Exploring the LCL Characteristics in GaN Based Single ... - IEEE Xplore

This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TIE.2017.2694379, IEEE Transactions on Industrial Electronics IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

Exploring the LCL Characteristics in GaN Based Single-L Quasi-Z-Source Grid-tied Inverters Yanjun Shi, Member, IEEE, Yuan Li, Member, IEEE, Thierry Kayiranga, Student Member, IEEE and Hui Li, Senior Member, IEEE Abstract— As more wide band-gap (WBG) devices are becoming commercially available, it is beneficial to use WBG device to increase the switching frequency in order to reduce the passive components. For quasi-Z-Source (qZS) grid-tied inverters, the reduction of passive components raises stability concerns as the coupling effect between the DC side and AC side of qZS inverter will increase. In this paper, the coupling effect between qZS impedance network and the output filter is analyzed by modeling both DC and AC side. Analysis reveals the resonant characteristic of qZS inverter. Controller parameter boundaries are derived, and a design method to improve stability is then proposed. Case studies for a 2.5kW 10kHz Si-based qZS inverter and a 1kW 100kHz GaN-based qZS inverter are presented. Circuit simulations and experimental verifications results are provided to assess analysis and the control design.

methods treat the ZS/qZS inverters as two-stage inverters [6]. The output current controller is designed based on the output filter, which is usually a naturally stable single L filter. Previous studies on ZS/qZS inverters were mainly focused on the design of the impedance network without considering the AC side influence. However, it has been observed that ZS/qZS inverters have a more complicated output behavior compared to voltage source inverter (VSI) with same output filter [20]-[26]. Therefore, designing the AC current control loop of qZS inverter as a VSI is not sufficient and sometimes causes stability issues.

Index Terms—current control, grid-connected inverters, gallium nitride (GaN), quasi-Z-source inverters (qZSI), stability I. INTRODUCTION

W

ITH an impedance network coupling the inverter main circuit to the DC source, the Z-Source (ZS) /quasi-Z-Source (qZS) inverters can achieve buck/boost and inversion in one power stage without the need for extra switching devices [1]-[3]. Among all types of ZS/qZS inverters, the voltage-fed qZS inverter, shown in Fig. 1, has advantages in grid-tied application [4]-[7], due to its continuous input current, reduced capacitor voltage stress, and relatively smaller passive elements [8]-[11]. In grid-tied inverter applications, the dynamic interaction between the inverter and the power grid has been a topic of continuous extensive study [12]-[19]. However, this topic has not brought much attention for the ZS/qZS grid-tied inverters. This is partially because conventional controller design Manuscript received November 29, 2016; revised February 20, 2017; accepted March 23, 2017. This work was supported by the National Science Foundation under Award No. ECCS-1125658. Y. Li is with the Department of Electrical & Information Engineering, Sichuan University, Chengdu 610065, China (email: [email protected]) Y. Shi, T. Kayiranga and H. Li are with the Center for Advanced Power Systems, Florida State University, Tallahassee, FL, 32310, USA (email: [email protected], [email protected], [email protected] ) Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org.

Fig. 1 voltage-fed qZS inverter

The influence of impedance network on the output of a standalone ZS inverter has been discussed for uninterrupted power supply (UPS) in [20]. The authors concluded that ZS impedance introduce additional resonant frequencies compared to the conventional VSI. Authors in [21] use Posicast damping and three step damping to reduce the output oscillation during step transient of boost ratio for a current-fed ZS inverter. Similar voltage oscillation issue under input voltage disturbance is also reported in [22] for a bidirectional qZS inverter. It is pointed out in [22] that the bidirectional qZS network is less damped by nature and specific consideration should be taken when designing the voltage control loop. In [23], a fixed frequency operating sliding mode controller is proposed for qZS inverter with battery. It can be observed that the output voltage and load current exhibited distortion under low input voltage. Although the aforementioned studies revealed possible stability issues, the effect of the output current on impedance network is still simplified as a fixed load or uncontrolled current source. Consequently, the interaction between output current control loop and the impedance network remains unclear. Different forms of model predictive control (MPC) have also been applied to solve the stability issue of qZS inverter [26][28]. However, the high computation requirement of the MPC

