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h

This is to certify that the
thesis entitled

Z-SOURCE INVERTER DESIGN, ANALYSIS, AND ITS
APPLICATION IN FUEL CELL VEHICLES

presented by

MIAOSEN SHEN

has been accepted towards fulfillment
of the requirements for the

Ph.D degree in Electrical Egoineeriry

 

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Date

MS U is an Affinnative Action/Equal Opportunity Institution

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PLACE IN RETURN BOX to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/07 p2/CIRC/DaIeDue indd-p,1

Z-SOURCE INVERTER DESIGN, ANALYSIS, AND ITS APPLICATION
IN FUEL CELL VEHICLES

By

Miaosen Shen

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2006

ABSTRACT

Z-Source Inverter Design, Analysis, and Its Application in Fuel
Cell Vehicles

By
Miaosen Shen

With its unique structure, the Z-source inverter can utilize the shoot through states to
boost the output voltage, which improves the inverter reliability greatly, and provides an
attractive single stage dc-ac conversion that is able to buck and boost the voltage. For

applications with a variable input voltage, this inverter is a very competitive topology.

This work starts with the PWM methods for the Z-source inverter. Three PWM methods
are proposed with different boost ratios. In order to reduce the switching loss, a modified
PWM scheme is proposed. A comparative study of the modified PWM method and a

commonly used PWM method is conducted.

To facilitate closed loop controller design, a small signal model for the Z-source inverter
is developed using the state space averaging and verified with simulation results. Based
on this model, a closed loop controller is designed using gain scheduling method. The

controller is verified with simulation and experimental results.

In order to show the strengths and weaknesses of the Z-source inverter, a comprehensive

comparison of the Z-source inverter and traditional inverters for fuel cell vehicle traction

drives is conducted. The switching device power, passive components requirement,
constant power speed ratio, reliability, and inverter efficiency are used as bench marks
for the comparison. The comparison shows that the Z-source inverter is very suitable for
applications with a moderate boost ratio needed, which most fuel cell/photovoltaic

sources reside in. A detailed design process of the Z-source inverter is also provided.

Besides the two basic operation modes of the Z-source inverter, new operation modes
have been found when the load power factor, modulation index, and the inductance of the
inductors are low. The critical conditions, voltage gain, and the consequence of these

operation modes are analyzed in detail.

Three configurations of Z-source inverter for fuel cell-battery hybrid vehicle traction
drives are presented with the basic control methods. With a battery in the system, the
effect of the undesirable operation modes is minimized. Also, control methods to totally
eliminate the undesirable operation modes are proposed. A simple comparison of the

three configurations is conducted.

ACKNOLEDGEMENTS

I would like to thank my advisor, Dr. Fang Z. Peng, for his guidance, encouragement, and
support during my graduate studies. His incredible knowledge and insightful view of
power electronics inspired me throughout my research. He has always been a model for

me to follow.

I am grateful to my committee members, Dr. Robert Schlueter, Dr. Elias Strangas, and Dr.
Rajan Mukherjee, for their inspiring classes, valuable discussion and suggestion, and for

their serving on my committee.

I would also like to thank Dr. Zhaoming Qian, my first mentor in power electronics. His

guidance and encouragement are always so valuable in the past and in the future.

It has been a great pleasure being a member of power electronics and motor drive
laboratory at Michigan State University. I would like to thank all members of the lab, in
particular, I would like to thank Dr. Jin Wang, Dr. Zhiguo Pan, Dr. Fan Zhang, Dr.
Haiping Xu, Mr. Lihua Chen, Mr. Alan Joseph, Mr. Honnyong Cha for delightful

discussions.

iv

I would like to thank Mr. Qingsong Tang for his help in the DSP programming and

experiments and Mr. Kent Holland for checking my thesis.

Finally and most importantly, I would like to thank my parents Bingshan and Qinxian for
their care and unconditional support. There is no word can express my appreciation to

them.

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................. ix
LIST OF FIGURES .............................................................. ' .............................................. x
CHAPTER 1 Introduction ................................................................................................... 1
1.1 Traditional Inverters .................................................................................................. 1
1.1.1 Voltage Source Inverter ....................................................................................... 1
1.1.2 Current Source Inverter ....................................................................................... 4

1.2 The Z-Source Inverter ............................................................................................... 7
1.3 Previous Works on the Z-Source Inverter ............................................................... 10
1.3.1 PWM Methods .................................................................................................. 10
1.3.2 Modeling and Control of the Z-Source Inverter ............................................... 12
1.3.3 Topology Derivation ......................................................................................... 13
1.3.4 Application of the Z-Source Inverter ................................................................ 16

1.4 Scope of the thesis ................................................................................................... 16
CHAPTER 2 PWM Schemes for the Z-Source Inverter ................................................... 18
2.1 Simple Control ......................................................................................................... 18
2.2 Maximum Boost Control ......................................................................................... 21
2.2.1 Description of the Control Method ................................................................... 21
2.2.2 Simulation and Experimental Verifications ...................................................... 27
2.2.3 Current Ripple ................................................................................................... 30

2.3 Maximum Constant Boost Control ................................................ . .......................... 32
2.3.1 Description of the Control Method ................................................................... 32
2.3.2 Simulation and Experimental Verifications ...................................................... 37
2.3.3 Voltage Stress Comparison ................................................................................ 40

2.4 Modified PWM Scheme .......................................................................................... 42
2.5 Comparison of Different PWM Schemes ................................................................ 46
2.5.1 Switching Loss. ................................................................................................. 46
2.5.2 Inductor Size ..................................................................................................... 48
2.5.3 Capacitor Size ................................................................................................... 50
2.6. Summary ................................................................................................................ 51
CHAPTER 3 Modeling and Control of the Z-Source Inverter with RL Load .................. 52
3.1 Modeling of the Z-source Inverter .......................................................................... 52
3.1.1 Model Simplification ........................................................................................ 52
3.1.2 Modeling by State Space Averaging ................................................................. 53
3.1.3 Model Verification ............................................................................................ 56

3.2 Controller Design .................................................................................................... 59
3.2.1 Gain Scheduling Controller .............................................................................. 59
3.2.2 Controller Design .............................................................................................. 60

3.3 Simulation Results ................................................................................................... 64
3.4 Experimental Verification ........................................................................................ 66

vi

3.5 Discussion ............................................................................................................... 69

3.6 Summary ................................................................................................................. 70
CHAPTER 4 Z-Source Inverter for Fuel Cell Vehicles—Design and Comparison with
Traditional Inverters .......................................................................................................... 71

4.1 Introduction ................................................................. ~ ............................................ 71

4.2 System Configurations for Fuel Cell Vehicle .......................................................... 72

4.3. Comparison Items, Conditions, Equations, and Results ........................................ 74

4.3.1. Total Switching Device Power Comparison .................................................... 75
4.3.2. Requirement of Passive Components Comparison .......................................... 78
4.3.3. CPSR Comparison ........................................................................................... 87
4.3.4. Reliability Comparison .................................................................................... 89

4.4 Comparison Example .............................................................................................. 89

4.5. Summary ................................................................................................................ 96
CHAPTER 5 Operation Modes and Characteristics of the Z-Source Inverter with Small
Inductance or Low Power Factor ...................................................................................... 98

5.1 . Introduction ............................................................................................................ 98

5.2. Operating Principle and Operation Modes Analysis ............................................ 100

5.3. Circuit Analysis and Characteristics ..................................................................... 104

5.3.1. Traditional SVPWM Control ......................................................................... 104
5.3.2. Maximum Constant Boost Control with Third Harmonic Injection .............. 107
5.4. Analysis Example ................................................................................................. 110
5.4.1. Voltage Gain ................................................................................................... 110
5.4.2. Relationship of Voltage Gain Versus Voltage Stress ....................................... 111
5.4.3. Design Guidelines .......................................................................................... 112

5.5. Simulation and Experimental Verifications .......................................................... 113

5.6. Method to Eliminate the Unwanted Operation Modes ......................................... 114

5.7. Summary .............................................................................................................. 117
CHAPTER 6 Application and Control of Z-Source Inverter for Traction Drive of Fuel
Cell — Battery Hybrid Electric Vehicles .......................................................................... 118

6.1 . Introduction .......................................................................................................... l 1 8

6.1.1 Fuel Cell-Battery Hybrid Vehicles .................................................................. 118
6.1.2. Fuel Cell and Battery Characteristics ............................................................. 121
6.1.3 Traditional power conditioner configurations ................................................. 122
6.2. Configurations and Control of Z-source Inverter for F CHEVs ........................... 124
6.2.1 Configurations ................................................................................................. 124
6.2.2 Control of the Inverter ..................................................................................... 126
6.2.3. Simulation and experimental verification ...................................................... 130
6.3. Undesirable Operation Modes and Control to Eliminate Them ........................... 139
6.3.1 Undesirable Operation Modes Without Control ............................................. 139
6.3.2 Control Method to Eliminate the Undesirable Operation Modes ................... 141
6.3.3 Experimental Verification of the Control Method .......................................... 143
6.4. Comparison of the Three Configurations ............................................................. 147
6.4.1 Configuration 1 ............................................................................................... 147

vii

6.4.2 Configuration 2 ............................................................................................... 148

6.4.3 Configuration 3 ............................................................................................... 149
6.4.4 Comparison Summary ..................................................................................... 150

6.5 Summary ............................................................................................................... 150
CHAPTER 7 Conclusions and Recommendations ............... -. ......................................... 152
7.1 Contributions ......................................................................................................... 152
7.2 Recommendations for future works ...................................................................... 153
APPENDIX 1 Switching Device Power Derivation ....................................................... 154
APPENDIX 2 Passive Components Requirement .......................................................... 160
APPENDIX 3 Critical Condition of New Modes Under SVPWM ................................ 162
APPENDIX 4 Critical Condition of New Modes Under Maximum Constant Boost
Control with Third Harmonic Injection .......................................................................... 164
APPENDIX 5 Characteristic Analysis of the Circuit Under CCM Condition ................ 166
References ....................................................................................................................... 168

viii

LIST OF TABLES

Table 1.1 Possible Shoot Though States ......................................................... 11

Table 2.1 Theoretical voltage stress and output voltage under different conditions ........ 28
Table 2.2 Theoretical voltage stress and output voltage under different conditions. . . . . ...38
Table 4.1 Switching device power comparison example ....................................... 89
Table 4.2 Required passive components ......................................................... 90
Table 4.3 Operation conditions at different power ............................................. 90
Table.6.1 Comparison of different configurations ............................................. 150

ix

LIST OF FIGURES

Figure 1.1 Voltage source inverter. .......................................... 3
Figure 1.2 Sketch map of PWM control of voltage source inverter. .................................. 3
Figure 1.3 Current source inverter ...................................................................................... 5
Figure 1.4 Waveforms of three-phase PWM current inverter ............................................. 6
Figure 1.5 Z-source inverter ............................................................................................... 7
Figure 1.6 PWM control of the Z-source inverter .............................................................. 7
Figure 1.7 Experimental results of Z-source inverter. ........................................................ 9
Figure 1.8 Modulation method using six shoot through states ......................................... 11
Figure 1.9 Examples of the existing controllers ............................................................... 13
Figure 1.10 Derived topologies ......................................................................................... 15
Figure 1.11 Configuration for Z-source inverter for photovoltaic systems ...................... 16
Figure 2.1 Simple boost control ........................................................................................ 20
Figure 2.2 Voltage gain of the simple boost control ......................................................... 21
Figure 2.3 Switches voltage stress versus voltage gain .................................................... 21
Figure 2.4 Sketch map of maximum boost control ...................... 22
Figure 2.5 Vac/0.5 V0 versus M ........................................................................................... 22
Figure 2.6 Voltage stress versus voltage gain ................................................................... 24
Figure 2.7 Sketch map of third harmonic injection control .............................................. 26
Figure 2.8 Vac/0.5Vo versus M .......................................................................................... 26
Figure 2.9 Voltage stress of switches versus voltage gain of proposed control method... 27
Figure 2.10 Simulation results with M=0.88 and input voltage 170 V DC ...................... 29
Figure 2.11 Simulation results with M=1 and input voltage 220 V DC ........................... 29
Figure 2.12 Simulation results with M=1.l with third harmonic injection and input
voltage 250 V DC ............................................................................................................. 29
Figure 2.13 Experimental result at M=0.88 and input voltage 170 V DC ........................ 30
Figure 2.14 Experimental result at ME=1 and input voltage 220 V DC ............................. 30

Figure 2.15 Experimental result at M=1.1 with third harmonic injection and input voltage
250 V DC .......................................................................................................................... 30

Figure 2.16 Maximum boost control sketch map ............................................................. 31

Figure 2.17 Model of the circuit ....................................................................................... 32
Figure 2.18 Sketch map of constant boost control ............................................................ 33
Figure 2.19 Vac/0.5 V0 versus M ........................................ 34
Figure 2.20 Sketch map of constant boost control with third harmonic injection ............ 36
Figure 2.21 Vac/0.5Vo versus M ........................................................................................ 37
Figure 2.22 Simulation results with M = 0.8 .................................................................... 38
Figure 2.23 Simulation results with M = 1 ....................................................................... 38
Figure 2.24 Simulation results with M = 1.1 .................................................................... 39
Figure 2.25 Experimental results with V0= 145V and M = 0.812 .................................... 39
Figure 2.26 Experimental results with V0 = 250V and M = 1 ........................................... 39
Figure 2.27 Experimental results with Vo= 250V and M = 1.1 ........................................ 40
Figure 2.28 Voltage stress comparison of different control methods ................................ 41
Figure 2.29 Switching states sequence of traditional PWM control ................................. 42
Figure 2.30 Modified PWM scheme ................................................................................. 43
Figure 2.31 Overall switching current comparison of different PWM ............................. 48
Figure 2.32 Six shoot through PWM when va=vc ....................................... _ ...................... 49
Figure 2.33 Capacitor voltage ripple comparison for different power factor ................... 51
Figure 3.1 Z-source inverter ............................................................................................. 53
Figure 3.2 dc equivalent circuit ........................................................................................ 53
Figure 3.3 Calculated bode diagram of Vc/D .................................................................... 57
Figure 3.4 Calculated bode diagram of V/M .................................................................... 57
Figure 3.5 Simulated bode diagram of Vc/D ..................................................................... 58
Figure 3.6 Simulated bode diagram of VO/M ..................................................................... 59
Figure 3.7 Simulation result of closed loop performance ................................................. 65
Figure 3.8 Circuit simulation results with parameter mismatch ....................................... 66
Figure 3.9 Experimental results ........................................................................................ 68
Figure 4.1 Typical PEM fuel cell polarization curve ........................................................ 72
Figure 4.2 Three inverter system configurations for fuel cell vehicles. ........................... 73

Figure 4.3 Comparison of SDP of different inverters at different operation conditions... 77

xi

Figure 4.4 Coupled inductors ............................................................................................ 80

Figure 4.5 Capacitor current ripple calculation ................................................................ 81
Figure 4.6 PWM scheme of different inverters at a certain interval ................................. 85
Figure 4.7 Capacitor current comparison of the inverters ........................................ 86
Figure 4.8 Capacitor voltage ripple comparison of the inverters ...................................... 87
Figure 4.9 Switching states sequence in one cycle ........................................................... 93
Figure 4.10 Calculated efficiency of inverters .................................................................. 94
Figure 4.11 Inverter efficiency calculated using software ................................................ 94
Figure 4.12 Efficiency testing and results ......................................................................... 96
Figure 5.1 The Z-Source inverter .................................................................................... 100
Figure 5.2 Possible operation modes of Z source inverter .............................................. 102
Figure 5.3 Inductor current waveforms for different operation conditions .................... 105
Figure 5.4 Z-source inductor current for different operation conditions ........................ 108
Figure 5.5 Simulation results when operating at CCM condition .................................. 115
Figure 5.6 Experimental results of CCM condition ........................................................ 116

Figure 5.7 Comparison of analyzed results and simulation results of capacitor voltage 116

Figure 5.8 Bi-directional Z-source inverter ................................................ _ .................... 117
Figure 6.1 Medium power operating mode 1 .................................................................. 120
Figure 6.2 High power operating mode 2 ....................................................................... 120
Figure 6.3 Low power operating mode 3 ........................................................................ 120
Figure 6.4 Regenerative braking operating mode 4 ........................................................ 121
Figure 6.5 Typical fuel cell polarization curve ............................................................... 122
Figure 6.6 Typical lithium-ion battery voltage versus SOC ........................................... 122
Figure 6.7 Traditional configurations of F CVs ............................................................... 123
Figure 6.8 Configurations of Z-source inverter for FCHEV. .......................................... 125
Figure 6.9 PWM scheme for the third configuration ...................................................... 126
Figure 6.10 Power control of fuel cell by controlling the voltage. ................................. 128
Figure 6.11 Model of the Z-source network considering parasitic parameters when the
fuel cell is turned off ....................................................................................................... 128
Figure 6.12 Simulation case 1 ......................................................................................... 132

xii

Figure 6.13 Simulation case 2 ......................................................................................... 133

Figure 6.14 Simulation case 3 ......................................................................................... 134
Figure 6.15 Experimental setup ...................................................................................... 135
Figure 6.16 Experimental results for case 1 ...................................... 136
Figure 6.17 Experimental results for case 2 .................................................................... 137
Figure 6.18 Experimental results for case 3 .................................................................... 137
Figure 6.19 Z-source inverter based fuel cell converter control system ......................... 138

Figure 6.20 Simulation results of configuration 3 with battery connected to the neutral

point of the motor ............................................................................................................ 141
Figure 6.21 Fuel cell V-I characteristics ......................................................................... 143
Figure 6.22 Experimental setup ...................................................................................... 144
Figure 6.23 Experimental result of case 1 ...................................................................... 145
Figure 6.24 Experimental results for case 2 .................................................................... 146
Figure A.1 Inverter model during shoot through ............................................................ 158

xiii

CHAPTER 1 Introduction

Power electronics has been widely used in various applications since it was born. The
three-phase inverter, which converts dc voltage/current into three-phase ac
voltage/current is one of its most important and popular converters. It has been widely
used in motor drives [1, 2], ac uninterruptible power supplies (UPS) [3, 4], induction
heating [5], ac power supplies [6, 7], active power filters [8, 9], and static VAR
generators or compensators [10, 11], etc. There are two types of traditional inverters,
namely voltage source (fed) inverters and current source (fed) inverters. However, both
inverters have some conceptual barriers, which will be discussed in detail later. The
newly presented Z-source inverter [12] has some unique features and it can overcome
some of these limitations. The purpose of this work is to investigate the properties of

the Z-source inverter and its applications.

1.1 Traditional Inverters

1.1.1 Voltage Source Inverter

1.1.1.1 Topology

Figure1.1 shows the topology of the voltage source inverter. The input to the inverter is
a dc voltage source usually with a capacitor in parallel to absorb the high frequency
current ripple. The inverter bridge consists of six switches with a freewheeling diode in

parallel with each of them.
1.1.1.2 Control Methods

Basically there are two kinds of control methods, one is six step control, which turns each

1

switch on and off only once per fundamental cycle creating a six step stair case waveform
for the output phase voltage; the other one is pulse width modulation (PWM), which
switches the devices at high frequency (hundreds of Hz to hundreds of kHz) to output a
voltage with very little low frequency harmonics as well as to control the output voltage

level.

For six-step control, the output line peak fundamental voltage [15] is

V, =-——2fiVi , (1.1)
7'!

where V," is the dc bus voltage. The six-step method has advantages such as simple,

low switching loss, on the other hand, it has limitations for it is unable to eliminate the

low frequency harmonics and unable to control the output voltage level.

The sketch map of the basic PWM control for voltage source inverter is shown in
Figure1.2. From Figure1.2, the two switches on the same phase leg are turned on and
off complementally. There are 8 switching states including 6 active states and two zero
states when the upper three or lower three switches are gated on shorting the load

terminals.
The output peak phase voltage can be expressed as

MV-
Vopk .—_ 2' , (1.2)

 

where V," is the input voltage, and M is the modulation index, which ranges from 0 to

1.15 with third harmonic injection or space vector PWM control.

 

Figure 1.1. Voltage source inverter.

       
   

Scn

110111110 000 1510‘ 111110

Figure 1.2. Sketch map of PWM control of voltage source inverter.
1.1.1.3 Limitations of Voltage Source Inverter
The voltage source inverter has several limitations as listed below.

From (1.2), the output voltage range is limited, the inverter cannot output a higher
voltage than the dc bus voltage. For many applications, when the input dc voltage is not
always constant, like a fuel cell, photovoltaic array, and during voltage sag, etc, a dc/dc
boost converter is often needed to boost the dc voltage to meet the required output
voltage. This increases the system complicity and the cost, and reduces the system

3

reliability.

The two switches on the same phase leg cannot be gated on at the same time, otherwise a
short circuit will occur and destroy the inverter. . The mistrigger caused by
electromagnetic interference (EMI) is a major killer of the inverter. For safety reasons,
there is always a dead time to make sure that the two switches will not be turned on
simultaneously. However, the dead time can cause output voltage distortion and
harmonic problems. The harmonic problem can be solved by implementing a

current/voltage feedback control [1 3], however, this increases the system complexity.

A LC filter is needed for the load side compared to current source inverter, which

increases the complexity and reduces the system efficiency.
1.1.2 Current Source Inverter

1.1.2.1 Topology

The current source inverter is shown in Figure 1.3. The input to the inverter is a current
source or a voltage source with an inductor in series. The inverter bridge consists of six
switches with a reverse blocking diode in series or switches with reverse blocking ability.
Three capacitors are connected at the ac side of the inverter to provide a leading power

factor load.

Because of the reverse blocking nature of the thyrister, it is often used in current source
inverter. However, due to the fact that it is unable to be turned off by the gate signal, a
leading power factor load is required to enable load commutation. This also limits the
control to be six-step only, which will cause lots of low frequency harmonics in the

output waveform.

Another group of current source inverters use self-commutated devices, such as GTO,
IGBT, with reverse blocking diode in series. PWM control can be implemented in these

inverters.

 

 

 

 

 

[ii-«>\
G)

 

 

 

 

 

 

 

 

 

 

Figure 1.3. Current source inverter.
1.1.2.2 Control Methods

The current source inverter is a dual circuit of the voltage source inverter, therefore the
dual of most of the PWM methods for voltage source inverter can be, used to control the
current source inverter. Generally the following rule has to be followed: there must be
at least one device in the upper three devices and one in the lower three devices gated on
at the same time to provide a current path. An example of the inverter current with
PWM control is shown in Figure 1.4. The PWM waveform is the inverter current, the

sinusoidal waveform is the current to the load after the capacitors.
1.1.2.3 Limitations of the Current Source Inverter

To keep the inductor average voltage zero, the maximum inverter dc side voltage, which
is the peak line-to-line output voltage of the inverter has to be greater than the input dc
voltage. Therefore, the inverter is basically a boost converter. For applications where
a wide voltage range is required, extra circuitry has to be used to obtain the required

5

voltage. The commonly used methods include using a dc/dc converter or a controlled
rectifier [14, 16]. However, these methods increase the circuit complexity and reduce

the efficiency as well as the reliability.

PWM controlled current
output current

/

Figure 1.4. Waveforms of three-phase PWM cru’rent inverter.

At least one switch in the upper three devices and one in the lower three devices has to be
turned on at the same time, or an open circuit will occur and destroy the inverter.
Mistrigger caused by the EMI noise could significantly reduce the inverter reliability.
To make sure that there will be no open circuit, overlap time is often needed, which will

cause output waveform distortion and low frequency harmonic problem.

The main switches have to have the reverse blocking capability, which requires an extra
diode in series with high-speed high-performance switches, such as an IGBT. This will
reduce the system efficiency. With the emergence of the reverse blocking IGBT [24],

this might not be a problem anymore.

1.2 The Z—Source Inverter

The newly proposed Z-source inverter [12] can overcome some of these limitations, it is

shown in Figure 1.5.

