Digital Geometric-Sequence Control Technique for

Thesis Titleby
Faculty of Electrical and Computer Engineering
in conformity with the requirements for
the degree of Master of Applied Science
Queen’s University
Kingston, Ontario, Canada
Abstract
Bidirectional DC-DC converters are widely used in different applications such as
energy storage systems, Electric Vehicles (EVs), Interruptible Power Supplies (UPS),
etc. In particular, future EVs require bidirectional power flow in order to integrate
energy storage units into smart grids. These bidirectional power converters provide
Grid to Vehicle (V2G)/ Vehicle to Grid (G2V) power flow capability for future EVs.
Generally, there are two control loops used for bidirectional DC-DC converters:
The inner current loop and The outer loop. The control of Dual Active Bridge (DAB)
converters used in EVs are proved to be challenging due to the wide range of oper-
ating conditions and non-linear behaviour of the converter. In this thesis, the precise
mathematical model of the converter is derived and non-linear control schemes are
proposed for the control system of bidirectional DC-DC converters based on the de-
rived model. The proposed inner current control technique is developed based on a
novel Geometric-Sequence Control (GSC) approach. The proposed control technique
offers significantly improved performance as compared to one for conventional control
approaches. The proposed technique utilizes a simple control algorithm which saves
on the computational resources. Therefore, it has higher reliability, which is essen-
tial in this application. Although, the proposed control technique is based on the
mathematical model of the converter, its robustness against parameter uncertainties
i
is proven.
Three different control modes for charging the traction batteries in EVs are in-
vestigated in this thesis: the voltage mode control, the current mode control, and
the power mode control. The outer loop control is determined by each of the three
control modes. The structure of the outer control loop provides the current reference
for the inner current loop.
Comprehensive computer simulations have been conducted in order to evaluate
the performance of the proposed control methods. In addition, the proposed control
have been verified on a 3.3 kW experimental prototype. Simulation and experimental
results show the superior performance of the proposed control techniques over the
conventional ones.
ii
Acknowledgements
I would like to thank my supervisor, Professor Alireza Bakhshai, for his ongoing
support and supervision. He has offered encouragement and guidance throughout my
research. Without his belief in my abilities, this work would not have been possible.
I would like to thank Dr. Majid Pahlevaninezhad for his mentorship, support, and
friendship throughout this project. His guidance had a large impact in completing
this research.
I am very grateful to have had such an amazing group of lab colleagues, friends.
Thank you all for your friendship and support in making this possible.
Thank you to Debra Fraser and the rest of the ECE department staff for always
having an open door and being there whenever I needed a little help.
Last but not least, I would like to thank my parents, Shahnaz Jafarian Abyaneh
and Hossein Askarian Abyaneh, and my brother Ehsan Askarian for all of their love
and support throughout this endeavour.
iii
Contents
List of Figures viii
List of Tables xiii
Chapter 1: Introduction 1 1.1 Thesis contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Thesis organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Chapter 2: Literature Review 7 2.1 Overview of Bidirectional DAB DC-DC Converters . . . . . . . . . . 7 2.2 Control of DAB Converters . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Control Parameters of DAB Converters . . . . . . . . . . . . . 9 2.2.2 Closed Loop Control schemes . . . . . . . . . . . . . . . . . . 10
2.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Chapter 3: Bidirectional Dual Active Bridge (DAB) DC-DC Con- verters 23
3.1 Lossless DAB Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Different Modulations . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 Phase Shift Modulation . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Alternative Modulation Methods . . . . . . . . . . . . . . . . 30 3.2.3 Optimized Modulation . . . . . . . . . . . . . . . . . . . . . . 36
3.3 Steady-State Model of the converter . . . . . . . . . . . . . . . . . . . 42 3.4 Loss Analysis of the DAB Converter . . . . . . . . . . . . . . . . . . 46
iv
3.4.1 Power Loss in Switch Converters . . . . . . . . . . . . . . . . 46 3.4.2 Transformer, Inductor . . . . . . . . . . . . . . . . . . . . . . 50 3.4.3 Total Losses - Predicted Efficiency . . . . . . . . . . . . . . . 53
3.5 Linear Control for DAB converters . . . . . . . . . . . . . . . . . . . 53 3.5.1 Closed Loop PI Control . . . . . . . . . . . . . . . . . . . . . 54 3.5.2 Digitalization of the PI control . . . . . . . . . . . . . . . . . 55
3.6 Simulation and Experimental Results . . . . . . . . . . . . . . . . . . 56 3.7 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Chapter 4: Controller Design 72 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 4.2 Modulation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.3 Digital Current Control in DAB Converters Based on Novel Geometric-
Sequence Control (GSC) Approach . . . . . . . . . . . . . . . . . . . 75 4.3.1 Oscillation Problem . . . . . . . . . . . . . . . . . . . . . . . . 76 4.3.2 Geometric-Sequence Current Control Approach . . . . . . . . 76
4.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 4.5 Robustness and Reliability . . . . . . . . . . . . . . . . . . . . . . . . 88 4.6 Outer Control Loop Design . . . . . . . . . . . . . . . . . . . . . . . 90 4.7 Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.8 Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
Chapter 5: Conclusions and Future Work 99 5.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Bibliography 102
CCM Continuous Current Mode.
CUL Counter Upper Limit.
DAB Dual Active Bridge.
DCM Discontinuous Current Mode.
G2V Grid to Vehicle.
PSM Phase-Shift Modulation.
UPS Interruptible Power Supplies.
V2G Vehicle to Grid.
ZVS Zero Voltage Switching.
1.1 Non-fossil energy sources for the utility grid . . . . . . . . . . . . . . 2
1.2 Typical Electricity load variation of the utility grid during 24 hours . 3
1.3 AC-DC Converter used in G2V/V2G applications . . . . . . . . . . . 4
2.1 Dual Active Bridge (DAB) converter . . . . . . . . . . . . . . . . . . 8
2.2 Negative feedback for Dual Active Bridge (DAB) converters . . . . . 11
2.3 Digitally controlled converter under an outer voltage and an inner cur-
rent loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.4 Closed loop control based with Predictive duty cylce modulation . . . 15
2.5 Investigated cascade control structure consisting of an outer voltage
loop and inner current loop . . . . . . . . . . . . . . . . . . . . . . . 16
2.6 control structure for the synchronous buck converter . . . . . . . . . . 17
2.7 Battery charging profile for Electric Vehicles (EV) . . . . . . . . . . . 18
2.8 A control structure for DAB DC-DC converter for aerospace application 19
2.9 Controller to operate converter with optimal efficiency . . . . . . . . 20
2.10 Self Sustained Oscillating Control Modulation structure . . . . . . . . 21
2.11 SSOC-PCM control system . . . . . . . . . . . . . . . . . . . . . . . . 22
3.1 Dual active bridge converter . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Lossless model of Dual Active Bridge (DAB) converter . . . . . . . . 25
viii
3.3 4 control parameters (, A, B, TS) to control single stage dual active
bridge converters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.4 Phase shift modulation waveforms . . . . . . . . . . . . . . . . . . . . 29
3.5 Voltage and current waveforms for power transfer from port A to port
B for triangular current mode modulator when VA > kVB . . . . . . 31
3.6 Voltage and current waveforms for power transfer from port B to port
A for triangular current mode modulator when VA > kVB . . . . . . 32
B for triangular current mode modulator when VA < kVB . . . . . . 34
A for triangular current mode modulator when VA < kVB . . . . . . 35
B for trapezoidal current mode modulator when VA > kVB . . . . . . 36
A for trapezoidal current mode modulator when VA > kVB . . . . . . 37
3.11 The 12 basic voltage sequences generated with DAB converter . . . . 39
3.12 Waveform of Dual Active Bridge Converter operating in Continuous
Current Mode (CCM) with Zero Voltage Switching (ZVS) . . . . . . 42
3.13 Area of the Current Waveform of DAB operating in CCM . . . . . . 45
3.14 3D ZVS region space for Va=400 Vb=350 . . . . . . . . . . . . . . . 51
3.15 2D-ZVS region space for Va= 400 V, Vb= 350 V . . . . . . . . . . . 52
3.16 PI controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.17 Conversion of Continuous PI controller to Discrete PI controller . . . 55
3.18 operation for Va=400 Vb=350 AB = 0.116 . . . . . . . . . . . . . . 57
ix
3.19 (A) depicts operation for Va=400 Vb=350 AB = 0.058 and (B) depicts
depicts operation for Va=400 Vb=350 AB = 0.029 . . . . . . . . . . 58
3.20 (A) depicts operation for Va=400 Vb=350 AB = 0.116 and (B) depicts
depicts operation for Va=400 Vb=350 AB = 0.063 . . . . . . . . . . 59
3.21 operation for Va=400 Vb=250 AB = 0.3, IBat = 12.1A With ZVS . . 60
3.22 operation for Va=400 Vb=250 AB = 0.2, IBat = 10A without ZVS . 61
3.23 operation for Va=400 Vb=250 AB = 0.2, A = 0.31, and B = 0.5
with IBat = 10A with ZVS . . . . . . . . . . . . . . . . . . . . . . . . 62
3.24 Steady state operation of the DAB converter with the operating con-
ditions: VA = 360, VB = 400, iBat = 6A . . . . . . . . . . . . . . . . 63
3.25 Steady state operation of the DAB converter with the operating con-
ditions: VA = 360, VB = 400, iBat = 8A . . . . . . . . . . . . . . . . 64
3.26 Transient response of the DAB converter VA = 360, VB = 400 I=8 A
to I=4 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
to I=6 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