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ωo L f iq

Lf

vgd

d d vC Lf

-

ωo L f id

+

A typical grid-tied voltage-fed qZS inverter consists a qZS impedance network, an inverter bridge, and an output filter, as shown in Fig. 1. By adding a shoot-through state to inverter modulation, a qZS inverter can achieve buck/boost function in a single power stage. Energy from DC source is stored in the qZS impedance network during the shoot-through state and then released during non-shoot-through states. A well-accepted modeling method for ZS/qZS inverters is to model the DC-stage and AC-stage separately. Similarly, in this research, the two stages are modeled separately first, then coupled through a virtual DC link. Therefore, the interaction between grid-side and DC-side can be modelled and analyzed. In order to create a virtual DC link, a virtual switching is inserted between the qZS network and the inverter bridge as shown in Fig. 2. The qZS inverter operation principle remains the same as long as this virtual switch performs the shootthrough and the inverter is bridge controlled under normal sinusoidal pulse width modulation (SPWM). From control and modeling perspectives, the virtual switch is used to simplify the analysis. More importantly, the virtual switch creates a DC-link that separate the qZS inverter into a DC-DC stage and a DC-

Fig. 2 qZS inverter with a virtual switch.

-

II. GRID-TIED QZS INVERTER OUTPUT CHARACTERISTICS MODELING

AC stage, with the two stages linked with a virtual DC link current idc. The introduction of a virtual DC link allows modeling the two stages separately, then coupled together. Taking the single-phase qZS inverter as an example, for the DC-AC stage, the inverter is first modelled in the dq frame as shown in Fig. 3. Then, an averaged model is derived, Fig. 4, where Lf is the equivalent output inductance in dq frame, dd and describes the dq are the duty ratios of inversion stage. relationship between output current and the equivalent DC-link capacitor voltage. If id is set as active current and iq is reactive current, then idc will only influenced by d-axis current. Here, idc is a controlled current source that contains the dynamic of the term is the coupling grid current and duty ratio. The coefficient between d-axis current and q-axis current [36]. The equivalent circuits of DC-DC stage is shown in Fig. 5. It should be noted that only two states are considered in the modeling of DC-DC stage. All the other switching states occurring during the non-shoot-through period are grouped together.

+

limits its application at higher switching frequency. More importantly, the mechanism of output current oscillation has not been explained clearly. A recent trend in the research of impedance source inverters is the application of wide bandgap (WBG) devices [10], [29][33]. As WBG devices become commercially available, it is beneficial to use these device to increase the switching frequency in order to reduce the impedance network [33]. However, the reduction of ZS/qZS network also increases the coupling effect between the impedance network and the output filter; thereby, raising concerns on stability especially in gridtied applications where the output current is regulated by utility standards such as IEEE 519. Since there is no energy buffer inside the inverter bridge, it is reasonable to infer that when qZS network parameters become smaller, the dynamic of qZS network will have a more significant influence on the output and cause stability issues. To make full utilization of increased switching frequency, these stability issues have to be investigated. The contribution of this research can be summarized follow. First, an LCL resonance is identified in the single-L qZS inverters for the first time. Second, a controller design method for qZSI is proposed to prevent the resonant and finally, the stability issue of WBG-based qZSI at higher switching frequency (> 100kHz) is investigated. Results show that, at higher switching frequency with reduced qZS network parameters, the stability of qZSI can be increased even when using conventional controller design. Therefore, this paper is to investigate the output characteristics of GaN based qZS grid-tied inverters as well as the dynamic interaction between the qZS network and the output filter. In section II, a virtual switch based modeling method is presented and the transfer function of the output current loop that includes qZS network dynamic is derived. Stability analysis of the output current loop is performed in section III based on the proposed model. A controller design method is then presented to increase the stability. Case study and simulation are presented in section IV. Experimental results on a 10 kHz 2.5 kW Si-based qZS inverter and a 100 kHz 1 kW GaN-based qZS inverter are provided in section V.

d qvC

vgq

Fig. 3 Single-phase qZS inverter with a virtual DC link and averaged output stage (dq frame).

v v

gd

d

d

ωLi o

f

ωLi o

f

d

_

1/ sL

d

+

i

d

f

+

q

v

G iv

c

c

i

dc

dc

d

v

q

v

q

+ +_

1/ sL

i

q

f

gq

Fig. 4 Averaged model of the qZS inverter AC side in dq frame.



Fig. 5 Equivalent circuits of qZS inverter DC stage: (a) shoot-through state and (b) non-shoot-through state.

