1
Center for Power Electronics SystemsA National Science Foundation Engineering Research Center
Virginia Tech, University of Wisconsin - Madison, Rensselaer Polytechnic InstituteNorth Carolina A&T State University, University of Puerto Rico - Mayaguez
Modeling and Control ofThree-Phase PWM Converters
Dushan BoroyevichVirginia Tech, Blacksburg, Virginia, USA
presented at:
The 2nd IEEE International Power & Energy ConferenceJohor Bahru, MALAYSIA
30 November 2008
PECon 2008
DB-2
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
2
DB-3
Diode Rectifier SCR Rectifier
First Three-Phase Converters
DB-4
Three-Phase Diode Rectifierwith Current Load
)cos( tVv ama ω=
)3
2cos( πω −= tVv amb
)3
2cos( πω += tVv amc0
300
600
0 60 120 180 240 300 360
Rectified Voltage
|vab| |vbc| |vca|
-600
-300
0
300
600
0 60 120 180 240 300 360
va vb vc
iaamV
amV⋅3
amV⋅5.1
Phase Voltage & Current
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
3
DB-5
Three-Phase Diode Rectifierwith Capacitive Load
Per phase load current
DB-6
Boost Rectifier Buck Inverter Voltage Source Inverter (VSI)
Buck Rectifier Boost InverterCurrent Source Inverter (CSI)
Three-Phase Pulse Width Modulated (PWM) Converters
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
4
DB-7
Three-Phase Applications
Power Factor Correction
Adjustable input displacement factorRegulated dc bus voltage
Three-phase PWM
Rectifier
InputFilter
DC Loadsor other
ConvertersFilter DC
Bus
AC Motor Drives
VSI with uncontrolled rectifier or CSI with SCR rectifierFirst and still the most common applicationRegulated output ac voltage or current (amplitude and frequency)
Rectifier FilterInputFilter
ACMotor
Three-phase PWM
Inverter
Usually only unidirectional power flow
DB-8
Three-Phase Applications
Uninterruptible Power Supply (UPS) – Parallel
Energy Storage (Battery)
Three-phase AC Load
Three-phase PWM
Inverter
Three-phase AC Load
Three-phase PWM
Converter
Three-phase PWM
Rectifier
Energy Storage (Battery)
Uninterruptible Power Supply – Series
• High efficiency• Power interruption• No power quality
improvement to source or grid
• Lower efficiency• No power interruption• Improvement source power quality• Input power quality improvement
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
5
DB-9
Three-Phase Applications
AC-AC Power Conversion
Cascade connection of Boost rectifier and VSI or Buck rectifier and CSIAdjustable displacement factor at input and outputBidirectional power flowUtility applications (e.g. UPFC)
Three-phase PWM
Rectifier
InputFilter
Three-phase AC Load
Three-phase PWM
Inverter
OutputFilter
Active Filters
Three-phase PWM
Converter
Three-phase Nonlinear
or Reactive Load
• Power factor control• Reduced current distortion• Improved damping• Utility applications
(e.g. STATCOM)
DB-10
source Input Filter
Controller
Switchingnetwork
Output Filter Load
Feedforward Feedback
• Switching network is discontinuous and nonlinear
?
Generalized Structure of A Power Converter
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
6
DB-11
Three-phase PWM converters below 100 kW operate with relatively high switching frequency (20 kHz - 100 kHz)
Elimination of audible noiseReduction of the size of reactive componentsSignificant improvement in waveform quality and closed-loop perfomance
Motivation
Only systems with switching frequency much higher than the line frequency will be studied!
1. MODELING
2. CONTROL DESIGN
DB-12
Modulator
CurrentControl
OutputControl
Three-phasePWM
Converter
CurrentFeedback
OutputFeedback
Small-signal modeling ofThree-phase PWM convertersThree-phase modulatorCurrent controllers
Modeling
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
7
DB-13
Control design ofCurrent controlOuter control loops
Modulator
CurrentControl
OutputControl
Three-phasePWM
Converter
CurrentFeedback
OutputFeedback
Control Design
DB-14
Steps in Modeling of Three-PhasePWM Converters
1. Switching model– Time-discontinuous– Time-varying– Non-linear
2. Average model in stationary coordinates– Time-continuous– Time-varying– Non-linear
3. Average model in rotating (synchronous) coordinates– Time-continuous– Time-invariant– Non-linear
4. Small-signal model– Time-continuous– Time-invariant– Linear
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
8
DB-15
Focus
• Power converter modeling for control design!• Only converters utilizing high-frequency synthesis:
• Minor emphasis on modulation• Only classical, small-signal control approach• No power stage design and optimization• No power device discussion• No topology evaluation, only control implications• No application considerations
mm ωω <<
DB-16
Outline
1. Introduction• Vector representation of three-phase variables
2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
9
DB-17
Y-connection Δ-connection
3-phase 4-wire
Three-Phase Circuits - Source
DB-18
Y-connection Δ-connection
3-phase 4-wire
Three-Phase Circuits - Load
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
10
DB-19
Y-connection Δ-connection
Three-Phase Variables
0≡++ cba iii
0≡++ cabcab vvv
0≠++ cnbnan vvv
0≡++ cba iii
0≡++ cabcab vvv
n
anv
bnvcnv
ai
bi
ci
a
b
c
abv
bcv
cavcai
bciabi
ai
bi
ci
a
b
c
0≠++ cabcab iii
baab vvv −=
cbbc vvv −=
acca vvv −=
abcaa iii −=
bcabb iii −=
cabcc iii −=
DB-20
-100
0
100
0 180 360 540 720
vb vc
cba vvv ++
va
Sinusoidal, Balanced, Symmetrical
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
11
DB-21
-100
0
100
0 180 360 540 720
vb vc
cba vvv ++
va
Non-sinusoidal, Balanced, Symmetrical
DB-22
-100
0
100
0 180 360 540 720
vb vc
cba vvv ++
va
Non-sinusoidal, Unbalanced, Symmetrical
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
12
DB-23
-100
0
100
0 180 360 540 720
vbvc
cba vvv ++
va
Non-sinusoidal, Balanced, Asymmetrical
DB-24
-100
0
100
0 180 360 540 720
vb
vc
cba vvv ++
va
Non-sinusoidal, Unbalanced, Asymmetrical
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
13
DB-25
Euclid vector representations
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)()()(
)(tvtvtv
tv
c
b
ar
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)()()(
)(tititi
ti
c
b
ar
Euclidean Space:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
001
aur
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
010
bur
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
100
cur
a
b
cvr
av
bv
cv
Vector Representations of Three-Phase Variables
DB-26
θ
vr
wr
222221
211
,|||| n
n
ii vvvvvvv +++==>=< ∑
=
Krrr
∑=
==>=<n
iii
TT wvvwwvwv1
, rrrrrr
||||||||max||||
0 xx
xr
r
rAA
≠∀=
Vector Multiplication andNorms in Euclidean Spaces
• Inner product:
• Vector norm (length):
θcos, ⋅⋅>=< wvwv rrrr• “Dot” product:
θsin|||| ⋅⋅=× wvwv rrrr• “Cross” product:
• Norm of a matrix:
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
14
DB-27
• Multiplication of a vector with any nonsingular matrix, T, of the same order:
abcxyz vv rr⋅= T
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅= −
001
1 in Tabcxur
ab
c
x
z
y
Change of Coordinates
is equivalent to the representation of the same vector in a different coordinate system (xyz), whose unit vectors have the following coordinates in the original coordinate system (abc):
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅= −
010
1 in Tabcyur
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅= −
100
1 in Tabczur
• If , new coordinates are also orthogonal.0,,, === xzzyyx uuuuuu rrrrrr
DB-28
tva cos=
)3
2cos( π−= tvb
)3
2cos( π+= tvc
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
ph
vvv
vr
ab
c
phvr
llv −r
Example: Balanced Three-Phase Voltages in abc Space
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
15
DB-29
0≡++ cba iii 0≡++ cabcab vvv
This defines a 2-dimensional subspace χ , perpendicular to the vectorin abc-space.
