Small Signal Modeling and Controller Design of a ... · functions of the dc-dc converter and VSI. It can realize the boost ... the voltage of a Quasi-Z-Source capacitor is controlled

Small Signal Modeling and Controller Design of A Bidirectional Quasi-Z-Source Inverter for Electric

Vehicle Applications Feng Guo, Lixing Fu, Chien-Hui Lin, Cong Li, and Jin Wang

Department of Electrical and Computer Engineering The Ohio State University Columbus, Ohio, U.S.A

Abstract— This paper presents the small signal modeling and controller design of a bidirectional Quasi-Z-Source inverter (BQ-ZSI) for electric vehicle (EV) applications. The derived small signal model shows that the Quasi-Z-Source network (QZSN) is prone to oscillate when there is a disturbance from the dc input voltage. Since the battery pack voltage in an EV is easy to fluctuate because of the battery’s internal impedance and rapid change of the load current, a dedicated voltage controller with feed forward compensation is designed to reject the disturbance and stabilize the dc-link voltage during non-shoot-through state. Simulation and experimental results are both presented to prove the effectiveness of the proposed controller.

I. INTRODUCTION The global push for greenhouse gas emission reduction

and higher fuel economy standards have made the development of electric vehicles (EVs) more urgent than ever. In an EV, the traction drive system is one of the key subsystems. Currently, the topology of a Voltage Source Inverter (VSI) cascaded with a dc-dc converter is widely used. Although the dc-dc converter increases the overall efficiency of the traction drive system by properly regulating the dc-link voltage, it brings additional cost. In 2001, the Z-Source Inverter (ZSI) [1] was proposed to combine the functions of the dc-dc converter and VSI. It can realize the boost function and dc-ac conversion in one active stage, which reduces the total cost and further improves the efficiency of the traction drive system [2]. However, the input current of the ZSI is not continuous, which will shorten the lifetime of the battery pack and degrade vehicle performance. By rearranging the components in the Z-Source network, a new topology called the Quasi-Z-Source inverter (QZSI) is proposed in [3]. The QZSI realizes the continuous input current, at the same time retaining all the merits of the ZSI. With an additional active switch, the bidirectional power flow can be achieved [4]. The characteristics of the bidirectional Quasi-Z-Source inverter (BQ-ZSI) make it a good candidate for EV applications [4].

On the control side, Li et al. [5] presented a control strategy for QZSI for Photovoltaic (PV) applications, where

the voltage of a Quasi-Z-Source capacitor is controlled to be constant. However, for the motor control in vehicle applications, it is better to keep a constant dc-link voltage, which can enable the decoupling between the dc side control and ac side control. In the BQ-ZSI, the dc-link voltage during non-shoot-through state (Unless otherwise stated, the dc-link voltage refers to the dc-link voltage during non-shoot-through state) is two times of the capacitor voltage minus the dc input voltage, thus a constant capacitor voltage cannot guarantee a constant dc-link voltage.

To accurately control the dc-link voltage, two issues need to be considered. First, the dc-link voltage in the BQ-ZSI is a pulsating voltage when there is shoot-through state, thus it cannot be used as a feedback. Second, the small signal analysis of the BQ-ZSI shows that, different from the ZSI [6], the BQ-ZSI is more vulnerable to the disturbance from the dc input source, which is more apparent at higher power ratings, when the ratio between the resistance and inductance in the system becomes very small. Liu et al. [7] presented a multi-loop controller that utilizes the capacitor voltage and shoot-though duty ratio to estimate the dc-link voltage in the QZSI, which is similar to the control algorithm of ZSI presented by Gajanayake et al. [8]. However, the disturbance rejection from the input voltage is not considered.

Differently from the above approaches, this paper proposes an improved dc side controller with feed forward compensation of the dc input voltage to solve the aforementioned issues. It estimates the dc-link voltage using the capacitor voltage and input voltage, and regulates the dc-link voltage to the reference value when the input battery pack voltage has rapid change; at the same time suppresses oscillation in the system. Since the dc-link voltage is regulated by the dc side controller, existing motor control algorithms, such as Field Oriented Control (FOC) or V/Hz control, can be directly implemented in the ac side controller without major modification, which will simplify the development of the control system, at the same time achieve a precise control of the electric motor.

