vref in sector I in the αβ reference frame is given as a constant voltage vector, the dwell times t1, t11 and t01 will be known constant quantities, as is shown in Equation (15) when using CPWM and in Equation (16) when using DPWM. Similarly, the CM voltage VCM_cpwm using CPWM and VCM_dpwm using DPWM is acquired in Equations (17) and (18). From Equation (17), it can be seen that VCM_cpwm can be controlled by adjusting the distribution factor k without needing to adjust dwell times t1, t11 and t01, which means that there is no coupling relationship between VCM_cpwm and Vref. In CPWM application, DM duty cycle which controls inductor DM current and CM duty cycle which controls inductor CM current are decoupled by a pair of small voltage vectors. In this example, the small voltage vectors are V01_POO and V01_ONN. (1 ) Energies 2020, 13, 1595 10 of 16 where, V ref is the DM voltage vector in the αβ reference frame, V CM_cpwm and V CM_dpwm are the CM voltage when using CPWM and DPWM respectively. T s is the switching period, T s = t 1 + t 11 + t 01 . t 1 , t 11 , and t 01 are the dwell time of voltage vector V 1 , V 11 , and V 01 in Figure 13, respectively. V 01_ONN and V 01_POO are a pair of small vectors V 01 . k is the distribution factor for V 01_ONN and V 01_POO , and 0 < k < 1.
In DPWM application, however, there is no a pair of small voltage vectors. According to Equation (16), if V ref is a given vector, then t 1 , t 11 , and t 01 are known constant quantities. According to Equation (18), if t 1 , t 11 , and t 01 are known constant quantities, V CM_dpwm will be a constant quantity, which means that there is a coupling relationship between V CM_dpwm and V ref by dwell time t 1 and t 01 . The only way to control V CM_dpwm is to adjust t 1 and t 01 , which means that V CM_dpwm can only be controlled by adjusting V ref . V ref can be adjusted by compensating one of the three specific compensating vector V cpi (i = A,B,C). The three specific compensating voltage vectors are shown in Figure 14a. As is shown in Figure 14b, when phase A is clamped to positive bus voltage, the vector V cpA can be used to compensate V ref to increase the dwell time t 1 of voltage vector V 1 in Equation (18) [21], which can decrease V CM_dpwm . In Figure 14c where, Vref is the DM voltage vector in the αβ reference frame, VCM_cpwm and VCM_dpwm are the CM voltage when using CPWM and DPWM respectively. Ts is the switching period, Ts = t1 + t11 + t01. t1, t11, and t01 are the dwell time of voltage vector V1, V11, and V01 in Figure 13, respectively. V01_ONN and V01_POO are a pair of small vectors V01. k is the distribution factor for V01_ONN and V01_POO, and 0 < k < 1.
In DPWM application, however, there is no a pair of small voltage vectors. According to Equation (16), if Vref is a given vector, then t1, t11, and t01 are known constant quantities. According to Equation (18), if t1, t11, and t01 are known constant quantities, VCM_dpwm will be a constant quantity, which means that there is a coupling relationship between VCM_dpwm and Vref by dwell time t1 and t01. The only way to control VCM_dpwm is to adjust t1 and t01, which means that VCM_dpwm can only be controlled by adjusting Vref. Vref can be adjusted by compensating one of the three specific compensating vector Vcpi (i = A,B,C). The three specific compensating voltage vectors are shown in Figure 14a. As is shown in Figure 14b, when phase A is clamped to positive bus voltage, the vector VcpA can be used to compensate Vref to increase the dwell time t1 of voltage vector V1 in Equation (18) [21], which can decrease VCM_dpwm. In Figure 14c, Vref is clamped to bus neutral point O. By compensating Vref with VcpA, the dwell time of voltage vector V03 is decreased, which can decrease VCM_dpwm. Similarly, when phase A is clamped to negative bus voltage, the vector VcpA can be used to compensated Vref to decrease the dwell time of voltage vector V4, which can decrease VCM_dpwm. To sum up, when phase A is clamped to DC bus positive, negative, or neutral point voltage, the vector VcpA can be used to compensated Vref to decrease VCM_dpwm.
Similar analysis can be implemented when phase B,C is clamped to DC bus positive, negative, or neutral point voltage. It can be arrived that VCM_dpwm can be decreased by compensating Vref with Vcpi (i = A,B,C) when phase i (i = A,B,C) is clamped to bus positive, negative, or neutral point voltage. Vref compensation is implemented in an abc reference frame. As is given in Equation (19), the compensating voltage vector Vcpi (i = A,B,C) in αβ reference frame can be converted to three-phase components dCM_DMa, dCM_DMb, dCM_DMc in the abc reference frame using inverse Clarke transform seen in Equation (20). dCM is the output CM voltage of Rv and determine the amplitude and sign of Vcpi (i = A,B,C). Kfa, Kfb, Kfc are coefficients which determine the angle of Vcpi (i = A,B,C). As is given in Equation V ref compensation is implemented in an abc reference frame. As is given in Equation (19), the compensating voltage vector V cpi (i = A,B,C) in αβ reference frame can be converted to three-phase components d CM_DMa , d CM_DMb , d CM_DMc in the abc reference frame using inverse Clarke transform seen in Equation (20). d CM is the output CM voltage of R v and determine the amplitude and sign of V cpi (i = A,B,C). K fa , K fb , K fc are coefficients which determine the angle of V cpi (i = A,B,C). As is given in Equation (21), K fa , K fb , and K fc is a value chosen from 1, −0.5, and −0.5 for simplification purposes. The value 1 is for the phase which is clamped to bus positive, negative, or neutral point voltage. The value −0.5 is for the remaining two phases. The judgment of clamped phase is based on sector detection algorithm in [21] and DPWM method in [16]. The final three-phase modulation quantities are given in Equation (22). The implementation diagram of VNRBAD for DPWM application is given in Figure 15.  Figure 15.