αβγ -space is traditionally defined by:• α -axis is chosen as
projection of the a-axis onto χ ,
• γ -axis is co-linear with vector
• β - axis is defined by right-hand rule.
[ ]T111
[ ]T111
a
b
c
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
111
χ
Change of Coordinates ( abc to αβγ )
DB-30
The transformation matrix
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
=
21
21
21
23
230
21
211
32
/ abcTαβγ
abcabc vTv rr⋅= /αβγαβγ
abcabc iirr
⋅= /Tαβγαβγ
1|||| / =abcTαβγ ab
c
χ α
γ
β
Transformation Matrix . abcT /αβγ
⇒⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
111
abcxr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
300
αβγxr
Example:
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
16
DB-31
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−−
−=== −
21
23
21
21
23
21
2101
321
/Tαβγ/abcαβγ/abcabc TTT αβγ
αβγαβγ vTv abcabcrr
⋅= /
αβγαβγ iTi abcabcrr
⋅= /
Transformation Matrix .αβγ/abcT
DB-32
tva cos= )3
2cos( π−= tvb )
32cos( π
+= tvc
abcabc vTv rr⋅= /αβγαβγ
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
abc
vvv
vr
Example: Balanced Three-Phase Voltages in αβγ Space
phvrllv −
r
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
17
DB-33
RL
av
bvcvR R
L L
ai
bi
ci
vidtid
dtidiv rr
rrrr 11 −− +=⇔+= LRLLR
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
vvv
vr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
iii
ir
vTv xxrr
⋅= iTi xx
rr⋅= such that xvr xi
rand are constant
xT is differentiable and invertible.
Example: State-Space Equations
are sinusoidal in steady-state! Find a coordinate transformation:vr ir
and
, ,
in steady state, and
DB-34
RL
av
bvcvR R
L L
ai
bi
ci
dtid
ivr
rr LR +=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
iii
ir
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+
−=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)3
2cos(
)3
2cos(
)cos(
πω
πω
ω
tV
tV
tV
vvv
v
m
m
m
c
b
ar
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
RR
R
000000
R
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
LL
L
000000
Lvidtid rrr
⋅+⋅−= −− 11 LRL
Example: State-Space Equations
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
18
DB-35
Natural Response
τττ dveietit
tt ⋅⋅⋅+⋅= −−⋅−⋅− ∫−−
)()0()( 1
0
)(11 rrrLRLRL
Forced Response
⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
⋅⋅−
+−+
⋅⋅+
+−−
⋅−−
⋅+⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⋅−
⋅−
⋅−
⋅−
tLR
tLR
tLR
mtLR
c
b
a
eZ
LRt
eZ
LRt
eZRt
ZVe
iii
ti
23)
32cos(
23)
32cos(
)cos(
)0()0()0(
)(
ωφπω
ωφπω
φω
r
RLω
φ arctan=222 LRZ ω+=φω jj eZLR ⋅=+=Z
where per-phase impedance, Z , at the source frequency, ω , is defined as:
Example: State-Space Equations – Solution
DB-36
vTv xxrr
⋅=
iTi xx
rr⋅=
where: xvr xir
and are constant in steady state
:xT differentiable and invertible
Want to find change of variables:
[ ]
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−+
−−
−
⋅=∞→
)3
2cos(
)3
2cos(
)cos(
)(lim
φπω
φπω
φω
t
t
t
Iti mt
r
RLω
φ arctan=
222 LR
VI mm
ω+=
0lim ≠⎥⎦
⎤⎢⎣
⎡∞→ dt
idt
r
Example: State-Space Equations –Steady State Solution
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
19
DB-37
A rotating vector in αβγ space can be a constant vector in a rotating space
α
β
d
q
θ
vr
dv
αvqv
βv ⎥⎦
⎤⎢⎣
⎡⎥⎦
⎤⎢⎣
⎡−
=⎥⎦
⎤⎢⎣
⎡
β
α
θθθθ
vv
vv
q
d
cossinsincos
)0()(0
θττωθ += ∫ dt
Where ω is the rotating speed
Transformation Matrix . αβ/dqT
DB-38
Transformation Matrix .αβγ/0dqT
Preserve the same third axis, that is 0-axis is the same as γ-axis
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
γ
β
α
θθθθ
vvv
vvv
0
q
d
1000cossin0sincos
Therefore
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
1000cossin0sincos
T / θθθθ
αβγdq0T
dq0dq0dq0αβγ TTT αβγαβγ /1
// == −
1|||| / =αβγdq0T
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
20
DB-39
Transformation Matrix .abcdqT /0
abcabcdq0dq0 vTv rr⋅= /
abcabcαβγdq0αβγαβγdq0dq0 vTTvTv rrr⋅⋅=⋅= /// αβγ
Therefore
where
abcdq0abcdq0 TTT /// αβγαβγ ⋅=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−−−
+−
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−=
21
21
21
)3
2sin()3
2sin(sin
)3
2cos()3
2cos(cos
32
21
21
21
23
230
21
211
1000cossin0sincos
32 πθπθθ
πθπθθ
θθθθ
Park’s Transformation
1|||| / =abcdq0T Tabcdq0abcdq0dq0abc TTT /
1// == −
DB-40
tva cos= )3
2cos( π−= tvb )
32cos( π
+= tvc
abcabcdq0dq0 vTv rr⋅= /
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
abc
vvv
vrphvr
llv −
r
Example: Balanced Three-Phase Voltages in dq0 Space
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
21
DB-41
[ ] ccbbaa
c
b
a
cba iviviviii
vvvivivp ++=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅==⋅=
rrrrr T
00qqddccbbaa iviviviviviviviviv ++=++=++ γγββαα
It can be easily proved that:
where v & i are corresponding voltages and currents in a three-phase circuit.