978-1-4673-0803-8/12/$31.00 ©2012 IEEE 2223

This paper is divided into five sections. Section II presents the small signal analysis of the BQ-ZSI. In Section III, the improved dc side controller is proposed. Section IV presents the simulation and experimental results. Section V concludes the paper.

II. SMALL SIGNAL MODEL OF BQ-ZSI The topology of the BQ-ZSI is shown in Figure 1. It has

two functional parts: Quasi-Z-Source network (QZSN) and VSI. There are three operation states: active state, zero state and shoot-through state [1]. In the shoot-through state, the BQ-ZSI allows the switches in the upper arm and lower arm of VSI conduct at the same time. This state is used to boost the voltage.

Figure 1. The Bidirectional Quasi-Z-Source Inverter.

By replacing the VSI and ac load with a switch S2 in parallel with load RL and LL, as shown in Figure 2, the dc equivalent circuit of BQ-ZSI can be obtained [9]. The shoot-through state can be represented by opening S1 and closing S2, while the active state can be represented by closing S1 and opening S2. Typically, when voltage boost is required, the shoot-through states are realized at zero states. Thus, non-shoot-through zero states are not considered in the following analysis.

Figure 2. DC equivalent circuit of the BQ-ZSI.

Since S1 allows bi-directional current, it can be assumed that the system is operating in continuous conduction mode (CCM). The capacitor voltages, inductor currents, and load current are defined as state variables, which can be written as a vector

1 2 1 2( ) [ ( ) ( ) ( ) ( ) ( )]TL L C C lx t i t i t v t v t i t= . (1)

When S1 is open and S2 is closed, the circuit is in the shoot-through state, the differential equations that describe the system can be written as

11

22

1 1

2 2

1

2

1

2

( )0 0 0 0 0 0 0 1 0( )0 0 0 0 0 0 1 0 0

0 0 0 0 ( ) 0 1 0 0 00 0 0 0 1 0 0 0 0( )0 0 0 0 0 0 0 0( )

( ) 1( ) 0( ) 0

0( )0( )

L

L

C

C

l ll

L

L

C

C

l

i tLi tL

dC v tdt

C v tL Ri t

i ti tv tv ti t

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⋅ +⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⎣⎣ ⎦

( )inv t

⎤⎥⎥⎥⎥⎥⎥⎦

, (2)

which can also be written in the state space form as 1 1( ) ( ) ( )Kx t A x t B u t= + .

Similarly, when S1 is closed and S2 is open, the circuit is in the non-shoot-through state, the differential equations that describe the system can be written as

11

22

1 1

2 2

1

2

1

2

( )0 0 0 0 0 0 1 0 0( )0 0 0 0 0 0 0 1 0

0 0 0 0 ( ) 1 0 0 0 10 0 0 0 0 1 0 0 1( )0 0 0 0 0 0 1 1( )

( ) 1( ) 0( ) 0

0( )0( )

L

L

C

C

l ll

L

L

C

C

l

i tLi tL

dC v tdt

C v tL Ri t

i ti tv tv ti t

−⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥= −⎢ ⎥⎢ ⎥ ⎢ ⎥−⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎣ ⎦

⎡ ⎤ ⎡⎢ ⎥ ⎢⎢ ⎥ ⎢⎢ ⎥ ⎢⋅ +⎢ ⎥ ⎢⎢ ⎥⎢ ⎥ ⎣⎣ ⎦

( )inv t

⎤⎥⎥⎥⎥

⎢ ⎥⎢ ⎥⎦

, (3)

which can also be written in the state space form as 2 2( ) ( ) ( )Kx t A x t B u t= + .