 

 

 

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To ac Load
Y or Motor

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Figure 1.5. Z-source inverter.

 

 

 

 

 

 

 

 

 

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31"."§.§(:L=‘¢3}5‘t;\5¢5~:~3£edeiz'nhFA—h. flu . _b. Aux-.11.. ataxia.

Figure 1.6. PWM control of the Z-source inverter.

The Z—source inverter employs an X shape LC network between the input voltage and the
inverter bridge. A simple PWM control scheme is shown in Figure 1.6. From Figure
1.6, there is one more switching state, the shoot through state, besides the eight switching
states (six active states and two zero states) for the traditional voltage source inverter.
With the unique LC network, we can intentionally add the shoot through state to boost
the output voltage. By utilizing this switching state to boost the output voltage, we can
output voltage higher or lower than the DC link voltage. Therefore, the inverter is a
buck-boost type converter and can output whatever voltage desired, and overcome the
voltage limitation of the voltage source inverter and current source inverter. With the
ability to handle the shoot through state, the reliability of the Z-source inverter is greatly
enhanced. Also, without the need of dead time, output waveform distortion can be
avoided. The detailed operation principle has been discussed in [12], the resulted output

peak phase voltage is

9,, = 143%, (1.3)

where M is the modulation index and B is the boost factor, which can be express as

B: , (1.4)

 

where To is the shoot through time per cycle and T is the switching cycle.

To demonstrate the Z-source inverter properties, some experimental results are shown

below.

. .(2
flllflmflfllfln '

   
      

 

cm i
ll 3

.
mt; -» . . :24:

3 vgzoo/ 11v):

       

    

. . ». ,
«a, .4...‘ ...mH—r.....,..++a.+ ..._

__ ,,, (200V/div) i

 

 

(a) (b)

V0: input dc voltage; VP”: inverter PN voltage; IL 1: Z-source inductor current; VLab: output

line voltage
Figure 1.7. Experimental results of Z-source inverter.

In Figure 1.7(a), the Z-source inverter produces 250 V ac when the dc voltage is only 145
V, while in (b), the inverter produces 300 V ac when the dc voltage is only 250 V.
Whereas, with traditional voltage source inverter with modulation index of 1, the
maximum output voltage obtainable for 145 V input is 89 V, and 153 V for 250 V dc
input.

In summary, the Z-source inverter has several unique features:

1. The inverter can boost and buck the output voltage with a single stage structure;

2. The shoot through caused by EMI can no longer destroy the inverter, which

increases the reliability of the inverter greatly.

3. Because of no dead time is required, perfect sinusoidal output waveform is

obtainable.

1.3 Previous Works on the Z-Source Inverter

Since the Z-source inverter was proposed in 2002 [12], lots of works have been done on
this subject. Mainly the previous works include four major topics: PWM methods;
modeling and control of the Z-source inverter; application of the Z-source inverter; and

modification of the topology for other applications.
1.3.1 PWM Methods

Different from traditional voltage source inverter, the Z-source inverter has an extra
switching state, the shoot through state, which occurs when turning on both switches in

the same phase leg. Theoretically there can be seven different shoot through states:

How to insert these switching states, and how much time the shoot through states should
be used are very important to the circuit performance and even to the efficiency of the

inverter.

Literatures [30, 50] presented several PWM methods based on different PWM methods
for traditional voltage source inverters. In those works, the shoot through time is
distributed between all traditional switching states, thus there are six shoot through states

in one switching cycle.

10

Table 1.1 Possible shoot though states

 

 

 

 

 

 

 

 

 

Switches S ap San pr Sbn Scp SC"
ST] 1 1 B E C E
ST2 A Z 1 1 C E
ST3 A X B E 1 l
ST4 l 1 l 1 C E
ST5 1 l B E 1 1
ST6 A Z 1 1 1 1
ST7 1 1 1 1 1 1

 

 

 

 

 

 

 

 

 
 
   

 

 

H Shoot-thrh wmillistate .

'3 Ref. Va(s1) -' '

 

1: I’ ::I

 

 

 

Z-Source
Inverter PWM
l

 

 

 

 

 

 

 

: 1 Ref Votes) Ref Vase :
l 55: : Rerbtse) ;:: :
'1 iii .1 i“ ' “L Remy?” iii 5
Viézz: :-: 13$:i; Ref vqsz, ;;; :
:‘ .1, 1:1T 'g. I ... “I
“an LtEcZ LtEca 7"“

Figure 1.8. Modulation method using six shoot through states.

Also, there are some other PWM methods for other purposes, such as to reduce the

common voltage by replacing the zero states with combination of active states [60]; and

11

PWM methods for multi-level Z-source inverters [52, 57, 73].
1.3.2 Modeling and Control of the Z-Source Inverter

Two small signal models have been developed for the Z-source inverter [40, 43, 51].
The models focus on the dc side of the inverter and only have one control variable, which
is the shoot through duty ratio. Also, several control methods for the inverter have been
developed [47, 59, 65], two examples of the controllers are shown in Figure 1.9. In
these controllers, the capacitor voltage is controlled by the shoot through duty ratio and
the output voltage is controlled by the modulation index. The two controllers are

designed separately, thus the whole system stability is not guaranteed.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

VDC‘
Z—Sourc
V AB“
C
t _
Vc + K Calculatq TS}:
. E +14 l—p Ts}, ‘ I... H
* MSV " E 1,
Vs + erf _. m a
_ f—* ._l_ 93 F a. m 3
V5,, P _, no
Calculat Vsmbm)
VSP
Load
(a)

 

 

 

 

 

Z-source Impedance l I l . Fiter net- Three phase
net-work PWM Inverter work H load

+ vcf

 

 

   
 

 

 

 

/
3

 

 

 

 

 

(b)

Figure 1.9 Examples of the existing controllers.
1.3.3 Topology Derivation

Since the first voltage fed Z-source inverter topology was proposed [12], a lot of
topologies have been developed based on it. The derived topologies spread the whole
spectrum of electric power conversion [23, 36, 41, 42, 46, 49, 54, 55, 62, 63, 72, 73], ie.

AC/AC, AC/DC, DC/DC, DC/AC.

l3

 

 

- Jill? -
,v / ,r'I‘oAC Load
orMotor

 

 

V

 

\I
II
\I
\

 

 

 

 

 

 

 

 

 

 

 

+
ll
44-?“

 

 

 

 

 

1::
r4

 

Q
Q
j ‘g
at"
('3
J.)\o—-o
0Q

 

1 lug

 

 

(a
3'

 

?
C?
I
5“
”Q
(p—o\o—< >——o\o—o

 

 

 

 

 

%
B
1
5h
0

00..

 

 

 

 

 

 

 

 

1&1??ch r,

(c) Single phase voltage fed ac/ac converter.

 

11.1 LI

I ‘\.>\.</; 11°

rLl 31 VCI‘ L2 /VC2 32 C3 ;:R::¢vo
m
<19

 

 

 

 

 

 

(d) Single phase current fed ac/ac converter.

Upper ZSource
Network (U)

Neutral—Point.
Clamped Inverter

  

 

 

 

 

Lower Z-Source
Network (G)

(e) Diode clamped multi-level Z-source inverter.

 

 

 

 

 

 

 

 

I -------------- \ z ‘‘‘‘‘‘‘‘‘ \
I - 1 l 1
l 3-phase rectifier I : Z-network 3
l 1
l 1 1 T m- ‘ ,1 i
l A 4' my I l 1
l l l + + .
1 11 I + <
' I 1 - ' x 8
I l l : - ‘P.
I l I '
I r l I I
l I l I I
l l I \.Q.Q_/ . l
1 1 1 + ,
l l I
r 1 1 l
\ I I

(f) Three-phase Z-source rectifier.

Figure 1.10. Derived topologies.

15

1.3.4 Application of the Z-Source Inverter

The application of the Z-source inverter also has been extensively examined. The most
popular application of the inverter is for fuel cell and photovoltaic applications [22, 28,
34, 45, 58, 67]. Also applications for integrated starter alternator have been reported in
[41, 46, 64], applications for general motor drives have been proposed in [25, 26, 72].

As an example, the configuration for photovoltaic system is shown in Figure 1.11.

C (”G J JG
_ q .__,,_ 120/240

’11‘ 60Hz
“ LSplitPhase
J «a «.2: ~
“To" - - -

Figure 1.11. Configuration for Z-source inverter for photovoltaic systems.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\l_

\I
ll

 

 

 

 

 

 

1.4 Scope of the thesis

Based on the previous works of traditional inverter and Z-source inverter, this thesis

focuses on the following subjects:

Chapter 2 discusses PWM methods for the Z-source inverter, and presents three PWM
methods to achieve simple boost, maximum boost, and maximum constant boost of the
Z-source inverter respectively. A modified method to achieve high efficiency is also
presented and a brief comparison between the proposed methods and an existing method

is conducted.

Chapter 3 presents a small signal model of the Z-source inverter for an inductive load,
and constructs a nonlinear controller for the feed back control using gain scheduling

method.

16

Chapter 4 conducts a comprehensive comparison of the Z-source inverter and traditional
inverters for fuel cell vehicles. The design procedure of the Z-source inverter is also

presented in detail.

Chapter 5 discusses possible operation modes of the Z-source inverter with small
inductances, low power factor, and low modulation index. The circuit performance is

also discussed under different conditions.

Chapter 6 focuses on the application of the Z-source inverter for fuel cell-battery hybrid
vehicles. A control method to eliminate the undesirable operation modes is presented.

Three configurations for fuel cell-battery hybrid vehicles are proposed and analyzed.

Chapter 7 discusses future work that needs to be done.

17

CHAPTER 2 PWM Schemes for the

Z-Source Inverter

Many PWM control methods have been developed and used for the traditional 3-phase
Voltage—source (V—source) inverter. The traditional V-source inverter has six active
vectors (or switching states), when the dc voltage is impressed across the load, and two
zero vectors when the load terminals are shorted through either the lower or upper three
devices. These total eight switching states and their combinations have spawn many

PWM control schemes.

The Z source inverter has an additional zero vector, the shoot-through switching state,
which is forbidden in the traditional V-source inverter. How to insert this shoot through
state becomes the key point of the control methods. It is obvious that during the shoot
through state, the output terminals of the inverter are shorted and the output voltage to the
load is zero. The output voltage of the shoot though state is zero, which is the same as
traditional zero states, therefore the duty ratio of the active states has to be maintained to
output a sinusoidal voltage, which means shoot through only replaces some or all of the
traditional zero states. Three different PWM methods will be presented in this chapter

and be compared.
2.1 Simple Control

As described in [12], the voltage gain of the Z—source inverter can be expressed as:

18

I?
——"C—=MB, (2.1)
V0/2

A

where Vac is the output peak phase voltage, V0 is the input dc voltage, M is the

modulation index, and B is the boost factor, which is determined by:

B = —l— (2.2)

1—2E
T

where To is the shoot-through time interval over a switching cycle T, or T7? 2 DO is the

shoot-through duty ratio.

Figure 2.1 illustrates the simple control method that employs a straight line equal to or
greater than the peak value of the three phase references to control the shoot-through duty
ratio in a traditional sinusoidal PWM. The Z-source inverter maintains the six active
states unchanged as the traditional carrier based PWM control. For this simple boost
control, the obtainable shoot-through duty ratio decreases with the increase of the
modulation index, M. The maximum shoot-through duty ratio of the simple boost
control is limited to (l-M), thus reaching zero at a modulation index of one. The thick
curve in Figure 2.2 shows the maximum obtainable voltage gain, M-B versus M, which
indicates no voltage boost and no voltage gain at M=1. The shaded area is the possible
operation region under the simple control. In order to produce an output voltage that
requires a high voltage gain, a small modulation index has to be used. However, small
modulation indices result in greater voltage stress on the devices. Based on the (2.1)

and (2.2), define the voltage gain G as

19

A

G=MB= Vac = M . (2.3)
V0/2 2M—1

 

For any desired voltage gain G, the maximum modulation index can be used is

20 -1 (2.4)

The voltage stress V, across the switches is BVO, The voltage stress under this voltage

gain can be calculated
VS = BVO = (20 —1)V0. (2.5)

The voltage stress across switches versus the voltage gain is plotted in Figure 2.3.
Using this control method, the voltage stress across the switches is quite high, which will

restrict the obtainable voltage gain because of the limitation of device voltage rating.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vp
V i >
Va \ Vc
,
/
<
\ \
Vn
Sap LIL] Ll ufl u 1‘ _
pri F1 J LHJ 1h-
scp Ufl_J u ‘ J‘l Ll U11
SanLl u 1‘1 LHI 1
sbn U ll]J LJHJ '1 I"
1 ‘ 1 um I um I

 

 

Figure 2.1. Simple boost control.

20

to

an

\I

O)

A

to

switch voltage stress / Vo
U'l

Voltage gain (MB)

 

    
        

0'3 1 1 2 3 4 5

0.2 0.4 0.6 .
Modulation index (M) Voltage galn

Figure 22. Voltage gain of the simple boost Figure 2.3. Switches voltage stress versus

control. voltage gain.

2.2 Maximum Boost Control

2.2.1 Description of the Control Method

Reducing the voltage stress under a desired voltage gain now becomes important to the
control of Z source inverter. As analyzed above, the voltage gain is defined as MB, and
the voltage stress across the switches is BVO, therefore, to minimize the voltage stress for
any given voltage gain, we have to minimize B and maximize M, with the restriction of
that their product is the desired value. On the other hand, we should maximize B for
any given modulation index, M, to achieve the maximum voltage gain. Consequently,

from (2.2), we have to make the shoot through duty ratio as large as possible.

Figure 2.4 shows the maximum boost control strategy. It is quite similar to the
traditional carrier-based PWM control method. The point is: this control method
maintains the six active states unchanged and turns all zero states into shoot-through zero

21

states. Thus maximum To and B are obtained for any given modulation index, M,

without distorting the output waveform.

7“ ”° l? “01 bl
)1

 

O<

 

7\

 

 

\\<

 

/ \T
/
cal-11----
Q
///\
alg'
\/7’

 

 

 

 

 

 

        

 

     

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

, w v 1 ~ 1 ‘2
sap l 0 fl _ g’
pr‘HJJ 1 r m J LL L J'1 g
509“ T T"CLLWJ_U L
San—TUTLTlm____7_rl_qJ “'1 H
:bnnflLJ-lUfl—L Ll—j—l —fl_Ll_‘iji—E:lf 0 0 0.2 0.4. 0.6 0.8
cn Modulation Index (M)

Figure 2.4. Sketch map of maximum boost

Figure 2.5. Vac/0.5 V0 versus M.
control.

As can be seen from Figure 2.4, the circuit is in a shoot through state when the triangular
carrier wave is either greater than the maximum of the references (Va, V1,, VC) or smaller
than the minimum of the references. The shoot through duty cycle varies each cycle.
To calculate the voltage gain, what we are interested in is the average shoot through duty

cycle.

The shoot through state repeats periodically everyg. Assuming that the switching
frequency is much higher than the fundamental frequency, the shoot-through duty ratio
over one switching cycle in the interval (%,%) can be expressed in the following

equation

22

2 — (M sinB — Msin(6 — 2375))

 

 

 

T0 (9)
__ = 2.6
T 2 ( )
The average duty ratio of shoot-through can be calculated by following equation
1 E E
2 2
[ZdB- ([Msin 6b'6- [Msin(6 — ——)d6)
_ 7: II
T * ‘* “ _
_Q ___ 6 6 6 = ___—.2” 36M . (2.7)
T _75 27r
2
[2d9
K
6
The boost factor B is obtained:
1 II
B = (2.8)
1_ 210 :3w/3M -7z'
T

With this type of control method, the voltage gain can be determined by the modulation

index M.

9a,; =MB= 7rM
V0/2 3fiM—n

 

(2.9)

The plot of—— V “/2 ——versus M rs shown by the thick curve in Figure 2. 5. The shaded area
0

in the figure is the possible operation region. As can be seen from Figure 2.5, the output

. 7r
voltage increases when M decreases. As M approaches — , the output voltage

373

increases to infinity.

Compared to Figure 2.2, the possible operation region of this control method is much

23

wider. On the other hand, for any given voltage gain, a higher modulation index can be

used, which means lower voltage stress across the switches.

From equation (2.9) and voltage gain defined in (2.3), the maximum modulation index

we can use for a given voltage gain G is

 

M = —”—G——. (2.10)
3x50 — 7r
Thus the voltage stress is
V ._
VS=BV0= ”0 =3‘/3G ”V0. (2.11)
3.5M —- 7r 7:

The voltage stress versus the voltage gain is shown in Figure 2.6.

Compared with the voltage stress in the simple control method, which is shown in Figure
2.3, the voltage stress in the maximum boost control method is much lower, which means

that for given devices, the inverter can be operated to obtain a higher voltage gain.

 

Switch voltage stress No

 

 

 

 

1.5 2 2.5 3 3.5 4 4.5 5
Voltage gain

Figure 2.6. Voltage stress versus voltage gain.

24

Third harmonic injection is commonly used in three-phase inverter systems to increase
the modulation index range. This can also be used here to increase the range of M so as
to increase system voltage gain range. The sketch map is shown in Figure 2.7. The
operating principle is identical to the above method, the only difference is that the

reference waveforms are changed. In this control, the maximum modulation index
2 . l . . . . .

M = — can be achieved at —th1rd harmonic Injection [17].
,5 6

Similar to the previous case, the shoot through duty cycle repeats every 1;. We can also

get the voltage gain through studying the behavior during (%,12r—) under this control

method. The shoot-through duty ratio in this period is described in the following

 

 

 

equation.
. 1 . . Zn 1 .
2 — (MsrnB + —Msrn36 —(Msrn(6———) + —Msrn36))
To (9) _ 6 3 6
_ (2.12)
T 2
From the above calculation, the average shoot-through duty ratio is:
E E E
2 2 . l . 2 . 27: 1 .
[2616 — ( [(M sm (9 + — M srn 3B)d6 —- [(M srn(B — —) + — M srn 3B)dB)
_ 71' 7f 6 7f 3 6
7‘6 = z a ‘6‘
71'
T "2‘ (2.13)
j2d6
Z
6
_ 27: — 3f3M
27r
Therefore:

25

 

1 7r
B=___=__ (2.14)
1_2Ig 3J3M—7r
T
9a.: W >
= =____ (2.15)
W2 3J3M—7r

The voltage gain is identical to the maximum boost control method for the same
modulation index. The curve of voltage gain versus modulation index is shown in
Figure 2.8, from which we can see that the possible operating region is extended with the
increase of modulation index. From (2.9) and (2.15), the two control methods have
identical voltage gain—modulation index relationship. Therefore they should share the
same voltage stress for any given voltage gain except that the range of voltage gain is
extended in the third harmonic injection method. The relationship of voltage stress

versus voltage gain is shown in Figure 2.9.

1% l

:5

 

 

 

 

 

 

 

 

 

V .
.1 l l [f .. .
= 1 Wk
l \
I 7: g j; A ___.
311M 1 i 17 i
I K :
\ N “a ./ —— - -__
“-1 -< -—-- —- a) 4
S v t v v H ,
SapJ'lJ mum _h_ .1
bPJflLMWJJ 1 g
Scp L rLLM‘lJJJ
San—“ULMLLM‘ r
em UTJ‘LZILT_TI_W
36,1374?me LUMT-

 

 

 

 

 

 

 

 

    

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

      

    

4 0 6 0.8 l

0 0.2 0. .
Modulation index

Figure 2.7. Sketch map of third harmonic

. . . Figure 2.8. Vac/0.5% versus M.
1nject10n control.

26

 

switch voltage stress {V0

 

 

 

 

Voltage gain

Figure 2.9. Voltage stress of switches versus voltage gain of proposed control method.
2.2.2 Simulation and Experimental Verifications

Simulation and experiments are conducted to verify the validity of the control strategies
with the following parameters: Z-source network: L1 = L; = lmH, C, = C2 = 1.3mF,
Switching frequency: 10 kHz. The simulation results with modulation index M=0.88,
M=l, and M=I.1 with third harmonic injection are shown in Figure 2.10, Figure 2.11, and
Figure 2.12 respectively, where V0 is the input voltage, Vpn is the DC bus voltage, and
V W is the output line to line voltage. The input voltages of these three cases are 170 V,
220V and 250 V respectively. Based on our analysis above, the theoretical voltage

stress (Vpn) and output line to line nns voltage is listed in Table.2.1.

27

Table 2.1 Theoretical voltage stress and output voltage under different conditions

 

 

 

 

 

Operation condition Voltage stress (V) Output voltage VL-L (V)
M=0.88, Vo=170V 373 . 200
M=1, V0=220 V 336 206
M=1.1, Vo=250 V 305 205

 

 

 

The simulation results in Figure 2.10 through Figure 2.12 are quite consistent with the

theoretical analysis, which verifies the above analysis and the control concept.

Experiments with the same operating conditions and system configuration as in the
simulation were conducted. The results are shown in Figure 2.13, Figure 2.14, and
Figure 2.15, where V0 is the input voltage, VpN is the voltage stress, IL; is the current
through the inductor in the Z-source network, and VIA], is the output voltage after the filter.
The figures show that the output voltage is kept nearly constant regardless of the wide
varying range of the input voltage and the output voltage agrees with the analysis and
simulation results very well. A lower input voltage requires a greater boost factor B,
and smaller modulation index. Because the output voltage is nearly constant, the lower
modulation index yields higher voltage stress, VpN. This can also be observed from the
experimental results. Based on these results, the validity of the control methods has

been proven.

28

 

(V) ”(5) (V) :1181
200.0 . V0 '

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2300' V0
A 180.0 1 "
160.0 ‘ 420.0“
140.0 ‘ (v) :t(s) 210-0 (V) :t(s;
V n 4000
400.0 9 Wm
’>‘ j: v
" 200.0 . ; 2000
J I .
0.0 (V) 21(8) (V) 208'
4000
V .~ _. VLab
ZOO-0i. Lab . $20M / \\ ,/ \g f
E 0.0 -< ‘ ‘.\ ['1’ \-\ 1,"; /-i “.\\ /’ /I
I, jzwol ‘\.__1
2000* \V/ \/ 4000 l 1 I
I I 1 1 _T——"
0.06 0.07 0.08 0.09 0.1

0.07 0.075 0.08 0.085 0.09 0.095
t(s) l(5)

F' 2.11.' 11' 1t 'thM=1
Figure 2.10. Simulation results with M=0.88 ‘gure S‘muam’esu 5“"

d' t It 220VDC.
and input voltage 170 V DC. an rnpu V0 age

 

 

 

 

 

 

 

(V) :t(s)
280.0 ‘ Va
E 260.0 "
240.0
220.0 ' (V) :t(s)
' v
400.0 [M
E __ __ __ _ __ ..L __
200.0
0.0 (V) :t(s)
200.0 ,/ fifi\ .’/ W“
E o. 0

 

-2oo.0 \ / \1 /l

0.07 0.075 0.08 0.085 0.09 0.095
t(s)

Figure 2.12. Simulation results with M=l .1 with third harmonic injection and input

voltage 250 V DC.

29

 

. v 1

V0 (250 V/(ii’v)

 

 

 

,. VPN (250 V/diy) . ' V0 (250 V/div)
' . ' ' VPN (250 V/div)

 

    

 

 

 

 

 

Figure 2.13. Experimental result at M=0.88 Figure 2.14. Experimental result at M=l

and input voltage 170 V DC. and input voltage 220 V DC.

 

 

 

 

 

 

Figure 2.15. Experimental result at M=1.1 with third harmonic injection and input voltage

250 V DC.
2.2.3 Current Ripple

This control method can achieve the minimal voltage stress across the switches for any
desired voltage gain. However, from Figure 2.16, the shoot through duty cycle is not

always a constant, and thus introduces current ripple through the inductors.