Power Definition in Three-Phase Circuits
DB-42
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+
−=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)3
2cos(
)3
2cos(
)cos(
πω
πω
ω
tV
tV
tV
vvv
v
am
am
am
c
b
ar
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−+
−−
−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
)3
2cos(
)3
2cos(
)cos(
φπω
φπω
φω
tI
tI
tI
iii
i
am
am
am
c
b
ar
RL
av
bvcvR R
L L
ai
bi
ci
RLω
φ arctan=where:
Example:Power in Sinusoidal Steady State
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
22
DB-43
Example:Power in Sinusoidal Steady State
φφ cos23cos|||||||||||| amam IVivivP =⋅=⋅=
rrrr
φφ sin23sin|||||||||||| amam IVivivQ =⋅=×=
rrrr
RL
av
bvcvR R
L L
ai
bi
ci
DB-44
)cos( φω += tVv amaφφ j
amama eVV ⋅=∠=V
ref
amV
φ
Phasors are defined ONLY for sinusoidal steady state!
aV
Phasor Representation
• Vector representation is NOT phasor representation!
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
23
DB-45
[ ] [ ])sin(j)cos(ReRe)cos( φωφωφω φ +++=⋅=+= tVtVeVtVv mmj
mma
φφ jrms
ma eVV
⋅=∠=2
V
Phasors are defined ONLY for sinusoidal steady state!
ref
rmsV
φ
aV
Phasors are very useful for the analysis of linear systems without transients, which are excited by constant single frequency (ω ) sinusoidal generators.
( )tjama etVv ωφω ⋅⋅=+= VRe2)cos(
Phasor Representation
DB-46
Outline
1. Introduction2. Switching Modeling and PWM
• Switching model of VSI & boost rectifier• Space vector modulation for VSI & boost rectifier• Other modulations for VSI & boost rectifier• Switching model and modulation for CSI & buck rectifier
3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
24
DB-47
• Boost Rectifierwhere Vm is the peakvalue of the line-to-lineinput voltage
mdc VV >
dcv
CAv ABvBCv
Laps bps cps
ans bns cns
C R
p
n
avbv
cv
• Voltage Source Inverter(VSI)
mdc VV > Load
aps bps cps
ans bns cns
L
C
p
n
dcvCAvBCv
ABvav
bvcv
Boost Rectifier / Voltage Source Inverter
vdc
Vm
VSI / BOOST RECTIFIERDC VOLTAGE RANGE
t|vab| |vbc| |vca|
DB-48
Switching function:
s =1, v = 0, if switch s is closed
0, i = 0, if switch s is open
• Voltage source or capacitor cannot be shorted• Current source or inductor cannot be open
Switching constraints:
s
iv
v
iv
i
Current bi-directional two-quadrant switch
Method of Modeling Switching Network
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
25
DB-49
• Three-Switch (Single-Pole-Double-Throw)
Boost RectifierVoltage Source Inverter
Allowed switching combinations:
;1=+ inip ss },,{ cbai ∈
p
n
aps bps cps
ans bns cns
avbv
cv
• Define Voltage-Unidirectional Single-Pole-Double-Throw Switch and switching function
;1 inipi sss −== },,{ cbai ∈
p
n
av
bv
cv
as
bs
cs
00
0
11
1
ai
bi
ci
dci
dcv
DC-Voltage-UnidirectionalThree-Phase Switching Network
DB-50
Find the relationships:
av
bv
av
bv
av
bv
av
bv
0=aps 0=bps 0=aps 1=bps
1=aps 0=bps 1=aps 1=bps
),( dcababab vsfv = ),( dcbcbcbc vsfv = ),( dccacaca vsfv =
),,,( cbaijidc iiisfi =
dcv dcv
dcv dcv
0=abv dcab vv −=
dcab vv = 0=abv
dcab
dcbpap
baab
vs
vssvvv
=
−=−=
)(
bbpaapdc isisi +=
dci dci
dci dci
ai
bi
ai
bi
ai
bi
ai
bi
Development of Switching Model(Boost rectifier / Voltage source inverter)
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
26
DB-51
Define phase-leg switching function
aps bps cps
ans bns cns
avbv
cv
ai
bi
ci
dci
dcv
;1 inipi sss −== },,{ cbai ∈
sa sb sc sa-sb sb-sc sc-sa idc vab vbc vca0 0 0 0 0 0 0 0 0 00 0 1 0 -1 1 ic 0 -vdc vdc0 1 0 -1 1 0 ib -vdc vdc 00 1 1 -1 0 1 ib+ic -vdc 0 vdc1 0 0 1 0 -1 ia vdc 0 -vdc1 0 1 1 -1 0 ia+ic vdc -vdc 01 1 0 0 1 -1 ia+ib 0 vdc -vdc1 1 1 0 0 0 ia+ib+ic 0 0 0
Development of Switching Model(Boost rectifier / Voltage source inverter)
DB-52
Development of Switching Model(Boost rectifier / Voltage source inverter)
Instantaneous voltage equation Instantaneous current equation
dc
ca
bc
ab
dc
ac
cb
ba
ca
bc
abv
sss
vssssss
vvv
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡[ ]
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=
c
b
a
cbadciii
sssi
Note that: baab sss −= baab vvv −=
p
n
av
bv
cv
as
bs
cs
00
0
11
1
ai
bi
ci
dci
dcv
……
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
27
DB-53
Relationship Between Line-to-Line Current and Phase Current
caaba iii −=
abbcb iii −=
abbccaababbccaabba iiiiiiiiii 3)(2)( =+−=−−−=−
)(31
baab iii −= Similarly )(31
cbbc iii −= )(31
acca iii −=
Assume 0=++ cabcab iii
ai
bi
ci
abi
bcicai
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=−+−+−=
−+−+−=++=
ca
bc
ab
cabcabaccacbbcbaab
bccacabbcbcaabaccbbaadc
iii
sssssississi
iisiisiisisisisi
)()()(
)()()(
bccac iii −=
DB-54
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ac
cb
ba
ca
bc
ab
ll
vvvvvv
vvv
vr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ac
cb
ba
ca
bc
ab
ll
ssssss
sss
sr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ac
cb
ba
ca
bc
ab
ll
iiiiii
iii
i31r
p
n
av
bv
cv
as
bs
cs
00
0
11
1
ai
bi
ci
dci
dcvdcllll vsv ⋅= −−
rr
llT
lldc isi −− ⋅=rr
where:
Boost Rectifier / Voltage Source Inverter Switching Model
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
28
DB-55
Outline
1. Introduction2. Switching Modeling and PWM