Assume the duty ratio of switch S2 is D, and the duty ratio of switch S1 is D'. In CCM, 1D D′+ = . Provided that the switching frequency is fast enough, the system can be linearized around the operation point and the dc steady state equations can be written as

1 2 1 20 ( ) ( )DA D A X DB D B U′ ′= + + + , (4)

where 1 2 1 2[ ], L L C C l inX I I V V I U V= = are steady state values.

By solving (4), the dc steady state value of each state variable can be obtained as

1 2, , C in C inD DV V V V

D D D D′

= =′ ′− −

2224

21 2 ( ) , in in

L L ll l

D V D VI I ID D R D D R

′ ′= = =

′ ′− −, (5)

which are exactly the same as the equations derived in [3].

To derive the small signal model, perturbations ˆ ( )inv t

and ˆ( )d t are added to the input voltage and duty ratio [10]. The perturbations result in variations in the state variables and the small signal state space equations can be obtained as

1 2 1 2 1 2 1 2ˆ ˆˆ ˆ( ) ( ) [( ) ( ) ] ( )KX DA D A X DB D B U A A X B B U d t′ ′= + + + + − + −

.( 6)

After Laplace transformation, (6) becomes

1 1 1 2 1 2

2 2 1 2 1 2

1 1 1 2 1 2

2 2 1 2

ˆˆ ˆ ˆ ˆ( ) ( ) ( ) ( ) ( ) ( )ˆˆ ˆ ˆ( ) ( ) ( ) ( ) ( )

ˆˆ ˆ ˆˆ ( ) ( ) ( ) ( ) ( ) ( )ˆ ˆˆ ( ) - ( )+ ( )

L C C in C C

L C C C C

C L L l l L L

C L L

sL i s D v s D v s v s V V d s

sL i s D v s D v s V V d s

sC v s D i s D i s D i s I I I d s

sC v s D i s D i s

′⋅ = − ⋅ + ⋅ + + + ⋅

′⋅ = ⋅ − ⋅ + + ⋅

′ ′⋅ = ⋅ − ⋅ − ⋅ + − − ⋅

′⋅ = ⋅ ⋅ − 1 2

1 2 1 2

ˆˆ ( ) ( ) ( )ˆˆ ˆˆ ˆ( ) ( ) ( ) ( ) ( ) ( )

l l L L

l l C C l l C C

D i s I I I d s

sL i s D v s D v s R i s V V d s

⎧⎪⎪⎪⎪⎨⎪ ′ ⋅ + − − ⋅⎪⎪ ′ ′⋅ = ⋅ + ⋅ − ⋅ − + ⋅⎪⎩

.(7)

Equation group (7) forms the small signal model of the BQ-ZSI.

To validate the derived small signal model, analysis results from the model are compared with circuit simulation results. Assuming the two QZSN inductors have the same inductance and the two capacitors have the same capacitance, circuit parameters shown in TABLE I are used in the study.

TABLE I. SIMULATION PARAMETERS

Input Voltage 356 V Load resistance 2 Ω

Inductance in QZSN 50 µH Load inductance 24 µH

Capacitance in QZSN 660 µF Switching frequency 10 kHz

Analysis and simulation results from both small signal model and detailed circuit model are shown in the same plot in Figure 3. A 20 V disturbance from input voltage is shown in Figure 3(a). The response of the voltage of C1 is shown in Figure 3(b), while the response of the current of L1 is shown in Figure 3(c). It can be noted that the values from the small signal model based analysis results are in line with the detailed circuit simulation results. Therefore, the small signal model is validated. The high frequency jittering in the detailed circuit model is at switching frequency.

III. CONTROLLER DESIGN The task of the BQ-ZSI controller is to boost the dc-link

voltage and drive the ac motor. Since the dynamics of the QZSN is much slower than the ac drive, a decoupled control

Figure 3. Comparison between the small signal model and detailed circuit model.

algorithm between the dc side and ac side of the BQ-ZSI can be realized. The dc side controller is used to regulate the dc-link voltage. In the ac side, existing motor control algorithms can be directly implemented without major modification. The next part will focus on the design of the dc side controller.