30

 

Qifls

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:1 ECIE
:I I3
FTP

 

Figure 2.16. Maximum boost control sketch map.

To calculate the current ripple through the inductor, the circuit can be modeled as in
Figure 2.17, where L is the inductor in the Z-source network, V, is the voltage across
the capacitor in the Z-source network, and V,- is the voltage fed to the inverter.

Neglecting the switching frequency element, the average value of V, can be described as
Vi=(1-—D0)*BVdc . (2.16)

From above analysis,

0 ° 2
2—(Msrn6—Msrn(6l-§7f)) f3—

D0(t9)= 2 =1——2—Mcos(B—%7r) (%<6<%) (2.17)

 

and (2.18)

72'
B=———— .
3J3M—7z'

As can be seen from Eq. (2.17), D0 has minimum value when 19:33!- and has

31

. 7r 7: .
maxrmum value when 6 = g or B = 5. If we assume the voltage across the capacrtor

is constant, the voltage ripple across the inductor can be approximated as a sinusoid with

peak-to-peak value of

,5
.6 6 <___-—-:i->M

 

VPkZPk : Vimax Vimin =(_— 2 MT'E’MC c—OS( 6))*BVdc 3J—M- 7r Vdc’ (2-19)
If the output frequency is f, the current ripple through the inductor will be
73 3
— —— MV
VPkZp" ( 2 4) dc (2.20)

 

 

*z*6f*L=12*(3J§M—7z)fr '

As can be seen from Eq. (2.20), when the output frequency decreases, in order to

maintain the current ripple in a certain range, the inductance has to be large.

 

 

 

Figure 2.17. Model of the circuit.

2.3 Maximum Constant Boost Control

2.3.1 Description of the Control Method

In order to reduce the volume and cost, it is important to keep the shoot-through duty
ratio always constant. At the same time, a greater voltage boost for any given
modulation index is desired to reduce the voltage stress across the switches. Figure 2.18

shows the sketch map of the maximum constant boost control method, which achieves

32

the maximum voltage gain while always keeping the shoot-through duty ratio constant.
There are five modulation curves in this control method: three reference signals, Va, Vb,
and Vc, and two shoot-through envelope signals, Vp and V". When the carrier triangle
wave is greater than the upper shoot-through envelope, Vp, or lower than the lower
shoot-through envelope, V", the inverter is turned into shoot-through zero state. In
between, the inverter switches in the same way as a traditional carrier-based PWM

control.

Because the boost factor is determined by the shoot-though duty cycle, the shoot-through
duty cycle must be kept the same in order to maintain a constant boost. The basic point is
to get the maximum B while keeping it constant all the time. The upper and lower
envelope curves are periodical and are three times the output frequency. There are two

half-periods for both curves in a cycle.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

A vp .
a \ Vc

~ - tvn — v
Sap—Ll‘lJ I Ll "1.1—l.
pr]_l 7_l—I_H_l .
Scp UFU L_l _l_l—l_JT ]_J Ifll
Sanlr Ll 'LJ 'l___l U—U
SbnT LflJ—Ll 7_l_b
San. l LHJ UTJ—l

 

 

 

 

 

 

 

 

 

 

 

Figure 2.18. Sketch map of constant boost control.

33

 

voltage gain (MB)

 

 

 

 

Figure 2.19. Vac/0.5 V0 versus M

For the first half-period, (0, KB) in Figure 2.18, the upper and lower envelope curves can

be expressed by Eqs. (2.21) and (2.22), respectively.

VPl =\/3M+sin(B——2—37£)M 0<9<§ . (2.21)

an .—. sin(6l — 2—3’5)M 0 < a «25 . (2.22)

For the second half-period (rt/3, 27t/3), the curves meet Eqs. (2.23) and (2.24)

respectively.
. 7: Zn
sz = sm(B)M —3— < 6 < —3— . (2.23)
Vnz = sin(B)M — J3M % < (9 < 237: . (2.24)

Obviously, the distance between these two curves is always constant, that is, «[3M .

Therefore the shoot-through duty ratio is constant and can be expressed as

34

_=_.___=1___ , (2.25)

The boost factor B and the voltage gain can be calculated:

1 l

B: = .
IQ f3M—1
1—2T

 

(2.26)

—VL—=MB= M
Vdc/2 fiM—l

 

(2.27)

The curve of voltage gain versus modulation index is shown in Figure 2.19. As can be

5

seen in Figure 2.19, the voltage gain approaches infinity when M decreases to —.

This maximum constant boost control can be implemented using third harmonic injection.
A sketch map of the third harmonic injection control method, with 1/6 of the third

harmonic, is shown in Figure 2.20. As can be seen from Figure 2.20, Va reaches its peak

value —2—3—M while Vb is at its minimum value -73M . Therefore, a unique feature

can be obtained: only two straight lines, Vp and V", are needed to control the

shoot-through time with 1/6 (16%) of the third harmonic injected.

35

 

 

 

_—

6
l
x—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—. 2: '
/
K
\h

SapUflJ 117.] U L
prl _ _lJ
Scp fill-LI L— T —l_.l Lil-ll
Sanif U E _l_l LJTJ
Sbnll l LI“ 1_ T — ‘1
Scnl —l__l Ll-lJ . UFUTI

 

 

 

 

 

 

 

 

 

 

 

Figure 2.20. Sketch map of constant boost control with third harmonic injection.

The shoot-through duty ratio can be calculated by

T_0=__2"/§M .—_1___‘/§M , (2.28)
T 2 2

As we can see, it is identical to the previous maximum constant boost control method.

Therefore, the voltage gain can also be calculated by the same equation. The difference is
that in this control method, the range of M is increased to g 3 . The voltage gain versus

M is shown in Figure 2.21. The voltage gain can be varied from infinity to zero smoothly
. . 75 2 . . . .
by mereasrng M from —3— to —— wrth shoot-through states (solrd curve 1n Frgure 2.21)

5

and then decreasing M to zero without shoot-through states (dotted curve in Figure 2.21).

36

 

voltage gain (MB)

 

 

 

 

Figure 2.21. Vac/0.5Vo versus M.

2.3.2 Simulation and Experimental Verifications

Simulation and experiments were conducted to verify the validity of the control strategies
with the following parameters: Z-source network: L I = L2 = 1 mH (60 Hz inductor), C I =
C 2 = 1,300 uF; switching frequency: 10 kHz; output power: 6 kW. The simulation results
with the modulation index M = 0.812, M = 1 without third harmonic injection, and M =
1.1 with third harmonic injection are shown in Figures 2.22 through 2.24, respectively,
where the input voltages are 145, 250, and 250 V, respectively. Table 2.2 lists the
theoretical voltage stress and output line-to-line rrns voltage based on the previous

analysis.

37

Table 2.2. Theoretical voltage stress and output voltage under different conditions

 

 

 

 

Operating condition Voltage stress (V) Output voltage VL-L (V)
M = 0.812, Vdc= 145V 357 . 177
M = 1, Vdc= 250 V 342 209
M=l.1 ,Vdc= 250V 276 186

 

 

 

 

 

The simulation results in Figures.2.22—2.24 are consistent with the theoretical analysis,

which verifies the previous analysis and the control concept.

150
...l
.l
400
a

 

 

 

 

 

 

 

 

 

 

 

Input 260 “ Input
Voltage 255 Vgltage
245
250 ,

 

 

DC Link
Voltage

DC Link 40°
Voltage
Vpn

   
 

E 200

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Inductor grductor
Current I urrent
3 3J L
0.0
400 ' 400 Load
Load -
200 /\ Voltage 200‘ /\ / \ / \\ ‘plttLge
S a . 1] l \/ vhb 2 °l ‘ ' ‘\ " \‘\ */
‘20 V "°°‘ V \/ \/
' 40 l I I l l . ‘00 I l I I
0.2 5 0.15 0.15 017 0.1 8 0.19
I") «I

Figure 2.22. Simulation results with M = 0.8. Figure 2.23. Simulation results with M = 1.

38

 

 

 

 

 

 

 

 

‘W Input
:55 Y," “86
E ‘w o
245
l”
*0” DC Link
Voltage
V n
8 200
Inductor
4o urrent
s ‘i
ll.

 

 

Load
a A /\ /\ me
M \ / \ /
\'\,,,/ \\_ /‘l \ _ //
115 im i111 i113 019
1(3)

 

 

 

Figure 2.24. Simulation results with M = 1.1.

The experimental results with the same operating conditions are shown in Figures 2.26,

2.27, and 2.28, respectively.

 

ass, ..aérfl...-. ...':. _. . .125: .g... _»;-.._...~.:;;-_.:WLL
1 V0 (200 V/cii’v'); . ,

   
 
     

   

[4.

l ”I“ (20 ‘A/diV)" "r' " '

   

m1 GMV

 

4» Mann: 129(3): Bower an «hm! 1:12 400m? hum-1295A 3M
{:4 skint A ff". / 455m V um "um .: Ch! «(MN I {M I 158! V

Figure 2.25. Experimental results with V, = Figure 2.26. Experimental results with V, =

145V and M=0.812. 250V and M= 1.

39

 

l
l" ‘ v,,,,( 200w WK"
F\ /r\\ /’l -

 

Figure 2.27. Experimental results with Vo = 250V and M = 1.1.

Based on these results, the experimental results agree with the analysis and simulation
results very well. The validity of the control method and the analysis of the voltage gain
and voltage stress are verified. Also from the simulation and experimental results, it is
evident that there is no low-frequency (6a)) ripple in the inductor current. With the
same input voltage, the output voltage is higher for the case with lower modulation index.

With a lower input voltage, the voltage stress has to be higher to output a similar voltage.
2.3.3 Voltage Stress Comparison

To examine the voltage stress across the switching devices, an equivalent DC voltage is
introduced. The equivalent DC voltage is defined as the minimum DC voltage needed
for the traditional voltage-source inverter to produce an output voltage, V0. Obviously

the equivalent DC voltage should be GVdc. The ratio of the voltage stress to the
equivalent DC voltage represents the cost that Z-source inverter has to pay to achieve

voltage boost.
The voltage stress across the devices, V5 can be expressed as
VS 2 BVdc- (2.29)

40

The ratios of the voltage stress to the equivalent DC voltage, VS/(G Vdc) for the simple
control, maximum boost control, and maximum constant boost control are summarized as

follows:

V
S = B Vdc = 2 _ _1__ for simple control (230)
GVdc GVdc G

 

V
S = BVdc = 3J3 —l- for maximum boost (2.31)
GVdc GVdc 7r

 

 

VS _ B Vdc
G Vdc G Vdc

 

_-. J3 —E1;- for maximum constant boost (2.32)

Figure 2.28 shows the voltage stress ratios. As can be seen from Figure 2.22, the
maximum constant boost method has a much lower voltage stress across the devices than
the simple control while having a slightly higher voltage stress than the maximum control

method. The ideal voltage stress ratio is 1.

 

1.8
1‘781mplecontrol/’ ............. 1
1.6 ' '
1.5
1.4
1.3
1.2

1.1 .

 

Voltage stress / Equivalent DC voltage

 

 

 

3
Voltage Gain

Figure 2.28. Voltage stress comparison of different control methods.

41

2.4 Modified PWM Scheme

The inverter with maximum constant boost control with third harmonic injection shoots
through twice in one cycle (triangular waveform cycle), the equivalent frequency to the
inductor is doubled, thus reducing the requirement to the inductors. However, it is
obvious from Figure 2.20 that the real switching frequency of the device also doubled,

which increases the switching loss.

In traditional PWM control, there is always a zero state after two active states as shown
in Figure 2.29. There are two types of zero states, Zero 1 and Zer02, Zero 1 occurs
when all upper three switches are turned on, and Zero 2 occurs when all lower three

switches are turned on.

 

 

Zerol Zerol

 

 

 

 

 

 

 

 

Figure 2.29. Switching states sequence of traditional PWM control.

The control of the Z-source inverter maintains the active states unchanged and shoots
through some or all of the zero states. The key point of the modified PWM control is to
turn half of the zero states (Zerol or Zero 2) into shoot through state, and leave the active
states unchanged. The duty ratio of that shoot through state equals to the shoot through

duty ratio of maximum constant boost control, which is

.5

D0 =1—7M. (2.33)

Therefore, the shoot through period lasts (l—l/g-Mfls in each switching cycle T5,

42

which means that the zero state (Zero 1 or Zero 2) turned into shoot through state lasts
J3 . . . . . .
(l — 7M)Ts. To realrze this function, there are two possrble schemes shown 1n Figure

2.30 (a) and (b) respectively.

 

 

 

 

“/6 “1)
35 5’”
an at:
bn b n
Cn cn
(a) (b)

Figure 2.30. Modified PWM scheme.
For the case in (a), assume the reference signals in traditional SPWM are

va = M sin(wt)

Vb = Msin(a)t —%7r). (2.34)

vc = M sin(a)t —§7r)

There are three different intervals in one line cycle in the modified PWM method, the

reference signals in the three intervals are respectively

43

va'=J§M—1
vb'= J3M —1+v,, —va 2k7r+% 3 wt < 2k7r +grr, k = o,1,2,.... (2.35)

vc'zx/3M—l+vC—va

va'=s/3M—l+va-vb
vb'zx/3M—l 2k7r+§6£_<_a)t<2k7r+%7r, k =0,1,2,.... (2.36)

vc'=\/3M—l+vC—vb

va'=\/3M-l+va—vc
vb'zx/SM—1+vb —vc 2k7r+127£5wt<2k7r+-lg3—7r, k=O,l,2,.... (2.37)

vc'zx/SM—l

where va, vb, and vc are expressed in (2.34). When the triangular waveform is higher

than the maximum value of the three, which is J3M —1 , the circuit is turned into a
shoot through state by gating on all the switches, otherwise, it operates the same as
traditional SPWM. It is obvious that the duration of each active state in a switching
cycle is kept the same as in traditional SPWM by keeping the distance between the
reference curves unchanged, therefore the fundamental element of the output voltage will

still be kept sinusoidal.

The second case is shown in Figure 2.30 (b), also there are three different intervals in one
line cycle for the modified PWM, the reference signals in the three intervals are

respectively

va'zl—fiM
vb'=l-\/3M+vb—va 2k7r+Zé£Swt<2k7r+l6l7zg k=0,l,2,.... (2.38)

vc'zl—x/3M4-vc—va

44

 

 

va'zl—x/3M+va—vb
vb'=1—J3M 2k7r+1—:—57£3a2t<2k7r+%7r, k=0,1,2,.... (2.39)

vc'=l—\/3M+vC—vb

va'=1—\/3M+va—vc

vb'=1—\/3M + Vb -—vc 2k7r —% S cot < 2k7r+—;-7r, k = 0,1,2,.... (2.40)
vc'=l—\/3M

where va, vb, and vc are expressed in (2.34). When the triangular waveform is lower

than the minimum value of the three, 1—\/—3M , the inverter is turned into a shoot
through state by gating on all the switches, otherwise, it operates the same as traditional
SPWM. Also the duration of each active state in a switching cycle is kept the same as

in traditional SPWM, therefore, the output waveform will still be kept sinusoidal.

From Figure 2.30, all six switches turn on and off 7 times in one cycle in total, which
reduces the switching actions quite significantly. The equivalent frequencies to the

inductors and to the load are both the triangular frequency.

The two different schemes achieve the same effect for the boost function of the inverter.
But the switching actions and the on time of the upper and lower switches in a phase leg
are different. From Figure 2.30.(a), all upper switches are turned on for 1/3 of the line
cycle taking turns, and all the lower switches are turned on and off every switching cycle.
It is the opposite for the case shown in (b). This results in unbalanced switching loss
and conduction loss between each of the switches inside the IPM, which could increase
the thermal stress of the switches. To minimize this unbalance, the two methods should

be used alternatively every line cycle.

45

2.5 Comparison of Different PWM Schemes

In this section, a comparative study of two different PWM schemes will be examined
using the switching loss, and passive components size as a benchmark. The two PWM
schemes are the methods shown in Figure 1.8 and the modified PWM mentioned above.
Basically, the first method shoots through six times in every switching cycle, and the

latter one shoots through only once per switching cycle.
2.5.1 Switching Loss.

Assuming the inverter power is P, input voltage is V,-,,, load power factor is PF. There
are two type of switching actions, one is the shoot through switching that is between
shoot through state and traditional state, the other is traditional switching state, which is

between traditional states.

a. Six shoot through method
For this method, every switching state is a shoot through switching state. Each
switching state current is twice of the inductor current, whose average value is the

average input current, thus the switching current is:

1, =2— (2.41)

b. Modified PWM

For this control method, there are two parts of switching losses: traditional switching
(switching actions between traditional states) and shoot through switching (switching
states between shoot through state and traditional states). The switching loss of

traditional switching can be calculated by:

46

1 . 1§£_ .
15w, =1peak 2—7‘_'(l(jtSIHde‘§ln6 alsrnx|dx)

g‘fl
1 1 5 1 7r 7:, (2.42)
I —— 2+—cos—7r—a —-—cos——a S—
: peak21fl( 2 (6 ) 2 (6 )) a 6

l 5 1 7r ' 7r
1 1+—cos—7r— +—cos——a >—
peak 27:} 2 (6 a) 2 (6 )) a 6

where [peak is the peak load current, a is the load phase angle. During shoot through
state, the current from the dc side is 211, , where 1;, is the inductor current. Assuming that
the current is evenly distributed in three phase legs, the average switching current of
shoot through state is 211/3. In each cycle, there are 3 shoot through switching states,

thus the shoot through switching loss of each IGBT is:

 

= , (2.43)

For power factor of c050 and modulation index of M, the load peak current is

P 3 MVdc

 

 

S = = — ————————— (2.44
0030: 2 pk 2(«/3M—1) )
Therefore,
_ 4(J3M — 1)P (245)

pk _ 3cosaMVdc

Putting all of this together, the overall switching current of the two different PWM
methods for different power factors is shown in Figure 2.31. As can be seen from the
figure, the switching current of the modified PWM is much smaller than in the six shoot

through PWM method, which means higher efficiency.

47

 

1 six shoot through method

.PF=0.5--1

, Modified PWM

0.4 e . .1
0.6 0.8 1 1 .2
Modulation index

Per unit switching current
.2;

9.0
once

 

 

Figure 2.31. Overall switching current comparison of different PWM.
2.5.2 Inductor Size

The inductor size is based on the current and the inductance. The average currents
through the inductors with different control methods are the same. The inductance is
designed based on the same current ripple. The inductor current rises linearly during

shoot through state and decreases linearly during other states.
a. Modified maximum constant boost control
The inductor current ripple is

Vc T0

L (2.46)

AIL:

where To is the total shoot through time in one switching cycle.

b. Six shoot through method

48

 

 

 

 

 

>

3°

Figure 2.32. Six shoot through PWM when va=va

The inductor current increases linearly during shoot through state and decreases linearly
during non-shoot through state with two different constant rates respectively. For the
six shoot through PWM method, the maximum current ripple occurs when the shoot
through periods are least distributed, ie, when two of the reference signal equals to each
other. Figure 2.32 shows an example when the reference signal va=vc. Assume that
1/6 of third harmonic is injected, and the modulation index is M, the reference signals can

be expressed as:

l

va = M sin a)! + A:— sin(3a)t)

< Vb = M sin(a)t — g 7:) + % sin(3cot) (2.47)

 

vc = M sin(cot — g7!) + }; sin(3a)t)
L

With maximum constant boost control, the shoot through duty ratio is
J3

DO =1-7M (2.48)

The total non-shoot through time in a cycle is

49

J3
Tm = 7MT,, (2.49)

where T 3 is the switching cycle. Assume that the middle reference signal is not shifted,

the traditional zero state T2 for this instant (va=vc) can be calculated by

1 J3 f 1

_ __ __1. _ __ =_3 __
T,_((1 2M 6M) (1 2M))TS/2 (4M 3M)T,. (2.50)

Considering that the total inductor current change in a switching cycle is zero, the current

drop in T, can be calculated as:

 

AT=IL*-V—CTO=3J§—4KQTO (2.51)
Tm L 673 L
The current change in T , can be calculated by
2 V , VCTO
Iripple =3-TO—LQ-Al =OH551T (2.52)

Therefore, compared to the modified PWM method, in order to keep the same inductor

current ripple, the required inductance with the six shoot through method is about 55%.
2.5.3 Capacitor Size

The capacitor size is determined by the capacitor current and the capacitance. The
capacitor current rating is the same regardless the modulation method. The capacitance
is selected so that the voltage ripple is small and the output THD is low. For a given
operating condition (power, load power factor, etc.) the capacitor charge ripple is shown
in the following figure for different control methods, different power factors, and

modulation indices. So the capacitance requirement is about the same.

50

Per unit capacitor voltage ripple

 

 

‘08 0.05 0.9 0.95 1 1.05 1.1 1.15
Modulation Index

Figure 2.33. Capacitor voltage ripple comparison for different power factor.
2.6. Summary

Three control methods: simple control, maximum boost control, and maximum
constant boost control have been presented in this chapter. Maximum boost control
achieves lowest voltage stress across the devices. However this method will also result
in line frequency current through the Z-source inductor, thus increases the requirement
for the inductor. The maximum constant boost control minimizes the voltage stress
across the device on the basis that there is no line frequency current ripple through the
Z-source inductor. A modified PWM method is proposed to minimize the switching
loss. A comparison between the modified PWM method and the existing six shoot

through PWM method is conducted.

51

CHAPTER 3 Modeling and Control of

the Z-Source Inverter with RL Load

There are several small signal models proposed [40, 43, 51, 59] for the Z-source inverter.
However in all the models, there is only one control parameter, which can not fully
represent the inverter characteristics. In this chapter, a complete small signal model for
the Z-source inverter will be proposed and a nonlinear controller will be designed.
Simulation and experimental results will be provided to confirm the validity of the

controller.

3.1 Modeling of the Z-source Inverter

3.1.1 Model Simplification

For a three phase inverter, the output power is always constant in steady state, thus the ac
side can be simplified as a dc load without loss of generality considering the dynamic
performance for three phase balance load. Figure 3.1 shows the three phase Z-source
inverter with a RL load. When operating at shoot through duty ratio of D and

modulation index of M, the equivalent circuit is shown in Figure 3.2.

In Figure 3.2, there are two switches, 81 is switching with a duty ratio of D, the duty ratio
of 82 is M, and D+M S l. The parameters of the Z-source network are the same As

the original circuit. For the Z-source inverter, the output power is

M l

2
1_2DVin ZZJZ) RLoad’ (3'1)

P=3(

52

where Z is the load impedance per phase.
While the load power of the dc equivalent circuit is

M 21
P: ——V- — ' 3.2
(1—2D m) R ( >

By equalizing the power of the two systems, we have

82

 

R (33)

=3cosgo'

LLoad RLoad

 

 

 

 

 

 

Figure 3.1. Z-source inverter.

L 82 x R
:— C C
Vin S]

v

 

 

Figure 3.2. do equivalent circuit.

where cosgo is the load power factor. Naturally X is determined so that the time

constant of the dc load is the same as the ac load, which means L/R is kept constant.
3.1.2 Modeling by State Space Averaging

To model the equivalent circuit, we have the following assumptions: the Z-source
network is symmetrical, so that the currents through the two inductors and the voltages

53

across the capacitors are the same. With this assumption, there are three state variables:
the current through the inductor in Z-source network, the voltage across the capacitor in

the Z-source network, and the current through the load inductor.
Before modeling the inverter, the following parameters are defined:
L: inductance of the inductor in Z-source;

C: capacitance of the capacitor in Z-source;

X: dc Load inductor;

R: dc Load resistance;

D: shoot through duty cycle;

M: duty ratio of the other switch (modulation index of the original system);
iL: current through the inductor in the Z-source;

ix: load current;

12,: capacitor voltage;

VdC: input voltage.