• Switching model of VSI & boost rectifier• Space vector modulation for VSI & boost rectifier• Other modulations for VSI & boost rectifier• Switching model and modulation for CSI & buck rectifier
3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
DB-56
Switching States forBoost Rectifier / Voltage Source Inverter
idc vab
Vdc
sa scsb0
1
00
00
0 00
0 00
0
11
11 1
1
1
1
111
icib
ib+ic
0
iaia+icia+ib
ia+ib+ic
Switching state
pnn
ppn
npnnpp
nnp
pnp
ppp
nnnvcavbc
Vdc
Vdc
Vdc
Vdc
Vdc
-Vdc
-Vdc
-Vdc
-Vdc
-Vdc
-Vdc
00
000
0
0000
00
p
n
av
bv
cv
as
bs
cs
00
0
11
1
ai
bi
ci
dci
dcV
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
29
DB-57
Vector Space of Line-to-Line Variables
a
c
b
ab bc
ca
α
β
ab
bc
ca
[1 1 1]T
χχ
• Phase variables (a, b and c) produceline-to-line variables (ab, bc and ca) in plane-χ
• Line-to-line variables (ab, bc and ca) do not haveγ-component in αβγ-coordinate system
DB-58
Line-to-Line Voltage Space Vector
where⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡
ca
bc
ab
abc
vvv
Tvv
/αββ
α
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−⋅=
23
230
21
211
32
/ abcTαβ
abα
βbc
ca
ρ
vα
vβ
θ
• Space vectorθ⋅ρ= jevr
22 vv βα +=ρ
⎟⎠
⎞⎜⎝
⎛=θ
α
β−
vv1tan
If Vm is the amplitude of balanced, symmetrical, three-phase line-to-line
voltages, then mV23
⋅=ρ
vr
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
30
DB-59
Switching State Vector [pnn]
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
⋅
⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡=
dc
dc
dc
dc
pnnca
bc
ab
abcpnn
pnn
V
V
V
V
vvv
Tvv
V
2123
0
23
230
21
211
32
/αββ
αr
θ⋅ρ== j1pnn eVVrr
dcV2 ⋅=ρ
°=⎟⎟⎠
⎞⎜⎜⎝
⎛=θ
α
β− 30vv1tan
ab, α
βbc
ca
ρ
vα
vβ
θ
1Vr
DB-60
Switching State Vector [ppn]
⎥⎦
⎤⎢⎣
⎡⋅
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡=
dcdc
dc
ppnca
bc
ab
abcppn
ppn VV
Vvvv
Tvv
V2
00
23
230
21
211
32
/αββ
αr
θ⋅ρ== j2ppn eVVrr
dcV2 ⋅=ρ
°=⎟⎠
⎞⎜⎝
⎛=θ
α
β− 90vv1tan ab, α
βbc
ca
ρ
vβ
θ
2Vr
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
31
DB-61
Switching State Vector [ppp]
⎥⎦
⎤⎢⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−
−−⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=⎥
⎦
⎤⎢⎣
⎡=
00
000
23
230
21
211
32
/
pppca
bc
ab
abcppp
ppp
vvv
Tvv
V αββ
αr
0VV 0ppp ==rr
ab, α
βbc
ca
0Vr
DB-62
Switching State Vectors
dcV2 ⋅
Sector I
IV
III
II
V
VI
at center point
30
150
90
-90
-150
0
-30
00
ρ θ (°)
ab, α
bc
ca
β
][ pnnV1
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
][ pnpV6
r
][ pppV0
r
][nnnV0
r
][ pnnV1
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
][ pnpV6
r
][][ nnnpppV0 ==r
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
32
DB-63
Reference Voltage Vector, Vref
ab, α
bc
ca
β
θvα
vβθ
β
α ⋅ρ=⎥⎦
⎤⎢⎣
⎡= j
refref e
vv
Vr
m22 V
23vv ⋅=+=ρ βα
tvv1 ω=⎟
⎠
⎞⎜⎝
⎛=θ
α
β−tan
where
In general,
( )( )( )⎥
⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
°+ω⋅°−ω⋅
ω⋅=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
120tV120tVtV
vvv
m
m
m
refca
bc
ab
coscos
cos
Assume
)()()( tjmref etV
23tV θ⋅⋅=
r
ρ ][ pnnV1
r
][ pnpV6
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
at center point][][ nnnpppV0 ==r
refVr
DB-64
Si
ii
T
0i
T
0ref TTdtVdtV
iS
=⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑∑ ∫∫ ,
rr
Vref (α)
V1(α) V2(α)
T1 T2 T0TS
t
Total area of = Area of
∫ ∫∫∫+
+
++=21
1
S
21
1S TT
T
T
TT02
T
01
T
0ref dtVdtVdtVdtV
rrrrFor example
vα
Definition of High Frequency Synthesis
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
33
DB-65
Step 1 : Choose desired switching state vectors to synthesize
Step 2 : Calculate the duty ratios of chosen switching state vectors
Step 3 : Make the sequence of chosen switching state vectors
refVr
Space Vector Modulation
DB-66
Switching State Vectors
ab, α
bc
ca
β
θ
vα
vβ ρ ][ pnnV1
r
][ pnpV6
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
][][ nnnpppV0 ==r
refVr
ia
ib
ic
va
vc
vb
sa
sb
sc
p
n
Vdc
idc
a
bc
1
0 1
0 1
0
I
IV
III
II
V
VI
φ
ρ
2Vr
0Vr
1Vr
11 Vdr
⋅
22 Vdr
⋅ refVr
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
34
DB-67
ab, α
bc
ca
β
Sector I
location Chosen vectors
Sector II
Sector VI
Sector III
Sector IV
Sector VV
VI
I
III
II
IV
• Minimize the number of switching
• Minimize the harmonic distortion
Choose minimum numberof switching state vectors
adjacent to .refVr
65 VandVrr
21 VandVrr
16 VandVrr
32 VandVrr
43 VandVrr
54 VandVrr
0Vandr
refVr
][ pnnV1
r
][ pnpV6
r
][npnV3
r
][nppV4
r
][ ppnV2
r
][nnpV5
r
at center point][][ nnnpppV0 ==r
refVr
Step 1 : Choice of Switching State Vectors
DB-68
∫ ∫∫∫+
+
++=21
1
S
21
1S TT
T
T
TT02
T
01
T
0ref dtVdtVdtVdtV
rrrr
2211Sref TVTVTV ⋅+⋅=⋅rrr
2211S T6060
VT01
VT ⋅⎥⎦
⎤⎢⎣
⎡°°
⋅+⋅⎥⎦
⎤⎢⎣
⎡⋅=⋅⎥
⎦
⎤⎢⎣
⎡φφ
⋅ρsincos
sincos
)sin( φ−°⋅ρ
⋅== 60V3
2dTT
11
S
1
φ⋅ρ
⋅== sin2
2S
2
V32d
TT
210 dd1d −−=φ
From HF synthesis definition,
Assume is constant in TS ,
where °−θ=φ 30
ρ
refVr
2Vr
0Vr
1Vr
11 Vdr
⋅
22 Vdr
⋅ refVr
Step 2 : Duty Ratio of Switching State Vectors at Sector I
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
35
DB-69
)sin( φ−°⋅ρ
⋅== 60V3
2dTT
NN
S
N
φ⋅ρ
⋅==+
++ sin
1N1N
S
1N
V32d
TT
1NN0 dd1d +−−=
Other sectors have the same results of duty ratio.