The state space equations of the DC-DC stage can be written as follow. For the shoot-through state in Fig. 5(a),

x' = A1x + B1u

(1)

where

x = [iL1 iL 2 vC1 vC 2 ]T , u = [idc Vin ]T  0 0  0  1 0  0 L2 A1 =  1  0 − 0 C1  − 1 0 0  C2

1 L1   0  0  0 

,

 0 B1 =  0  0 0

1 L1  0  0 0

because d-axis is locked with phase A, q-axis current is reactive current, which has no influence on the capacitor voltage. Therefore, in Fig. 7(b) there is no coupling between the DC side and AC side. The q-axis block diagram in Fig. 7 (b) is the same as the one in a single-L VSI. The interaction between qZS network and output filter is reflected in the d-axis block diagram of Fig. 7 (a) i.e. the influence of on Lf. Fig. 7(a) can be further transformed into Fig. 8 where Gcid is the current loop controller. G1 and G2 are defined in (5): VC G1 = VC + I d Dd Gidc  1  G2 = L s − D 2GVC f d idc 

(5)

Fig. 9 (a) shows the open loop frequency response of a VSI and a qZS inverter. Both inverters have the same output filter value. The control parameters are designed with conventional linear control theory and a phase margin (PM) of 30° is chosen as the design target with kp=0.005, ki= 36.3. However, when applying the control parameters to qZS inverter, the PM is only 5°. This means the qZS inverter has a worse stability margin and is also poorly damped and therefore sensitive to sudden changes, as shown in Fig. 9 (b). In

and for non-shoot-through state in Fig. 5(b):

x' = A2 x + B2u

(2)

where 1    0   0 0 −L  0 1   1  0 0 −  0 0 L2  , B =  1 A2 =  1 − 2  0 0 0   C1  C1   1 0 1  − 0 0    C2 C2

1 L1  0  0   0 

Applying state space averaging,

x' = Ax + Bu , y = Cx

(3) + 1− , ds is the shoot-through duty where = ratio, and y = vC = ( vC1 + vC 2 ) / 2 , C = [ 0 0 1 1] . Equation (1)-(3) are similar to conventional qZS inverter state-space model [6]. The difference is that idc is used as input = instead of a load current. In this research, idc is defined as: × where dd is the d-axis duty ratio and id is d-axis grid current. By introducing perturbation in the averaged model, the transfer function from the DC-link current to the capacitor voltage is derived in (4), which represents dynamic influence from the inverter stage to the qZS network:

Fig. 6 Control diagram for three-phase qZS inverter PV system.

GivdcC ( s) =

−0.5 (1 − DS ) L1L2 ( C1 + C2 ) s 3 − 0.5 (1 − DS )( L1 + L2 ) s 2 2 L1L2C1C2 s 4 + ( DS − 1) ( L1C1 + L2C2 ) + DS2 ( L1C2 + L2C1 ) s 2 + ( 2 DS − 1)  

(4)

where DS is the steady state shoot-through duty ratio. has been shown in Fig.4. Based on the mathematic model, the output characteristics and the control methodology can be developed. III. OUTPUT STABILITY ANALYSIS AND CONTROL DESIGN METHOD

The overall control diagram of the three-phase qZS inverter is shown in Fig. 6. In Fig. 7, the diagram for d-axis and q-axis are shown in (a) and (b) respectively. It should be noted that,

Fig. 7 Control system small signal representation of the (a) d-axis and (b) q-axis.



~ vgd

G1 ~* id

-

G cid

+-

VC

+

+

+

Gid _ open = Gcid ⋅ G1 ⋅ G2 2 V LCs 2 − I d Dd (1 − DS ) Ls + VC (1 − 2DS ) = Gcid C 2 L f LCs3 + (1 − 2DS ) L f + Dd2 (1 − DS ) L  s  

G2 +

1/ sLf

~ id

+

By separating the resonant part from the other, (6) can be rewritten in (7).

Dd2 G ivdcC

Id DdGivdcC

(6)

 V  s 2 − 2δ s + γω r2 Gid _ open =  Gcid ⋅ C  ⋅ = G L ⋅ Gr  sL f  s 2 + ω r2 

Fig. 8 control block diagram of the direct axis.