There are three states, when 81 is turned on and S2 is turned off, the state equation is

, 0 i 0 -
SIL 1 L 1L
svc = C 0 0 ye (3.4)
St _ 1
x 0 0 —R-"
._ x—

 

 

The duty ratio of this state is D.

54

The second state is when both switches are turned off, the state equation is

0 :1 0 - _Kctc:
siL 1 L ’L L
svc = C O 0 vc + 0 (3.5)
si _ i 0
x 0 0 J: -"
1— x _. .- ‘

 

 

 

 

The duty ratio of this state is l-D-M.

The third state is when 81 is turned off and $2 is turned on, the state equation is

 

o 11— 0 - _Kd; 1
SiL 1 L 1 iL L
svc = — 0 :— v, + 0 (3.6)
. C C .
31x 0 2 ___]: _lx ”Vdc
_ x x _ - x -

 

 

 

The duty ratio of this state is M.

Using state averaging, the system model can be obtained by
(3.4)*D+(3.5)*(l-D-M)+(3.6)*Mas shown in (3.7).

— q

. O _2D—1. O _. —V_d£_(l_D)

51L L ’L L
l—2D —M

svc = -— 0 ~-—— vc + 0 (3.7)

. C C . V

_ 1 -
S1x 0 2M _3 _x dc M
_ x x - - x _

 

 

 

 

The output variables to be controlled are
1.L
0 l 0 0
vc = vc — . (3.8)
vx 0 0 R . 0

’x

55

The small signal model of the circuit becomes

[8+ (1—21))2 + 2M2 1'17
CSL C(XS+R) C
= (1 —-2D)(2Vc -Vdc)—-2SLiL 543+ M(2VC — Vdc))47
CSL C C(XS+R)

 

(3.9)

 

 

Thus the transfer functions of the capacitor voltage over the shoot through duty ratio and

modulation index for a given operating point are respectively:

(XS + R)(Vdc — ZSLiL)

 

 

lie: 3 2 2 2 2 (3.10)
D CLXS +CLRS +(2M L+X—4DX+4D X)S+R—4DR+4D R
f2 = —LS(2MVC —MVdc +ixR+ixXS) (3 11)
M CLXS3 +CLRs2 +(2M2L+X—4DX+4D2X)S+R—4DR+4D2R

3.1.3 Model Verification

To confirm the small signal model, the bode diagram of the above transfer function is

compared with simulation results.

The following figures show the calculated bode diagrams of the inverter at M=0.75,

D=0.2, R=3.33Q, L=lmH, C=1.32mF, X=lmH.

56

Bode Diagram

  

 

Magnitude (dB)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3 ---
I; 279 H H

in

(B
S
0- 189

N‘~‘!-
an
101 1o2 103 10‘ 105
Frequency( radlsec)
Figure 3.3. Calculated bode diagram of V/D.
Bode Diagram

3

E

0

1:

3

'E

m

G

E

‘5

0

3

0

W

N

.C

n.

Frequency( radlsec)

Figure 3.4. Calculated bode diagram of V/M.

Normal simulation software is not able to perform small signal simulation for switching

57

The PSIM 6.0 [74] provides the ac sweep function for small signal

mode circuits.

Figures 3.5-3.6 show the simulated bode diagram of V/D and V/M

analysis.

The simulation software

respectively, which confirms the calculated bode diagrams.

uses signal injection and calculates the gain and phase point by point on a switching

circuit, therefore the results on some points is not very accurate.

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.
uuuuu 111111.1111111111 111117111.4
_
lllll L 11111 111 nflloln
. .
. .
iiiii 111111 11111111110111 1111114
.
.
11111 L11111 1111;
.
—
.
IIIII I‘IIIII Illtl 1.1111
.
.
.
.
v 11111 . 111111111111111111111111111
-
.
.
.
u
.
.
r-.--4 ...............
.
.
.
.
. .
_ .
. .
. .
.
. .
a -
wv1uul 111111111111111111111111111
. . _
11111 111111.111 11111101111111111.
. . .
w IIIIIIIIIIIIIIIIIIIII Q lllllllllll
. . .

. .
11111111111111111111111111111111
. .

. .

1 nnnnnnnn r 11111111111111111 f 11111

. . .
. . .
. —
.1 1111111111111111111n 11111 . 111111
. . .
. . .
1 11111111111 . 11111 h 11111 €111:
T . . . .
. .
u _ -
. .
. . . _ .
1111.11-11“ 11111 flint-q 11111 . 111111
n - p g n
_ . . . .
. . . . .
. . . . .
. _ _ _ .
. . _ _ .
11111 1111111111117.:11141111111111A
. . . . .
. . . _ .
. . . . .
. . . . .
. . . u .
o p n — .
. . . . .
. . . . .
. . . . .
. . . . .
. . .
F P b b
0 0 0 m 0 0
0 0 0 . 0 0
0 0 0 0 0 0
5 U 5 0 5 0
. 4 4.. 9.. o. 3

3.00

'2 .00

0.80 0.80 1 .00

0.40

0 .08

0 .06

0.02

0.01

Frequency (KHz)

Figure 3.5. Simulated bode diagram of Vo/D.

58

.47.... .

emp1‘ V0 7 )

 

  
 
    

1 l l I 0 l l ‘ ' l l l I I
i 1 1 1 1 I : 1 1 1 1 1 1 1
$001 1 .1 1 1 1 1 . 1 1 . 1
_P"r"1"1"-‘"""“"""‘r"“"‘""" ._. ....'___l.
i 1 1 1 1 1 1 1 1 1 '1 1
1 1

«mo *4 ............. g

i
3000 l
, .

l

' I l
. 1 1 1 1 . : 1 1 1 1 I 1 1' I 1
1 p l I y l l l I l I I I ' 1 - v
20001 1 1 11 1 1 I 1 1 1 1 11 1 I 1
. 1-mp»‘.J}x¢e67T-..u..._r.n..__1_..__ -_.,w.1.-fl._,.'-,_-.-----..n-,h-.-_e.r_.--1e-.e
“#1 I 1 1 1 1 I 11 1 1 1
1 1 1 1 1 1 1 1 1 I 1 1

10,00l.-:..:_.:._; ____________ f ________ i“”_.'___i

 

1
1
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 l '1 1 I 1 I f V"‘1-_‘1-'T‘YTI ““““““““““ 1 """" l' """ "'1""_-"'I“"""1“‘1"f'" I "'W‘
l 1 1 1 1 1 1 I 1 1 1 1 1 1 1 I 1 1 1 I 1 1
1
00 l 1 1 I 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - 1 1 1 a
0 .H.L J. L.u.._-.....-..L..-..-J-.-..k-..1..4..J-.L.1_1_.-..--.----J..-.--_l--_.J..-J.--I.-L-J_.k1:-.-.--._-.-uu.-.-_...
. 1 1 1 1 1 1 1 1‘ I 1 1 1 1 1 I - 1 - 1 1 1 1
I - 1 1 I
i 1 1 1 1 1 r I 1 1 1 1 1 1 1 1 1 r 1
-1000l 1 1 1 I A ; 1 1 1 l 1 1 1 1 . 1 1 1
. .__ 4 L 1 1 4 n 1 1 - L A 4

 

phese(V07)
5°00: . I T .'

 

 

400.001 T f f :
450.00
.20000 1: 5:

-250 .00

 

 

 

300.00 3;:3 : " ‘Iz'ii: L l ' --~'. .
0005 00000.01 0.02 0.04 0.06 0.00 0.10 0.20 0.40 0.50 0.80 1.00 2113 4L”
Frequency (KHz)

 

Figure 3.6. Simulated bode diagram of Vc/M.

From the above results, the bode diagram matches well, the proposed small signal model

is verified.

3.2 Controller Design

3.2.1 Gain Scheduling Controller

As can be seen from (3.7), the system is a nonlinear system. A very common way to
design a controller for nonlinear system is to linearize the system around an equilibrium
point and design a linear controller for it. However, for tracking purpose, the Operating
point changes with the reference, the linear controller can not guarantee the stability of
the system when the operating point is far from the equilibrium point the controller was

designed for.

A natural way to design a nonlinear controller then would be a gain scheduling controller

59

[75, 76], which changes the controller gain along with the operating point. There is a
limit in this control method, the operating point has to change relatively slow so that the
system is always at or close to the operating point the controller is designed for,

otherwise the system could be unstable.
3.2.2 Controller Design

In order to design a gain scheduling controller, one has to design a series of linear
controllers for all possible equilibrium points. Also the linear controllers must be
parameterized by a certain parameter that is related to the targeted output. The variable

parameters at equilibrium points of the system for a targeted output values (V633, Vm) are:

 

szs
53
2Vcss " Vdc
_ Vcss — Vdc
ss _
2Vcss " Vdc
2
szs
1 = _ 3.12
lss R Vdc ( )
Vcss = Vcss

The system has to be linearized at this equilibrium point. For a general form of system:
i=f(x,u), (3-13)
To linearize the system around an equilibrium point of (x55, u”), fix, u) is expressed in

Taylor series for the operating point of (x 53 + 365.1155 + u 5) ,

d(xss + x5)

0’ d
= f(xss,uss)+——f(xss,uss)x5 +—f(xss,uss)u5 +0(x5,u5). (3.14)
dt dx du

As a result, the following equation can be derived:

60

5:

which is a linear system.

6f

6f

5 = afocsssussho +af(xssauss)u§9

(3.15)

Based on the above method, the model can be linearized at any given equilibrium point,

the linearized ABCD parameters are as follows.

 

 

 

 

 

 

 

 

 

0 — Vdc 0
(zvcss " Vdc)L
Azglxzxssmzuss = Vdc O ‘szs (3.16)
(3x (2Vcss - Vdc )C (zvcss "' Vdc )C
0 2szs ___R
(2vcss — Vdc )x x
-2Vcss " Vdc 0 -
L 2
6f VXSS vxss
B - —— = , = = - 2— -— 3.17
au lx x” u "SS CR Vdc CR ( )
0 2vcss _ Vdc
_ x -
6g 0 1 O
C=Ex=xss,u=uss=[0 O R] (3.18)
g 0 0
D=Ex=xss,u=uss= O O (3.19)

 

The input voltage is assumed to be varying relatively slowly, so it is not counted in the
dynamic response of the system. To simplify the system, the following variable

transformation is implemented,

(3.20)

61

The ABCD parameters after the transformation become

1

 

V
(2 4535— — 1)C
Vdc

 

2 VCSS

Vdc
L

vxss 2

B: _ (Vdc

 

—1

 

V
(2 fl - 1)L
Vdc

2 vxss
Vdc

 

 

V
(2 £3— — 1)x
Vdc

—1

0
L

CR Vdc CR

2&4

“13 31

The second step in designing a gain scheduling controller is to design the controller for a

series of possible equilibrium points.

_ vxss

 

 

o
_ ”_xsg
v Vdc (3.21)
(2 fl — 1)C
Vdc
:13
x
(3.22)
(3.23)
(3.24)

To make sure that the output variables are

controlled to the desired values, integral control for the output variables are used. Thus

the new state equations are:

62

£=A§+Bu
o"=y-—r , (3.25)
y=C§+Du

where r is the tracking reference signal. Thus the new system becomes

, A 0 B
2 = z + u. (3.26)
C O D
Design a controller u = Kz so that the closed loop system
' ( A O + B K) (3 27)
z = z .
C 0 D

is stable, which means the poles of the closed loop system locates in the left half plain.
MATLAB command place can achieve this goal. As a result, we get the controller gain
K for one equilibrium point. There are two output variables, if we parameterize the
controller with the two variables with n possibilities for each, we have to design n*n
controllers. In order to simplify the design process without loss of generality, we can fix
the relationship of the targeted values of the two output variables as long as the output
voltage is the only variable the system is required to control. From Figure 3.2, when S]

and 82 are turned on and off complementally, the following equation will be reached:

Vcss = szs - (328)

Transfer back to the three phase system, this means the inverter is controlled with simple
boost and M+D=1. However, for the purpose of generality and to improve the transient

response, we choose M+D<l, thus the following equation is derived:

V

CSS

= kratio szs 2 (3 29)

63

 

where kmu-o 21 .

Now we can use only one parameter to parameterize the controller for a given kmuo, after
we design a series of linear controllers and parameterize all the controller gains with

Vm/Vdc, the gain scheduling controller design is finished.

It is noteworthy to mention that in all above processes, the input voltage is considered as
a given constant value or varying very slowly, this is because that in real system, the
input voltage usually changes relatively slow if it is not constant, it will not affect the

dynamic performance.
3.3 Simulation Results

Simulation of the closed loop system is performed with the following circuit parameters
and the configuration shown in Figure 3.2: L=1m, C=1.32m, Rf3.33ohm, X=lmH,
V,-,,=200V, kra,)0=l.05 the closed loop system poles are placed to [-100+5i -100-5i -200+i
-200-i -300]. Figure 3.7 shows the simulation results using the model in MATLAB.
From the simulation results, the capacitor and output voltage/current follows the reference

quite well in a wide range with acceptable amount of overshoot.

64

450;
400 (
3505
300‘
250
200;; , A

 

100 »

 

 

Figure 3.7. Simulation result of closed loop performance.

To examine the robustness of the controller, circuit simulation with mismatched
parameters using SABER is performed. The controller is the same one as above, the
parameters of the circuit in simulation are L=O.85m, C=1.12m, R=3.20hm, X=1mH,
V,,,=200V, k,a,,~o=1.05. The reference output voltage has a slow step change from
Vm/ 14n=1.1 to 1.8 and back to 1.1. The simulation result is shown in Figure 3.8, from
which we can see that the output current and the capacitor voltage follows the reference,
and the inductor current is off by certain value because of the parameter mismatch and
circuitry loss. Therefore, satisfactory performance is obtained even though the circuit

parameter is not exactly what the controller is designed for.

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Grapho
(A) = «5)
40-0 10.11)
20.0
S. 0.0
-20.0
40.0“ (NM-'1)
30.0 but
EW~ ,. -.. _, A L
25.0“ 3* +1
.— . ioref
$20.0“ ““- ~..- ..
15.0“ q r “- 55,224.32
may "‘_:'w".."fii
10.0“ (V) = «5)
240.0 ' -L - vc
220.0 II 22 500 .
200.01 200.0i
2130.0“ Z1750. (V): 1(3)
1 60.0 “
150.0 vet“
1 40.0 “
120.0. 125.0“ (A) : tts)
70.0“ 70.0 .L
60.0 J 60.0 313%; fixerrxzrwfaznwzuxtr mm
A 50.0 ‘ g 50.0 :_ ..‘g
$40.0“ $40.0 .. (A): KS)
30.0“ 30.0 met
20.0“ 20.0
10.0“ 10.0 f
0.4 0.6
t(s)

 

Figure 3.8. Circuit simulation results with parameter mismatch.

3.4 Experimental Verification

 

To verify the effectiveness of the controller, experiments were conducted. The

parameters of the set up are listed as follows:

L=lmH, C=1.32mF, LLoaflmH, R=3.33ohm, V,-,,=37 V, k,a,,-0=1.15 (capacitor

voltage/output voltage), the reference output voltage has a slow step change from

V,,,/ Vm=1 .1 to 1.8 and back to 1.1. The experimental results are shown in Figure 3.9.

66

10.0nU/diu 0.500 U/dw 0.500 U/dlu
DCSO Ful Full

| DC Full 13.”

 

(a) Step up response

Hill!

all” low

10.011041“) 0.500 U/dlu 0.500 U/div
11850 Full DC Full DC Full

 

(b) Step down response

67

205/1 1/02

nun

RI “mat in

 
  
 
 
 
 

m EMU/MU 10.0nU/div 0.500 U/diu H.504} Univ! Ht Pleasure
11'7")?! Full DCSB Full Full 1". “iii? Range m 21

(c) Steady state operation when szs/Vm =1 .8

menu/din 10.0nU/diu 04110 a. 1m. ”,IHfl'
51030 Full 065% Full DC Full 0:; 1:"

(d) Steady state operation when szs/ Vm=1 .1

Figure 3.9. Experimental results.

Figure 3.9 (a) shows the performance of the controller with a step up change. The
capacitor voltage, Vc, and the load current, [Lam], follow the reference very well. Only
very small overshoot occurs. The inductor current, IL, has to provide extra energy to

68

charge/discharge capacitor besides the steady state current, thus the it follows the
reference less well, however, the overshoot/undershoot is acceptable. Similar results are
obtained in Figure 3.9 (b) when a step down change happens. Figure 3.9(c) and (d)
show the circuit performance under steady state operation, from where perfect sinusoidal
output current can be observed, and the ratio of the capacitor voltage and the input
voltage is exactly the value the controller is supposed to regulate. In the experiment, the
error in the controller is limited to a certain range in the program to eliminate out of range
sensing error, which also results in the transient response being slightly different from the
simulation results. In these experiments, the output voltage increase/decrease transition

only takes 25-50mS, which is enough for most applications.
3.5 Discussion

In the Z-source inverter, there are two control freedoms: the modulation index, M, and the
shoot through duty ratio, D. However, the two freedoms are related, which is M+Dsl
for simple boost control. In the above controller design, this limit is not considered and
the two variables are controlled independently. During the transient, it is possible that
the controller outputs a large M and D, which can not be implement, this might result in
the system becoming unstable. Also, the sensing error might have the same effect. To
make the system stable, it is preferred to use a large kmo, which results in a small M+D
during steady state. However, this also causes higher voltage stress across the switches.
Another way to solve this problem is to limit the variable errors in the program as we did
in the experiment. By doing this, the output of the feed back controller is limited during
a short period of time, thus the feed forward controller dominates during the step change,

and the feed back controller will regulate the output afterwards.

69

3.6 Summary

In this chapter, a small signal model of the Z-source inverter with a RL load is developed. .
Based on this model, a nonlinear closed loop controller is also designed using gain
scheduling. The detailed modeling process and controller designing process are
presented. Simulation results are used to confirm the small signal model. The

controller is verified with simulation and experimental results.

70

CHAPTER 4 Z-Source Inverter for
Fuel Cell Vehicles—Design and

Comparison with Traditional Inverters

The Z-source inverter is a good candidate for applications like fuel cell vehicles and solar
panel systems. In this chapter, comparisons of the Z-source inverter and traditional
inverters are provided to show the potential of Z-source inverter for these applications.

A detailed design process of the Z-source inverter for fuel cell vehicle is also presented.

4.1 Introduction

Fuel cells are very promising as the emerging energy sources in the near future. They
have been used in a variety of areas, such as domestic applications, utility applications,
and traction applications [18-21]. Unlike batteries that have a fairly constant output
voltage, the fuel cell has a unique V-I characteristic and wide voltage range [77]. Figure
4.1 shows the V-I characteristic of proton exchange membrane (PEM) fuel cell, which is
most promising for transportation applications. This unique V-I curve imposes
challenges on the conditioning/interface circuits. For example, for fuel cell vehicles,
this results in difficulty for high-speed, and high-power operation to achieve a great
Constant Power Speed Ratio (CPSR). In addition, because the voltage drops at high

power, the inverter has to be oversized.

71

 

 

 

 

0.8 ‘ “““““ r- ------- : ————————
a, 0.6 _, --------------- I _______
CD I
.‘P. .
(_D 0.4 ---------------- , _______
> I

0.2 ‘1' ‘‘‘‘‘‘ r- ------- . ________

0.0 T I

0.0 0.5 1.0 1.5
Current Density (Alcmz)

Figure 4.1. Typical PEM fuel cell polarization curve.

Currently, there are two existing inverter topologies used for hybrid electric and fuel cell
vehicles: the conventional 3-phase Pulse Width Modulation (PWM) inverter and a
3-phase PWM inverter with a dc-dc boost converter. Because of the wide voltage range
and limited voltage level of the fuel cell stack, the conventional PWM inverter topology
imposes high stresses to the switching devices and the motor, and limits the motor’s
CPSR. The dc/dc boosted PWM inverter topology can alleviate the stresses and
limitations, however suffers problems such as high cost and complexity associated with

the two-stage power conversion.
4.2 System Configurations for Fuel Cell Vehicle

As previously mentioned, three different inverter systems are to be investigated: the
conventional PWM inverter, the dc/dc boosted PWM inverter, and the Z-source inverter.
Their system configurations for fuel cell vehicles are shown in Figure 4.2 (a), (b), and (c),

respectively.

72

129

 

 

 

 

fuel
cell _
stack

 

\

l

O
‘00-.
B

 

 

 

 

 

 

 

 

 

 

 

(a) System configuration using conventional PWM inverter

SZU
D

__.___r‘o'l?’\ -
lsdc _l .1 J16}

fuel i(L-a)

cell _T_ J S _ yr.
A
JG‘VVsian JG

(b) System configuration using dc/dc boosted PWM inverter

 

 

A

 

\l

 

 

 

 

 

 

 

 

 

 

 

 

D1IL L1

 

 

fuel
cell
stack

 

 

1|”, “11L

 

 

 

.33
,wgfuve 113

(c) System configuration using the Z-source inverter

 

 

 

 

 

 

 

 

Figure 4.2. Three inverter system configurations for fuel cell vehicles.
In the traditional PWM inverter, the dc bus voltage, which is also the output voltage of
the fuel cell stack, varies with the load. The modulation index has to be controlled to
achieve the required output voltage. The boost converter in the dc/dc boosted PWM

inverter system boosts the dc voltage only when the required output voltage is higher than

73

the fuel cell voltage. The Z-source inverter outputs a required voltage by adjusting the
shoot through duty ratio with the restriction to keep the voltage across the switches not to

exceed their limits.
4.3. Comparison Items, Conditions, Equations, and

Results

For all comparisons, the conventional PWM inverter and the dc/dc boosted PWM
inverter are controlled by SPWM with third harmonic injection to achieve the maximum
modulation index possible when necessary. The Z-source inverter is controlled with
maximum constant boost control with modified PWM scheme [69]. To compare

different inverters, some parameters that are necessary for derivation are listed below.
P0 The maximum output power;

me The maximum output voltage of the fuel cell stack, open circuit voltage;
cosgo, PF The power factor of the motor;

V, The fuel cell stack output voltage at maximum power;

M The modulation index of inverters, defined by the ratio of the amplitude of

the reference waveform and that of the carrier waveform for traditional SPWM;

Vdc The output dc voltage of the boost converter in the dc/dc boosted PWM
inverter;
k Boost ratio, defined by Vmax/V, for conventional PWM inverter; Vd/K, for

dc/dc boosted PWM inverter; and k=B in (A122) for Z—source inverter. Basically, this

is the ratio of the maximum dc voltage across the inverter and minimum fuel cell output

74

voltage;

D Duty ratio of the dc/dc converter in dc/dc boosted PWM inverter;
1m", Capacitor rrns current;

T, Switching cycle;

[peak Peak load current.

4.3.1. Total Switching Device Power Comparison

In an inverter system, each switching device has to be selected according to the
maximum voltage impressed and the peak and average current going through it. To
quantify the voltage and current stress (or requirement) of an inverter system, switching
device power is introduced. The SDP of a switching device/cell is expressed as the
product of voltage stress and current stress. The total SDP of an inverter system is
defined as the aggregate of SDP of all the switching devices used in the circuit. Total
SDP is a measure of the total semiconductor device requirement, thus an important cost

indicator of an inverter system. The definitions are summarized as follows:

N
Total Average SDP = (SDP)av = Z Vm1m_ave,age , and

m=l

N
Total Peak SDP = (SDP)pk = Z Vm 1m_ peak , where N is the number of devices used,

m=l
Immmge and 1m flak are the average and peak current through the device respectively, and
V”, is the peak voltage induced on the devices.
In our comparison, the input end diode D1 in the traditional PWM inverter and the
Z-source inverter are not considered, because it’s difficult to compare the cost of a diode

and a switch of the same rating. The average and peak SDPs of conventional PWM

75

inverter are:

V
(SDP)av: 8 maxPo ,
cos¢V,-7rM
(SDP)pk =M.
cosgoViM

The average and peak SDPs of dc/dc boosted PWM inverter are:

 

SDP = +—*V ,

( )av coswrM V, DC
8P P

SDP = 0 +—0-*V

( )pk COMM V DC

The average and peak SDPs of Z-source inverter are:

2P (2 f M) 47-1D
(73114—1) +cosgozr’

 

(513mm,:

4P0 8P0

 

(SDP) pk_ — max(

The detailed derivation is in appendix 1.