The response of the state variable can be expressed as a linear combination of each perturbation. In particular, the transfer function of the voltage of capacitor C1 can be derived from (7) as

1 2 2ˆˆ ˆ( ) ( ) ( ) ( ) ( )C v d v in inv s G s d s G s v s= ⋅ + ⋅ , (8)

where

22

1 2 1 2 1 23 2 2 2 2

( )=

( ) [ ( )+ ( )][ ( ) 2 ] ( )

v d

l L L l l in C C l l L L in l

l l l l

G s

I I I L L s LV D L V V R L I I I s V RL LC s R LC s L D D D L s D D R

′− − ⋅ + + + − − ⋅ +′ ′ ′⋅ + ⋅ + − + ⋅ + −

(9)

is the control-to-output transfer function, and

23 2 2 2 2

2 3 2 2 2 2

( )=

[( ) ] ( )( 1) [ ( ) 2 ] ( )

v in

l l l l

l l l l

G s

D L LC s D R LC s D D D L D L s D D D RLC s L LC s R LC s L D D D L s D D R

′ ′ ′ ′ ′ ′ ′⋅ + ⋅ + − + ⋅ + −′ ′ ′⋅ + ⋅ ⋅ + ⋅ + − + ⋅ + −

(10)

is the input-to-output transfer function.

Using the parameters in TABLE I, the pole-zero maps of transfer function Gv2d(s) and Gv2in(s) are plotted in Figure 4.

From Figure 4(a), it can be noticed that there is a right-half-plane (RHP) zero in the transfer function Gv2d(s), which makes the system a non-minimum-phase system and limits the system dynamics response. However, since the dc side controller is usually designed to be much slower than the ac side to avoid oscillation, this issue is not severe in EV applications. From Figure 4(b), it can be noticed that two

0.095 0.1 0.105 0.11350

360

370

380

0.095 0.1 0.105 0.11380

400

420

440

460

0.095 0.1 0.105 0.11100

200

300

400

Small Signal ModelDetailed Circuit Model

Small Signal ModelDetailed Circuit Model

2225

complex poles are located on the imaginary axis of Gv2in(s), which means that the system does not have enough damping to suppress the disturbance from the input voltage and oscillation will happen, as shown in Figure 3(b) and (c). This characteristic is unique in the BQ-ZSI topology and does not exist in the ZSI topology [6]. Since in EV applications, the battery pack usually has impedance more than several tenth of one ohm, the battery voltage can fluctuate a lot with fast dynamics of the load current. This issue needs to be properly addressed in the controller design.

(a) Gv2d(s) (b) Gv2in(s) Figure 4. Pole-Zero maps of the transfer functions.

Thus, a dedicated closed loop voltage controller is proposed to realize the dc-link voltage control. Instead of using the dc-link voltage, which is a pulsating signal, the capacitor voltage vC1 is fed back to the controller. Based on (5), it is easy to derive that

1 ( ) / 2C in dcv v v= + . (11)

Therefore, since the input battery voltage is also fed back to the controller, which is necessary for the safe operation of the system, the reference value of vC1 can be calculated based on the desired dc-link voltage and the input voltage of the system. By properly adjusting this reference value according to (11), the dc-link voltage can be accurately controlled. Moreover, this approach creates a feed forward loop in the controller, so the performance of the input voltage disturbance rejection can be greatly improved. The simplified control diagram is shown in Figure 5.

Figure 5. Simplified control diagram for QZSN.

A simple Proportional-Integral (PI) controller is used in the control loop. When designing the controller, parameters are tuned carefully to ensure that 1): the transfer function from *ˆ ( )dcv s to 1ˆ ( )Cv s has good dynamics response and; 2)

the transfer function from ˆ ( )inv s to 1ˆ ( )Cv s has enough damping at the oscillation frequency.

The complete system level control algorithm is shown in Figure 6. Without losing generality, the current regulator under synchronous frame is implemented in the ac side controller.

Figure 6. System level control strategy.