Experiments and Discussions
In order to verify the effectiveness of the proposed VNRBAD method, experiments are implemented in a string inverter produced by TBEA Xi'an Electric Technology Co., Ltd. Grid

Experiments and Discussion
In order to verify the effectiveness of the proposed VNRBAD method, experiments are implemented in a string inverter produced by TBEA Xi'an Electric Technology Co., Ltd. Grid simulators are used as load and PV simulators are used as DC power supply in the experiment. The specification of the inverter and experiment conditions are given in Table 2. Note that the inductance value of L f is variable with current conducting through it. The value of 95 µH is a measurement value of L f at 80A. The model number of devices used in the experiment are shown in Table 3.

The Value of R v
As is shown in Figure 11, the smaller R v value it is, the better CM current resonance suppression performance it is. However, R v cannot be too small for better CM current resonance suppression, because it will degrade DM current injected to grid. Since the CM voltage is controlled by compensating DM voltage, the CM current resonance will be suppressed better when R v is smaller, while the DM current will be overcompensated and is degraded. As is shown in Figure 16, when R v is changed from −3 to −1.5, the CM current in i o is worse, while the inductor current i L and grid current i g are better.
Although the damp performance of i o is better when R v = −3, an apparent current distortion can be observed in i L and i g , which is because the amplitude of the compensated vector is too big. There must be a compromise between the damping performance of i o and THD of i g .
Energies 2020, 13, x FOR PEER REVIEW 13 of 16 because it will degrade DM current injected to grid. Since the CM voltage is controlled by compensating DM voltage, the CM current resonance will be suppressed better when Rv is smaller, while the DM current will be overcompensated and is degraded. As is shown in Figure 16, when Rv is changed from -3 to -1.5, the CM current in io is worse, while the inductor current iL and grid current ig are better. Although the damp performance of io is better when Rv=-3, an apparent current distortion can be observed in iL and ig, which is because the amplitude of the compensated vector is too big. There must be a compromise between the damping performance of io and THD of ig.
In practical application, there exist delays in the control loop which will degrade the CM current resonance suppression performance, so it is suggested that the value of Rv can be adjusted from -3 to 0 by hand. Figure 16. Waveform of io, iL, ig, and vc with Rv varied from -3 to -1.5.

Comparison Between VNRBAD and Without VNRBAD Under Different iL
Based on the analysis of Rv above, -2 is chosen as the value of Rv in the experiments. The waveform comparison of io, iL, ig, and vc between VNRBAD and without VNRBAD under different values of iL is given in Figures 17 and 18, respectively. Note that the CM current in iL resonates heavily under 80A iL when without VNRBAD, and the protection program of the control system is triggered to shut down the inverter. The proposed VNRBAD can suppress the CM current resonance effectively, although not completely. The harmonics of iL is shown in Figures 19 and 20, respectively. The harmonics at CM resonance frequency is reduced effectively. In practical application, there exist delays in the control loop which will degrade the CM current resonance suppression performance, so it is suggested that the value of R v can be adjusted from −3 to 0 by hand.

Comparison between VNRBAD and Without VNRBAD under Different i L
Based on the analysis of R v above, −2 is chosen as the value of R v in the experiments. The waveform comparison of i o , i L , i g , and v c between VNRBAD and without VNRBAD under different values of i L is given in Figures 17 and 18, respectively. Note that the CM current in i L resonates heavily under 80A i L when without VNRBAD, and the protection program of the control system is triggered to shut down the inverter. The proposed VNRBAD can suppress the CM current resonance effectively, although not completely. The harmonics of i L is shown in Figures 19 and 20, respectively. The harmonics at CM resonance frequency is reduced effectively. Figure 16. Waveform of io, iL, ig, and vc with Rv varied from -3 to -1.5.