734M0—1 +cos¢M’ cos¢M '

(4.1)

(4.2)

(4.3)

(4.4)

(4.5)

(4.6)

Assume that the motors for different inverters are all in constant torque region, thus the

motor current almost maintains constant for different power.

The conventional PWM

inverter and the dc/dc boosted PWM inverter are operated with modulation index of 1.15

with third harmonic injection or SVPWM at maximum power condition to minimize the

required SDP. Based on the above equations and operation conditions, the required SDP

of each inverter can be calculated for different load power factor and boost ratio, k, the

results are shown in Figure'4.3. As mentioned above, the boost ratio is defined by

Vmax/V) for the conventional PWM inverter; VdC/V), for the dc/dc boosted PWM inverter;

and k=B for the Z-source inverter.

76

Traditional PWM inverter

dc/dc boosted PWM inverter

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

------------ Z-source invencr — . — - -—
30 30
/
25 - . ,. 25 /
o PF=0.7 / o PF=O.8. ‘
s. 20 ° ~ .. s 20 y + /
on / Ac... 0. ‘ ."
On 0 ' ' a. . / .0 .
D15 ' / .‘O' " 4 Q 15 “ “v .v". ' i
(I) ...O' U) / ..'o.
10 9"“— - 10 refit"
5 . . . 5 i . .
l 1.5 2 2.5 3 l 1.5 2 2.5 3
Boost ratio Boost ratio
25 25
/
20 ' . 3 20 . <
0 PF=O.9 / o _ . /
$ ...01 $ PF_1.0 / ...1)
£15 / ...." . , a. 15 ,. .. "‘o" .
D ' o '0’. D A’.'
m /',.- m 3".
10 -" 4 10 ’4'“
5 r . A 5 . A
l 1.5 2 2 5 3 l 1.5 2.5 3
Boost ratio Boost ratio

(a) Peak SDP for different power factors

 

 

SDPav/Po

 

 

SDPav/Po

 

 

 

 

 

 

Boost ratio

2.5 3 l 1.5

2.5 3

 

 

 

 

O"

SDPav/Po

 

 

 

 

 

1 .5 2
Boost ratio

2.5 3 l 1.5

 

2.5 3

Boost ratio

(b) Average SDP for different power factors

Figure 4.3. Comparison of SDP of different inverters at different operation conditions.

77

For the conventional PWM inverter, it is obvious that the required SDP is proportional to
the apparent power, therefore the SDP should increase linearly with the decrease of the
load power factor. For the dc/dc boosted PWM inverter, the SDP of the inverter
increases with the apparent power and the SDP of the dc/dc converter stays constant. It
is similar for the Z-source inverter, the SDP contributed by shoot through doesn’t change
with the apparent power. Therefore, as load power factor reduces, the SDP of
conventional PWM inverter increases quicker than the other two inverters, which is
verified from Figure 4.3. As can be seen from Figure 4.3, if the same boost ratios are
used for the three inverters, the Z-source inverter requires low SDP at low boost ratio
region (1-2), especially average SDP, which is also an indicator of thermal requirement
and conversion efficiency. When the boost ratio is above 2, the dc/dc boosted PWM
inverter is the best in terms of SDP requirement. For very low boost ratio (1-1.25) and
high power factor, the SDP of the conventional PWM inverter is very similar or even

smaller than the Z-source inverter.
4.3.2. Requirement of Passive Components Comparison

Passive components—inductors and capacitors are also important parts determining the
inverter cost and volume. The inductors are designed to limit the current ripple, and the

capacitors are designed based on current capacity and capacitance requirement.
4.3.2.1 Inductor Design and Comparison

For an inductor, the size is determined by the inductance and the current level. The
average current, 10., through the inductor in the dc/dc boosted PWM inverter at maximum

power is:

78

P
1 =——9-. 4.7
av Vi ( )

For the Z-source inverter, the average current through the inductor equals to that through
the input end diode, based on power balance, the average inductor current can also be

calculated by (4.7).

The purpose of the inductors is to limit the current ripple through the semiconductor

devices, therefore, the inductance can be determined by a given current ripple level.

The inductor current ripple, A1 L, in the dc/dc boosted PWM inverter can be calculated

by:

V- V —V-
AIL : 1( DC I)Tsa
LVDC

 

(4.8)

where L is the inductance of the inductor.

The inductor current ripple in the Z-source inverter using maximum constant boost

control is:

 

L (4.9)

= VI‘EM (1—3/3MJTS,
2L(73M-1) 2

where L is the inductance of the inductors.

The detailed derivations of the equations are in appendix 2. In (4.8) and (4.9), all other
parameters are known for maximum power operation condition, therefore the inductance

requirement can be calculated by limiting the current ripple within a certain range.

79

For the two inductors in the Z-source inverter, the current through them and the voltage
across them are exactly the same, therefore they can be built on the same core, with the

same size of one inductor with doubled inductance as shown in Figure 4.4.

 

' 1.1.... 1:" TIN Pawnlxit 31131
. a; K.” -
”1» A i '
wire ‘33.
V1 coil 5"
1‘-
‘36:,
“a!"
it:
.- . pa
12 fits
. l‘.
Ari-P
wure “C
I 3 .I.
v 0011 if: N
2 “2%”
.a-
:37
Wart-w '“u L -11 3. " . ,5
-. an} 11.»: st, my a. d 3 were . a".
EmasuMR "‘1' in“ “ii-‘1 “Va“: 1‘11

Figure 4.4. Coupled inductors.
For a single coil on one core, the flux through the core is

¢= PNi, . (4.10)

where P is a constant related to the core material and dimension, N is the number of turns

of the coil, and i is the current through the coil. The inductance of the coil is

L=M=PN2. (4.11)

I

For the two inductors in the Z-source inverter, because of the symmetry of the circuit, the
current through the inductors is always exactly the same. For two coils on one core with

exactly the same current, i, the flux through the core is

,3 = 2PNi. (4.12)

80

The resulted inductance of each coil when supplying exactly the same current to the two

coils is
L=—_—=2PN2._ (4.13)

The inductance of each coil is doubled.
4.3.2.2 Capacitor Requirement Comparison

The current through the capacitor is an important factor determining the capacitor size.
It is obvious that the current through the capacitor repeats every 1/3 of the fundamental

cycle. From Figure 4.5, the capacitor current can be calculated by the following

equation:

 

l
y
I...... = 3f I (i..(r>—i..(t»2dt. (4.14)
0

where f is the fundamental frequency.

. 1
1c oc

 

 

_L

T

v

Figure 4.5. Capacitor current ripple calculation.

For the conventional PWM inverter, the input current to the capacitor, 1)C, is assumed to
. P 0

be constant, which equals to the average current, —. The output current of the
i

capacitor changes with the time, and it can be different for different PWM schemes.

8l

However, the rrns current stays the same [78, 79]. For the switching cycle with SPWM

scheme shown in Figure 4.6 (a), the instant current can be described as:

0 TO<tST1,T3<t.<_T5,T7<tST8
i000): ia T2<tsT3,T5 <tST6' (4.15)
—ib T1<tST2,T6<tST7

where ia and ib is the load current of phase a and b.

As a result, the following equation can be derived:

 

9 73M
[mm2 = GEM2 cosgp2 + II cosgo2 +—4'7—r—flpeak2 (4.16)

Based on this equation, the capacitor current for different modulation indices and load

2

[Cf ms

power factors are shown in Figure 6 (a). K,- is defined by K ,- = 2

, where [peak is
1 peak

the output peak current.
For the dc/dc boosted PWM, assuming that the control of the dc/dc converter and the

inverter share the same carrier, for a certain interval with modulation scheme shown in

Figure 5 (b), the input and output current of the capacitor can be described as:

0 TO<t$T2,T8<tSTIO
“Cm: 5’— T2<t<T8 (4'17)
i
0 TO<tST1,T4<tST6,T9<tST10
i0c(t)= ia T3<tsT4,T6 <tST7 (4.18)
—ib Tl<tsT3,T7<tST9

The resulted current depends on the relative positions of the reference signals. It is

82

easier to use numerical calculation than analytical expression. The dc/dc boosted PWM
inverter operates as a conventional PWM inverter when the input voltage is higher than
the required output voltage with D=O. When the input voltage is not high enough to
output the required voltage, to minimize the switching loss, the inverter is always
operated with maximum modulation index possible, 1.15, and the dc/dc converter
operates with D greater than zero to output the required voltage. Figure 4.7 (b) shows
the calculated capacitor current for the dc/dc boosted PMW inverter for these two
conditions. The solid curves show the capacitor current when the D=0 with different
modulation indices, the dotted curves show the capacitor current when D>0 and M=1.15

for different load power factors.

Maximum constant boost control with third harmonic injection shown in Figure 4.6 (c) is
used to control the Z-source inverter. As seen from Figure 4.6 (c), when the carrier is
higher than Vp or lower than V", the inverter is in shoot through state, while in between, it

operates as traditional SPWM. The input current to the capacitor is the inductor current,

which can be considered as constant, %. During traditional zero states and active

1

states, the output current of the capacitor is the same as the conventional PWM inverter.
During shoot through state, the two capacitors are charging the two inductors separately
through the inverter bridge, therefore the current through the inverter bridge is twice of
the inductor current. More detailed description of the output current of the capacitor at

the interval shown in Figure 4.6 (c) as an example is as follows:

83

O T1<tST2,T4<tST5,
T7 < t S T8,T10 < t _<_ T11(traditional zero state)

 

 

ioc(t)=<ia T3<tST4,T8<tST9 ) (4.19)
-—ib T2<tST3,T9<tST10
12%)— TO < t _<_ T1,T5 < t _<_ T7,T11 < t .<_ T12 (shoot through state)
1 .

From this equation, we can calculate the current through the capacitors in the Z-source
inverter. Similarly, the Z-source inverter can be operated as a conventional PWM
inverter when the input voltage is high enough to output the required voltage, and it will
be operated with shoot through only when the input voltage is lower than the required
output voltage. The resulted capacitor currents for these two different operation

conditions are shown in Figure 4.7(c).

It is note worthy to mention that [peak is different for different inverter systems. For the
same fuel cell stack and same power, [peak 0f the conventional PWM inverter will be

higher than the other two because of lower available voltage.

 

 

 

 

 

Vb

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

i g E 1
T0 T1 T2T3 T4 T5T6 T7 T8 T0 T1 T2T3T4 T5 T6 T7 T8 T9 T10

(a) Conventional PWM inverter (b) dc/dc boosted PWM inverter

84

 

 

Vc

 

vb

TOT1T2 T3 T4 T5 T6 T7T8 T9 T10T11T12

 

 

 

 

 

 

 

 

 

 

 

(c) Z-Source inverter

va, vb, vc: reference signals of inverter PWM; vd: reference signal of the dc/dc converter;

vp, v": reference signals controlling the shoot through
Figure 4.6. PWM scheme of different inverters at a certain interval.

The capacitance can also be an important factor determining its size depending on the
type of the capacitor. The capacitance is designed based on limiting the voltage ripple
across the capacitor to limit the output current harmonic. From the above analysis of the
current through the capacitor, it is easy to numerically calculate the maximum voltage
ripple of the capacitors under different conditions. The calculated results are shown in

Figure 4.8. The dotted curves in (b) and (c) show the results when boost function is

AQ

needed. The result is defined by K v, = ——}— , where AQ is the charge causing the
peak“ 3

maximum voltage ripple, [peak is the peak load current, Ts is the switching cycle. Also,

[peak is different for different inverter systems.

85

Ki

 

 

 

0 0.2 0.4 0.6 0.8 1
Modulation index M

(a) Conventional PWM inverter

Modulation indexM

(b) dc/dc boosted PWM inverter

    

3.11 with 41.501111151312111; r 7
1 w—,—,il_vith9_ulish09tlthughr -.
3 i ‘ PF=I',0
0. . ' -“ ' ‘ '-
90.
0.
0 , ‘ -
1 0.2 I
00

 

0,2 0.4 0.6 0.8 1
Modulation index M

(c) Z—source inverter

2
Ki = Icrms

I 2 , [peak is the peak load current of the inverter
peak

 

Figure 4.7. Capacitor current comparison of the inverters.

86

(a) Conventional PWM inverter

0T’”0.6"0‘.8 “ ‘1 *0
Modulation Index M

 

 

     

 
 

 

 

 

   

0.2 0.4 0.6 0.8 I 0 0.2 0.4
Modulation index M dc/dc duty ratio D

(b) dc/dc boosted PWM inverter

 

0.35r -
; -... with shoot through .
. -—.- without shoot thnou n t
0'3 g '. PF=1 0
.1
0,25. 1‘. 0
0.8 “ 0‘
0.2 a ‘ t 7
E 0.6‘ ‘ ..O
O. '
0.15» 1’...
‘4'
0.1» \
‘\‘ '0'
005» v
0 1 1 n __4 _- _2_ ._
0.2 0.4 0.6 0.8 1

Modulation index M

(c) Z-source inverter

Figure 4.8. Capacitor voltage ripple comparison of the inverters.

4.3.3. CPSR Comparison

CPSR is limited mainly by available dc voltage of the PWM inverter.

The fuel cell

voltage decreases as the current drawn increases, which greatly limits the motor’s power

output and efficiency at high speed. For the dc/dc boosted PWM inverter and the

Z-source inverter, the available output voltage is theoretically infinite.

To compare the

CPSR limited by the inverter output voltage, certain operating conditions have to be

87

specified. To make a fair comparison, the same fuel cell stack and the same device
voltage ratings are used. This is accomplished by limiting the maximum output voltage of
the dc/dc converter, Vdc, the maximum voltage across the device of Z-source inverter,

A

vpn , and the open circuit voltage of the fuel cell, Vmaj, to the same values. Thus the

maximum voltage across the device and the boost ratio are the same for all inverters.
The modulation indices of the conventional PWM inverter and the dc/dc boosted PWM
inverter are both 1.15, to output the maximum voltage. The maximum obtainable output

phase peak voltage of conventional PWM inverter at peak power is

i (4.20)

zzfi

.. M
Vac = Vi_—

The maximum obtainable output voltage of the dc/dc boosted PWM inverter is

A M [(V-
vac =VdC—="_I“ (4-21)

2 15

For the Z-source inverter, the modulation index used to boost the voltage is determined

by [7]:

 

= k . (4.22)

The resulted obtainable output voltage is:

. :(k+1)V,.. (4.23)

v _—
(1C 2J3
Define the CPSR of conventional PWM inverter to be 1 per unit. The CPSR of dc/dc

boosted PWM inverter and the Z-source inverter are k and I?! respectively.

88

4.3.4. Reliability Comparison

Compared to traditional PWM inverter and the Z-source inverter, the dc/dc boosted PWM
inverter uses one more active device, which inevitably reduces the reliability of the
inverter. Also, the Z-source inverter can handle shoot through state, thus momentary
shoot through caused by EMI will not affect the inverter operation. While for traditional

inverters, this might trigger protection and cause unexpected system shut down.

4.4 Comparison Example

In this section, a comparison example will be conducted based on a given fuel cell and

system with the following parameters:
V,-=250 V; Vmax=420 V; P0=50 kW; cosgo=0.9;

The conventional PWM inverter and the dc/dc boosted PWM inverter are operated with
modulation index of 1.15 with third harmonic injection or SVPWM at maximum power
condition. For a fair comparison, the same device voltage ratings are used, which

means the boost ratios of the three inverters are kept the same.

With these assumptions, the SDP of different inverters can be calculated and shown in

 

 

 

 

Table.4. 1.
Table 4.1 Switching device power comparison example
Inverter Systems Total Average SDP (kVA) Total Peak SDP (kVA)
PWM inverter 207 650
dc/dc boosted PWM inverter 207 470
Z source inverter 19] 577

 

 

 

 

 

89

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Table 4.2. Required passive components
Inverter Number Inductance Average Number Capacitance Capacitor Capacit—
Systems of (11H) inductor of of the ms or
inductor current capacitor capacitor current voltage
(A) (PF) (A) rating
(V)
Conventi- 0 N/A N/A 1 296 l 12 420
onal PWM
inverter
dc/dc l 510 200 l 51 l 105 420
boosted
inverter
Z-source 2 (1) 339 200 2 428 110 420
inverter
Table 4.3. Operation conditions at different power
Power Rating 50 kW 40kW 30 kW 20 kW 10 kW
Fuel cell voltage (V) 250 280 305 325 340
Conventional PWM inverter 101.7 81.4 61 40.7 20.3
Motor Phase dc/dc boosted PWM inverter 170.8 136.6 102.4 68.3 34.2
It V
W 3g“ ) Z—source inverter 136.8 109.4 82.1 54.7 27.4
Conventional PWM inverter 182 182 182 182 182
Motor current (A) dc/dc boosted PWM inverter 108.4 108.4 108.4 108.4 108.4
Z — source inverter 135 135 135 135 135

 

 

 

 

 

 

 

 

 

The Z-source inverter’s average SDP is the lowest among the three while the

conventional PWM inverter’s SDPS are the highest in both average and peak values.

90

Usually, the selection of the switching device is based on the nns/average current rating,

and also the average SDP is a measurement of thermal requirement.

Table 4.2 shows the passive component comparison with the requirements to limit the
inductor current ripple to be less than 10% of its average value, and capacitor voltage
ripple less than 3% of the maximum voltage, 420 V @ switching frequency of 10 kHz.
The Z-source inverter’s two inductors can be built on one core to minimize the size and
weight. In general, the required L and C of the Z-source inverter are slightly greater

than those of the dc/dc boosted PWM inverter.

For the conventional PWM inverter with the fuel cell model described above, the fuel cell
voltage is the dc voltage of the inverter, which drops to 250 V at 200 A. From the 250 V
dc, the conventional PWM inverter can only yield 176 V to the motor. This low motor
voltage limits CPSR and lowers mechanical output power and efficiency. The PWM
inverter with dc-dc boost can keep the dc voltage to 420 V, which in turn increases CPSR
by a factor of 1.68. Theoretically the Z-source inverter can output whatever voltage as
required. By the restriction of the same switch voltage stress as the traditional PWM
inverter and dc/dc boosted PWM inverter, the Z-source inverter can increase the CPSR
by 1.34 times over the traditional PWM inverter. In other words, the motor voltage
produced by the Z-source inverters is 1.34 times that produced by the conventional PWM
inverter, thus the same motor can output 1.34 times the power than when driven by the

conventional PWM inverter.

Efficiency is an important criterion for any power converter. High efficiency can reduce

thermal requirements and cost.

The inverter system is assumed to be operated in constant torque region and space vector

91

control is applied to all inverters. Based on this assumption, the output voltage and
current of the inverter can be derived in Table.4.3. From the resulted voltage and
current, the following devices are chosen for effrciency calculation: the switches for the
main inverters of dc/dc boosted inverter and Z-source inverter~ are Powerex 1PM
PM3OOCLAO60, the switch for the dc/dc boost converter is Powerex PM300DSAO60, the
switch for the traditional inverter is Powerex 1PM PM450CLAO60, the input end diode of

the traditional PWM inverter and the Z-source inverter is IXYS MEO 500-06DA.

For the efficiency comparison, only the semiconductor devices losses are considered.
For conventional PWM inverter and dc/dc boosted PWM inverter, the IGBT loss consists
of switching loss and conduction loss, the diode loss includes conduction loss and reverse
recovery loss, the detailed calculation method can be found in [80]. For the Z-source
inverter, the loss of the inverter consists of the following parts: conduction loss of the
switches during non-shoot through states, switching loss between non-shoot through
states, conduction loss of the switches during shoot through state, switching loss of the
shoot through state, free wheeling diodes conduction loss and reverse recovery loss
during non-shoot through states. The modified PWM sequence of maximum constant
boost control is implemented for efficiency comparison, in which one shoot through
period is employed in a switching cycle as shown in Figure 4.9. The switching loss for

the modified PWM method can be found in chapter 2.

The reverse recovery loss of the free wheeling diodes is reduced because some of the turn
off states of the diodes are turn into shoot through turn off, the reverse recovery loss can

be calculated accordingly.

92

 

 

Shoot

through Actrvet Active2

Active2 Activet

 

 

 

 

 

 

 

 

Figure 4.9. Switching states sequence in one cycle.

The conduction losses of the IGBTs and the diodes also change because of the shoot
through states. Assuming the shoot through duty ratio is Do and the corresponding
conduction losses of IGBTs and diodes for traditional PWM inverter under the same load
current are Pan/GET and Pondiode respectively, which can be calculated based on [80], the

conduction losses of the Z-source inverter during traditional states become:
PoniGBTtraditionaI = (1- Do)PonICBT (4-24)
Pondiodetraditional =(1— D0 )Pondiode (4-25)

Assuming that the inductor current is high enough so that all IGBTs are on during shoot

through state, the average current through the IGBTs during shoot through is Z—gé- , the

conduction loss of IGBTs during shoot through is

2
PonIGBTs/roothrough = D O VCE (sat) * 3 I L a (4-26)

where chmu is the saturation voltage of the IGBT. From all above discussion, total
loss of the inverter bridge can be calculated. Also, the conduction loss and reverse
recovery losses of the input end diode of the Z-source inverter and the traditional PWM
inverter are considered. The inverter efficiency calculation results are shown in Figure

4.10.

93

 

 

 

 

 

 

 

 

 

 

 

1
dc/ dc boosted
0' 98 PWM 1nverter
5‘ 0.96 1 .
5 O 94 conventronal
'8 . + PWM inverter
3:? 0.92
LL]
0-9 Z source
0 88 inverter
0.86 0 ‘ ‘
20 40 60
Power( kW)
Figure 4.10. Calculated efficiency of inverters.
1
0.983 -W 2%” -. -m_____
5‘ 0.96.
5 conventional
15’ 0'94‘ PWM inverter
5 0.9 -- —
0.88 .- -.
0.86

 

0 10 20 30 40 50 60
output power (kW)

Figure 4.11. Inverter efficiency calculated using software.

From the above comparison, the Z-source inverter provides the highest efficiency in most

regions of the power range of the inverter itself.

To verify the efficiency calculation of the traditional inverters, Mitsubishi average loss
simulation software [81] is used. The efficiencies of the conventional PWM inverter

and dc/dc boosted PWM inverter calculated from the software under the same operation

94

conditions are shown in Figure 4.11. As can be seen from Figure 4.10-11, the calculated
efficiencies and simulated efficiencies are very close. However, because of the different
operating principle, the software can’t be used to simulate the efficiency of the Z-source
inverter. To verify the efficiency calculation of the Z-source inverter, efficiency
measurements on a SOkW Z-source inverter shown in Figure 4.12 (a) are conducted.

Developed for a different project, the inverter bridge is Powerex 1PM PM600CLAO6O to

meet high temperature requirement, the inductor is 25 11 H each and built on one core,
which is equivalent to a 50 11 H inductor, also one film capacitor is connected in parallel

with the input voltage source just to minimize the current loop. However, the loss
characteristics of the two IPMs are quite similar. The test was carried out on a RL load
test and with power factor of 0.8. The resulted efficiency is shown in Figure 4.12 (b).
Because of the load, the current at low power becomes smaller than the motor current
when operated in constant torque mode. This results in higher efficiencies than the
calculated value at low power range. Also, because of the lower. power factor, and the
losses on the passive components, the efficiency at high power is slightly lower than as
expected from the calculated values. Considering all these factors, the calculated

efficiencies of the inverters are quite accurate.