where ( ) °−°⋅−−θ=φ 30601NN : sector number ( 1 ~ 6 )
ρ
φ
1NV +
r
1N1N Vd ++ ⋅r
0Vr
NVr
NN Vdr
⋅
refVr
Duty Ratio of Switching State Vectors
DB-70
)sin( φ−°⋅= 60VVd
dc
mN
φ⋅=+ sindc
m1N V
Vd
1NN0 dd1d +−−=
Define the modulation index
For all the switching state vectors, dcN V2V ⋅=
dc
m
VVM =
)sin( φ−°⋅= 60MdN
φ⋅=+ sinMd 1N
1NN0 dd1d +−−=
and mV23
⋅=ρ
Modulation Index
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
36
DB-71
ab, α
bc
ca
βAssume d0 = 0, then dN + dN+1 = 1
The trajectory of makesa hexagon.
( )( )
( )
1
30VV
30M60Mdd
dc
m
1NN
=
φ−°⋅=
φ−°⋅=φ+φ−°⋅=+ +
cos
cossin)sin(
( )φ−°=∴
30VV dc
m cos
refVr
][ pnnV1
r
][ pnpV6
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
at center point][][ nnnpppV0 ==r
refVr
Maximum Amplitude of Vref
DB-72
Maximum Amplitude of Vref
ab, α
bc
ca
βAssume M = 1, then
The trajectory of makes
a circle whose radius is
1VV
dc
m =
dcm VV =∴
dcV⋅23
This trajectory of representsthe largest 3-phase sinusoidalvoltage that can be synthesized.
θ⋅⋅= jdcref eV
23V
r ][ pnnV1
r
][ pnpV6
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
at center point][][ nnnpppV0 ==r
refVr
refVr
refVr
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
37
DB-73
Step 3 : Sequence of Switching State Vectors
TS
Sa
Sc
Sb
V1 V2 V0
T2T1 T0
Sa
Sc
Sb
V1 V2
T0
V2
T12
T22
T12
T22
V1V0
TS
Asymmetricalsequence
Symmetricalsequence
3-phasecommutation
2-phasecommutation
Feedback
Feed forward
=⎟⎟⎠
⎞⎜⎜⎝
⎛== ∑ ∫∫
=1i
T
0i
T
0ref
iS
dtVdtVrr
DB-74
TS
T2T1
Sa
Sc
Sb
T0 /2 T0 /2
TS
T2T1T0 /2 T0 /2
V0 V0V1 V2 V0 V1 V2 V0
• Use both zero switching state vectors
• Asymmetrical sequence
• Six commutations per switching cycle
< Example in sector I >
• 3Φ-LA has same characteristics
Sequence of SSVs – SVM 1(Three-Phase – Right Aligned: 3Φ-RA)
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
38
DB-75
TS
Sa
Sc
Sb
TS
V0 V1 V2 V0
• Use both zero switching state vectors
• Symmetrical sequence
• Six commutations per switching cycle
Low THD
V1V2
< Example in sector I >
T04
T12
T22
T02
T22
T12
T04
T04
T12
T22
T02
T22
T12
T04
V0 V0V1 V2 V0 V1V2
Sequence of SSVs – SVM 2(Three-Phase – Centered: 3Φ-C)
DB-76
TS
T2T1
Sa
Sc
Sb
T0
V1 V2 V0 [ppp]
• Use zero vectors alternatively in adjacent switching cycle
• Asymmetrical sequence in TS , but symmetrical in 2·TS
• Three commutations
< Example in sector I >
50 % switching loss reduction
V1V2 V0 [nnn]
TS
T1T2 T0
• Introduction of harmonics around ST2
1⋅
Sequence of SSVs – SVM 3(Three-Phase – Double-Period: 3Φ-2T)
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
39
DB-77
• Use a zero vector in one switching cycle
• Asymmetrical sequence
< Example in sector I >
Reduced switching losses
TS
Sa
Sc
Sb
V1 V2 V0[ppp]
TS
• Four commutations
Sector I, III, V : [ppp]Sector II, IV, VI : [nnn]
T2T1 T0 T2T1 T0
V1 V2 V0[ppp]
• 2Φ-LA has same characteristics
Sequence of SSVs – SVM 4(Two-Phase – Right Aligned: 2Φ-RA)
DB-78
• Use a zero vector in one switching cycle
• Symmetrical sequence
< Example in sector I >
Reduced switching losses
TS
Sa
Sc
Sb
V1 V2 [ppp]
• Four commutations
Sector I, III, V : [ppp]Sector II, IV, VI : [nnn]
T0
V1
Low THD
V2
T12
T22
T12
T22
TS
T0T12
T22
T12
T22
V2 [ppp]V1 V2
Sequence of SSVs – SVM 5(Two-Phase – Centered: 2Φ-C)
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
40
DB-79
• Possible sequences: 2Φ-RA-mL, 2Φ-LA-mL, or 2Φ-C-mL
• Reduce the switching losses up to 50% compared to 3Φ modulations, assuming that switching losses are proportional to the current
• Choose a zero switching state vector to avoid switching the phase with the highest instantaneous current
|ia| > |ic| ?No
YesV0 = [ppp]
V0 = [nnn]
Choice of zero vector in sector I (pnn, ppn, and ppp or nnn)
Sequence of SSVs – SVM 6 (Minimum-Loss SVM)
(Two-Phase – Right Aligned – minimum Loss: 2Φ-RA-mL)
DB-80
Sector Ivab
vbc
vca
va
vc
vbia
ic
ib
nnnppp nnn
δ
vab
vbc
vca
va
vc
vb
ia
ic
ib
δ
ppp
• Switching loss reduction = 50%°≤ 30δ °> 30δ
• Switching loss reduction < 50%• •
(pnn, ppn)
Sector I
(pnn, ppn)
LineVoltages
PhaseVoltages
PhaseCurrents
Example of 2Φ- x -mL SVM in Sector Ifor balanced, symmetrical, sinusoidal, steady-state case
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
41
DB-81
THD of line-to-line voltage THD of phase currentwith L + VABC load/source
With the Fourier series ,1
2n
2n
V
VTHD
∑∞
==∑∞
=
ω⋅=1n
tjnn eVV
Modulation index, M Modulation index, M0 1 0 1
3Φ
3Φ-C
2Φ
3Φ or 2Φ3Φ-2T
3Φ-C
RA or LA 3Φ-2T
RA or LA
Calculated and switching-model simulation results for .lineS ff ⋅>100
Comparison:Total Harmonic Distortion (THD)
DB-82
For RA or LA modulations:
Relative peak-to-peak current ripple
where TP = TS , exceptTP = 2TS , for 3Φ-2T.