(7)

where: ωr = Bode Diagram

Amplitude

Magnitude (dB) Phase (deg)

Step Response

1.5

qZSI

xx x VSI xx x x 50 xx x x xx xx x x 0 xxx x x x x x x x x x -120 x x x x x PM=30 -150 x x x x PM=5 x x x -180 x x x x x

x qZSI xx xx x VSI xx x x x x x 1 xx xx xx x x x x x x x x x x x x x x x x x x x x x x x x x xx x x x x x 0.5 x x x x x x x

0 0

Frequency (Hz)

0.004 0.006 Time (seconds)

(a)

(b)

2

4

10

10

0.002

0.008

0.01

Fig. 9 Comparison of qZS inverter and VSI. (a) open-loop frequency response of qZS inverter and VSI; and (b) closed-loop step response of qZS inverter and VSI.

6000

100μH

(seconds-1)

0 -2000 -4000

0

50

100 (seconds-1)

(a)

150

1000μ F

2000

1000μH

200

Magnitude (dB)

1000μH (seconds-1)

80 100μF

4000

1000μF 0

60 40 20

-2000

0

-4000

-20 0

-6000

0

100

200

300

400

500

(seconds-1)

(b)

Fig. 10 Pole/zero trajectories of the duty ratio-to-output transfer function (G1·G2) with qZS parametric variations. (a) inductor varies; and (b) capacitor varies.

, γ ,

α1 L f α1 L f + α 2 L

=

G L = G cid

,

Gr =

s 2 − 2δ s + γω r2 s 2 + ω r2

x L x x x r x x x x -20log(γ) x x x x x x x x x x x x x x x x x x x x x x x x

G G

x x x x x x x x x x x x x x x

PM = π+∠GL+∠Gr

-90

x x x

-180

x x

-270

x x x x x x x x

-360

0

2

10

4

10 Frequency (Hz)

10

Fig. 11 Open-loop frequency response of symmetrical qZS inverter divided into a single-L VSI + controller (GL) and a resonant stage (Gr).

Magnitude (dB)

80 60

x

GL Gr

40 20

-20log(γ)

0 -20 360 270

Phase (deg)

Fig. 10, the pole-zero trajectories of the duty ratio-to-output transfer function (G1G2) are presented. With L=L1=L2 varying from 100 μH to 1000 μH, the zeros and poles change vertically as in Fig. 10(a). With C=C1=C2 varying from 100μF to 1000μF, the poles change vertically and the zeros change diagonally, as shown in Fig. 10(b). Similar trends can also be observed in qZS inverter with asymmetrical parameters. The location of poles and zero in Fig. 10 also show that a single-L qZS inverter has similar resonant feature as an LCL filter based VSI. This similarity between qZS inverter and LCL-VSI implies that LCL-VSI damping methods can be adapted to the qZS inverter. Passive damping will not have digital control issues, but will cause undesirable power loss. Active damping methods will lead to a more complicated controller design and additional sensors when applied to qZS inverter. In this paper, the stability of qZS inverter is improved by defining a safety boundary for the control parameters. For symmetrical qZS inverter parameters (L1=L2=L, C1=C2=C), the open-loop transfer function of d-axis current (from reference to sample) is presented in (6).

VC sL f

, α1 = (1 − 2 DS )2 , α2 = Dd2 (1 − DS )

The transfer function in (7) the is divided into the transfer function of a single-L VSI and controller (GL) cascaded with a pure resonant stage (Gr), as shown in Fig. 11. It is interesting to find in (7) that the frequency response of a symmetrical qZS inverter is similar to that of a LCL-VSI where ωr is the resonant angular frequency. The only difference is that, the resonant frequency of the qZS inverter changes with the operating point. The safety boundaries of the controller parameters can be derived with the method shown in Fig. 11. With γ being the steady state value of Gr, the boundary of controller parameters under a given phase margin (PM) is defined in (8):

100μF

4000

-6000

I d D d (1 − D S ) 2VC C

6000

100μH

2000

δ =

L f LC

Phase (deg)

100

α1 L f + α 2 L

PM = π+∠GL+∠Gr

180 90 0 -90 -180 0 10

10

1

2

3

10 10 Frequency (Hz)

4

10

Fig. 12 Open-loop frequency response of asymmetrical qZS inverter divided into a single L VSI (GL) and a double resonant stage (Gr).