95

0.975
0.97
0.965 —

Efficiency

0.96
0.955
0.95

 

0.945 __._1_

   

(a) 50kW Z-source inverter (b) Measured efficiencies

Figure 4.12. Efficiency testing and results.
4.5. Summary

A comprehensive comparison of the three inverter systems has been performed.
General equations and curves of switching device power, passive component requirement,
and CPSR have been derived for comparison. A comparison example with detailed
specifications has been carried out. For this comparison example, efficiencies of the
different systems have also been compared and verified. The comparison results show
that the Z-source inverter has lowest average SDP in low boost ratio range (1-2). Also it
provides higher efficiencies in most operating ranges. It can increase the CPSR over the
conventional PWM inverter significantly. It does slightly increase the passive

component requirement.

It is note worthy to mention that this is a purely theoretical comparison. In practical
cases for the dc/dc boosted PWM inverter, the associated cost and volume increase of the
extra heat sinking effort, and the required gate drive for an extra switch, is also significant.
Also, the great reliability enhancement of the Z-source inverter is a very important
advantage. In general, the Z-source inverter is very competitive in moderate boost ratio

96

range (1-2), in which most fuel cells reside. In cases when a low voltage fuel cell is
used, and boost ratio much higher than 2 is needed, the dc/dc boosted PWM inverter is
the best configuration. In this chapter, the comparison is limited to a fuel cell vehicle
without a battery, all configurations can be modified to fuel cell-battery hybrid vehicles
by adding a battery [12] to incorporate the regenerative braking function without

significant change of the main circuit.

97

CHAPTER 5 Operation Modes and
Characteristics of the Z-Source
Inverter with Small Inductance or Low

Power Factor

All of the descriptions and analysis done in previous chapters are based on an assumption
that the inductance of the inductor in the Z-source network is large, and the load power
factor is high so that the inductor current is almost constant. In some applications, the
inductance should be minimized in order to reduce cost, volume, and weight. In this
chapter, we will analyze the Z-source inverter operation when the inductance is very
small and the inductor current has high ripple or is discontinuous. The analysis reveals
new operation modes and characteristics. As a result, design guidelines and

considerations will be provided.
5.1. Introduction

The traditional voltage source inverter (VSI) is a buck converter which can only produce
a voltage lower than the DC bus voltage. A 3-phase VSI has 6 active states and 2 zero
states. A zero state is produced when the upper three or lower three switches are turned
on at the same time shorting the output terminals. The upper and lower two switches of

any phase leg can never be gated on at the same time, otherwise a shoot-through short

98

circuit would occur and could destroy the inverter. The shoot-through is a forbidden

switching state for the traditional VSI. The Z-source inverter shown in Figure 5.1,

instead, uses the shoot through state to boost the output voltage. However, under certain

load condition and inverter parameters, there could be more operating modes.

In order to make the description of the converter easier, a list of symbols are listed below.

Nomenclature

V0: Input voltage of the Z-source inverter;

v,: Voltage across the inverter bridge;

L: Inductance of inductors in the Z-source network;

C: Capacitance of the capacitors in the Z-source network;

VS: Voltage stress across the device, equals to maximum value of v,;
VC: Voltage across the capacitors in the Z-source network;

i L: Current through the inductors in the Z—source network;

1'): Current fed to the inverter bridge;

im: Input current to the Z-source inverter;

T0, T7; Zero states in SVPWM/SPWM;

T 0.7: Total traditional zero state in one switching cycle;

T1, T2; Active states in SVPWM/SPWM;

G: Voltage gain, defined by the ratio of output peak phase voltage and half of the

input voltage;

99

M: Modulation index;

cos¢ : Load power factor;

T3,: Shoot through period in one switching cycle;
Ts: Switching cycle;

Z: Load impendence per phase;

ka: Load peak current;

V0,“: Output phase rms voltage.

 

 

 

 

V 2

 

>

0——>

_| To ac
Load

Figure 5.1. The Z-Source inverter.

 

 

_I
vwv

 

 

5.2. Operating Principle and Operation Modes Analysis

The Z-source inverter utilizes the shoot through zero states to boost voltage in addition to
the traditional 6 active states and 2 zero states. The basic operating principle described
in [12] assumes that the inductor current is relatively high and almost constant. When
the inductance is small or the load power factor is low, the inductor current can have high
ripple or even become discontinuous. Instead of having the two operating modes
described in [12], the Z-source inverter may have five different operation modes as

shown in Figure 5.2 when viewed from the Z-source network. Modes 1 and 2 have been

100

described in [12], whereas modes 3, 4 and 5 are new modes that may exist for small

inductance and low power factor cases.

[Mode 1]: The circuit is in a switch shoot-through zero state when the two switches in
any of the three phase legs are turned on at the same time, the sum of the two capacitors’
voltage is greater than the dc source voltage (VC1+ VC2 > V0), the diode is reverse biased,

and the capacitors charge the inductors. The voltages across the inductors are:
VL1 = VCIaVLZ = VC2- (5-1)

The inductor current increases linearly assuming the capacitor voltage is constant during
this period. Because of the symmetry (L1=L2=L and CI=C2=C) of the circuit, one has

VL1=VL2=VL, iL1=iL2=iL, and VCI = VC2 = VC-

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

J

__ +

_1_[_— V0 Vcl _VCZ +GD
J

x
(a) Mode 1
Off iu
(c) Mode 3 ((1) Mode 4

101

I

 

 

T

 

 

 

 

(e) Mode 5

Figure 5.2. Possible operation modes of Z source inverter.
[Mode 2]: The inverter is in a non-shoot through state (one of the 6 active states and 2

traditional zero states) and the inductor current meets the following inequation,

. 1 .
’1. >51). (5.2)

Again because of the symmetry of the circuit, the capacitor current i0 and 1°C; and the
inductor current iu and 1'” should be equal to each other respectively. In this mode, the

input current from the dc source becomes:
1'," =iL1 +1},1 =1“ +(iL2 —z',-) = 2iL —i,- > 0 (5.3)
Therefore, the diode is conducting and the voltage across the inductor is
vL = V0 — VC , (5.4)

which is negative (the capacitor voltage is higher than the input voltage during boost
operation when there are shoot through states), thus the inductor current decreases

linearly assuming the capacitor voltage is constant. As time goes on, the inductor

102

current keeps decreasing to a level that no longer the condition of (5.2) can be met and
the input current, i)”, or the diode current is decreased to zero, Mode 2 ends and the

inverter enters to a new mode.

[Mode 3]: The inverter is in one of the 6 active states, and the inductor current equals to
half of the inverter dc side current, i,-. As a result, the input current becomes zero and
the diode becomes reverse-biased. Assuming that the inverter load is inductive and has
a much larger inductance than that of the inductor L I and L 2, the voltage drop across L I

and L 2 are negligible and the inductor current and inverter voltage v. are respectively

1L =—12-i,- and (5.5)

v,- = V6. (5.6)

[Mode 4]: The inverter is in one of the 2 traditional zero states (i,=0) and at the end of
Mode 2, the inductor current decreases to zero, thus a new operation mode appears. In
Mode 4, the diode stops conducting and the inverter is an open circuit to the Z-source
network because of i,=0. The inductor current becomes zero and maintains zero until the
next switching action. Therefore, in this mode, the Z-source circuit is isolated from both
the dc source and the load. The load terminals are shorted by the upper three switches or

the lower three switches as shown in Figure 5.2 ((1).

[Mode 5]: Freewheeling diode shoot through state: The inverter is switched to an active
state and the inductor current at that instant is less than i/Z. After having been switched
to an active state, the inverter cannot enter the active state immediately because that the
available current from the dc side 21'L is not enough to supply the load current, i), the

inverter enters a free-wheeling state described in Figure 5.2 (e). The two diodes in the

103

equivalent circuit are the free wheeling diodes of the inverter phase legs. This diode free
wheeling state turns the inverter into a shoot through state. During this shoot through
state, all the equations of Mode I hold true and the inductor current increases linearly.
This mode continues until the inductor current reaches ’i/Z or another switching action
happens. The difference between this mode and Mode 1 is that this mode is not
intentionally created by the control signal and depends on the load current and the inductor

current at the time of switching.
5.3. Circuit Analysis and Characteristics

The Z-source inverter can be operated with both traditional SPWM/SVPWM
without any shoot through states and with shoot through to boost the output voltage.
Under both cases, the new operation modes may or may not occur depending on the
inductance and load conditions. The combination and sequence of the operation modes
can be different for different circuit parameters. In this section, the circuit will be
analyzed under both traditional SVPWM control and with boost by giving the operation
conditions including the combination and sequence of the new modes and the critical

conditions for the new modes to happen.
5.3.1. Traditional SVPWM Control

Under traditional SVPWM control and simple operation, when only the Mode 1 and Mode
2 occurs, the capacitor voltage always equals to the input voltage and the inductor current

is a pure dc. However, under certain condition, new operation modes can also occur.
5.3.1.1. Operation Conditions
For SVPWM control, one switching cycle consists of 4 switching states, two zero states To

104

and T7 and two of the six active states, T1 and T2 for instance. During To and T7, the load
terminals are shorted by the upper or lower three switches, and there is no current feeding
the inverter bridge from the dc side. To simplify the analysis, assume that the current
from the dc side to the inverter bridge, 1'), during the two active states are the same, and the
capacitor voltage is always constant. Under these assumptions, there can be two
different operation conditions termed as CCM and DCM with the inductor current shown
in Figure 5.3 (a) and (b) respectively. In both cases, because of Mode 5, the capacitor
voltage is higher than the input voltage. For the CCM condition, the inductor current
decreases in the zero state To. When active state T1 starts, the inductor current is less
than i/2, thus freewheeling diode shoot through mode (Mode 5) happens, and the inductor
current increases linearly until it reaches i/2. The inductor current then maintains by
entering Mode 3 afterwards. The inductor current will decrease again when the inverter
enters the second part of the zero states T7. In some cases when the inductance is
extremely low, the inductor current can reach zero during the zero'states and stay at zero
in Mode 4 until the next switching action as shown in Figure 5.3 (b), which forms a
DCM operation condition.

’1.

A 1.

I: _I

2 \/ \ 2
>

O t. : : : : O ration: :— 1 :
$533332 35:: 3 :2 26.8.. §2§4§5§ 3 2

 

 

 

 

 

 

(a) CCM (b) DCM
Figure 5.3. Inductor current waveforms for different operation conditions.

105

It is note worthy to mention that the currents to the inverter bridge at different active states
are actually different, which will make the operation condition more complicated and
there also could be additional operation conditions, however the basic principle and the
possible operation modes are the same. Because of the variation of the line current in a

line cycle, more than one operation condition can occur in one line cycle.

Under simple operation condition, when only Mode 1 and 2 occur, the capacitor voltage
always equals to the input voltage. With these new operation modes, the capacitor
voltage is boosted and the voltage across the device is also increased. Because some
parts of the active states becomes a freewheeling diode shoot through state, the output
voltage can be higher or lower than the output voltage under simple operation with the
same modulation index. Also, the time duration of the freewheeling diode shoot through
state depends on the current to the inverter bridge, which changes over a line cycle, this

can cause slight harmonics in the output voltage.
5.3.1.2. Critical Condition

As discussed above, the new modes will bring different characteristics to the inverter, it is
important to know under what condition these new modes will happen. The critical
condition of the new modes to happen under traditional SVPWM without any controlled

shoot through is:
2
M cosrp < 3 (5.7)

As from (5.7), when the load power factor is low, there will be new operation modes.

The detailed derivation can be found in appendix 3.

106

5.3.2. Maximum Constant Boost Control with Third Harmonic Injection

When the circuit starts to boost the voltage by introducing shoot through states, the new
modes can also happen depending on the load condition and the inductance of the
inductors. As stated previously, different control methods might yield different circuit
characteristics, therefore a specific control method is required for the analysis. The
maximum constant boost control method with third harmonic injection is employed here
to analyze the circuit characteristics as an example. Circuit characteristics under other

control methods can also be derived by following the provided procedure.
5.3.2.1. Operation Conditions

The maximrun constant boost control with third harmonic injection is quite similar to
traditional SVPWM. In each switching cycle, there are two active states, T1 and T2 for
instance, two zero states, To and T7, and one shoot through state, Tst, with the following
sequence: T51, T0, T1, T2, T7. To simplify the analysis, the currents feeding to the inverter
bridge in the two active states are assumed to be the same, and the capacitor voltage is
assumed to be constant. Under these assumptions, there are two possible operation

conditions as shown in Figure 5.4 (a) and (b) respectively.

107

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. 1T5, T0 Tl & T, T,

‘1. A
, T2 T7

pk E ' Ta] . ' To) .

L

2 \

In

. > .

0px;?! 1 2 3 2 Ofififnl 1 2 3 . 2 l4 ’

(a) CCM Operation (b) DCM Operation

Figure 5.4. Z-source inductor current for different operation conditions.

Figure 5.4 (a) shows the CCM operation condition. In each switching cycle, the circuit
starts with a shoot through zero state, T5,, during which the inductor current increases
linearly. After T5,, the inverter switches into traditional zero state, To, and then active
states with operation Mode 2, during which the inductor current keeps decreasing and is
reduced to half of the i,- in the middle of active states. Then Mode 3 comes into play by
keeping the inductor current almost constant and turning off the diode D. When the
active states are over and the circuit is turned into traditional zero state, Mode 2 appears
again, and the inductor current decreases linearly until another switching action turning
the circuit into switch shoot through state again (Mode 1). During this whole cycle, the
inductor current is continuous, therefore this operation condition is termed as CCM
condition. In the second half of the traditional zero state, it is also possible that the
inductor current decreases to zero before another switch shoot-through state starts, and
stays in Mode 4 for the rest of the cycle until the next switching action. Figure 5.4 (b)

shows this operation condition, which is termed as DCM condition.

108

In the practical case, the current fed to the inverter bridge is different for different active
states, which will make the operation condition more complicated, however, the basic
principles and the possible operation modes are the same, and it will be shown in the
following example that the analysis with the simplification still predicts the circuit
behavior very well. Also, there can be more than one operation condition during one line

cycle because of the variation of 1',- over a line cycle.
5.3.2.2. Critical Condition

Using a small inductor can reduce the cost, volume, and weight, but at the same time new
operation modes might occur. On the other hand, using a small inductor causes higher
current stress to the switches, and more importantly, under the new operation conditions
with new operation modes, the voltage to the inverter v,- is no longer always a constant
during active states, which is always 2Vc-Vo in the simple operation condition. This
might cause some unexpected output harmonics. Thus in some applications where
harmonic regulation is very crucial, one might not want the new operation modes to
appear. The critical condition of the new operation modes becomes important to the

design of the inductors.

The critical condition of the new operation modes under maximum constant boost control

with third harmonic injection can be described by the following inequation,

W

 

 

 

 

i 3Mcos¢2 _73TS, <1 cosgozl
< 22(73M — 1) L z 2 i (5 8)
3M cosqp 73Tst 1 7r 1
— < —cos((o -— —) cosqp < —
[22(73M — 1) L Z 3 2 J

109

When this inequation is met, the circuit starts to have new operation modes. The detailed

derivation of the critical condition can be found in appendix 4.

5.4. Analysis Example

The circuit characteristics will be different when the new operation modes happen. Also
it can be different for different operation conditions with different combinations and
sequences of the new modes. In this section, we will analyze the circuit characteristics
under CCM condition under maximum constant boost control with third harmonic
injection as an example. A detailed process will be provided, which can also be used to

analyze the circuit under other conditions.
5.4.1. Voltage Gain

Assume the circuit is operated under CCM condition throughout the whole line cycle.

The voltage across the capacitors in the Z-source network satisfies equation (5.9)

2
VcV0TO+7Tst _ Ts! VcVo '- T0+7TstVo2 = 0455M 005$ TSVC " 2T0+7Vc '1' T0+7V0
LTSVO — 2T0+7LV0 + 2T0+7VCL - VCLTS Z Ts - Ts! — T0+7

 

(5.9)

With this equation and all known parameters, one can calculate the output voltage and

voltage stress across the inverter switches based on the following equations:

2V —V T V T
Output rrns phase voltage: Vow =( c 0) “1 + C 02
2J§(Tal “1“ T02)

 

M, (5.10)

Switch voltage stress: VS = 2V6 — V0 , (5.11)

where Ta, and T 02 are shown in Figure 5.4 (a) and can be calculated by (A51) and (A52)

in appendix 5.

110

5.4.2. Relationship of Voltage Gain Versus Voltage Stress

From the previous discussion in chapter 2, one important criterion to judge the inverter
performance is the relationship of voltage gain versus the voltage stress. To evaluate the
new operation conditions, a comparison of the voltage stress versus voltage gain
relationship of the CCM condition with the simple operation condition under maximum

constant boost control with third harmonic injection is provided.

For simple operation condition, the voltage stress, Vs, versus voltage gain, G, relationship

is provided in chapter 2, which is
V, = (730 —1)V0 . (5.12)

For the CCM condition under maximum constant boost control, where T,,=(1-0. 866M) TS,

To+7=0. 0386MTS, from (5 .10), the voltage gain can be calculated by:

 

G = 72V0u, = 27r(Vc(1— 0.0772M) + 0.0386MV0). (5.13)

V0 /2 373V0

With this voltage gain, the value of (736 —1)V0 is,
(73G —1)V0 = (2.09 — 0.16M)Vc + (0.08M —1)V0. (5.14)

The modulation index M ranges from 0.58 to 1.15, the voltage stress under CCM

condition can be approximated as
V, = 2V2. — V0 e (.50 —1)V0. (5.15)

Therefore the voltage stress versus voltage gain relationship of CCM condition under

maximum constant boost control with third harmonic injection is about the same as in the

111

simple operation condition.
5.4.3. Design Guidelines

In some applications, one might want to avoid the new modes. When the load power
factor and modulation index are not very low, one could design the inductor of the
Z-source network according to (5.8) to avoid the unwanted operating modes. However,
from (5.8), it is obvious that when the load power factor is very low, say 0, the critical
condition is always met and the new operation condition is inevitable. Under these
extreme conditions, it is still possible to avoid the DCM by properly choosing the
parameters of the Z-source network. The basic design method of the Z-source inverter
for simple operation condition is given in chapter 4. Further rules have to be followed
to avoid the DCM condition. In all above analysis, the voltage across the capacitor is
assumed to be a constant, thus as long as the capacitance is large enough so that the
voltage ripple across the capacitor is reasonably low, the capacitor doesn’t have any effect
on the operation modes. From Figure 5.4, the main difference between CCM and DCM
is whether or not the inductor current decreases to zero during T 7. The inductor current
during T02 when it almost maintains constant in CCM condition can be calculated from
(5.16).

2 2
£1;_ VcV0TO+7Tst “Ts! VcVo ”TO+7TstV0

_. 5.16
2 LTSVO —2T0+7LV0 + 2T0+7VCL - VcLTS ( )

 

The detailed derivation can be found in the appendix 6.

Assuming that the inverter is operating under CCM condition, during T7, the inverter is

under operation mode 2 and the voltage across the inductor is Vo— Vc. The current drop

112

during this period is:

V —V V —
C 0T7s C V070”. (5.17)

A]:
T7 L L

 

 

T7 changes over the time, however, it never exceeds To+7.
The following inequation is a sufficient condition to avoid the DCM condition,

I,- Vc—Vo
—> T . 5.18
2 1. 0+7 ( )

 

However, this is not very straightforward way to design the inductor, because the
capacitor voltage needed in the equation is inductance related too. A simple try and
error procedure has to be followed in the design: start from the initial inductance value
from design process provided in chapter 4, for a given load and operation condition,

calculate the capacitor voltage Vc according to (5.9). With the resulted Vc, one can

calculate % and AIT7 from (5.16) and (5.17), and check whether (5.18) is met. One
needs to increase the inductance and go through the process again if (5. 1 8) is not met.

5.5. Simulation and Experimental Verifications

To verify the analysis, simulation of the Z-source inverter with L=50uH, Vo=100V,
f,=10kHz, f=60Hz (output frequency), Z=IOQ+1mH per phase Y connection under
maximum constant boost control with third harmonic injection and modulation index of
0.9 is performed. This circuit operates in CCM condition under these parameters. The
simulation results are shown in Figure 5.5 (a) and (b). Experimental results with the
same set up are shown in Figure 5.6 (a) and (b). In both cases, the output line to line

voltage, VLab, is the voltage across the load resistors. Comparison of simulation results

113

and calculated value of the capacitor voltage at different modulation indices is shown in
Figure 5.7. From the simulation and experimental results, we can see that the inductor
current stays constant while the dc bus voltage across the inverter bridge decreases to the
capacitor voltage in some period, which clearly demonstrates the new operation mode,
Mode 3. Also, the analysis results meet the simulation and experimental results very well
in terms of voltage boost. The actual operation of the circuit is more complicated than
the CCM condition analyzed above because of the inverter current variation over a whole
line cycle. However, as from Figure 5.7, the simplified analysis also provides quite

accurate estimation of the real performance.

Figure 5.7 also shows that the circuit characteristics of the CCM conditions is quite
different from that of simple operation condition, the voltage boost effect of CCM
condition is significantly higher than the simple operation condition with the same

modulation index and the same control method.

5.6. Method to Eliminate the Unwanted Operation Modes

From the above analysis, by proper design of the Z-source network and proper control,
one can avoid the operation Mode 3 to Mode 5 to certain degree in time of unwanted, e. g.
avoid the DCM condition. However, it is impossible to completely avoid these modes
with the configuration shown in Figure 1. Figure 8 shows a configuration where the
input diode is replaced by a switch, by using this configuration, the inverter is able to
completely avoid the unwanted operation modes by turning on the switch S during all
active states and traditional zero states. Further more, this configuration provides the

circuit bi-directional power flow function.

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V0: input voltage; 1in inductor current; V,-: dc link voltage across the inverter bridge; VLab:
output line to line voltage; Vc: capacitor voltage; I(La): Load current

Figure 5.5. Simulation results when operating at CCM condition.

115

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1 1 i r— calculatedresflsi

23 """ f """ I” """ f'l é simulation results

. simple operation
I

  
   

Capacitor voltage (V)
M
8
l
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Figure 5.7. Comparison of analyzed results and simulation results of capacitor voltage.

116

 

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from" V-' ”3}qu g L880

 

 

 

 

v

Figure 5.8. Bi-directional Z-source inverter.
5.7. Summary

Analysis of the Z-source inverter with small inductance or low power factor is given.
Three new operation modes are discussed. It is noted that with a small inductor in the
Z-source and low load power factor, there can be very different operation conditions.
Circuit operation conditions under traditional SVPWM control and maximum constant
boost control with third harmonic injection are both analyzed. The paper also analyzed
the Z-source inverter performance with small inductor under maximum constant boost
control with third harmonic injection as an example, and provided the capacitor voltage,
output voltage and voltage stress —voltage gain relationship. The analysis is verified by

simulation and experimental results.

It is note worthy to mention that the circuit can have different combination and sequence
of operation modes with different control methods and different circuit parameters, which
yields different circuit characteristics. However the analysis method and the detailed
derivation method for the equations in the appendices can also be applied to analyze the
circuit these conditions. A simple method to totally eliminate the new operation modes
by replacing the diode with a switch is proposed as an option. This configuration also

provides bi-directional power flow ability.

117

CHAPTER 6 Application and Control
of Z-Source Inverter for Traction Drive
of Fuel Cell — Battery Hybrid Electric

Vehicles

6.1. Introduction

6.1.1 Fuel Cell-Battery Hybrid Vehicles

Fuel cells (FCs) have achieved global attention as an alternative power source for hybrid
electric vehicles (HEVs) [82]. Fuel cell vehicles (FCVs), are being developed by auto
manufacturers [83-88], and have generated interest among induStry, environmentalists,
and consumers. A FCV promises the air quality benefits of a battery-powered electric
vehicle, with the driving range and convenience of a conventional internal combustion

engine vehicle.