( ) Sdc
pp TMML
VI ⋅⋅−⋅⋅
⋅= 1
32
Modulation index, M0 1
For centered modulations:
( ) Pdc
pp TMML
VI ⋅⋅−⋅⋅
= 13
2Φ or 3Φ-C
2Φ or 3ΦRA or LA
or3Φ-2T
M : modulation indexL : load inductance
Comparison:Peak-to-Peak Current Ripple
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
42
DB-83
• Number of commutationsper switching cycle:
3Φ-x : 6
2Φ- x -mL avoids switching the phase with highest current between -30° and 30°.
3Φ-2T : 32Φ-x : 4
SVM2
SVM1,3,53Φ-x
3Φ-2T
min Loss
2Φ-xMax Loss
Comparison: Switching Losses
DB-84
Outline
1. Introduction2. Switching Modeling and PWM
• Switching model of VSI & boost rectifier• Space vector modulation for VSI & boost rectifier• Other modulations for VSI & boost rectifier• Switching model and modulation for CSI & buck rectifier
3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
43
DB-85
Pulse WidthModulation
Feedforwardscheme
Feedbackscheme
Sinusoidal PWM
Third-harmonic injection PWM
Space vector modulations(SVM1, 2, 3, 4 and 5)
Hysteresis current control
Space vector modulations(SVM 6)
Pulse AmplitudeModulation
Modulation Methods
DB-86
Sinusoidal Pulse Width Modulation(SPWM)
vavref
vcar
sap
san
2Vdc
2Vdc
• Determine the switching state by magnitude of vref and vcar
If |vref| > |vcar|, then sap on,If |vref| < |vcar|, then san on
• Switching frequency is the same as carrier wave frequency, fc
fsin
fc
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
44
DB-87
va
2Vdc
Va(1)
-Va(1) 2Vdc
va(1) : Fundamental of v a
refav
refaV
vcar
Vcar
Vcar
t
t
refaV−
dc
1a
dc
1a
car
refa
VV2
2V
VVV )()( ⋅
=⎟⎠⎞
⎜⎝⎛
=car
refadc
1a VV
2VV ⋅=)(
Waveforms of Single-Phase SPWM
DB-88
refbvref
av refcv
ia
ib
ic
sa
sb
sc
p
n
2Vdc
2Vdc
vcvbva
vcar
refV
refV−
Vcar
Vcar
t
Three-Phase SPWM
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
45
DB-89
refbv
refav
refcv
V1 V1V2 V2V0V0 V0
pnn
ppn
ppp
nnn
pnn
ppn
t
vcar
TS
t
t
t
sa
sb
sc
nnn
T0N T0P T0N
Assume and areconstant in a switching cycle.
refc
refb
refa vvv ,
Symmetrical (Center-based)Three-phase commutation
SVM 2
)( carrefa
car
SN0 Vv
V4TT −⋅⋅
−=
)(
)(
carrefc
car
S
carrefc
car
SSP0
VvV4T
VvV4T
2T
2T
+⋅⋅
=
−⋅⋅
+=
Modulation Example of SPWM in Sector I
DB-90
V1 V1V2 V2V0V0 V0
pnn
ppn
ppp
nnn
pnn
ppn
TS
t
t
t
sa
sb
sc
nnn
T0N T0P T0N
V1
V2
d1·V1
d2·V2
φV0
ρ
)sin( φ−°⋅⋅== 60VV2d
TT
1
m1
S
1
φ⋅⋅== sin2
m2
S
2
VV2d
TT
210 dd1d −−=)cos( φ−°⋅⋅−=
==
304
TVV
4T
4TT
2T
S
dc
mS
0N0
P0
refvr
Modulation Example of SVM2 in Sector I
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
46
DB-91
10 20 30 40 500
0.05
0.1
0.15
60
T0NT0P2
T0N+ T0P2
0.05
0.1
0.15
φ in sector I ( o )
TTS
T0N
T0N+ T0P2
TTS
10 20 30 40 500 60
φ in sector I ( o )
• In SPWM, assume Vref = Vcar
• In SVM2, assume Vm = Vdc
)cos( φ−°⋅−=
==
304
T4
T4TT
2T
SS
0N0
P0
0.125
0.067
0.067
)( refrefaref
SN0 Vv
V4TT −⋅⋅
−=
)( refrefcref
SP0 VvV4T
2T
+⋅⋅
=
φ⋅= cosrefrefa Vv
( )°+φ⋅= 120Vv refrefc cos
where
Zero Vector Timings in Sector I
DB-92
Maximum AC Voltage of SPWM
dcm V23V ⋅=
By definition: θ⋅⋅= jm
ref eV23vr
The maximum AC voltage:
The trajectory in SPWM:refvr θ⋅⋅= jdc
ref eV22
3vr
1551
2V3
VVV
dc
dc
SPWMm
SVMm .=⎟⎠
⎞⎜⎝
⎛ ⋅=
The maximum AC voltage of SVM: dcm VV =
SVM produces 15.5% highermaximum output than SPWM !