to I=12 A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.29 instability in some operating conditions for PI controller . . . . . . . 66
3.30 transient response when VB = 400V , I = 6A VA = 360V to VA = 400V 66
3.31 steady state operation at VA = 100V , VB = 90V without ZVS . . . . 68
3.32 steady state operation at VA = 100V , VB = 120V without ZVS . . . . 69
3.33 steady state operation at VA = 100V , VB = 120V , I = 2.2A with
achieved ZVS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
x
3.34 steady state operation at VB = 100V , VA = 90V with achieved ZVS . 71
3.35 transient response at VB = 100V , VA = 120V with achieved ZVS . . . 71
4.1 Sawtooth counter created based on the digital counter for PSM . . . 74
4.2 switching instants created based on their respective sawtooth counter 75
4.3 Oscillation in CCM mode in response to perturbation . . . . . . . . . 77
4.4 Oscillation in CCM mode in response to poor control scheme . . . . . 77
4.5 Effect of Change in AB on the current waveform . . . . . . . . . . . 78
4.6 Transient and steady-state waveforms in one half-cycle . . . . . . . . 81
4.7 Overall procedure of the applied control method . . . . . . . . . . . . 84
4.8 Inner control block diagram . . . . . . . . . . . . . . . . . . . . . . . 85
4.9 Overall procedure of the waveform when a = 1 . . . . . . . . . . . . . 87
4.10 Outer loop control in order to set the current reference for the inner
current loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.11 GSC control approach for 50% step change in current for Va=360 V
and Vb=400V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
and Vb=400V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
and Vb=400V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
and Vb=250V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
and Vb=250V with LS% = 20% . . . . . . . . . . . . . . . . . . . . 94
xi
and Vb=400V with LS% = −20% . . . . . . . . . . . . . . . . . . . 95
4.17 GSC control approach for change in primary voltage: Va=360 V to
Va=400 V and fixed Vb=400V . . . . . . . . . . . . . . . . . . . . . 95
4.18 3kW bidirectional AC/DC converter prototype . . . . . . . . . . . . . 97
xii
CATED RANGE OF DUTY RATIOS; * DENOTES NO OSCILLA-
TION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.1 The criteria required to enable each modulation . . . . . . . . . . . . 40
3.2 The Power levels of DAB with respect to the each modulation for the
applied DC voltages, duty cycles and the phase-shift . . . . . . . . . 40
3.3 Phase-shift required to achieve a certain power level . . . . . . . . . 41
3.4 The RMS current IL with respect to the considered voltage sequences 41
3.5 DAB waveform details for VA = 360 VDC and VB = 400 VDC with
different power levels . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 DAB converter Specifications . . . . . . . . . . . . . . . . . . . . . . 67
3.7 DAB system parameters . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.1 Elimination of the error current by a geometric progression procedure
with a common ratio of 1 2
. . . . . . . . . . . . . . . . . . . . . . . . 84
xiii
1
Human activities, since the industrial revolution, have constantly changed the natural
composition of Earth’s atmosphere. The revolution had an enormous impact on the
concentration of greenhouse gases in the atmosphere. This has resulted in global
warming and an increase in air pollution, which continues on its trend with even a
steeper slope everyday. There is indisputable evidence that the conversion of forests
to agricultural land, the emission of industrial chemicals, and the consumption of
fossil fuels are the major contributing factors to air pollution [1].
According to [1], the Earth’s surface temperature has risen by about 1F in the
past century, with accelerated warming occurring in the past three decades. Accord-
ing to statistical reviews of the atmospheric and climatic records, there is abundant
evidence that global warming over the past 50 years is directly attributable to human
activities [2].
In order to mitigate air pollution and the CO2-caused global climate change,
mankind has to increase the efficiency of their uses, and shift to non fossil energy
sources as depicted in Figure 1.1.
One of the greatest achievements of modern technology was the development of
2
EV Power Conditioning Systems
internal combustion engine vehicle, especially auto-mobiles. The mass usage of auto-
mobiles has caused carbon dioxide to aggregate in the atmosphere which significantly
accounts for the catastrophic problem of global warming [3]. Recent research and
development suggest a replacement of high efficiency, clean and safe transportation
such as Electric Vehicles, Hybrid Electric vehicles, and fuel cell vehicles over conven-
tional vehicles which use combustion engines [3]. Hybrid Electric Vehicles utilize an
optimized internal combustion engine with electric machines to improve the efficiency.
For CO2 free emissions, electric vehicles powered by battery packs and fuel cell are
used [4, 5].
In order to have a conventional way to charge the traction batteries of electric vehi-
cles, it is desirable to have the option of charging the batteries from inside our homes
with the utility grid -the so-called Grid to Vehicle (G2V)-. It is also beneficial to
have a bidirectional power flow between the grid and the EV batteries. This allows
the Electric Vehicle (EV) to act as a local Distributed Generator (DG) and help the
3
Valley filling
Peak Shaving
Figure 1.2: Typical Electricity load variation of the utility grid during 24 hours
power grid [6–9].
Uncontrolled recharging of Purely Electric Vehicle (PEV) could cause significant
impacts on power systems [6,9,9–13]. Basically, the electricity load in the grid load is
desirable to have a flat line shape. However, as depicted in Figure 1.2 the electricity
load base lines typically demonstrate a valley and a rising electricity load during
specific times of the day. The valley happens during midnight and the peak is around
19:00 to 22:00 [14]. Moreover, there is a high correlation between the electricity load
and the cost of electricity generation [15]. Therefore, a valley-filling pattern is ideal
for utilizing the idle capacity of power generators and minimizing recharging cost [10].
Recent trends in the automotive industry toward EVs has create the need for
highly compact, lightweight, and efficient power converters to exchange electrical
power between the power grid and EVs [16].
Figure 1.3 depicts the schematic structure of a bidirectional AC-DC converter used
for the purpose of power transfer between the grid and the EV. The AC-DC con-
verter used in this thesis is comprised of a full-bridge AC-DC converter followed by
a bidirectional DC-DC converter. In the G2V application, the grid applies 240 RMS
1.1. THESIS CONTRIBUTIONS 4
Figure 1.3: AC-DC Converter used in G2V/V2G applications
voltage to the input of the AC-DC converter. The inductor Lg makes the AC-DC
converter act as a boost converter meaning the output DC voltage (VA) has to be
above 350 V DC. In order to have a low RMS current at the high frequency network
(transformer winding) of the DC-DC converter, it is desirable to have a flexible DC
bus (Side A) voltage ranging from 350 VDC to 450 VDC. The traction battery volt-
ages in the Electric Vehicles (EVs) are typically between 250 V DC to 430 V DC.
Therefore, In order to have a converter applicable for charging all the traction electric
vehicles, a DC-DC converter is introduced which converts the DC bus voltage of side
A to a range of 230 VDC and 450 VDC at side B.
1.1 Thesis contributions
The main contribution of this thesis are summarized as follows:
1. Design and development of bidirectional DC-DC converter with V2G/G2V ca-
pability
2. Development of the precise discrete-time domain model for steady-state opera-
tion and during transients.
1.2. THESIS ORGANIZATION 5
3. digital current programmed control technique for Dual Active Bridge (DAB)
converters based on a novel Geometric-Sequence Control (GSC) approach.
4. Development of a variable structure outer loop for three different modes of
operations during charging process of the traction batteries.
5. Stability Analysis of the closed loop control schemes.
1.2 Thesis organization
chapter 2, Literature Review
In this chapter a brief description of the degree’s of freedom to control DAB converters
are explained. Moreover, some of the literature control structures for DC-DC convert-
ers are explored. The benefits and drawbacks of the conventional control structures
are also discussed.
chapter 3, Bidirectional DAB DC-DC converter
Steady state, soft switching, and loss analysis of the system is evaluated. Also, a PI
control is designed and used to control the power flow between the two DC Buses.
chapter 4, Proposed control
To solve the drawbacks of conventional linear controllers, a digital current control
based on the proposed geometric-sequence control approach is proposed. The perfor-
mance, stability, and robustness of the control is explored in this chapter. Moreover,
1.2. THESIS ORGANIZATION 6
the modulation scheme specifically made for the proposed control method is presented
to create the switching instants of the semi-conductors. Simulation and experimen-
tal results are provided to verify the behaviour of the proposed geometric-sequence
control approach.
chapter 6, Conclusion
A conclusion of the thesis along with future work in the field.
7
Bidirectional DC-DC converters with galvanic isolation are used in Plug-in Hybrid
Electric Vehicles (PHEVs), Purely Electric Vehicles (PEVs) and energy storage sys-
tems for the purpose of charging batteries from the grid and releasing the battery
charge back to the grid. Dual Active Bridge (DAB) converters are one of the most
prominent bidirectional DC-DC converters [17, 18, 18–22]. The DAB topology offers
the low number of passive components, the evenly shared currents in the switches,
and its soft switching properties which make it a good candidate for bidirectional
DC-DC converters [23–25]. In this chapter, a literature review of DAB converters
and their control schemes are explained.