Because of its nature, a fuel cell prefers to be operated under constant power to prolong
its lifetime and increase the efficiency. However, the traction power the vehicle
demands is ever changing. To balance the difference of these two and also to handle the
regenerative energy, a battery is often used as an energy storage device in FCVs, which
forms a Fuel Cell-battery Hybrid Electric Vehicle (FCHEV). Therefore, basically the

traction drive system of a FCHEV consists of a fuel cell stack, a battery pack, a controller

118

(power inverter), and a traction motor. The main source of the vehicle’s power is the
fuel cell. The secondary power source is the battery, which also stores excess energy
from the fuel cell, and from regenerative braking. The four utilized operating modes

and the power flow diagrams are outlined in the following.
Mode 1, medium power (Figure 6.1)

Under medium power, the vehicle traction motor only receives power from the fuel cell.

The fuel cell can also provide power to the battery if its state of charge (SOC) is low.
Mode 2, high power (Figure 6. 2)

During acceleration, or uphill driving, both the fuel cell and the battery provide power to
the traction motor. The battery speeds up the vehicle’s response time for a request of
acceleration, because the fuel cell typically has a slow response time. This also allows the

fuel cell to maintain a safe and efficient operating point.
Mode 3, low power (Figure 6. 3)

Because of the parasitic loads, such as the air compressor, associated with the fuel cell,
the fuel cell system efficiency decreases when operated under low power [85]. Thus the
vehicle will be operated strictly as a battery powered electric vehicle under low power by

turning off the fuel cell stack.
Mode 4, regenerative braking (Figure 6. 4)

During regenerative braking, the fuel cell produces no power, and the electric motor acts
as a generator, using the wheels to apply torque to the motor to generate electrical power,
this torque in turn slows the vehicle down. The electrical energy generated during

regenerative braking is stored in the battery until needed.

119

Mechanical Electan
- ' [:2] connection

 

 

connection
“gem 1J5 Conditional
V Power Flow “1"} power flow
""‘Tw‘.
. Fuel
iii i Cell J ”i
l_._._. .._..-.-. i.”
I?) l-
l."“"'? ”*1 1J3
Vii»: ...eJ 132’ '
2 ower
Batte ”‘
l W: J Inverter‘l
ll

 

  

Figure 6.2. High power operating mode 2.

Fuel

a LFESU

.._. ..... "Jr..-

a.-- ...- ..
[WW ml E Povirerhi
é . l l
i Battery _l‘ l Inverter |

 

 

E w “"‘W‘s. ' l
1.. _L .--—_—>_‘~—— ‘ :3 I .
l; 1

Q,

;_ Wheel L Motor Wheel l

.\__»’/

 

 

Figure 6.3. Low power operating mode 3.

120

n l Cell 4
r_. ___..- __.L_..
i Battery l+ J Power :
E l

J: ’ (Inverter l

__ .V. WT r__;_.1

 

Figure 6.4. Regenerative braking operating mode 4.

It is important to mention that in any of the operating modes, if the SOC of the battery

becomes too low, the fuel cell will provide power to charge the battery.
6.1.2. Fuel Cell and Battery Characteristics

Although there are many complex subsystems and parasitic loads associated with a fuel
cell, we are mainly concerned with the voltage and current. The fuel cell’s voltage (and
power) is determined by two main factors. First the rate at which hydrogen flows through
the fuel cell establishes the level of the V-I polarization curve. Second the amount of
current drawn by the inverter determines the point on this curve where the fuel cell will
operate. Thus, by controlling the amount of current drawn by the inverter, the fuel cell
power can be controlled for given hydrogen flow rate. The typical steady state V-I
polarization curve of the fuel cell is shown in Figure 6.5. As can be seen from Figure
6.5, the output voltage of the fiiel cell is heavily dependent on the load current as so does

the power.

121

 

'o

 

 

 

/ Ii ‘\
/
O 8‘L' / /’ ' [I \\
curve

3 0.6-- / 1.
O
00 V l
S / \
'5' 0.4““ / ‘
> I \

0.24- l/ I

/ 1
01“ l l
0.0 0.5 1.0 1.5
Current Density (A/cmz)

Figure 6.5. Typical fuel cell polarization curve.

On the other hand, the output voltage of a battery is relatively less current dependent
because of much smaller internal resistance. The voltage of a battery changes with the
SOC of the battery. A typical curve of voltage versus SOC of a 330 V lithium-ion

battery is shown in Figure 6.6.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

400.0 g ' l
E 3000-
u '
.553” . i . .
o l l l
> 200.0 g r . .. . !....... . .g.. , .} ..- . ‘e. .... .e- . . t
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0 I r l
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g ' ' l
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0-0 F I I I I I
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

State of Charge

Figure 6.6. Typical lithium-ion battery voltage versus SOC.
6.1.3 Traditional power conditioner configurations
As can be seen from above analysis, the power inverter is the key component in the

system to handle all power flow control. The inverter in FCHEV has to output the

122

requested power to the traction motor, capture excess power from the fuel cell, and to
absorb energy from regenerative braking. There are typically two configurations
available for this application shown in Figure 6.7. The FCHEV using the conventional
inverter (Figure 6.7 (a)) must use a bi-directional dc-dc converter to control the SOC of
the battery, because the modulation index is the inverter’s only control freedom. Also, the
conventional inverter is a buck (step-down) inverter, the output ac voltage is limited
below the fuel cell voltage. Because of the wide voltage range of the fuel cell, the
conventional inverter imposes high stresses to the switching devices. The dc-dc boosted
inverter (Figure 6.7(b)) can reduce these stresses, at the price of higher cost and
complexity. The dc-dc boost converter is used to boost (step-up) the voltage from the fuel
cell, to a steady dc bus voltage, and the inverter’s output ac voltage is controlled by the
modulation index. The system configuration using the dc-dc boosted inverter typically

uses a bi-directional dc—dc converter to control the SOC of a low voltage battery [83].

 

 

 

 

 

 

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cell 1'5 1 DC] ;:C E
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(a) System configuration using a conventional inverter

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lBoost; [___] DC 1 l

L l . 1 J Jig} JG

(b) System configuration using a DC-DC boosted inverter

 

   

 

 
  

 

 

 

  

 

Figure 6.7. Traditional configurations of FCVs.

123

Both configurations use an inverter bridge and at least one dc/dc converter, which

increases the cost and system complexity and reduces the system reliability.

In this chapter, several configurations of fuel cell-battery hybrid vehicles using the
Z-source inverter will be presented, the control strategy will be discussed. The
undesirable operation modes will be eliminated by control. A simple comparison of

these configurations will be provided.
6.2. Configurations and Control of Z-source Inverter for

FCHEVS

6.2.1 Configurations

With two control freedoms: shoot-through duty cycle and modulation index, it is possible
to use the Z-source inverter in fuel cell-battery hybrid vehicles. Three possible

configurations are shown in Figure 6.8.

 

 

 

 

Dl
G A
“it J t}
fuel Batte Q
+ XCl
cell 7 V0

 

l l
E
//
L5

Stack " _ D2 cz

 

 

 

 

 

 

 

 

 

(a) Configuration with battery connected in parallel with one of the capacitors

124

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

———-l>l— / e
14}
mill “ C C]
CC T o "‘ v
stack-_- _ D? c C2
JG
wt, - -
L2
— Bi-dir
Dcldc

 

 

 

2. Battery

 

 

(b) Configuration using a dc/dc converter to interface the capacitor and battery

 

Machine

 

 

 

 

 

 

 

(c) Configuration connecting the battery to the neutral point of the machine

Figure 6.8. Configurations of Z-source inverter for‘FCHEV.

For the first configuration, the capacitor voltage becomes the battery voltage. In the
second configuration, the capacitor voltage is controlled by the dc/dc converter
interfacing the battery. For the third configuration, the inverter bridge also serves as a
dc/dc converter interfacing the capacitor and the battery. With the PWM scheme shown

in Figure 6.9, where VI) and Vn control the shoot through duty ratio, and the three

reference signals are shifted from the center by V,.

The voltage relationship between the

capacitor and the battery under this PWM method can be derived as:

2(1—130)
Vc:—_—
l—DO-V,

125

b9

where Va is the capacitor voltage, Vb is the battery voltage, Do is the shoot through duty
ratio. Therefore, the capacitor voltage can also be controlled by the battery and shift of

the reference signals.
/\ r. /\ /\

1111!“

0'16"." V'"

 

””V V \

Figure 6.9. PWM scheme for the third configuration.
6.2.2 Control of the Inverter

From above discussion, the capacitor voltages of all three configurations are all
controlled and we still have two other freedoms: modulation index and shoot through

duty ratio, to control output power and battery state of charge.

In this system, there are three power sources/consumers: fuel cell, battery, and the motor,
as long as we can control the power flow of two of them, the third element automatically
matches the power difference. For Z-source inverter, the relationship of the capacitor

voltage and the input voltage is:

 

l-D0
= V, 6.2
C 1-21)0 0 ( )

where D0 is the shoot through duty ratio, V0 is the fuel cell voltage, Vc is the voltage

across the capacitor in Z-source network. From above analysis, the capacitor voltage Vc

126

is controlled by the battery and/or other circuitry, thus, the fuel cell voltage is controlled

to be

V0=1—2DO
1—00

 

Vc. . (6.3)

For given hydrogen and air flow rates, the V-I characteristic of the fuel cell is determined.
As a result, the fuel cell voltage determines the output current and power of the fuel cell.
Figure 6.10 shows the V-I curve of a typical 30kW fuel cell, with the controlled fuel cell

voltage, V0, the shaded area illustrates the output power of the fuel cell.

At the same time, the output power can be controlled by manipulating the modulation
index to produce the desired output voltage. The output peak phase voltage of the

inverter is

. M
Vphase = (ZVC — V0) * —2—, (6.4)

where M is the modulation index defined as the ratio of the magnitude of the reference
waveform and the triangular waveform in traditional SPWM.

The output power can be expressed as

pout 7.3.111? 1, (6.5)

\[2- phase

where I is the rms load current and pf is the load power factor.

Therefore the system is able to control the fuel cell output power and the output power to

the motor at the same time, as a result, the power charging the battery is

Pb = V010 — Pow. (6.6)

127

Thus we are able to control the SOC of the battery and drive the vehicle at the same time.

420

 

336 + i0, V0) - '

V)

v248~~
168-5

Voltage

Pin. .
84 -.. . _

 

 

 

O + l
O 1 00 200 300
Current (A)

Figure 6.10. Power control of fuel cell by controlling the voltage.

The above method can be used to control the converter when the fuel cell is turned on.
When the vehicle is in low speed cruising mode and not much power is needed, the fuel
cell will be turned off and the vehicle becomes a full electrical vehicle. Under this
condition, the diode in parallel with the fuel cell bypasses the fuel cell. The equivalent

circuit is shown in 6.11.

|L1 .
RL Ii
I I
A C Vc
I. iC ‘
R RC
0 1
RL 'L2

Figure 6.11. Model of the Z-source network considering parasitic parameters when the

fuel cell is turned off.

128

As discussed above, the capacitor voltage is controlled by the battery voltage, therefore,
the capacitor voltage is assumed to be Vc. assuming the voltage across the diode when
conducting is Vd and the voltage across the inverter bridge when conducting is V,, the

voltage across the inductor Ll during shoot through can be calculated by:
VLSI=VC_iLl(RL+RC)_VS' (6.7)

With the similar method, we can calculate the inductor voltage during non-shoot through,

which is:
VLnst = "-(Vb “ ib(Rb — RC» -(iLl 'ii)Rc "iLlRL — VD- (6-8)

Based on the average voltage across the inductor should be zero during steady state, for

shoot through duty ratio of Do, the following equation should be met:
VLstDO + Vb“, (1 — D0) = O , (6.9)
from which, the average inductor current can be calculated by (6.10).
1 ilt

= RC +RL
((2D0 —1)VC +ic(Rb —RC)+iiRC +D0(icRC -iiRC —2icRb —iCRL —VS '1“ VD)_VD))

iLl

(6.10)

From above analysis, the current through L1 and L2 can be controlled by adjusting shoot
through duty ratio around 50%. Because RC and RL are very small, the inductor current
will be very sensitive to the shoot through duty ratio. However, this can be easily

relieved by employing a simple current feed back control.

Analogous to the four vehicle operation modes shown in Figure 6.1, the inverter has

different operation methods too. For mode 1 and 2, the inverter operation is very

129

similar: the fuel cell power is controlled by shoot through duty ratio, the output power is
controlled by the output voltage and current. The only difference is that the output
power is higher than the fuel cell power and the battery is being discharged in mode 2,
the fuel cell power can be slightly higher/lower than or equal to the output power to
charge/discharge or maintain the battery based on the battery SOC in mode 1. For mode
3, the fuel cell is turned off and the diode D2 bypasses the fuel cell. To maintain the
inductor current at certain level, the shoot through duty ratio has to be slightly higher than
50%, and the modulation index is still used to control the output voltage/power. For
mode 4, to maintain a certain inductor current, the shoot through duty ratio also has to be

around 50%, and the power is being charged back to the battery.
6.2.3. Simulation and experimental verification

To verify the above mentioned feature of the Z-source inverter for FCHEVs, three cases
are examined and simulated for the first configuration. In these cases the circuit
parameters are Ll=L2=200uH, C1=400uF, C2 has been replaced (or connected in
parallel) with a 6.5Ah lithium-ion battery with a nominal voltage of 330V, switching
frequency of 10 kHz, and using constant boost control with third harmonic injection.
The characteristics of the battery and fuel cell are shown in Figure 6.6 and Figure 6.10.
An RL load is used in the simulation. The legends in the simulation results can be

found in Figure 6.15.
Case 1

The fuel cell voltage is kept constant at 300V (P=30kW), and the load power is varied
from 30kW, to 55kW, to 5kW, back to 30kW. As one would expect the battery SOC
should remain constant while the load is at 30kW (Pin==Pout). When the load is increased

130

to 55kW (Pin<Pout) the battery should supply the additional power requested by the load,
thus the SOC will decrease. When the load is decreased to SkW (Pin>Pout) the additional

power provided by the fuel cell will charge the battery, increasing the SOC.

These results are verified by simulation, Figure 6.12, starting from the top, the fuel cell
voltage is constant, and the fuel cell current is fairly constant. Next are the battery voltage,
SOC, load voltage, load current, and load power. Initially the load absorbs 30kW, and the
SOC stays constant. The load is then increased to 55kW and the SOC decreases. Next the
load is decreased to 5kW, and the SOC increases. Finally the load is returned to 30kW

and the SOC remains constant.

This simulation shows that we can operate the fuel cell at an efficient operating point,
while the battery handles the load dynamics. This also verifies the Z-source inverter can

be used to provide the medium, and high power operating modes.
Case 2

The load power is kept constant at 30kW, and the fuel cell power is varied between
30kW, 50kW, 20kW. Again the battery SOC should remain constant while the fuel cell is
producing 30kW. The battery will be charged when the fuel cell power is increased to
50kW (Pin>Pout), increasing the SOC. When the fuel cell power is decreased to 20kW
(Pin<Pout), the battery will supply the additional power requested by the load, decreasing

the SOC.

This can be verified in Figure 6.13. Starting from the bottom, the load power, current,
and voltage are constant, where the power is at approximately 30kW. Next are the

battery SOC, battery voltage, fuel cell current, and fuel cell voltage. Initially the fuel cell

131

produces 30kW, and the SOC stays constant. Then the fuel cell power is increased to
50kW and the SOC increases. Again the fuel cell produces 30kW, and the SOC stays
constant. Next the fuel cell power is decreased to 20kW, and the SOC decreases. Finally,
the fuel cell again produces 30kW, and the SOC stays constant. Case 2 shows that we

can control the fuel cell power, thus controlling the battery SOC.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

..-r . r“
r—\ - - -.....‘n, , h . P— -— ""
.. ...... .‘_.__~_. _ . A“ ...—-
I ‘ “‘*~" M ...-0"."
-—s.-~_ ‘

-.. 323:: r—\//— Battery
0.84 .. SOC

40001....“ .1

 

 

 

 

 

 

E o_o ' t Vl,oad

l 1 l
pun]. . . . 1.
.5. .. .—

 

 

 

$ 00‘ ‘

 

-200.0

60000.0 - _

A 40000.0 ‘

5. —-| P
20000.0 - l_‘ Load

_] Wfirr
0-0 I l I I l I I

0.0 5.0 10-0 15.0 20.0 25.0 30.0 35.0 40.
Its}

 

 

 

 

 

 

 

 

Figure 6.12. Simulation case 1.

132

 

 

 

 

 

 

 

 

5 320.0 ‘ "mm Vb

3 072 j /\ Battery
0-7 SOC

 

 

 

 

 

 

(V)
O
O
s
8
Q.

 

(A)

0.0“):

 

 

 

 

-200.0 ‘
40000.0 '

 

 

E 20000.0 “ PLoad

 

 

 

 

0-0 ' I I I 1 I I I

0.0 5.0 10.0 15.0 20.0 25.0 30.0 35.0 40
t(s)

Figure 6.13. Simulation case 2.
Case 3

The fuel cell operation is stopped, and the load power is varied from 5kW, t0 20kW. As
one would expect the battery SOC should decrease. When the fuel cell is turned off, D2
provides a current path for any possible current. These results are verified by simulation
as seen in Figure 6.14, starting from the top, the fuel cell power is zero. Next are the
battery voltage, SOC, load voltage, load current, and load power. Initially the load
absorbs 5kW, and the SOC decreases slowly. The load is then increased to 20kW and the
SOC decreases more rapidly. Case 3 verifies that the vehicle can be operated without

using the fuel cell, strictly as an electric vehicle, as in the low power mode. This also

133

demonstrates the ability of the inverter to capture power during regenerative braking,

when the fuel cell is also turned off and the output voltage and current is out of phase.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

GraphO

1.0 ll - . . - - .
E: 0'0 l] i f i I 3 I I Pfuelcell
E
S
g
$200000 1
" 0.0 ---:--:--l : : ; PLoad

 

Figure 6.14. Simulation case 3.

A lOkW scaled down prototype has been constructed with the first configuration as
shown in Figure 6.15. The battery used in the experiment is a 201V/6.5Ah NiMH
battery used in Toyota Prius HEV. The circuit parameters are LI=L2=1mH,
C1=C2=1.3mF, Rad—“SQ, LLoaflmH, switching frequency of 10 kHz, and using
constant boost control with third harmonic injection, a RL load is used. The fuel cell was
replicated by a 210V dc voltage source in series with a 1.350 resistor (Vinit=210V,
R=1.35 Q) to mimic the current dependent output voltage characteristic of the fuel cell, a

capacitor of lmF is connected in parallel with the “fuel cell”.

134

 

Battery J K} J G J LLoad RLoad

S C] \\ ILoad‘ 1 1.
C2

 

 

 

 

 

 

 

 

 

 

Figure 6.15. Experimental setup.

In the case 1 experiment, the load power is varied from 1.5kW (M=0.626, Do=0.065) to
2kW (M=0.73, Do=0.061) to 3kW (M=0.91, Do=0.054) back to 2kW. The fuel cell power
is designed to 2kW constant. The experimental result is shown in Figure 6.16. As
from the results, the load current changes as desired, the fuel cell voltage, V0, and the fuel
cell current, 10, are quite constant, and the fuel cell power is about 2kW. The battery
voltage is also pretty constant as it supposed to be. The battery current changes to meet
the power difference of the load power and the fuel cell power: when the load power is
1.5kW, the battery is charged; when the load power is 2kW, the average battery current is
about 0; when the load current is 3kW, the battery is being discharged. Figure 6.16

shows that the battery can be used to handle the load dynamics.

135

V0: 50V/div

“mm-mm
m

 

Figure 6.16. Experimental results for case 1.

In the case 2, the load is designed to 2kW constant. The fuel cell power is varied from
1.5kW (M=0.756, Do=0.037) to 2kW (M=0.73, Do=0.061) to 3kW (M=0.694, Do=0.l)
back to 2kW. The experimental result is shown in Figure 6.17. As from the results,
the load current is constant, the fuel cell voltage and fuel cell current changes, which
changes the fuel cell output power as desired. Also the battery current changes to

compensate the power difference between the load and the fuel cell power.

136

 

Figure 6.18. Experimental results for case 3.

In case 3, the fuel cell is disconnected, the battery is the only power source powering the
load. Shoot through duty ratio of around 50% is used to control the inductor current.

The experimental result is shown in Figure 6.18. The load current is pure sinusoidal.

137

As from the experimental results, the battery is being discharged and the load power is

2.4kW.

In practical application, the battery can only be charged when it is not fully charged, a
battery management system is necessary to estimate the state of charge of the battery and
feed back to the system to control the fuel cell power. Also, the fuel flow rate should be
controlled based on the required system power. There can be several different possible
controller configurations. Figure 6.19 shows one of the possible configurations. In
this system, the required power is given by the vehicle controller, and battery voltage is
measured to estimate the state of charge. With these information, the system
coordinator calculates the amount of energy that should be charged/discharged and
provide the corresponding shoot through duty ratio to the motor drive controller, also the
amount of power that should be supplied by the fuel cell is calculated and sent to the firel

cell controller to regulate the fuel flow rate.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fuel cell power System Shoot through duty ratio
I coordinator I
Power '53“er
command 5 informatlon Motor ‘— motor current &
Fuel cell Vehicle Battery soc WW 1: torque mend
controller controller estimator
A I I I I I I Gate signals
’5
a [ r——"
'E ’60"
.. ...... _. ”GHQ
g, loads
13 i
>~ \
I M
_> Fuel cell [
stack
1 J 00

 

 

 

 

 

 

 

 

Figure 6.19. Z-source inverter based fuel cell converter control system.

138

6.3. Undesirable Operation Modes and Control to

Eliminate Them

6.3.1 Undesirable Operation Modes Withorit Control

As from the previous chapter, when the load power factor, modulation index, and the
inductance of the inductors are low, there could be undesirable operation modes.
However, in these systems with batteries, the capacitor voltage is controlled by the
battery voltage, therefore the voltage stress is limited. Simulation is performed with
configuration 3 to examine the effect of undesirable modes. In this simulation, a RL
load is used with LLoad=3mH; R=0.4936Q at 60Hz, which yields power factor of 0.4, the
parameters of the LC network are L=lOOuH, C=lmF, the fuel cell is modeled with a
voltage source of 290 V with a 0.67 0 resistor in series, the battery is at nominal voltage
of 160 V and internal resistance of 0.45 (2, modulation index of 0.3 and shoot through

duty ratio of 0.18 are used. Figure 6.20 shows the simulation results.

As from the simulation results, the inverter outputs a perfect sinusoidal current, the
voltage across the inverter bridge is well limited. The battery is being charged and there
is only very little undesirable operation modes. Therefore, when a battery is used in the
system, the undesirable operation modes are not severe problems. However, the

undesirable operation modes can also be totally eliminated with a little additional control.