To achieve a good system level control, the dynamics of the ac side should be designed to be much faster than the dc side to avoid the oscillation. Since shoot-through states are always overlapped with the zero states [11], the change in one control parameter will impose limitation on the other. The controller usually will perform better with higher end of the input voltage range.

IV. SIMULATION AND EXPERIMENTAL RESULTS An 85 kW BQ-ZSI prototype is built in the lab. The

prototype picture is shown in Figure 7. Simulations are performed with Matlab/Simulink. Both the simulation and experimental results are shown to verify the circuit analysis and controller design.

(a) (b)

Figure 7. Hardware Development of the BQ-ZSI. (a) Mechanical Layout Design; (b) The 85 kW prototype.

A. Functionality of the BQ-ZSI The operation of the BQ-ZSI with high voltage and high

current is first tested. In the setup, the output of the BQ-ZSI is directly connected to a three-phase reactor. On the dc side, a three-phase diode rectifier is used as the dc source. The dc input voltage is set as 225 V. With the shoot-through duty ratio of 0.25, the dc-link voltage is boosted to the designed maximum value of 450 V. The output current

-10 -5 0 5x 104

-5000

-4000

-3000

-2000

-1000

0

1000

2000

3000

4000

5000Pole-Zero Map

Real Axis

Imag

inar

y A

xis

-10 -5 0 5x 104

-6000

-4000

-2000

0

2000

4000

6000Pole-Zero Map

Real Axis

Imag

inar

y A

xis

ˆ( )d s 1ˆ ( )Cv s( )CG s

*1ˆ ( )Cv s

2 ( )v dG s

2 ( )v inG sˆ ( )inv s

12

ˆ ( )dcv s∗

S7

L1

C1Vin

L2

C2

S1 S3 S5

S4 S6 S2

ABC/dq

PI *qI

*dIPI

dI qI

dq/ABC

Modulator

dcV ∗ 12

1cV

1cV ∗

inV

PI

7 ChannelsPWM

Motor

2226

achieves 300 A. The experimental results are shown in Figure 8. Due to the limitation of the ac power supply, there is no large active power in the circuit, so the average input current is small. Under this maximum voltage test condition, the inductor current ripple is at the designed maximum value. With a small 220 uF input capacitor, the ripple current in the inductor is partly filtered out and a smooth dc input current is obtained. Compared to the rated dc current of 378 A, the maximum input current ripple is small. Since the inductor windings are made of copper sheet, the inductor currents cannot be directly measured and is not shown in the results.

(a)

(b)

Figure 8. Experimental results with high voltage and high current. (a) Input voltage, capacitor voltage, ac output current, and dc input current. (b) Input voltage, capacitor voltage, voltage stress on S7, and dc-link voltage.

The reverse power flow capability of the BQ-ZSI is then examined. The BQ-ZSI is connected to a three-phase adjustable voltage source through a three-phase reactor. A high voltage electronics load is used at the dc input side to emulate a battery pack. The inverter is controlled to operate in the regeneration mode and send energy from the grid to the dc load. The shoot-through duty ratio is fixed at 0.15 and the ac side voltage is 70 V. The experimental results are shown Figure 9. The results show that the reverse power

flow is achieved. The dc-link voltage boost function is also realized.

(a)

(b)

Figure 9. Experimental results during reverse power flow. (a) Input voltage, dc-link voltage, ac output current and dc input current. (b) Input

voltage, capacitor voltage, voltage stress on S7, and dc-link voltage.

B. Closed loop control of the BQ-ZSI The simulation results of the control strategy are shown

in Figure 10. In the simulation, at 0.3 s, there is a 50 V step change from the input voltage. The QZSN side controller stabilizes the dc-link voltage by adjusting the shoot-through duty ratio. After a short period of transient time, the dc-link voltage changes back to its initial value. Meanwhile, with the help of the ac side current controller, the ac load barely sees this disturbance and the current are kept constant. Moreover, compared to the results in Figure 3, the oscillation in the capacitor voltage is damped. This demonstrates the effectiveness of the controller for rejecting the disturbance from the dc input voltage.