2.1 Overview of Bidirectional DAB DC-DC Converters
Figure 2.1 depicts a typical full-bridge DAB converter that is widely used in Vehicle
to Grid (V2G)/ Grid to Vehicle (G2V) applications. DAB converters consist of two
.
Ls K:1
Figure 2.1: Dual Active Bridge (DAB) converter
• CA and CB absorb the high frequency current ripple produced by the two active
bridges. This results in smooth terminal DC voltages and currents with low
ripple at the ports A and B.
• The DC-AC inverter converts the DC voltage on the input bus to a quasi square
wave high frequency voltage. This voltage is then applied to the high frequency
transformer. The AC-DC bridge converts the high frequency AC voltage to the
DC voltage at the output bus. Bidirectional power transfer can be achieved by
interchanging the role of the bridge A and the bridge B.
• The high frequency transformer allows large voltage and current transfer ratio
as well as providing galvanic isolation [26]. The transformer and the filter com-
ponents become smaller when the switching frequency is high. The transformer
leakage inductance (Ls) is used to control the power flow in DAB convert-
ers [26, 27].
A typical DAB converter has 4 degrees of freedom in terms of control to adjust
the power transfer:
(t) within −π < < π,
2.2. CONTROL OF DAB CONVERTERS 9
• The duty cycle, DA, of vACA (t) within 0 < DA < 1/2 or equivalently the phase
shift between the leading and lagging leg of bridge A,
• The duty cycle, DB, of vACB (t) within 0 < DB < 1/2 or equivalently the phase
shift between the leading and lagging leg of bridge B,
• The switching frequency fs
2.2.1 Control Parameters of DAB Converters
The most common and conventional way to control the power transfer between the
two DC ports A and B in a DAB converter is to utilize the phase-shift between the
two active bridges, , as the only control parameter out of the 4 control parameters
mentioned above. This method is called the Phase-Shift Modulation (PSM) scheme
for DAB converters [28]. The advantage of PSM scheme is its simplicity and easy
control since it only has one degree of freedom in terms of control. Additionally,
due to the symmetric circuit topology on the primary and the secondary sides of the
transformer, fast and smooth bidirectional power flow for G2V and V2G applications
can be achieved by simply using positive or negative phase-shifts. However, the main
disadvantage of the conventional PSM is the poor efficiency at light loads. This is due
to high RMS currents in the high frequency and the high switching losses transformer
when DAB is operated in wide voltage ranges operations and the hard switching of
the semiconductors specially at light loads.
In order to overcome the aforementioned issues for DAB converters with PSM
scheme, complex control structures and alternative modulation methods have been
proposed in literature (cf. Section 3.2) [29,30].
The proposed modulation schemes take advantage of the multiple degrees of free-
dom offered by DAB converters in order to optimize the performance.
The modulation schemes investigated in [31] extend the zero voltage switching of
the DAB converter and reduce the transformer RMS current. Detailed investigation
of the behaviour of the control parameter in [31] with either D1 ≤ 0.5 and D2 = 0.5
or D1 = 0.5 and D2 ≤ 0.5 is given in [23, 32]. Therefore, these modulations are
faced with a one dimensional (1-D) optimization problem to improve the converter
efficiency since either D1 or D2 changes [23].
[32–34] represent a 2-dimensional (D1 and D2 change simultaneously) optimiza-
tion approach in which highly efficient operation of the DAB converter is reported.
However, compared to the 1-d problem, the 2-d problem is considerably more complex
to solve. More on modulations are investigated in Section 3.2.
In addition to an optimized modulation scheme, a robust control structure is of
great importance for an optimal performance of DAB converters. Here, we are going
to introduce some of the control structures presented in literature.
2.2.2 Closed Loop Control schemes
Figure 2.2 depicts a simple negative feedback structure for controlling a DAB con-
verter. The output load of a DAB converter depends on the input and output volt-
ages, the high frequency inductance (transformer leakage inductance), transformer
ratio and the 4 control parameters. The objective in DC-DC power converters is
to maintain a constant output voltage/ current, in spite of the disturbances in the
Switching Dual Active Bridge (DAB) DC-DC Converter
v(t)=f(vg ,iload ,d) vg(t)
iload(t)
Figure 2.2: Negative feedback for Dual Active Bridge (DAB) converters
system. Therefore, in order to obtain a given constant output voltage under all con-
ditions, a negative feedback has to be built in the system to automatically adjust the
control parameters as necessary.
The negative feedback controls of the DAB converter can typically be categorized
into single measurement controllers (voltage controller/ current controller) or a cas-
caded control consisting of an inner current control loop and an outer voltage control
loop which provides a current reference for the inner loop.
The current programmed control can be controlled analogy or digitally.
• Analogue current programmed control for DAB converters can be used in wide
applications and power factor correction application [35–42] . Analog current
programmed control is categorized into peak or valley current control. Since the
inductor current is controlled tightly, the converter dynamics becomes simpler
and consequently resulting in simple and robust wide-bandwidth control in DC-
DC converter. Moreover, the peak current control provides an over-current
switch protection.
• Digital control offers advantages as such lower sensitivity to parameter vari-
ations, programmability and possibilities to improve performance using more
advanced control structures [43]. However, as compared to analog control, dig-
ital control suffers from a smaller control loop bandwidth due to the presence
of time delays the digital control structure and the computation.
There are two ways to observe the current feedback of the system; One way is to
observe the input or output DC current. This way includes the capacitor dynamics
and that might result in a slow inner current loop. Another way is to detect the high
frequency current. Sampling the high frequency current can make the inner current
control loop in cascade control very fast compared to the outer voltage loop. Two
very common control methods based on these current observations are the average
current control [44–47] and the peak current control [48–51].
• In average current control the outer voltage loop gives the desired Iref and the
inner current loop produces the phase shift to match the average current equal
to Iref . The advantage of average controller is its simplicity.
• In Peak Current Mode, the outer voltage loop outputs the desired peak current
of the high frequency network. This current reference is then compared with
the high frequency current. The output of this comparison is then given to the
modulation scheme to produce the switching instants
Power Stage
A/D
Figure 2.3: Digitally controlled converter under an outer voltage and an inner current loop
In the following we are mainly reviewing some of digital control structures in lit-
eratures:
Sampling delays (e.g. Analog to Digital Conversion (ADC) or Digital to Analog Con-
version (DAC)) and digital calculation processes of the micro-controllers (e.g. Digital
Signal Processing (DSP) and/or Field Programmable Gate Arrays (FPGA)) can com-
promise control performance, especially in high-frequency applications. One way to
improve the digital control performance is to use predictive technique by calculating
the duty cycle for the next switching cycle based on the sensed or observed state and
input/output information in each switching cycle, such that the error related to the
controlled variable is minimized in the next cycle or in the next several cycles. pre-
dictive and deadbeat digital current programmed control are investigated in [52–55]
. In [52] a predictive digital control for valley, peak or average current is discussed
Table 2.1: PERIOD-DOUBLING OSCILLATIONS OCCUR FOR THE INDI- CATED RANGE OF DUTY RATIOS; * DENOTES NO OSCILLATION
Modulation Valley Peak Average
Trailing Triangle * D > 0.5 *
Leading Triangle D < 0.5 * *
in [52] for three basic converters: buck, boost, buck-boost. It is shown in [52] that
the current controller in predictive valley control under trailing edge modulation, is
inherently stable for all operating points where in predictive average current control
and predictive peak current control oscillations occur under the operating conditions
when the duty cycle is greater than 0.5. This is exactly the same as in analogue
current-programmed control, where usually a slope-compensation ramp signal to the
sensed current signal is used to suppress the instability. [52] summarizes the corre-
lation between different modulation methods and the controlled variables of interest
can be organized as shown in Table 2.1
In [56–59] predictive current mode control is used in bidirectional isolated DC-DC
converter. [58] presents predictive phase shift current mode coontrol and predictive
duty cycle mode (Figure 2.4) of control for single phase high frequency transformer
isolated DAB DC-DC converter. The predictive control algorithm increases the band-
width of the current loop of the converter which enables tracing of the current refer-
ence within one switching cycle.
[59] proposes a valley-peak current control for the dual active bridge (DAB)
converter to improve dynamic responses. With this control approach, the reference
current, can be achieved in one switching cycle. The valley-peak current control
+ PI
ITs/2
d2
I0
d1
Figure 2.4: Closed loop control based with Predictive duty cylce modulation
strategy offers a fast over-current switch protection, and meanwhile eliminates the
possible saturation of the high-frequency transformer.
In [60,61] an accurate small-signal model for a galvanic isolated, bidirectional DC-DC
converter and the implementation of a corresponding digital controller are detailed.
Figure 2.5 depicts a cascaded digital control loop block diagram which is used in
[60, 61]. The voltage controller GC,V and the current controller GC,I is implemented
based on the precise small signal model derived in the papers.