1   GL (ωx ) = γ  ∠GL (ωx ) + ∠Gr ( ωx ) + π = PM

(8)

Where ω is the cross over frequency of the open loop = 1+ transfer function in (7). For the PI controller, 1/ , if the integration time constant Ti is designed to be longer than the resonant period 2π/ωr , i.e., approximately 10 times, then (8) can be rewritten as:  k pVC 1 =   ωx L f γ  2δω π   2 x 2 = tan  − PM  γω ω − 2    r x T = 20π  i ω  x

(9)

Lf   2 δ 2 + γω r2 − 2δ   k p ≤ k p max_ 45 =  γ VC    20π Ti ≥ Ti min = 2 δ 2 + γω r2 − 2δ 

(10)

Given that in most cases 2δ is much smaller than !" , the constrain for Ti can be further simplified in (11), where #! is the resonant frequency. γ ki 1 = ≤ fr (11) Ti

10

The maximum proportional gain kpmax_45 in (10) gives the safety boundary for a PM of 45°. If kp exceeds this boundary, an increment in kp will cause a large decrease in phase margin resulting an output current loop that is very sensitive to step change in reference and disturbances. For applications where less overshoot and lower tracking speed is preferred, the value of kp for phase margin of 60° is given in (12): k p max_ 60 =

Lf  3δ 2 + γω r2 − 3δ   γ VC 

(12)

From another perspective, if kp is designed so that the zero crossing frequency of magnitude response is much smaller than the resonant frequency, then the lower boundary of kp can be found as in (13). Smaller than this boundary, any further decrement in kp will not increase stability, but will only decrease the bandwidth. k p min =

Lf 10γ VC

ωr

(13)

Equations (10) – (13) give the parameter boundary of the symmetrical qZS inverter. For asymmetrical qZS inverter, the same method can be adapted, as shown in Fig. 12. Considering that the asymmetrical qZS inverter response is a 6th-order system, as expressed in (14), numerical analysis is recommended.  1  b4 s 4 + b2 s 2 + b1s + b0 Gid _ open = Gcid ⋅ G1 ⋅ G2 = k p 1 +  5 3 2  sTi  a4 s + a2 s + a1s + a0 s

where: 2 a0 = (1 − 2 DS ) L f + Dd2 (1 − DS )( L1 + L2 ) 2 a1 = Dd (1 − DS ) L1 L2 ( C1 + C2 ) 2 a2 = (1 − DS ) ( L1C1 + L2C2 ) L f + DS2 ( L1C2 + L2C1 ) L f a4 = L f L1 L2C1C2 2 b0 = VC (1 − 2 DS )

IV. CASE STUDY AND SIMULATION RESULTS

A three-phase grid-tied qZS inverter is selected as the base case of this study. The case study is used to verify the theoretical analysis of qZS inverter output characteristics, to validate the proposed design method effectiveness over a wide range of qZS impedance values, and to study the qZS inverter stability at high frequency with smaller qZS impedance value. In this study, three cases are selected for comparison. Case 1 is set as the base case, with the parameters as listed in Table I. Table I BASE CASE PARAMETERS

For instance, the parameters safety boundaries when PM=45° are:

kp

b1 = − I d Dd (1 − DS )( L1 + L2 ) 2 b2 = VC (1 − DS ) ( L1C1 + L2C2 ) + VC DS2 ( L1C2 + L2C1 ) − I d Dd (1 − DS ) L1L2 ( C1 + C2 ) b4 = VC L f L1L2C1C2

(14)

Parameters

Values

Nominal power Switching frequency fs Input voltage Vin Equivalent dc-link voltage VC1+VC2 Grid voltage vg(l-l) Line frequency Impedance network L1=L2 Impedance network C1=C2 Output filter Lf Steady state shoot-through ratio Ds Steady state modulation ratio M Controller gain kp

2.5 kW 10 kHz 170 V 212 V 104 V rms 60 Hz 500 μH 400 μF 1 mH 0.1 0.8 0.06

Table II Key PARAMETERS FOR CASE STUDY

Case 1 Case 2 Case 3

fs

L 1, L 2

C1, C2

Lf

10 kHz 25 kHz 50 kHz

500 μH 200 μH 100 μH

400 μF 160 μF 80 μF

1 mH 0.42 mH 0.21 mH

The other two cases have same input/output values but different switching frequency. The passive elements are modified using (15) to keep the ripple ratio constant, as shown in Table II. In (15), $ and r& are ripple ratios for input current and capacitor voltage respectively. J0 is the zero order Bessel function used to calculate the maximum voltage harmonic near the switching frequency, and rig is the required ratio of the harmonic current to fundamental current.  π  2VC J 0  M   2  L f = 2 π ⋅ I g ⋅ rig % ⋅ f s   Vin DS (1 − DS )   L1 = L2 = 4 (1 − 2 D ) ⋅ I ⋅ r % ⋅ f S L1 i s  I L1DS C = C = 1 2  4VC 2 ⋅ rv % ⋅ f s