Experimental result is shown in Figure 11. It can be seen that after the input voltage increases from 200 V to 250 V (25 %), the dc side controller maintains the dc-link voltage constant. At the same time, the capacitor voltage increases since the shoot-through duty ratio decreases. During this

Vin: 250V/div

VC1: 250V/div

Iin: 25A/div

Ia: 500A/div

Vin: 100V/div

VC1: 100V/div

VS7: 250V/div

VPN: 250V/div

2227

interval, the ac side current controller also maintains constant ac current. There is very little disturbance on the output current. The designed control algorithm is proven to be effective.

Figure 10. Simulation results of the closed loop control.

Figure 11. Experimental result of the controller design.

V. CONCLUSION This paper presents the small signal analysis and

controller design of the BQ-ZSI for EV applications. The small signal model of the BQ-ZSI shows that the BQ-ZSI topology is easy to oscillate when there is a disturbance from the input voltage. Since this issue does not exist in the traditional ZSI topology, it has not been addressed before.

The proposed controller utilizes the input voltage and capacitor voltage to stabilize the dc-link voltage. This approach applies a feed forward loop to enhance the input voltage disturbance rejection and oscillation suppression capability of the controller. An 85 kW BQ-ZSI prototype is built and tested in the lab. The simulation and experimental results have verified the proposed control strategy.

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[3] J. Anderson and F.Z. Peng, “Four Qusai-Z-Source inverters,” in Proc. IEEE PESC’08, Rhodes, Greece, June 2008.

[4] H. Xu, F. Z. Peng, L. Chen, and X. Wen “Analysis and Design of Bi-directional Z-Source inverter for electrical vehicles,” in Proc. IEEE Twenty Third Applied Power Electronics Conference and Exposition, APEC 2008, Austin, Texas, Feb. 2008.

[5] Y. Li; F.Z. Peng, J.G. Cintron-Rivera, and S. Jiang, “Controller design for quasi-Z-source inverter in photovoltaic systems,” in Proc. IEEE Energy Conversion Congress and Exposition (ECCE), pp. 3187-3194, Sept. 2010.

[6] J. Liu, J. Hu, and L. Xu, “Dynamic modeling and analysis of Z Source converter -Derivation of AC small signal model and design-oriented analysis,” IEEE Trans. Power Electron., vol. 22, no. 5, pp. 1786-1796, Sept. 2007.

[7] Y. Liu, B. Ge, F.Z. Peng, A.R. Haitham, A.T. de Almeida, and F.J.T.E. Ferreira, “Quasi-Z-Source inverter based PMSG wind power generation system,” in Proc. IEEE Energy Conversion Congress and Exposition (ECCE), pp. 291-297, Sept. 2011.

[8] C. J. Gajanayake, D. M. Vilathgamuwa, and P. C. Loh, “Development of a comprehensive model and a multiloop controller for Z-Source inverter DG systems,” IEEE Trans. Industrial Electronics, vol. 54, no. 4, pp. 2352-2359, Aug. 2007.

[9] D. M. Divan, “The resonant DC link converter-a new concept in static power conversion,” IEEE Trans. Ind. Appl. , vol. IA-25, no. 2, pp. 317–325, Mar./Apr. 1989.

[10] R. W. Erickson and D. Maksimovic, Fundamentals of Power Electronics, 2nd ed. Norwell, MA: Kluwer, 2001.

[11] F. Z. Peng, M. Shen, and Z. Qian, “Maximum boost control of the Z-Source inverter,” IEEE Trans. Power Electron., vol. 20, no. 4, pp. 833–838, Jul. 2005.

0.2 0.25 0.3 0.35 0.4 0.45150

200

250

300

Vin

(V)

0.2 0.25 0.3 0.35 0.4 0.45

200

400

Vc1

(V)

0.2 0.25 0.3 0.35 0.4 0.450

200

400600

Vpn

(V)

0.2 0.25 0.3 0.35 0.4 0.45

-1000

100

Time (S)

Iabc

(A)

2228