abα
bc
ca
β
][ pnnV1
r
][ pnpV6
r
][ ppnV2
r
][npnV3
r
][nppV4
r
][nnpV5
r
refvr
at center][][0
nnnpppV
==
r
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
47
DB-93
78504
2V42
V
V
V
dc
dc
Square1
SPWM1 .max)(
max)(
=π=⎟⎠⎞
⎜⎝⎛ ⋅
π
⎟⎠⎞
⎜⎝⎛
=
2Vdc
Over modulation region isnon linear with more harmonics
2V4 dc⋅
π
)(1VVcar
t
1 3.24 car
ref
VV
Overmodulation
Squarewave
switching
• Vref ≤ Vcar : Linear modulation• Vref > Vcar : Over modulation
Vref
Over Modulation of SPWM
DB-94
ref1v )(
t
ref3
ref1
ref VVV )()( +=
t3VtVv ref3
ref1
ref ω⋅+ω⋅= sinsin )()(
ref3v )(
refv
ref3V )(
refV
0
ref1V )(
ref1V )(−
Third-Harmonic Injection PWM
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
48
DB-95
The maximum AC voltageis 15.5 % more than SPWM
1V ref =
⎟⎠⎞⎜
⎝⎛ θ⋅+θ⋅= 3
61Vv ref
1ref sinsin)(In general,
1551
23
1V ref1 .)( =
⎟⎠⎞
⎜⎝⎛
=
1
ref1v )(
t
refv
1551V ref1 .)( =
0
Assume ,
then
Maximum AC Voltage of Third-Harmonic Injection PWM
DB-96
( ))60sin(
sin60sin22
021
0
φφφ
+°=+−°=
+++−=dddd
Vv
dc
a
1
-1
1
-1
1
-1
4T0
2T0
av
TS
bv
cv( )
)30sin(3
sin60sin22
021
0
°−⋅=
+−°−=
++−−=
φ
φφ
ddddVv
dc
b
4T0
2T1
2T1
2T2
2T2
a
dc
c
v
ddddVv
−=+°−=
+−−−=
)60sin(22
021
0
φ
Average Values of Phase-to Neutral Voltage for SVM 2 (3Φ-C) in Sector I
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
49
DB-97
( )
)90sin(3
)30sin(3
sin60sin22
021
0
θ
φ
φφ
−°⋅=
−°⋅=
−−°=
+−+−=dddd
Vv
dc
a
( )
)60sin()60sin(
sin60sin22
021
0
°−−=+°−=
−−°−=
+−−−=
φφ
φφ
ddddVv
dc
a
• In sector II, • In sector III,
0 120 180 240 300 36060
0
-1
1
θ ( o )
va
Sector I Sector IVSector II Sector III Sector V Sector VI
23
Average Values of Phase-to Neutral Voltage for SVM 2 (3Φ-C)
DB-98
1va =1
-1
1
-1
1
-1
av
TS
bv
cv)sin( φ−°⋅−=
⋅−=++−=
6021d21
dddv
1
021b
2T1
2T1
2T2
2T2
)sin()(
φ+°⋅−=+⋅−=+−−=
6021dd21dddv
21
021c
T0
• If ia is the largest current,
• If ic is the largest current,
( )( ) 1602
1602dddv 210a
−θ+°⋅=−φ+°⋅=
++−=
sinsin
Average Values of Phase-to Neutral Voltage for SVM 6 (2Φ-C-mL) in Sector I
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
50
DB-99
0 120 180 240 300 36060
0
-1
1
θ ( o )
va
Sector I Sector IVSector II Sector III Sector V Sector VI
0
-1
1
va
0 120 180 240 300 36060θ ( o )
0.60.2-0.2-0.6
• M = 1
• M = 0.8
Average Values of Phase-to Neutral Voltage for SVM 6 (2Φ-C-mL)
DB-100
iref
sap
san
2Vdc
2Vdc
Load
i
β
• Switching frequency is varyingin one switching cycle.
If , then sap on,
If , then san on
iiref >β
−2
iiref <β
+2β
iref
it
Hysteresis Current Control
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
51
DB-101
Pros and Cons of Hysteresis Current Control
• Simple to implement• Excellent dynamic performance
• Strong harmonics lower than the switchingfrequency (Subharmonics)
• No intercommunication betweenthe individual hysteresis controllers
Increase the switching frequencyat lower modulation index
• A tendency at lower speed to lock intolimit cycle of high-frequency switching
• Not strictly limit the current error
Pros:
Cons:
in time domain
in frequency domain
DB-102
Outline
1. Introduction2. Switching Modeling and PWM
• Switching model of VSI & boost rectifier• Space vector modulation for VSI & boost rectifier• Other modulations for VSI & boost rectifier• Switching model and modulation for CSI & buck rectifier
3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
52
DB-103
Buck Rectifier / Current Source Inverter– similar approach but different results –
• CSI
• Buck Rectifier
Av
Bv
Cv
aps bps cps
ans bns cns
RC
L
dcv
p
n
av
bv
cv
aps bps cps
ans bns cns
L
C
p
n
dcv
Av Bv Cvav
bvcv
R
{ }|})(||,)(||,)(max{|min tvtvtvV cabcabtx∀
=
⎟⎟⎠
⎞⎜⎜⎝
⎛=< mxdc VVV
23
•
• Unidirectional DC current
• Bi-directional DC voltage
Vx
vdc
t|vab| |vbc| |vca|
CSI / BUCK RECTIFIERDC VOLTAGE RANGE
DB-104
• Two single-pole-triple-throw (SPTT) current-unidirectional switches
• Allowed switching combinations:
aps bps cps
ans bns cns
p
n
avbv
cv
;1=++ ckbkak sss },{ npk ∈
• Topology: • Three-phase terminals are
voltage controlled• DC port is current controlled• Six current-unidirectional,
voltage-bi-directional, switches
Av
Bv
Cv
p
n
avbv
cv
Av
Bv
Cv
dci
dci
DC-Current-UnidirectionalThree-Phase Switching Network
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
53
DB-105
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
abc
vvv
vr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
cncp
bnbp
anap
c
b
a
abc
ssssss
sss
sr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
c
b
a
abc
iii
ir
dcabcabc isi ⋅=rr
abcTabcpn vsv rr
⋅=
where:
p
n
avbv
cv
Av
Bv
Cv
dciai
bi
ci
as bs cs
pnv
Buck Rectifier / Current Source Inverter Switching Model
DB-106
Switching State Vectors
dcI2 ⋅
30
150
90
-90
-150
-30
0
ρ θ (°)
0
][abI1
r
][acI2
r
][bcI3
r
][baI4
r
][caI5
r
][cbI6
r
][aaI0
r
][bbI0
r
][ccI0
r at center point][][][ ccbbaaI0 ===r