139

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Graph
3 0.0
-100.0
600.0'
... 400.0: ,
ca 2000‘ .~
0.0‘
000.0“
380.0‘ ; I
3 360.0 ‘ g;;,,;;;,j,;;§.;j,;;;w
3400*"; """" F """
3200‘ i l
0.05 0.1
1(5)

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200.0 II. I :

A 180.0 2" """" i """""
160.0 II '1 """"" i """""
140.0 ' ' '

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(V) 31(5)

v_luelcell

 

(-) :t(S)
i_luelce|l

 

140

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Graph
0 :t(s)
100'0 : 1r i_m0tor
+ i
3 0.0 L------: ------
(-) :t(S)
600'0 v_inverter
A 400.0‘ ‘
4’ 200.0‘
0.0“
200.0 ‘ (‘l 2 [(3)
: : v_capcitor
1~ 350-0 "C;:1;1111:;3_I;I; "
340.0” """ 1 """ r """
320.0‘I ; i 1
0.0926 0.0927 0.0928 0.0929
1(6)

 

 

 

(C)

Figure 6.20. Simulation results of configuration 3 with battery Connected to the neutral

point of the motor.
6.3.2 Control Method to Eliminate the Undesirable Operation Modes

Basically, the critical condition for undesirable operation modes to happen is when sum
of the two inductor currents is less than the current to the inverter bridge. With a battery,
the inductor current can be increased by charging the battery. There are some slight
differences for the three configurations. Take configuration 1 for an example, assuming
for a given fuel cell load current If, the fuel cell output voltage, V0, can be expressed as

follows:

141

V0 =f(If). (6.11)

From eq. (6.3), one can control the fuel cell voltage, so as to control the fuel cell current.
The fuel cell voltage is determined by (6.3), therefore, the fuel cell output current, 1;, and

power, Pf, can be calculated by
If =f"(Vo) (6.12)
Pf =Ifo. (6.13)

The average current through the inductor L1 equals to the fuel cell output current.
Assuming the output power of the inverter is P, then the power being charged to the

battery is PfP, as a result, the average battery current i1, is

 

. Pf — P
lb = (6.14)
Vb
The average current through L2 is
iL2=iL1—ib=if—ib (6.15)

Therefore, by controlling the shoot through duty ratio, one can control iu and in, and the
battery charging/discharging current. Also, this can also be used to eliminate the
unwanted operation modes when the load power factor and/or modulation index is low.
As from (1), the unwanted operation modes will only appear when IL1+IL2<I,-. One can
increase the shoot through duty ratio, D0, to reduce the fuel cell output voltage and
increase the fuel cell output current, thus increasing iL 1+iL 2. At the same time, the
inverter output voltage can be controlled by manipulating the modulation index from

(6.4). As a result, the fuel cell output power is increased to be higher than the load

142

power and the battery will be charged during the operation. For example, the initial
operating point is (If), Vfl) in Figure 6.21, by increasing the shoot through duty ratio, the
fuel cell operating point moves to (112, Vfl) assuming the battery voltage is almost
constant. Thus the inductor current, IL 1, whose average equals to fuel cell current,
increases, and so does IL 1+IL 2. At the same time, by controlling the modulation index,
the output voltage is maintained and so is the output current. Thus the sum of the two
inductor currents can be higher than the load peak current to avoid the unwanted

operation modes.

 

 

 

 

 

 

 

 

42 §
336 In 0 any”)
5.: 252~ ’
Fe"
g 168-
82-- .
0.0 I
0.0 100 200 ‘ 300
Current(A)

Figure 6.21. Fuel cell V-I characteristics.

In the above discussion, the method to eliminate the undesirable operation modes when
the fuel cell is turned on is presented. When the fuel cell is turned off, the inductor
current can be controlled from (6.10), therefore the undesirable operation modes can also

be eliminated.
6.3.3 Experimental Verification of the Control Method

To demonstrate the effectiveness of the proposed control, experiments are carried out

based on the setup shown in Figure 6.22. In this setup, L1=L2=1mH, Cl=C2=1320uF,

143

a NiMH battery with nominal voltage of Vb=201V is connected in parallel with C1. To
mimic the characteristic of a fuel cell stack, a voltage source of V,-,,,-—-225V connected in
series with a resistor of 0.99 is used, a capacitor of lmF is connected in parallel with the

fuel cell simulator.

 

 

 

 

L1 1
| | . JG JG JG
FY LLoad RLoad
R + W
A V. W»
Vim -

 

 

 

 

 

 

 

 

 

Figure 6.22. Experimental setup.

As analyzed in chapter 5, the unwanted operation modes are most likely to happen under
low modulation index and low power factor. Therefore, two sets of experiments were

set up to confirm the ability to eliminate the unwanted modes under these conditions:

1) Low power factor, and low output current. The fuel cell is on, and the battery is
being charged to increase the inductor current to eliminate the undesirable operation
modes. This operation condition is similar to low speed cruising for induction machine
based vehicles. In the experiment, an extreme case is used: a pure inductive load of
2mH per phase, and very low modulation index of M=0.1 with the fundamental
frequency of 60Hz. Because the battery is connected in parallel with the fuel cell
through a inductor and a diode, the output voltage of the fuel cell can not be higher than

the battery, thus the minimum output current of the fuel cell is more than 20 A assuming

144

the battery is 201V. With this low load current condition, the minimum fuel cell current
is enough to prevent the unwanted modes from happening. Therefore, in this case, no

shoot through is introduced. The experimental result is shown in Figure 6.23.

-VB. -60Wdlv

 

 

°“ V0; 50Vldiv
5 VPN:100VIdiv

10A/dlil3:

 

 

 

 

Figure 6.23. Experimental result of case 1.

As from the experimental results, the output current is a pure sinusoidal waveform and
VpN is a constant dc voltage that equals to the battery voltage and fuel cell voltage, which
clearly shows that only operation modes (a) and (b) appeared. Also, the battery is being

charged

2) In the second case, the load power factor is relatively high and the load current is
high too, and the load voltage is still low, thus the modulation index is low. This
operation is trying to mimic the low speed acceleration mode, where speed is still low
and load torque is high so the load voltage is low and load current is high with high
power factor. In this experiment, the load is 2mH+1.2£1 per phase at 60 Hz. The fuel
cell is on, to take more current out of the fuel cell, shoot through duty ratio of 0.09 is used,

and modulation index of 0.34 is used. The experimental results are shown in Figure

145

6.24 (a) and (b).

("an I)

 

 

 

. . m'
www— ae—s 4 h.‘

w “J, . .
.. IL':20NdIV"" . ..f .. ..

 

(a) in fundamental cycles

 

(b) in switching cycles

Figure 6.24. Experimental results for case 2.

146

From the experimental results, the battery is being charged to produce more inductor
current to eliminate the unwanted operation modes. The PN voltage clearly shows that

only mode (a) and (b) appeared. Pure sinusoidal output current is achieved.

The experimental result when the fuel cell is turned off is shown in Figure 6.18. As can

be seen from the result, there are no undesirable operation modes.
6.4. Comparison of the Three Configurations

All of these configurations can be used in fuel cell hybrid vehicles. Each configuration
has its own pros. and cons.. In this section, two main factors are considered for purpose
of comparison: the cost and output voltage range. In order to make a fair comparison,

the maximum allowable voltage stress across the inverter bridge is assumed to be Vinverter-
6.4.1 Configuration 1

For this configuration, there are two control freedoms, the shoot through duty ratio and
the modulation index. The capacitor voltage equals to the battery voltage, Vb, the
maximum voltage stress across the inverter bridge is 2Vb when the fuel cell is turned off.
Therefore, for a given allowable inverter bridge voltage stress, the battery voltage is
limited to 0.5 14mm”. For a shoot through duty ratio of D, the fuel cell voltage is:

V0 : (1 _ 2D)Vinverter (616)
2 — 2D

Using maximum constant boost control, the maximum modulation index available for

shoot through duty ratio of D is

2(1 — D)

.5

M = (6.17)

147

The available output phase peak voltage becomes

V

_ inverter

1 1
V =—M 2*—V- —V _———. 6.18
out 2 ( 2 1nverter o) 2J3 ( )

This is the available output voltage when the fuel cell is turned on. When the fuel cell is
turned off, the shoot through duty ratio is 0.5, which is also covered in the above

equation.

The cost of this configuration is low because there is no extra component required.
6.4.2 Configuration 2

In the second configuration, an extra dc/dc converter is used to interface the battery and
the capacitor. Thus the capacitor voltage can be controlled independently without using
the two freedoms of the Z-source inverter. Therefore, the voltage across the inverter
bridge can always be controlled to its maximum limit by the dc/dc converter. As a

result, the obtainable output voltage for a given shoot through duty ratio becomes

1 (1 _ D )Vinverter
V = —MVJ = ,
out 2 1nverter J3

 

(6.19)

which is higher than the previous case. This also includes the condition when the fuel
cell is turned off and the shoot through duty ratio becomes 0.5, which is the same as the

previous case.

With an extra dc/dc converter, the cost of the inverter system is increased.

148

6.4.3 Configuration 3

With the battery connected to the neutral point of the machine, the inverter has an extra
control freedom: the shift of the reference signals, however, this freedom is related to the

shoot through duty ratio and the modulation index.
To maximize the available output voltage, we choose the pn voltage to be the maximum

value that the IGBT module can handle, Vimmr, the firel cell voltage should be

4(1—D)Vb
V =2V —V- , =——— - , . (6.20)
0 C mver er l-D—VS mver er

To get this relationship of the capacitor voltage and fuel cell voltage, the shoot through

duty ratio should be

_ _ 2(1 - UV];
(1 "' D _ Vs )Vinverter

 

(6.21)

Therefore the shift of the reference can be expressed as a function of the battery voltage
and the shoot through duty ratio in order to keep the inverter bridge voltage to its

allowable level:

— V
Vs: 2” +1—D (6.22)

inverter

 

Assume that the maximum constant boost control is used, the available modulation index

under this condition is

2(1—D—|V,|) _ 2

.5 ‘73

2V],

 

 

). (6.23)

(l—D-Il—D—

 

inverter

Thus the available output voltage is

149

 

 

l 1 2V
Vout = ‘2' M * Vinverter = To _ D — I1 _ D _ b )* Vinverter (6'24)
3 inverter

Therefore, the available output voltage is related to the shoot through duty ratio as well as
the battery voltage. In general, the voltage range is between the first two configurations.
For this configuration, there is no extra dc/dc converter is needed. However, there
might be some dc current flowing through the machine, which increases the motor copper

loss, therefore the machine needs to be oversized a little bit.
6.4.4 Comparison Summary

From all above discussion, a summary of the comparison is given in the Table 6.1.

Table.6. 1. Comparison of different configurations

 

 

 

 

Configurations Battery Voltage Output voltage Cost
range
1 High Small 1 Low
2 Low ~ High Wide High
3 Medium Medium Medium

 

 

 

 

 

 

The battery voltage is included here because it is also a consideration for safety issues.
6.5 Summary

In this chapter, three configurations of fuel cell-battery hybrid vehicles using the
Z-source inverter are presented. Basic control methods are described. It is shown that

with a battery in the system, the effect of the undesirable operation modes can be

150

minimized. A control method to totally eliminate the undesirable operation modes is

also presented. A simple comparison between the three configurations is provided.

151

CHAPTER 7 Conclusions and

Recommendations

7.1 Contributions

This work has the following contributions:

0 Three PWM methods are proposed to achieve simple boost, maximum boost, and
maximum constant boost respectively. A modified PWM is also proposed to reduce the
switching losses. A comparison with an commonly adopted PWM method shows that

the modified PWM method can reduce the switching loss significantly.

0 A small signal model for the Z-source inverter is developed, based on this model a
closed loop controller is also designed using gain scheduling method. The validity of

the model and the controller are verified with simulation/experimental results.

0 A comprehensive comparison of the Z-source inverter and traditional inverters for
fuel cell vehicle application is conducted. The comparison uses switching device power,
passive components requirement, constant power speed ratio, reliability, and inerter
efficiency as bench marks. The comparison shows that the Z-source inverter is very

promising for applications with low boost ratio.

0 Three new operation modes of the Z-source inverter are found and analyzed in

detail. The critical condition and consequences when they happen are presented.

0 Three configurations of the Z-source inverter for fuel cell-battery hybrid vehicle
applications are proposed with the basic control methods. The effect of undesirable

152

operation modes is minimized in these configurations. Also control methods are

proposed to eliminate the undesirable operation modes.
7.2 Recommendations for future works

The modified PWM method proposed in this work reduces the switching loss, but also
requires a relatively larger inductor. Further work should be done on the PWM scheme
to see if it is possible to double the frequency on the inductor, keeping the switching loss

within a reasonable amount, so that the inductor size can be reduced significantly.

The controller designed in this work is for a RL load, further work should be done to
design the controller for motor load. The load comes into the picture of the whole
system, a comprehensive model of the motor is also needed. Another future job in
designing the controller is to consider the limit of the two control freedoms and its effect

on the system stability and transient response.

In this work, only the voltage fed Z-source inverter is considered. As a counterpart, the

current fed Z-source inverter should also be studied.

153

APPENDIX 1 Switching Device Power

Derivation

a. Traditional PWM Inverter

For the traditional PWM inverter, the output phase RMS voltage at peak power is

V.
V =——’—M. (Al.1)
p 2.5

With motor power factor of cosm, the output line RMS current is

P0

=———. Al.2
p 3cos¢Vp ( )

Because the line current is evenly shared by two switches in a line cycle, the average

current through each switch is:

P0 2 4P0

*

av — 3cos (0V,- /(2\[2_)M E _ 3cos goVinM °

 

 

1 (A13)

The maximum voltage stress of the switches occurs when the output power is zero, and

the fuel cell voltage reaches its maximum value, which is
Vs = Vmax - (Al.4)

The total average switching device power of the circuit can be derived based on its

definition by

8Vmax P 0

. (A1.5) ‘
cos gal/inM

(5010),, =6*V,. *Im, =

154

The peak current through the switches is the peak line current, from (A12),

4P
1pk=./§1p=———0—. (A16)
3 cos goViM
The total peak switching device power of the traditional PWM inverter is
8V P
SDP =6*V *I =——'flx—°. Al.7
( ) pk 3 pk COS (0V,- M ( )

b. dc/dc Boosted PWM Inverter

For the switch in the boost converter, treating the switch and the diode as a switching cell,
the maximum voltage it sustains is VDC, based on power balance, the average current

through it during maximum power is
P
Iavs = 7:. (A18)

The average switching device power of the dc/dc converter isthe product of the current

and voltage, which is

P
(501’ch = 179' * VDC- (A19)

1'
Assume the current through the inductor in the boost converter is constant, the peak

current through the switch is the same as the average current. The peak switching

device power is

P
(SDP) Dcpk = 7/9- * VDC. (A1.10)

l

The voltage stress of the inverter switches is VDC, The RMS phase voltage at

155

modulation index of M is

 

VDCM
V = . (A1.11)
” 2J5
The RMS line current under load power factor of cos W is:
Ip = L. (A1.12)
3V p cos (p
The average current through switches under maximum power is
1 J5 4
— P — P0 (Al . 13)

 

(mm - n - 3cos¢7VDCnM '

The average switching device power of the system is the sum of SDP of the dc/dc

converter and the inverter, which is

8P0 P

 

(SDP)a,, =6*VDC *Iavim, +(SDP)DCm, =———+—0*VDC (A1.14)
cos (071'M - V,-
The peak switch current of the inverter is
4P
1 - = .51 = 0 . A1.15
pkmv ‘0 3 cos (oVDCM ( )

The peak switching device power of the system is the sum of the SDP of the dc/dc

converter and the inverter

 

8P P
SDP =6*V *1 - + SDP = 0 +—£*V . Al.16
( ) pk DC pkmv ( )DC pk COS 99M Vi DC ( )

c. Z-Source Inverter
For the Z source inverter, the current through the inverter switches consists of two

156

elements, one is the current to the load and the other is the current through them during
the shoot through state. Because of the symmetrical structure of the inverter, the current
during shoot through in terms of average is evenly distributed in three parallel paths.
The current through the inverter during shoot through is twice of the inductor current.

Therefore, the average current value in shoot through period through each switch is

2
10m=31u (A1.l7)

where IL is the inductor current. From the input end, the average current through the
diode is equal to the sum of the average current through inductor L1 and capacitor C). In
steady state, the average current through the capacitor is zero, the average current through
the inductor equals to that of the diode. The output power of the fuel cell stack under
maximum power is Po, therefore, the average current through the diode as well as the

inductor is:

”‘0

id=1L=—£. (A1.18)

1
While in traditional switching states, the average current is the same as a conventional

PWM inverter, therefore the overall average current through inverter switches is the sum

of the current in both states, which is

3, .I0,_JZL()__T_0)=Z,L.I0+MO_I0), (A1.l9)

I :
“vs 3 L Ts 3V0 cosgozr T, 3 Ts 3V,- cosgozzM TS

where T 0 is the shoot through period in a switching cycle T 5, V0 is the RMS output phase

voltage. With the maximum constant boost control, To and V0 can be expressed as

157

To =[1—1/5— ITS. (A120)

M V-
V=____l. (A121)
0 «GM—1245 '

Voltage stress of the inverter switches with constant boost control under modulation

 

 

 

index of M is
V - BV- - Vi (A1 22)
S ' (EM -1 ° '
The average switching device power of the inverter is
P _.
(SDP)... =6Im.,V, =4ILV,- T0 +3__0_ 1__T0 = 2Po(2 15M) + 4.510,
TS («B—M — 1) cos (pnM Ts (JEM —1) cos (pn
(A123)

The peak current through the switches occurs during shoot through. To calculate the
peak current through the switches, we assume that when the sWitches are on they are pure

resistors with the same resistance, which is shown in Figure A. 1.

 

l
21L R R R
IS]
I
132 a 1G
R R

Figure A. 1. Inverter model during shoot through

Based on this model and voltage balance of three paths, we can have the following

158

equations

132+151=§1L9 (A124)

Isl — 132 = Ia. (A125)
where la is the phase a load current. From which we can get
1 2
Isl =—Ia+—IL. (A126)
2 3
The peak current through the switch S1 occurs when the line current of phase A is at its
peak, which is

1,): ‘5’? = “"0 . (A127)
p 3cosgoVo 3cos¢MBV,-

 

Thus the peak switch current is

1 1 =_;4_L_,Z,L =J__+2Po (A128)
p 3cos¢MBV,- 3 3cos¢MBVi 3V,-

 

The peak switching device power of the inverter is

V) 2 P0 1 4P0 4P0 4P0
——+— )= + .
«[3M—1 3 V,- 2 3cosgoMBV) fiM—l cosgoM

 

 

 

(SDP) p1 = 61,,k‘V, = 6 (A129)

159

APPENDIX 2 Passive Components

Requirement

a. dc/dc Boosted PWM Inverter

For the inductor in the boost converter, the current ripple can be calculated by the current

increase when the switch is on.

 

V VV V
N):40n_ “DC 5)“ m2n
L Lnx

where D is the duty cycle.
b. Z—Source Inverter

For the Z-source inverter, when the inverter is in a shoot through state, the voltage across
the inductor is the voltage across the capacitor. Therefore the current ripple of the

inductor can be calculated as
A1 = —£ T 0 , (A22)

where V, is the voltage across the capacitor C,

 

VC = £0 + B) = K'— fiM . (A23)
2 zfiM-l
and To is the shoot through duty ratio, which is
To = (1 - i23— M)TS , (A2.4)

160

where T, is the switching period.

We have

 

V,./§M [ «5

= 1————'MT. A2.5
2L(J§M-l) 2 IS ( )

161

APPENDIX 3 Critical Condition of New

Modes Under SVPWM

In simple operation when only mode 1 and 2 occur, the inductor currents are pure dc.
The new modes only happen when the inductor current is less than half of the current to
the inverter bridge, 1'). From the circuit, it is obvious that the average inductor current in
the Z-source equals to the average input current. Based on the power balance, the

average inductor current can be calculated by

P
11. =70 (A31)

For modulation index of M, load peak current of ka, and power factor of cosgo, the

inductor current is

P 3
IL =7=2Mcosgolpk. (A32)
The maximum current fed to the inverter bridge is

[pk c0502 2%
[W = 1 (A33)
71'
1 pk cos((o — 3) cosgo < 5

The new modes happen when IL < él'l’k . As a result, the critical condition is

162

2 1
Mcos <— cos 2—
‘0 3 q) 2

Mcowgmg) (A34)

However, when COS(p<—;-, for 05¢sg- and M<l.15, the inequation is always true,

therefore the critical condition is reduced to M cosgp <i:- .

163

APPENDIX 4 Critical Condition of New
Modes Under Maximum Constant
Boost Control with Third Harmonic

Injection

Under maximum constant boost control with third harmonic injection, the shoot through
time, T5,, in one cycle is always constant. For simple operation condition, the inductor

current increases linearly during shoot through

and decreases linearly otherwise. The current ripple is
(A4.1)

The average inductor current can be calculated by (A32), the minimum inductor current

with shoot through time of Ts, is

[Lmin =—--——-, (A42)

For modulation index of M in simple operation condition, the output phase rms voltage,

V0,”, can be calculated by

(A43)

 

V = MVO
out 2x/—2_(x/§M-l)

The output power, P, can be expressed as

164

2

3cos ——L—
3 ¢(2(\/3M — 1)) 3cosgo MVO ,
P=_Vou11pk COS<0= : ( ) '
42 22 82 JEM —1

 

(A4.4)

Under this control method, the relationship of modulation index and shoot through period,

T3,, is

2 7
M = __ 1- ___S’ . A45

f

The new modes happen when the minimum inductor current is less than half of the

maximum current feeding to the inverter bridge expressed in (A33).

Put all these equations together, we can get the following critical condition:

r )

 

 

 

 

3M cosgo x57}, 1 l
— < — cosgo Z —
22(J3M — 1) L z 2 (A4 6)
< .
3M cosgo JETS, 1 7r 1 k
— <—cos((o——) cosgp <—
(2Z(J§M — 1) L Z 3 2 J

165

APPENDIX 5 Characteristic Analysis

of the Circuit Under CCM Condition

For the Operating condition shown in Figure 5.4 (a), Ta] is an active state with operation

Mode 2 and Tag is active state with operation Mode 3.

For a given control method and a

modulation index, T,,, T 0, and T7 are already determined. To simplify the analysis,

assume T3,, T0, T a], Tag, and T 7 are constants throughout a whole line cycle, and T 0=T7,

and the current to the inverter during active states, it, is always a constant.

For the CCM

operation condition shown in Figure 5.4 (a) and voltage second balance of the inductor,

we have following equations

1,,L+ VCTS, JEIL

T = —T
al Vc—Vo 0

 

T02 =Ts ”Ts! ‘TO—Tal —T79

_i_i__T0+T7 Vc-Vo
2 2 L

 

In:

3

where 1,, is shown in Figure 5.4 (a).
In shoot-through state, the current to the inverter is

ii =2iL.

(A5.1)

(A52)

(A53)

(A5.4)

In steady state, the capacitor voltage is constant, and the average current goes through the

capacitor is zero, therefore,

I66

1V i-
1n(Tst +T0 +T7 +Ta1)+§-iC-Tsz(Tsz +To +T7 +Tal)+‘2LT02

(A55)
Vc 2 -
= 2InTst + '2‘ st + (Tal + Ta2)li
Put all equations together and solve for I), we have
2 2
i- _ 2(VcVoTO+7Tst 'Tst VcVo _TO+7TstVo ) (A5.6)

 

’ _ LTSVO - 270,71. V0 + 2T0+7VCL — VCLTS '
The dc link voltage of the inverter during Ta; is 2Vc- V,-, and Vc during Ta), therefore, the

output phase rms voltage of the inverter can be calculated by:

= (2V6 — V0)Tal + VcTaZ M
2550211 + Taz)

 

(A5.7)

out

Based on the output voltage, we can calculate the average 1',- in active states from the load
side. The average current to the inverter during active state can be calculated as in

(A5.8).

011!

N

If

— . . . 4
3‘5], srn(a)t — (o)[srn a)! — srn(cor — —3— 71)]
If

6

sin a)! -sin(a)t — g7!) (A53)

sin(a)t — w — Z 7r)[sin(a)t — 3 7r) - sin(a)t — 3 7:) V
+ 3 3 3 )dt = -———' 0‘” cosgo
I I Z
sm wt — srn(a)t - 3 7r)

 

Putting (A5.7) into (A5.8), and equalizing (A56) and (A5.8), one can get (A5.9). One
can solve (A59) to get capacitor voltage Vc under this operating condition with
maximum constant boost control with third harmonic injection.

2 2
VcVoTO+7Tsl _Tst VcVo _TO+7TslV0 = 0455M 00560 Tch '2T0+7Vc + T0+7
LTSVO —2T0+7LV0 +2T0+7VCL—VCLTS z T, —Ts, 40,7

 

 

V0 (A59)

167

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