(15)

In each case, three converter configurations are studied. The first one is a VSI that has the same Lf value as that of qZS inverter. The value of kp_VSI for the VSI is designed with a crossover frequency of 1 kHz and a PM of 89°. It should be noted that kp will vary from case to case as the value of Lf is different. The second configuration is a qZS inverter with kp_VSI as the controller gain. The last configuration is a qZS inverter with kp=kpmax_60 using the proposed method in (12). The ki in all three configurations are given by (11). Fig. 13 shows the open-loop root locus and closed-loop step response of Case 1. According to (7), the VSI and qZS inverter share the same locus on the real axis. However, the qZS inverter has two extra locus near imaginary axis which depicts the resonant part. It is shown in Fig. 13 (a) that if qZS inverter employs kp_VSI as controller gain, the converter is nearly



unstable, while kpmax_60 can give the converter a better damping. The step response in Fig. 13 (b) also shows the qZS inverter with kpmax_60 has better damping and shorter settling time than those of conventional design. Simulation results are provided to verify the theoretical analysis and to compare the proposed method with the conventional strategy. It should be mentioned that the qZS inverter is in grid-tied mode and the current reference had a step change from 12A to 18A. Fig. 14 shows the simulation result of Case 1, the qZS inverter operating at 10kHz with conventional design, along with the closed-loop bode plot of the output current loop and the harmonic spectrum of d-axis current. Fig. 15 also shows the simulation result of Case 1. However, the controller was designed using the proposed method i.e. kp=kpmax_60.

18.

Fig. 16 Case 2 step response. (a) kp = kp_VSI (conventional design) and (b) kp = kpmax_60 (proposed design).

Fig. 13 Case 1, fsw=10 kHz. (a) root locus, and (b) step response.

Fig. 17 Case 3 step response. (a) kp = kp_VSI (conventional design) and (b) kp = kpmax_60 (proposed design).

Fig. 14 Case 1 step response, with kp = kp_VSI (conventional design).

In Fig. 18(a), given the same ripple ratio, it is shown how circuit parameters (L, C, Lf) and the corresponding resonant frequency defined in (7) change with switching frequency. Fig. 18 (b) gives the value of kp vs. switching frequency under different design methods, where kp_VSI is obtained using the conventional design and kp_max45, kp_max60, kp_min defined by (10), (12), and (13). It can be seen that the conventional design gives a much larger kp at low switching frequency. When the switching frequency is high enough, kp_VSI is within the safety boundaries. This explains why Case 2 and Case 3 designs were very stable. It can be observed that the range where kp_VSI > kp_max60 is when the switching is greater than 30 kHz or when the resonant frequency is close to 1 kHz. This result provides further simplification to the proposed design method as in (16) k p = k p _ qZSI =

Fig. 15 Case 1 step response, with kp = kpmax_60 (proposed design).

The resonant frequency in these simulations are consistent with the theoretical analysis in Fig. 11. It also should be noted that, in both simulations, there is no resonance in q-axis current, which is also consistent with the small signal model in Fig. 7. In Case 2, shown in Fig. 16, conventional design and the proposed design give nearly identical results. In Case 3, shown in Fig. 17, both designs have short settling time while conventional design gives smaller overshoot. To further investigate the stability analysis and the proposed controller design method, a parameter sweep was performed based on the theoretical analysis in section III, the results are shown in Fig.