[62] proposes a new digital control solution for bidirectional DC-DC converters
for energy storage (figure 2.6). the charging algorithm of the battery is divided into
two states; the first state, when the battery is discharged the converter must supply a
constant current. This current is maintained constant until the voltage on one storage
cell reaches a certain limit (e.g. 4.2 V for Li-ion cell), after that in the second state
the voltage is kept constant and the current decreases. At the end of the charging
+ + Voltage
Figure 2.5: Investigated cascade control structure consisting of an outer voltage loop and inner current loop
process a minimum current is supplied to the battery to compensate the self discharge
phenomenon. The charging characteristic which the converter follows to ensure higher
life expectancy for battery is depicted in Figure 2.7. One important issue of this
implementation that must be considered for this digital control topology is the windup
effect. In the direct transfer mode, in the first charging cycle when the current
is constant, the voltage PID controller is saturated to its maximum output value.
During this time the integral element increases. If the calculations are implemented in
a fixed point format this element can reach high values. When reaching the predefined
threshold the voltage PID controller must come out of saturation to keep the battery
voltage constant. This is impossible because of the high value of the integral element.
In the reverse mode, the converter must supply to the DC bus the amount of power
that is demanded by the energy management master control system.
[63] presents a controller for bidirectional control of a DAB DC-DC converter
PWMCin
S1
S2
Cin
Figure 2.6: control structure for the synchronous buck converter
which uses the current at the high frequency network of the DAB as a control pa-
rameter to meet the dynamic power and regeneration demand of advanced aircraft
electric loads using ultra-capacitors. (figure 2.8)
[30] analyses the performance of a high current DAB DC-DC converter when oper-
ated over a wide operating range. [30] shows that the high currents on the battery side
cause significant design issue in order to obtain a high efficiency. The conventional
phase shift modulation can have high conduction and switching losses. Therefore,
a combined triangular and traapezoidal modulation method is used to reduce losses
over the wide operating range. The control modulation was implemented on a fuel
cell vehicle application where a bidirectional DAB converter is used as an interface
between a 12 V battery and a high voltage DC bus; the result was 2% improvement
4
5
3
2
1
Charge Time/h
C ha
rg e
V ol
ta ge
in efficiency compared to phase shift modulation.
[64–66] proposes a multi-variable control system for an efficient Zero Voltage
Switching (ZVS) full-bridge DC-DC converter used in a (Plug-in Hybrid Electric Ve-
hicle (PHEV)). This converter processes the power between the high voltage traction
battery and low voltage (12V) battery. Generally, Phase-shift between the two legs of
the full-bridge converter is the main control parameter to regulate the output power.
However, the zero voltage switching cannot be guaranteed by merely controlling the
phase-shift particularly for light load conditions. Efficient operation of the converter
is crucial in order to maintain the energy of traction battery for a longer time and for
increasing driving distance. Therefore, In order to extend the soft switching operation
of the converter for light loads, asymmetrical passive auxiliary circuits are used to
FBC1 FBC2
ioiL Ls
Polarity
Figure 2.8: A control structure for DAB DC-DC converter for aerospace application
provide reactive current. However, the auxiliary circuits increase extra current bur-
den on the power MOSFETs, leading to lower efficiency. To obtain the optimal power
transfer, the duty cycle of bridge legs (as another control parameter) is also controlled
to minimize the conduction losses of the converter. Basically, the multi-variable con-
troller adjusts the phase shift angle to mainly serve as the output regulation control
parameter while duty cycle control of bridge legs are varied to keep converter in the
soft switching region in such a way that the circulating currents are kept at their
minimum level which helps in reduction of conduction losses.
In [64] a modified DAB topology with a modulation technique is proposed for bidi-
rectional DC-DC conversion that improves the soft switching range of the converter
and reduces the large current ripples at low voltage side (figure 2.9). Phase shift
and duty cycles of active bridges on two sides, (DA, DB, ), are used to control the
converter in order to extend the soft switching range against wide range of operating
Voltage and current
? Lookup table for (VA,VB)
Figure 2.9: Controller to operate converter with optimal efficiency
voltages on both ports while reducing the circulating current to obtain an optimal
efficiency for DAB converter. The converter operation is analysed and the soft switch-
ing conditions are extracted.
[67] presents Peak Current Mode Self Sustained Oscillating Control (PCM-SSOC)
technique for DAB DC-DC converter. The proposed control improves the performance
of the bidirectional DAB DC-DC converter over wide operating conditions. Basically,
the proposed PCM-SSOC technique adaptively regulates the frequency and the peak
current of the high frequency network for a triangular modulation to achieve an op-
timal performance (figure 2.10).
Bridge A DC/AC
Figure 2.10: Self Sustained Oscillating Control Modulation structure
In [68] a control approach for a current-driven full-bridge DC-DC converter, which
significantly improves the converter efficiency over a very wide range of operating con-
ditions is presented. The proposed control approach is based on the Self Sustained
Oscillating Control (SSOC) scheme, which adaptively changes the phase shift and
the switching frequency of the converter for different operating points. In this control
technique, the switching instants of the power mosfets are determined by the primary
current feedback and the timing signal produced based on the zero crossing instants
of the transformer primary current. Therefore, for different operating conditions the
control systems automatically tunes the the control variable in order to achieve an
improved converter performance as depicted in Figure 2.11.
2.3. SUMMARY 22
2.3 Summary
Overall, the dynamical equations of DAB converter has non-linear characteristics.
Therefore, a linear control can be utilized by linearisation of the dynamical equation
at a particular point -small signal model of the system-. This method is selected
when the converter is operating at a particular point in which the DC voltage and
DC current remain constant. However, linear control of the DAB converter over wide
range of operating conditions will show poor performance performance since it lacks
the required control flexibility to regulate the output of the DAB converter. To solve
this issue, in this thesis, a digital current control based on the mathematical model
of the system is proposed which offers improved transient response, higher reliability,
and robustness against parameter uncertainties.
23
Converters
3.1 Lossless DAB Model
The DAB converter introduced in the previous chapter is redrawn in Figure 3.1. The
two DC voltages in the input and output DC ports are converted to quasi square wave-
forms (VACA and VACB
with assuming ideal conversion -no losses-, ideal transformer transformer magnetiz-
ing and parasitic capacitance are neglected-, and assuming constant supply voltage
VA and VB, the full-bridge circuits can be replaced by the respective square-wave
voltages(Figure 3.2).
For lossless converter with no switching losses or conduction losses, the quasi
3.1. LOSSLESS DAB MODEL 24
square waveform of vACA (t) can obtain the following three different voltage levels,
vACA (t) =
+VA for state I : TA1, TA4 on, TA2, TA3 off
0 for state II : TA1, TA3 on, TA2, TA4 off
0 for state III : TA2, TA4 on, TA1, TA3 off
−VA for state IV : TA2, TA3 on, TA1, TA4 off
(3.1)
By replacing every A in (3.1) with B, the different voltage levels of vACB (t) is similarly
determined. To avoid the high frequency transformer from saturating, it is crucial
that in steady state the average values of vACA (t) and vACB
(t) evaluated over one
switching cycle becomes zero. The resulting voltage across the inductor vL(t) is:
Ls K:1
VL(t) = vACA (t)− kvACB
(t) (3.2)
The High Frequency (HF) inductor current at time t1, with respect to an initial
current of iL(t0) is derived as follows:
iL(t1) = iL(t0)− 1
k .VAC-B
LsiL=iAC-A
Figure 3.2: Lossless model of Dual Active Bridge (DAB) converter
The produced/ received instantaneous power is calculated by PA(t) = vACA (t).iL(t)
and PB(t) = kvACB (t).iL(t). Since the Dual Active Bridge (DAB) is lossless, PA(t) =
PB(t). To simplify the calculations in (3.3), ti and ti+1 are selected such that vACA and
vACB remain constant during that respective time intervals.
The average power over on switching cycle, Ts = 1/fs, is computed as
PA = 1
PB(t)dt (3.5)
for side B. Thus, one switching period, t0 < t < t0 + Ts, is split up into m time
intervals (numbered with the index counter i) with constant voltages vACA (t) and
vACB (t) to further simplify the average power calculation,
3.1. LOSSLESS DAB MODEL 26
vp
vS
t
t
TS
φA TS
φB TS
Figure 3.3: 4 control parameters (, A, B, TS) to control single stage dual active bridge converters
time interval I: t0 < t < t1
time interval II: t1 < t < t2
.
final time interval : tm−1 < t < tm = t0 + Ts
According to (3.4) and (3.5), PA is determined by vACA (t) and iL(t) and similarly, PB
is determined by kvACB (t) and iL(t) .
3.2. DIFFERENT MODULATIONS 27
Figure 3.3 depicts 4 control parameters that adjust power flow in DAB converters:
• The phase-shift, , between vACA (t) and vACB
(t) with −π < < π,
• The duty cycle, DA, of vACA (t) with 0 < DA < 1/2 / also the phase-shift
between the leading and lagging leg of bridge A (A),
• The duty cycle, DB, of vACB (t) with 0 < DB < 1/2 / also called the phase-shift
between the leading and lagging leg of bridge B (B),
• The switching frequency fs
3.2.1 Phase Shift Modulation
Phase Shift Modulation (PSM) is the most common modulation principle. This
modulation only uses one of the 4 control parameters which is the phase shift, ,
between the two full bridge converters to adjust the transferred power. It operates
the DAB converter with a fixed switching frequency and keeps the duty cycles at its
maximum (DA = DB = 1/2). Therefore, out of the three voltage levels for vACA (t)
and vACB (t) based on (3.1), the zero voltage level is eliminated and the resulted
voltage is square wave voltage with only positive and negative values (Figure 3.4).