Comparison Between VNRBAD and Without VNRBAD Under Different iL
Based on the analysis of Rv above, -2 is chosen as the value of Rv in the experiments. The waveform comparison of io, iL, ig, and vc between VNRBAD and without VNRBAD under different values of iL is given in Figures 17 and 18, respectively. Note that the CM current in iL resonates heavily under 80A iL when without VNRBAD, and the protection program of the control system is triggered to shut down the inverter. The proposed VNRBAD can suppress the CM current resonance effectively, although not completely. The harmonics of iL is shown in Figures 19 and 20, respectively. The harmonics at CM resonance frequency is reduced effectively.

Discussions
The PD method with a 0.61 ohm resistor connected in series with the inductor in the CM circuit is also implemented. The waveform of io, iL, and ig using PD method is shown in Figure 21. The RMS value comparison of io of three methods under different iL is shown in Figure 22. It can be seen that PD has a better CM current suppression performance compared with the proposed VNRBAD method.
Two reasons limit the CM resonance suppression performance of the proposed VNRBAD. One is that there is a compromise between the CM current control performance and DM current control performance. If Rv is too small, the CM resonance current damping performance is not good. On the contrary, too small values of Rv overcompensate DM voltage Vref in αβ reference frame and degrade the quality of DM current ig. Another reason is the delay in the control loop which can be studied further. Although the damping performance of the proposed VNRBAD method is not as good as PD, VNRBAD can suppress the CM current resonance within an acceptable range to let the inverter operate normally in DPWM application.

Discussion
The PD method with a 0.61 ohm resistor connected in series with the inductor in the CM circuit is also implemented. The waveform of i o , i L , and i g using PD method is shown in Figure 21. The RMS value comparison of i o of three methods under different i L is shown in Figure 22. It can be seen that PD has a better CM current suppression performance compared with the proposed VNRBAD method.

Conclusions
In this paper, VNRBAD is proposed to suppress the CM current resonance for a three-level gridtied inverter with DPWM. The harmonics comparison of zero sequence quantity between DPWM and CPWM is presented, which shows DPWM has more harmonics than that of CPWM. In DPWM application, the CM voltage and DM voltage is coupled. VNRBAD is implemented by compensating the DM voltage to control the CM voltage. Experimental results demonstrate the effectiveness of VNRBAD. Although VNRBAD is not as good as PD in the experiments, VNRBAD can suppress the CM current resonance within an acceptable range and does not need additional PD power resistor. Future work will focus on the reduction of the delay in the control loop which can enhance the CM current resonance suppression performance of the proposed VNRBAD method.
Author Contributions: S.Z. provided guidance and supervision. Q.L., H.Z., J.Z., and J.L. implemented the main research, performed the experiment, wrote the paper, and revised the manuscript as well. All authors have Two reasons limit the CM resonance suppression performance of the proposed VNRBAD. One is that there is a compromise between the CM current control performance and DM current control performance. If R v is too small, the CM resonance current damping performance is not good. On the contrary, too small values of R v overcompensate DM voltage V ref in αβ reference frame and degrade the quality of DM current i g . Another reason is the delay in the control loop which can be studied further. Although the damping performance of the proposed VNRBAD method is not as good as PD, VNRBAD can suppress the CM current resonance within an acceptable range to let the inverter operate normally in DPWM application.

Conclusions
In this paper, VNRBAD is proposed to suppress the CM current resonance for a three-level grid-tied inverter with DPWM. The harmonics comparison of zero sequence quantity between DPWM and CPWM is presented, which shows DPWM has more harmonics than that of CPWM. In DPWM application, the CM voltage and DM voltage is coupled. VNRBAD is implemented by compensating the DM voltage to control the CM voltage. Experimental results demonstrate the effectiveness of VNRBAD. Although VNRBAD is not as good as PD in the experiments, VNRBAD can suppress the CM current resonance within an acceptable range and does not need additional PD power resistor. Future work will focus on the reduction of the delay in the control loop which can enhance the CM current resonance suppression performance of the proposed VNRBAD method.
Author Contributions: S.Z. provided guidance and supervision. Q.L., H.Z., J.Z., and J.L. implemented the main research, performed the experiment, wrote the paper, and revised the manuscript as well. All authors have equally contributed to the theoretical analysis, experiment, and result discussions. All authors have read and agreed to the published version of the manuscript.