ωr L f VC

(16)

It should be noted that in all case studies, the controller gains given by the proposed design method remained constant under different qZS parameters. This is because in this study the passive elements are assumed to vary linearly with switching frequency as given by (15). In practice, when other issues are considered, the controller gain may not be constant. However, the phase margin given by proposed design will remain constant. The phase margin vs. switching frequency under conventional design (kp_VSI) and proposed design (kp_qZSI) is given in Fig. 19. (16) can also be applied to asymmetrical qZS inverter. The above parameter sweep study shows that the proposed design method can provide a constant phase margin



over a wide range of qZS network parameters, while the conventional design has a poor damp effect in some parameter range. Table III PROTOTYPE 1 PARAMETERS

Parameters Nominal power Switching frequency fs Input voltage Vin Equivalent dc-link voltage VC1+VC2 Grid voltage vg(l-l) Line frequency Impedance network L1=L2 Impedance network C1=C2 Output filter Lf Steady state shoot-through ratio Ds Steady state modulation ratio M Controller gain kp

Values 2.5 kW 10 kHz 170 V 212 V 104 VRMS 60 Hz 500 μH 400 μF 1 mH 0.1 0.8 0.06

Table IV PROTOTYPE 2 PARAMETERS

Parameters Nominal power Switching frequency fs Input voltage Vin Equivalent dc-link voltage VC1+VC2 Grid voltage vg Line frequency Impedance network L1 Impedance network L2 Impedance network C1 Impedance network C2 Output filter Lf Steady state shoot-through ratio Ds Steady state modulation ratio M Controller gain kp

Fig. 18 Parameter sweep. (a) qZS parameters and resonant frequency vs. switching frequency, and (b) kp vs. switching frequency.

Values 1 kW 100 kHz 150 V 214 V 120 V rms 60 Hz 330 μH 215 μH 2200 μF 20 μF 0.6 mH 0.15 0.8 0.04

Fig. 20 - 21 show experimental results of prototype I. In Fig. 20, where kp = kp_VSI, the qZS inverter system is nearly unstable. It can be observed that the grid current (ia, ib, ic) waveforms have distortions and the shape of the waveforms are different in each line period. It should be noted that when kp was increased, current protection was triggered. Fig. 21 shows the result when integration gain, ki, is larger than the maximum integration gain given by (11). It can be seen that low frequency oscillation can be observed in Fig. 21.

vgab 250V/div ia ib ic 16.7A/div Fig. 19 Parameter sweep. phase margins vs. switching frequency under conventional design (kp_VSI) and proposed design (kp_qZSI). V. EXPERIMENTAL RESULTS

Two sets of experiments were performed on two laboratory prototypes. The first one is performed on a three-phase qZS inverter switching at 10 kHz with the parameters listed in Table III. The second is performed on a GaN-based single phase asymmetrical qZS inverter switching at 100 kHz with the parameters listed in Table IV. These two prototypes are chosen to verify the inherent resonance phenomenon under high frequency and low frequency. Both prototypes have single-L output filter. The controller gains were increased to push qZS inverter over the stability boundary in order to verify the resonance in single-L qZS inverters and the controller gain designed using the theoretical analysis.

Vin 100V/div Vdc 100V/div

Fig. 20 Experimental result of a three-phase 10 kHz grid-tied qZS inverter, with kp=kp_VSI, ki=0.



Vin 100V/div

harmonics frequencies match the resonant frequencies. Compared to the case study for three phase qZS inverter shown in Fig. 14, results in Fig. 23 are not a perfect match. This is because in the 100 kHz single-phase prototype, in addition to the conventional shoot-through and non-shoot-through states, an unwanted state namely LCC is triggered, as shown in Fig. 24. The existence of this uncontrollable state brings complications to the qZS inverter modeling. For three- phase qZS inverter topology, the LCC state can be avoided since there is no double frequency component propagating throughout the system. However, for single-phase qZS inverter with reduced impedance network parameters, i.e. prototype II, it is difficult to avoid LCC state. The issues and analysis on LCC state are presented in details in [40].

vgab 100V/div

ia ib ic 10A/div

Fig. 21 Experimental result of a three-phase 10 kHz grid-tied qZS inverter, with kp=0.3*kp_VSI, ki=20.

Fig. 23 shows the experiment results of prototype II where kp =0.04, which is 3.65 times kp_qZSI. From the root locus in (b), it can be seen that the kp chosen in the experiment makes the system marginally stable. It is not easy to observe the distortion in grid current waveforms as the output filter is relatively large. However, it can be seen from Fig. 23(a) that there are highfrequency ripples on DC side capacitor voltage VC2 indicating that the system has poor stability. As the resonant frequency is defined in dq frame, performing the Fast Fourier Transform (FFT) of the grid current will give ±60Hz mismatch in resonant frequency. 4

1.5

x 10

1

Fig. 24 Experimental demonstration of the unwanted LCC state in the 100 kHz signal phase qZS inverter prototype. Ch1: DC-link voltage Vdc; Ch3: input voltage Vin; Ch4: output voltage vO.