At steady-state operation, the phase-shift time T and the DC supply voltage VA and
VB remain constant during one cycle. This property results in vACA (t), vACB
(t), and
vACA (t+
Ts 2
) = −vACA (t)
vACB (t+
Ts 2
) = −iL(t)
Therefore, the power flow can be recalculated and derived by evaluating one half-cycle
as follows:
PA = 1
with t0 = 0.
To determine iL(t) in the time interval 0 < t < Ts/2, t needs to be broken into
time intervals I and II (Figure 3.4). In each of these intervals voltage across the
inductor remains constant. For a positive phase shift, 0 < < π, the instantaneous
inductor current can be written as:
time interval I: iL(t) = il,0 + 1
L (VA + kVB)t ∀ 0 < t < t1
time interval II: iL(t) = il(t1) + 1
L (VA + kVB)t ∀ 0 < t < t2
(3.9)
With consideration of PA = PB and (3.8), and by extending the results to the full
phase-shift range (−π < < π) the transferred power is
P = PA = PB = kVAVB(π − ||)
2π2fsL ∀ − π < < π (3.10)
where P > 0 (positive ) denotes a power transfer from side A to side B and P < 0
Figure 3.4: Phase shift modulation waveforms
(negative ) denotes a power transfer from side B to side A. Maximum power transfer
occurs for = ±π/2 with the solution
|PPS,max| = kVAVB 8fsL
(3.11)
The resulting expression for the phase-shift needed to obtain a given power transfer
is derived by rearranging (3.10):
= π
) sgn(p) ∀ |P | < |PPS,max| (3.12)
The wide usage of phase-shift modulation is because of its simplicity to adjust the
transferred power. Drawbacks of DAB converters operated under phase-shift modu-
lation are high switching losses at some operating conditions and large RMS currents
in the HF transformer for most operating conditions when operated in wide voltage
ranges. Effective transformer utilization is obtained only when VA is close to VB.
3.2.2 Alternative Modulation Methods
In this section, some alternative modulations are investigated, which use not only the
phase shift between vACA (t) and vACB
(t), but also change the duty cycles of vACA (t)
and vACB (t) to overcome the phase-shift modulation problems. The alternative mod-
ulation schemes bring the following advantages as compared to the phase-shift modu-
lation: Minimum RMS HF inductor current (IL) that results in low conduction losses;
soft switching over a wide range of operating conditions.
For simplicity, a more intuitive method is typically used to determine DA and DB,
where DA and DB are selected in order to achieve a triangular or trapezoidal shape
of the transformer current which results in low switching losses and low conduction
losses. With the triangular and trapezoidal current mode modulation schemes, con-
siderable efficiency improvements are reported [30,69].
Triangular Current Mode modulation
This modulation scheme provides zero current switching for some switches and reduces
the transformer RMS current to achieve low conduction losses. The typical voltage
and current waveforms for the triangular current mode modulation are depicted in
Figure 3.5 and Figure 3.6. From Figure 3.5 it can be seen that at t=0 the inductor
current, iL, is zero. Therefore, zero current switching is achieved at t = t0. According
to this Figure 3.5, the inductor current during time interval 0 < t < t1 increases as:
iL(t) = 0 + VA − kVB
III
t2
Figure 3.5: Voltage and current waveforms for power transfer from port A to port B for triangular current mode modulator when VA > kVB
At t = t1, the ZVS condition for side A full-bridge is fulfilled, and during t1 < t <
t2, VACA (t) changes its value to zero while the voltage at side B remains constant
(VACB = VB). Therefore, the inductor current changes according to:
iL(t) = iL(t1)− kV2
L (t− t1) ∀ t1 < t < t2 (3.14)
At t = t2 the inductor current is zero and voltage at side B changes to vACB = 0.
Consequently, the inductor current remains zero during t2 < t < Ts/2
iL(t) = 0 ∀ t2 < t < Ts 2
(3.15)
The transferred power in the triangular modulation is calculated according to
P = kVB TsLs
III
t2
Figure 3.6: Voltage and current waveforms for power transfer from port B to port A for triangular current mode modulator when VA > kVB
Moreover, T2 depends on T1 in order to achieve iL(t2) = 0
T2 = T1 VA − kVB kVB
(3.17)
) = πfsT2 (3.18)
where is the phase-shift. From equations 3.16, 3.17 and 3.18, the transferred power
can be recalculated as
π2fsL(VA − kVB) ∀ VA > kVB and 0 < < ,max (3.19)
where is the only control parameter.
To achieve a given power level, , T1, and T2 are calculated as:
= π
T1 =
T2 =
πfs (3.22)
The maximum power transfer in triangular current mode modulations is restricted
by the maximum phase-shift angle (,a,max). This is because increasing the power
transfer (P ) results in the reduction of T3 = TS/2−T1−T2. Consequently, the upper
power limit for this triangular current mode modulation is achieved for T3 = 0 and is
equal to
,a,max = (P,a,max) = π
2
VA
) (3.24)
The general power flow for different voltage conditions can be written as:
P =
−2V 2 A(kVB)
(3.25)
Similar to the power transfer in the positive direction, maximum power transfer for
the combined triangular current mode modulation schemes (Figure 3.5, 3.6, 3.7 and
t2
Figure 3.7: Voltage and current waveforms for power transfer from port A to port B for triangular current mode modulator when VA < kVB
3.8) is achieved when T3 = 0, which results in t2 = TS/2 and is equal to
P,max =
,b,max = (P,b,max) = π
2
t2 ½TS
Figure 3.8: Voltage and current waveforms for power transfer from port B to port A for triangular current mode modulator when VA < kVB
Trapezoidal Current Mode Modulation
A disadvantage of using triangular current mode modulation schemes is that there
exists an upper limit to the power transfer (|P | < P,max). This is particularly un-
desirable when VA ≈ kVB where P,max is very close to zero and thus there is no
power transfer at these operating conditions. The operation of DAB for |P | > P,max
can be achieved by using a trapezoidal current mode modulation. One of the main
disadvantages of this modulation is that it does not maintain zero current switching
at side B.
Typical waveforms of trapezoidal current mode modulation is depicted in Figure
3.9 and Figure 3.10. Three different intervals can be recognized for the trapezoidal
current mode modulation. In the first interval (during time 0 < t < t1), vACA (t) =
VA and vACB (t) = 0 and the absolute value of the HF transformer current, |iL(t)|,
increases; starting its trajectory from zero. In the second interval (during time t1 <
t < t2)vACA (t) remains constant and vACB
(t) changes to vACB (t) = kVB. In the third
T2 T3
Figure 3.9: Voltage and current waveforms for power transfer from port A to port B for trapezoidal current mode modulator when VA > kVB
time interval (during t2 < t < Ts/2) vACA (t) changes to vACA
(t) = 0 and vACB (t)
remains constant at vACB (t) = kVB . The respective power levels are calculated as
follows:
The trapezoidal and triangular current mode modulations mentioned above have been
selected intuitively due to the low switching losses and the low RMS current, which
results in lower conduction losses. In order to obtain the lowest conduction losses,
T2 T3
½TS
Figure 3.10: Voltage and current waveforms for power transfer from port B to port A for trapezoidal current mode modulator when VA > kVB
switching losses, and/or magnetizing losses, a more systematic approach is needed
in order to find the modulation scheme that leads to the lowest total losses. In this
section, the discussion focuses merely on the minimization of RMS current (IL).
As depicted in Figure 3.11, DAB converters can be operated with 12 different
basic voltage sequences which form 12 different modulation schemes. These voltage
sequences are distinguished with respect to the different sequences of rising and falling
edges of vACA and vACB
depicted in Figure 3.11. However, to find the modulation
scheme with the lowest IL, only the 6 sequences 1, 9, 2, 11, 12, and 8 are considered.
This is because in the remaining 6 options, an increase in IL does not necessarily
result in a higher DAB power transfer level.
Table 3.1 shows the criteria needed to implement each of the selected 6 voltage
sequences. Power levels , phase-shifts required to obtain specific power levels, and
the inductor RMS current (IL) for different voltage sequences are given in Table 3.2,
Sequence 9 Sequence 10
Sequence 11 Sequence 12
Figure 3.11: The 12 basic voltage sequences generated with DAB converter
Table 3.3, and Table 3.4, respectively.
To achieve the minimal IL for |P | < P,max, triangular current mode modulation
is suggested [69] . For |P | > P,max, however, a modulation scheme different to
the trapezoidal current mode modulation is obtained: According to [30], optimal
modulation schemes for high power levels are voltage sequence 11 for P > P,max and
12 for P < −P,max.
The respective values of the optimal duty cycles depend on the ratio VA/(kVB).