0.5 0 -0.5 -1 -1.5 -2000

-1500

-1000

-500

0

500

Magnitude (dB)

60

VI. CONCLUSION

40 20 0

Phase (deg)

-20 0 -180 -360 -540 -720 10

1

2

10

10 Frequency (Hz)

3

10

4

Fig. 22 A single-phase GaN based 100 kHz grid-tied qZS inverter with kp = 8.8. (a) Experiment waveforms: blue: VC2, red: ig, green: vg; (b) root locus; (c) open loop bode plot of d-axis current.

(d)

Fig. 23 FFT of the experiment result of VC2 for a single-phase GaN based 100 kHz grid-tied qZS inverter.

This paper investigates the output characteristics of grid-tied qZS inverters considering of the coupling effect between qZS network and output filter. By using a virtual switch, the output current transfer function that contains the dynamics of qZS network is derived. A resonant frequency of the qZS inverter is defined and controller parameter boundaries and design methods to improve stability are then presented. Through case studies, circuit simulation and experiment results, the theoretical analysis is verified and a simplified controller design method is found. A single-L qZS inverter behaves like an LCL based VSI at the output terminal. The qZS impedance network will resonate with output filter of qZS inverter, and cause potential stability issues. To guarantee enough stability margin under all circumstances, the influence of the qZS impedance network should be considered. For WBG-based qZS inverter with increased switching frequency and reduced passive elements, the stability of the grid current loop is improved. This is because the ratio of resonant frequency to switching frequency (fr/fs) is rather low in qZS inverters, 1/30 in this paper. When switching frequency increases, the resonant frequency is further away from the controller crossover frequency and making the system more stable. This effect also indicates that, considering LCL grid-tied inverters can have up to 0.5 fr/fs ratio, the passive elements in qZS inverters can be further decreased.

Thereby FFT of VC2 is presented in Fig. 23 to compare if the



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Yanjun Shi (S’11 - M’ 13) received the B.S. degree in electrical engineering and the Ph.D. degree in power electronics from Huazhong University of Science and Technology, Wuhan, China, in 2007 and 2012, respectively. He is currently a Research Faculty at the Center for Advanced Power System, Florida State University, Tallahassee, FL, USA. His research interests include grid-connected PV system, high power density PV inverter, high-penetration PV integration, Wide-Band-Gap device application, modeling, and control of power electronics converters. Yuan Li (M’10) received the B.S., M. S. and Ph. D. degrees in electrical engineering from Wuhan University, Wuhan, China, in 2003, 2006 and 2009, respectively. She is an Associate Professor at Department of Electrical & Information Engineering, Sichuan University, Chengdu, China. She was a visiting scholar at Department of Electrical & Computer Engineering, Michigan State University, East Lansing, MI, and a visiting professor at Department of Electrical and Engineering, Northeastern University, Boston, MA, from 2007 to 2009 and 2015 to 2016, respectively. She is an Associate Editor of IEEE Transactions on Power Electronics from 2015. Her research area includes Z-source inverter, photovoltaic inverters, PWM rectifier, DC microgrid, power quality analysis and control in active distribution network, etc.



Thierry Kayiranga (S’12) received the B.S. degree in electrical engineering from Florida State University, Tallahassee, FL, USA, in 2013. He is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, College of Engineering, Florida State University, Tallahassee, FL, USA. His research interests include bidirectional dc– dc converters, wide-bandgap devices applications in renewable energy, and cascaded multilevel inverters. Hui Li (S’97 - IM’00 - SM’01) received the B.S. and M.S. degrees from Huazhong University of Science and Technology, Wuhan, China, in 1992 and 1995, respectively and the Ph.D. degree from the University of Tennessee, Knoxville, TN, USA, in 2000, all in electrical engineering. She is currently a Professor with the Department of Electrical and Computer Engineering, College of Engineering, Florida State University, Tallahassee, FL, USA. Her research interests include photovoltaic converters applying a wide-bandgap device, bidirectional dc-dc converters, cascaded multilevel inverters, and power electronics applications in hybrid electric vehicles.