1. Phase shift modulation
Mode Criteria
2 |D1 −D2| < π < min
( D1 +D2, 1− (D1 +D2)
) 8 |D1 −D2| < −
π < D1 +D2
12 1− (D1 −D2) < − π < D1 +D2
Table 3.2: The Power levels of DAB with respect to the each modulation for the applied DC voltages, duty cycles and the phase-shift
Mode DAB power level
eP
2 eP = −1 4
])
])
2. Optimal transition mode (D1,opt is calculated and D2,opt = 0.5) for
VA > kVB ∩ P,max < |P | < Pa,max
3. optimal transition mode (D2,opt is calculated and D1,opt = 0.5) for
VA < kVB ∩ P,max < |P | < Pb,max
Table 3.3: Phase-shift required to achieve a certain power level
Mode
4 − 2fSL|P |
)
Table 3.4: The RMS current IL with respect to the considered voltage sequences
Mode Inductor RMS current IL
ALL IL = 1 2fSL
A
])
])
Pa,max : DA,opt(Pa,max) = 0.5 ∩ P,max < Pa,max < PPS,max
Pb,max : DB,opt(Pb,max) = 0.5 ∩ P,max < Pb,max < PPS,max
3.3. STEADY-STATE MODEL OF THE CONVERTER 42
t TS
i2 i3
Figure 3.12: Waveform of Dual Active Bridge Converter operating in Continuous Cur- rent Mode (CCM) with Zero Voltage Switching (ZVS)
3.3 Steady-State Model of the converter
Figure 3.12 depicts the voltage and current waveforms of sequence 11 in Figure 3.12.
For simplicity, let
iL(t0) = iL(0) = −i0
(3.29)
By considering the half-cycle symmetry of the current waveform in DAB converters
(3.7)
is derived. From Figure 3.12, i0 is calculated in terms of the two DC voltages and
the time interval between each switching.
i1 = 1
i2 = 1
i(1/2Ts) = 1
Lω (−kVB)TIV + i3 (3.35)
by adding the two sides of (3.32), (3.33), (3.34), and (3.35), the following equation is
derived:
] =>
] (3.36)
i1 = −1
] (3.37)
] (3.38)
] (3.39)
As discussed before in the literature review dual active bridge converters are controlled
by 4 control parameters; the phase shift () between the two bridges, the duty cycle,
DA, of side A, the duty cycle, DB, of side B, and the switching frequency (fS).
Therefore, it is desired to calculate the edge currents (i0, i1, i2, and i3) as a function
of the DC voltages, transformer ratio, and the 4 control parameters instead of the 4
different time intervals (TI , TII , TIII , and TIV ) depicted in Figure 3.12.
To do this first the relation between the control parameters and the time intervals
are presented as follows:
TI + TII + TIIITIV = π
TI = (DA +DB + − 1)π (3.41)
TII = (1− 2DB)π (3.42)
TIV = (1− 2DA)π (3.44)
Replacing (3.36), (3.37), (3.38), and (3.39) with (3.41), (3.42), (3.43) and (3.44)
results in:
i0 = π
t TS
i2 i3
A3A2 A1
Figure 3.13: Area of the Current Waveform of DAB operating in CCM
From Figure 3.13 the area of the current in each interval is calculated as:
A1 = (i1 − i0)t1
iL(t)dt = A1 + A2 + A3 (3.50)
By replacing (3.45), (3.46), (3.47), (3.48), (3.49), and (3.50) into (3.8) the power
flow for sequence 11 is derived as:
P = πVAkVB Lω
and the average output current is calculated by P VB
. Therefore,
3.4 Loss Analysis of the DAB Converter
In this thesis, the lossless electric DAB model (Figure 3.2) is used to evaluate the loss
analysis of the DAB converter and to determine the converter stress values mentioned
below.
• RMS currents related to the semiconductor switches, the transformer windings,
and the inductor windings (which result in conduction losses) ;
• Instantaneous currents during switching (which result in switching losses);
• peak inductor currents (which result in inductor core loss);
• voltage-time areas applied to the transformer core (which result in transformer
core loss).
The calculation of the power dissipated in DAB converters considers the windings,
semiconductors, inductor and transformer, and the surrounding parasitic components
(e.g. PCB stray inductances). The parasitic components are neglected in the simple
loss model evaluated in this section.
3.4.1 Power Loss in Switch Converters
the total dissipated power in switch converters consist of conduction losses and switch-
ing losses.
3.4. LOSS ANALYSIS OF THE DAB CONVERTER 47
Conduction losses of each switch is determined based on their respective RMS
current. Due to the half-cycle symmetry of the current and voltage waveforms in
DAB converters and that every switch conducts a total of half a switching cycle in
each period (TS), the semiconductor RMS current of side A (ISA) and side B (ISB)
are derived as follows
ISA = IL√
2 (3.53)
ISB = kIL√
2 (3.54)
Where IL is the RMS value of iL(t). Therefore, switches in side A and side B generate
the total conduction losses of
PSA,cond = 4RSAI 2 SA (3.55)
PSB,cond = 4RSBI 2 SB (3.56)
In this application, Litz wire is used in the high frequency network, which helps reduce
the skin effect by a noticeable amount. Therefore, the influence of high frequency skin
and proximity effect are neglected.
The DC switch resistance of the MOSFETs are obtained from the data sheet values:
RSA = RSB = 0.11 VGS = 10V, ID = 12.7A, Tj = 150C (3.57)
Calculation of the switching losses is obtained from Ploss,sw = ∫ v(t)i(t) over the
switching interval. Therefore, if the switching transition occurs when the voltage is
leaning towards 0 (Zero Voltage Switching)/ the current is almost 0 (Zero current
switching), the power dissipated as switching losses can be neglected.
Soft Switching Conditions for DAB Converters
In power converters, Soft switching of a semiconductor device happens when the
switching process occurs with considerably low power dissipation. This phenomena
can occur in two ways: either the voltage across the semiconductor is kept at zero
(Zero Voltage Switching (ZVS)) or the current passed through the switch remains
near zero (Zero Current Switching (ZCS)) while the switching takes place.
In DAB converters, turn-on zero voltage switching of MOSFETs is achieved when
the body diode is on and the voltage across the switch is almost zero (body diode
voltage). Turn off ZVS occurs when the current of the MOSFET just before turn off
passes through the switch rather than the body diode. This way when the switch
is turned off the transformer stray inductance won’t allow sudden changes in the
current. Subsequently, the current will have no choice but to alter its way through
the inherent capacitor across the transistor. The capacitor slows down the trend
of the rising voltage across the MOSFET, making it stay at almost zero while the
switching takes place. It is evident that the use of snubber capacitors can help keep
the voltage at zero for a longer duration; Thus, improving the ZVS at turn off when
needed.
For the specific case of DAB converters, the ZVS can be determined from the cur-
rent and voltage waveforms of the HF network of the conveter rather than analyzing
each bridge at every switching instant. When two switches are arranged as a leg on
a dc bus (e.g. SA1 and SA2 or SA3 and SA4 in Figure 2.1), the ZVS condition for
both switches on each leg is met when the net current leaving the leg pole (centre
of the leg) lags the voltage of the pole.. A short amount of time should be set aside
between the turn-off and turn-on time of the two switches on single leg; This time
interval is called the dead time and its duration plays a major role in achieving a
better ZVS for turn-on of the MOSFETs. Basically, the dead time allows the current
to have enough time to discharge the snubber capacitor with the DC bus voltage and
charge the other snubber capacitor from 0 to the DC bus voltage. Dead times should
not be too short to allow complete charge/discharge of snubber capacitors nor should
they be too long to let the current alter its direction, which results in reversing the
charge/discharge process of the capacitor. These charge and discharges are basically
a resonant between snubber capacitors and the stray inductance of the high frequency
transformer. With the assumption of a given dead time (td) and a constant current
during the dead time, the minimum current required to achieve ZVS in each leg is
calculated from
(3.58)
Operating under ZVS results in very low switching losses, since SA1 is turned off with
vDS,T1 ≈ 0 (zero voltage turn-off), and with the assumption of sufficient Tdeadtime, SA2
is turned on with vDS,T2 = −vD ≈ 0 (zero voltage turn-on; typically, the losses due
to the forward voltage drop vD of the body diode can be neglected during the short
time the diode conducts). Therefore, this switching operation is termed Zero Voltage
Switching (ZVS) or soft switching.
Optimal Transition Mode for High Power
As explained previously the optimal transition mode of the converter uses sequence
11 in Figure 3.11 for power transfer higher than P,max. In this modulation, the
current leaving the centre pole of each leg lags the corresponding voltage of the node.
Therefore, the condition for a turn-off and turn-on ZVS is satisfied. However, to
achieve the rising and falling sequence of the voltages VACA and VACB
for the desired
modulation depicted in Figure 3.12, the discrimination characteristics explored in
Table 3.1 must be satisfied. Figure 3.14 depicts the 3D control space (DA, DB, and
AB) where the conditions for ZVS is satisfied for Va = 400V and Vb = 350V . This
plane depicts the combination of all the three phase-shifts at each point required to
achieve ZVS for certain switching frequency (here, fS = 300kHz) and DC voltage
levels. Therefore, this plane is called the 3D ZVS region.
Figure 3.15 depicts the 2-D ZVS plane for the same DC voltage levels. However,
in this figure, the DA −DB plane for different values of AB is depicted. The planes
in Figure 3.15 show the ZVS regions of the converter at certain operating conditions
(DC voltages) and switching frequency; Hence, they are termed the soft switching
zones. In order to achieve a high efficiency converter operating at high frequencies,
it is essential to follow these ZVS regions.
3.4.2 Transformer, Inductor
The power dissipated in the high frequency transformer can be categorized into:
conduction losses and core losses. The transformer copper losses are obtained from
Ptr,cond = (RtrA + k2RtrB)I2 L (3.59)
Figure 3.14: 3D ZVS region space for Va=400 Vb=350
Where RtrB and RtrB are the respective resistance of the side A and B windings.
The transformer core losses are derived from Steinmetz equation,
Ptr,core ≈ Vtr,corekf α SB
Figure 3.15: 2D-ZVS region space for Va= 400 V, Vb= 350 V
with the Steinmetz parameters k, α, and β, the total core volume Vtr,core, and the
peak magnetic flux density Btr,peak,
Btr,peak = max[Φtr(t)−min[Φtr(t]
2
1
3.5. LINEAR CONTROL FOR DAB CONVERTERS 53
and
Φtr(t) =
∫ t
0
vM(tint)
NA
dtint + Φ(0) (3.62)
(NA is the number of turns of side A winding, Atr,core is the core cross sectional area,
and vM is the voltage applied to the magnetizing inductance.)
3.4.3 Total Losses - Predicted Efficiency
The auxiliary power losses is not calculated in this thesis. Therefore, the total losses
Pt are calculated with
Pt = PSA,cond + PSA,sw + Ptr,cond + Ptr,core + PSB,cond + PSB,sw (3.63)
The most simple loss model evaluates all required characteristics (e.g. RMS current
values) at a given input power, Pin, in order to include the impact of the losses on
these quantities. The efficiency η = Pout/Pin is then calculated.
3.5 Linear Control for DAB converters
In this section, a closed-loop PI control loop is introduced for the DAB converter. The
control loop consist of a modulator, look-up table, linear compensator, and a feedback
which are explained in more detail in this section. Moreover, the digitalization of the
control will be explained.
Ls k:1
3.5.1 Closed Loop PI Control
As depicted in Figure 3.16, the closed loop PI control used in this section is con-
structed with a modulator, look-up table, linear compensator, and a current feedback.
The modulator of the DAB converter creates the switching instants for both of the
bridges A and B based on a set of inputs: A, B, AB, and the frequency. Therefore,
in order to control the DAB converter, the 4 control parameters A, B, AB, and the
frequency need to be given to the modulator. As explained previously, it is desirable
to have ZVS for all the switching. Thus, in order to achieve ZVS a look-up table is
constructed that for every given power, outputs the 4 control parameter in such way
that the discrimination characteristics explored in Table 3.1 is satisfied.
An average current mode control -which is the current at the DC ports- is used
to regulate the current at the output of the DAB converter. As depicted in Figure
3.16 the current at the output is given back to the controller as a feedback and the
3.5. LINEAR CONTROL FOR DAB CONVERTERS 55
k1 +

Figure 3.17: Conversion of Continuous PI controller to Discrete PI controller
error of the current is passed through a PI controller that creates a reference power
for the look-up table.
3.5.2 Digitalization of the PI control
Digital control is based on discrete control rather than continuous control. In digital
control the feedback to the control system is converted from analog to digital values
via Analog to Digital Conversion (ADC) with a specific sampling rate. The sampling
rate of the ADC used in this section is 400 kHz.
As for the compensator, a conversion has to be made from continuous transfer function
to discrete transfer function -s domain to z domain-. Figure 3.17 (A) depicts a PI
controller with the transfer function k 1+sT sT
. By converting the s-transfer function to
z-domain transfer function via Tustin’s method, k1 + k2 TS 2
1+z−1
1−z−1 is achieved (Figure
3.17 B). To simplify the digital coding into the micro-controller (in this project FPGA
is used), a conversion is made from the block diagram illustrated in Figure 3.17 (B)
3.6. SIMULATION AND EXPERIMENTAL RESULTS 56
to the block diagram in Figure 3.17 (C) K11 and K22 is calculated as follows:
Y (z)
(3.64)
The above equation shows that k11+k22 represents the integral coefficient and k11−k22
corresponds to the gain of the controller.
3.6 Simulation and Experimental Results
In this section, Simulation results are provided to demonstrate phase-shift modula-
tion with A = 0.5 and B = 0.5, or with either A or B set to 0.5 and the other
variable. Moreover, the transient behaviour of the linear control system is examined
for the step load change at the output of the converter and for step voltage change
at the input of the converter. It will be shown that the linear controller is unstable
at some points.
Figure 3.18 shows the the steady state operating points for different cases with dif-
ferent voltage and power levels.
Table 3.5 shows different cases of phase shift control for VA = 360 VDC and VB = 400
VDC. The estimate current (I∗) and the estimate power (P ∗) in the table is calculated
via (3.52) and (3.51), respectively.
Figure 3.18 depicts the waveforms of the switching instant for Cases 3 and 6
0.000102 0.000104 0.000106 0.000108 0.00011 Time (s)
0
-200
-400
200
400
Figure 3.18: operation for Va=400 Vb=350 AB = 0.116
in Table 3.5. Figure 3.18 (A) depicts case 6, where P = 2137W and ZVS is fully
achieved. Figure 3.18 (B) depicts case 3, where P = 1195W and ZVS is critically
achieved. The minimum Required transformer current (ip) in order to achieve ZVS
is derived by replacing Coss = 160pF , VBus = 360V , and td = 41ns into (3.58). With
the assumption of a constant current while switching (I1 ZV S = I2
ZV S), the minimum
current is calculated as IminZV S = 2.76A. However, the assumption of the constant
current is while switching does not hold in this case. Figure 3.18 A and B both
demonstrate that the magnitude of ip decreases during the dead-time. Therefore, in
order to achieve ZVS the charge of the mosfet output capacitor ∫ td ip(t)d(t) has to
be above 0.1152µ A.sec. The charge current in case 6 is much higher than 0.1152µ
A.sec and consequently it can be seen in Figure 3.18 (A) that turn on for SA2 occurs
much after transformer primary voltage (Vp) has altered its value form −VA to VB.
Table 3.5: DAB waveform details for VA = 360 VDC and VB = 400 VDC with different power levels
Case 1 2 3 4 5 6 7 8 9
A 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.44
B 0.5 0.5 0.5 0.5 0.5 0.5 0.5 0.44 0.5
AB 0.029 0.058 0.063 0.069 0.087 0.116 0.232 0.232 0.232
IB 1.17 A 2.66 A 3 A 3.28 A 4.1 A 5.33 A 9.18 A 9 A 8.96 A
IestB 1.45 A 2.80 A 3.07 A 3.33 A 4.09 A 5.28 A 9.17 A 9 A 9 A
P 470 W 1066 W 1195 W 1317 W 1647 W 2137 W 3691 W 3634 W 3600 W
P est 580 W 1126 W 1231 W 1335 W 1638 W 2114 W 3671 W 3601 W 3602 W
I1 ZV S 1.32 A 3.22 A 3.54 A 3.97 A 5.16 A 6.9 A 13.66 A 13.67 A 6.95 A
I2 ZV S 0.45 A 1.12 A 1.29 A 3.29 A 4.1 A 3.9 A 10.3 A 10.39 A 3.23 A
ZVS 7 7 Critical3 3 3 3 3 3 3
0
-200
-400
-600
200
400
Time (s)
Time (s)
ZVS
td
(B)
Figure 3.19: (A) depicts operation for Va=400 Vb=350 AB = 0.058 and (B) depicts depicts operation for Va=400 Vb=350 AB = 0.029
The Charge current in Figure 3.18 (B) is exactly 0.1152µ A.sec. Therefore, critical
ZVS is achieved. Turn on of SA2 occurs exactly when transformer primary voltage
(Vp) reaches +VA.
Figure 3.20 A and B show the case where ZVS is not fully achieved because the output
capacitors across the mosfet did not have enough time to discharge and charge. In this
case the charge of the output capacitance is less than 0.1152µ A.sec. It can be seen
that in both cases switch SA2 turns on before Vp reaches +VA. Next, we are going
0
-200
-400
-600
200
400
Vs
Time (s)
Time (s)
I2 ZVS
(A) (B)
Figure 3.20: (A) depicts operation for Va=400 Vb=350 AB = 0.116 and (B) depicts depicts operation for Va=400 Vb=350 AB = 0.063
to investigate the case where either A or B is variable. In the previous chapters
we explained that it is desirable to have A and B as close to 0.5 as possible. This
results in less circulating current which means higher efficiency. Therefore, as long as
the desired power level is achieved with conventional phase shift control, there is no
need to modify A or B.
Let us consider different voltage levels for the DAB converter. Figure 3.21 depicts the
steady state operation of the DAB converter with VA = 400 VDC, VB = 250 VDC.
In this figure, conventional phase-shift A = B = 0.5 is used to transfer the power
for high load (IBat = 12A). It can be seen that the operation takes place with full
achieved ZVS for all the switching instants. However, Figure 3.22 depicts another case
for the same voltage levels, where the conventional phase-shift modulation is used for
lower power transfer (IBat = 10A). Here, it can be depicted that the switching in
bridge B do not achieve ZVS while Vs changes its state from -VB to + VB. This is
because the inductor current at this switching state is still negative -the circle area in
Figure 3.22- and this prevents ZVS from happening while switching takes place. To
go around this, some modifications can be done to achieve ZVS for the same power
level. Figure 3.23 shows the case where B remains at 0.5, however A is set to 0.3.
0.000118 0.00012 0.000122 0.000124 0.000126
Time (s)
Vp
Vs
Figure 3.21: operation for Va=400 Vb=250 AB = 0.3, IBat = 12.1A With ZVS
This modification, enables ZVS operation for the DAB converter for lower battery
current (IBAT = 10A) -lower power levels-. As explained previously, the values are
achieved from the ZVS regions for different power level.
Figure 3.26, 3.27, and 3.28 depict different transient responses of the converter.
The gain and the time constant -k and T in Figure 3.17 (A), respectively- is tuned
at k = 6 and T = 0.000028. Using Tustin’s method k1 and k2 in Figure 3.17 (B) is
calculated to be:
k1 = 6 (3.65)
k2 = 600000 (3.66)
0.000118 0.00012 0.000122 0.000124 0.000126
Time (s)
Vp
Vs
Figure 3.22: operation for Va=400 Vb=250 AB = 0.2, IBat = 10A without ZVS
The sampling frequency of the ADC used in this application is 400 kHz (TS =
400kHz). By substituting (3.65), (3.66) and TS = 400kHz in (3.64), coefficients
of Figure 3.17 (C) are calculated as follows:
k11 = 6.75 (3.67)
k22 = 5.25 (3.68)
One of the major problems of using a PI controller in DAB converters is that due to
the non-linearity of the system-dynamics, it is not always stable over a wide voltage
range and load range. Figure 3.29 depicts a case of instability in the system.
0.000118 0.00012 0.000122 0.000124 0.000126
Time (s)
Vp
Vs
Figure 3.23: operation for Va=400 Vb=250 AB = 0.2, A = 0.31, and B = 0.5 with IBat = 10A with ZVS
0.00025 0.000252 0.000254 0.000256 0.000258 0.00026 Time (s)
0
-200
-400
-600
200
400
10*ip Vp Vs
Figure 3.24: Steady state operation of the DAB converter with the operating conditions: VA = 360, VB = 400, iBat = 6A
0.000258 0.00026 0.000262 0.000264 0.000266 0.000268 0.00027 Time (s)
0
-200
-400
200
400
10*ip Vp Vs
Figure 3.25: Steady state operation of the DAB converter with the operating conditions: VA = 360, VB = 400, iBat = 8A
4
5
6
7
0
-200
-400
-600
200
400
600
Figure 3.26: Transient response of the DAB converter VA = 360, VB = 400 I=8 A to I=4 A
6
6.5
7
7.5
0
-200
-400
-600
200
400
600
11.5
12
12.5
13
13.5
14
I(BAT)
0
-200
-400
-600
200
400
600
0
5
10
Time (s)
Figure 3.29: instability in some operating conditions for PI controller
360
370
380
390
400
V_A
6
6.2
6.4
I_BAT
0
-200
-400
200
400
Figure 3.30: transient response when VB = 400V , I = 6A VA = 360V to VA = 400V
3.7. EXPERIMENTAL RESULTS 67
3.7 Experimental Results
A 3kW prototype is implemented to verify the performance of the proposed converter.
The proposed high voltage DC/DC converter is a part of the 3KW AC/DC converter.
The converter specifications are shown in Table 3.6. Table 3.7 shows the passive
components used to implement the DAB converter.
Figures 3.31 and 3.32 depict the steady state operation where the discrimination
Table 3.6: DAB converter Specifications
Symbol Parameter Value
fsw Frequency 200 - 350 kHz
Iin(max) Maximum input current 10 A
Table 3.7: DAB system parameters
Symbol Parameter Value
k Transformer Turn’s Ratio 0.9:1
CA Filter Capacitance 2*80 uF
CB Filter Capacitance 2*80 uF
S Mosfets IPB65R110CFD
of the ZVS region is not taken into account. It can be seen in the figures that this
results in hard switching for some switches causing voltage spikes and ringing on the
transistors. Switching without ZVS will cause dissipation in the semiconductors and
at high voltage levels, the spikes can burn out the switches. Figures 3.33 and 3.33,
ZVS not achieved
Figure 3.31: steady state operation at VA = 100V , VB = 90V without ZVS
however, depict the steady state operation of the DAB converter with the necessary
calculation required to achieve ZVS. It can be seen that to maintain ZVS, A and B
is modified.
Transient response for some operating conditions of the converter is depicted in
Figure 3.35.
ZVS not achieved
Figure 3.32: steady state operation at VA = 100V , VB = 120V without ZVS
ipVp
VsIBAT
Figure 3.33: steady state operation at VA = 100V , VB = 120V , I = 2.2A with achieved ZVS
Vp Vs
Figure 3.34: steady state operation at VB = 100V , VA = 90V with achieved ZVS
Figure 3.35: transient response at VB = 100V , VA = 120V with achieved ZVS
72
Chapter 4
Controller Design
4.1 Introduction
The employed controller, in this thesis, is similar to that of a traditional 2-loop
controller, where the inner current loop shows a very fast dynamic response and
the outer loop control (Voltage loop, Current loop, and Power loop) has very slow
dynamics due to the large capacitances on the DC buses A and B.
The inner current loop is a digital current control designed based on a novel
Geometric-Sequence Control (GSC) approach for Dual Active Bridge (DAB) convert-
ers. The control variable in the GSC approach is calculated such that the error reduces
with a geometric-progression trend. Although the GSC control is derived based on the
mathematical model of the converter, it is very robust against parameter variations
and system uncertainties.
In this chapter, first the modulation scheme is described. After that, the proposed
GSC approach is explained for digital current control and Robustness of the control
will be investigated. The outer voltage loop that creates the current reference for the
fast inner current loop is also explained. Simulation and Experimental results are
4.2. MODULATION SCHEME 73
provided to verify the validity behaviour of the proposed control structure.
4.2 Modulation Scheme
The modulation scheme used for the proposed control in this thesis is based on Phase-
shift Control. Phase-shift modulation creates switching instants of the DAB
converter based on 4 input parameters: A, B, AB and the switching frequency as
described in Chapter 3. The method used in this thesis constitutes of digital counter,
sawtooth counters, and construction of the switching instants based on the sawtooth
counters.
First a digital counter is created based on the switching frequency of the converter.
The digital counter counts to the highest value (Counter Upper Limit (CUL)) before
it resets. The CUL is defined to be:
CUL = fClk fSW
(4.1)
where fClk is the clock frequency of the micro-controller and fSW is the switching
frequency.
After creating the counter, 4 sawtooth counters with the same CUL as the counter are
created (Figure 4.2). The starting point of the sawtooth counters are created based
on the remaining 3 input parameters of the modulation: A, B, AB. However, in
order to have a simpler modulation design P,AB replaced AB.
P,AB = 1
( A − B
) + AB (4.2)
Figure 4.2 demonstrates how the switching instants of the converter is created based
4.2. MODULATION SCHEME 74
VA
φA
φB
φP_AB
Counter
t
t
t
t
t
t
B_Sawtooth_I
B_Sawtooth_II
A_Sawtooth_I
A_Sawtooth_II
Figure 4.1: Sawtooth counter created based on the digital counter for PSM
4.3. DIGITAL CURRENT CONTROL IN DAB CONVERTERS BASED ON NOVEL GEOMETRIC-SEQUENCE CONTROL (GSC) APPROACH 75
t
t
SA1
SA2
t
A_Sawtooth_I
(A)
t
t
SA3
SA4
t
A_Sawtooth_II
(B)
t
t
SB1
SB2
t
B_Sawtooth_I
(C)
t
t
SB3
SB4
t
B_Sawtooth_II
(D)
Figure 4.2: switching instants created based on their respective sawtooth counter
on each sawtooth counter. It can been seen that each of the sawtooth counter are
responsible for creating two of the semiconductor switches which correspond to the
switches of each individual leg. The rising and falling edges of the switching takes
place when the sawtooth counters reach half switching cycle (TS/2) or when the
sawtooth is reset to zero.
4.3 Digital Current Control in DAB Converters Based on Novel Geometric-
Sequence Control (GSC) Approach
In this thesis, a digital current control technique is used to determine the control
parameter of the next switching cycle based on a sensed or observed state and in-
put/output information. A Novel GSC approach is proposed to design a discrete
4.3. DIGITAL CURRENT CONTROL IN DAB CONVERTERS BASED ON NOVEL GEOMETRIC-SEQUENCE CONTROL (GSC) APPROACH 76
control law for current programmed control technique. Basically, the proposed con-
trol scheme acts in such way that the magnitude of the feedback error is reduced by
a geometric-sequence trend in each cycle.
4.3.1 Oscillation Problem
Figure 4.3 depicts open loop control of the DAB converter when there is current
perturbation. The current error, i(n), is defined as the current difference between
the desired steady state current and the actual instantaneous current at that spe-
cific half-cycle (i(n)). Therefore, perturbation of the current can be denoted as i
at the beginning of any half-cycle. In can be seen from Figure 4.3 that if −i oc-
curs, provided that the phase-shift remains constant during the whole switching cycle
(open-loop), the error oscillates between +i and −i in the each half switching cy-
cle. This creates an oscillatory behaviour in the system. Typically, in basic converters
such as buck, boost, etc the control is designed such that the current error, i, is
compensated in one iteration. However, as depicted in Figure 4.4, compensation of
the error in one half-cycle results in oscillation of the current waveform.
To avoid these oscillations, the progression of the current waveform in response to
a change in the phase-shift is investigated. Through that, a digital control approach
is proposed to improve the state space current waveforms and re

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