θ
iα
iβρ
a, α
b
c
β
refIr
][abI1
r
][acI2
r][baI4
r
][caI5
r
][bcI3
r
][cbI6
r
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
54
DB-107
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling
• Average model of boost rectifier• Average model of VSI• Average models in rotating coordinates
4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
DB-108
Boost Rectifier Switching ModelState-Space Equations
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡+
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ca
bc
ab
ca
bc
ab
ac
cb
ba
ac
cb
ba
CA
BC
AB
vvv
iii
dtdL
vvvvvv
iiiiii
dtdL
vvv
3
Rv
dtdvCi dcdc
dc +=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
ca
bc
ab
CA
BC
AB
ca
bc
ab
vvv
Lvvv
Liii
dtd
31
31
RCvi
Cdtdv dc
dcdc −=
1
dtdiLv
dtdiLv b
aba
AB −+=p
n
C R
L dcv
dci
ai
bi
ciCAv ABv
BCv
0
1av
bv
as
csbs0
10
1
cv
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
55
DB-109
Boost Rectifier Switching ModelState-Space Equations
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
CA
BC
AB
LL
vvv
vr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ca
bc
ab
ll
vvv
vr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ca
bc
ab
ll
iii
ir
llLLll v
Lv
Ldtid
−−− −=
rrr
31
31
RCvi
Cdtdv dc
dcdc −=
1
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ca
bc
ab
ll
sss
sr
dcllll vsv ⋅= −−rr
llT
lldc isi −− ⋅=rr
dcllLLll vs
Lv
Ldtid
⋅−= −−− rrr
31
31
RCv
isCdt
dv dcll
Tll
dc −⋅= −−
rr1
DB-110
Applying an average operator to switching model ττ= ∫−
dxT
txt
Tt
)(1)(
• Switch duty cycle
• Phase-leg duty cycle
• Line-to-line duty cycle
dττsT
tsdt
Ttapapap ∫
−
== )(1)(
anapa ddd −== 1
ba
t
Ttababab dddττs
Ttsd −=== ∫
−
)(1)(
• KVL and KCL
• Linear components
0=Σi0=Σv
RR iRv =dtidLv L
L =dtvdCi C
C =
Average Circuit Modeling
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
56
DB-111
dcabab vsv ⋅=
dcabdcab
t
Ttdcabab vdvsτdτvτs
Tv ⋅=⋅≈⋅= ∫
−
)()(1
Averaging of Quadratic Terms
if maximum-frequency components of are .)(tvdc T21<<
dclldclldcll vdvsvs ⋅=⋅≈⋅ −−−rrr
llT
llllT
llllT
ll idisis −−−−−− ⋅=⋅≈⋅rrrrrr
DB-112
ττττττ dvsL
vLT
ddt
idT
t
Ttdcllll
t
Tt
ll ∫∫−
−−−
− ⋅−= ))()(31)(
31(1)(1 rr
r
ττττττ dRC
visCT
ddt
dvT
t
Tt
dcll
Tll
t
Tt
dc ∫∫−
−−−
−⋅= ))()()(1(1)(1 rr
Applying average operatordcllLL
ll vsL
vLdt
id⋅−= −−
− rrr
31
31
RCvis
Cdtdv dc
llT
lldc −⋅= −−
rr1
dcllLLll vd
Lv
Ldtid
⋅−= −−−
rrr
31
31
RCvid
Cdtvd dc
llT
lldc −⋅= −−
rr1⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ca
bc
ab
ll
ddd
dr
Development of Boost Rectifier Average Model
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
57
DB-113
dci
dcab vd ⋅
abi
bci
cai
Averaging
Lai
bi
ciABv
BCv C
Rdcv
Switching Model
Average Model
dcllLLll vs
Lv
Ldtid
⋅−= −−− rrr
31
31
RCvis
Cdtdv dc
llT
lldc −⋅= −−
rr1
dcllLLll vd
Lv
Ldtid
⋅−= −−−
rrr
31
31
RCvid
Cdtvd dc
llT
lldc −⋅= −−
rr1dcbc vd ⋅
dcca vd ⋅
CAv
p
n
C
RL dcv
dci
ai
bi
ciCAv ABv
BCv
0
1av
bv
as
csbs0
10
1
cv
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=−
ac
cb
ba
ll
ssssss
sr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=−
ac
cb
ba
ll
iiiiii
i31r
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
CA
BC
AB
LL
vvv
vr
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=−
ca
bc
ab
ll
ddd
dr
Average Model of Three-Phase Boost Rectifier
DB-114
dcv
dcab vd ⋅
dcbc vd ⋅
dcca vd ⋅
ABv
BCv
CAv
abi
bci
caiabab id ⋅ bcbc id ⋅ caca id ⋅
L3
L3
L3
C R
• Third order system due to degeneration
Another Equivalent Circuit forBoost Rectifier Average Model
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
58
DB-115
dcab vd ⋅
dcbc vd ⋅
ABv
BCv
abi
bci
L3
L3
dcVabab id ⋅
bcbc id ⋅
caca id ⋅
C
R
Steady-State Operation under Balanced Sinusoidal Excitation
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
π+π−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)3/2cos()3/2cos(
)cos(
ωtVωtV
ωtV
vvv
m
m
m
CA
BC
AB
Given:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
π+π−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)3/2cos()3/2cos(
)cos(
ωtIωtI
ωtI
iii
m
m
m
ca
bc
ab
Goal:
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−π+−π−
−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)3/2cos()3/2cos(
)cos(
θωtDθωtD
θωtD
ddd
m
m
m
ca
bc
ab
Assume:
⎥⎦⎤
⎢⎣⎡ −
π+
π++−
π−
π−+−= )
32cos()
32cos()
32cos()
32cos()cos(cos θωtωtθωtωtθωtωtIDi mmdc
dci
⎥⎦⎤
⎢⎣⎡ −
π+++−
π−++−+= )
342cos(cos)
342cos(cos)2cos(cos
2θωtθθωtθθωtθIDi mm
dc
const.cos23const.cos
23
==⇒== θIRDVθIDi mmdcmmdc
DB-116
Steady-State Operation under Balanced Sinusoidal Excitation
Can use positive sequence phasors for steady state:
dcabABAB VωL ⋅+⋅= DIV 3j
11]1,0[,, ≤≤−⇒∈−≡−= abbpapbpapbaab dddddddd
)cos(,const. θωtVDvdVv dcmdcabdcdc −⋅=⋅⇒==
2m
ABV
=V2m
ABI
=I
θVDθVDV dcmdcmdcab sin
2jcos
2⋅
⋅−⋅
=⋅D
θRDVI
θDVV
RDωLθ
m
mm
m
mdc
m222
1
cos32,
cos,4sin
21
⋅==⎟⎟⎠
⎞⎜⎜⎝
⎛= −
mdcm VVD ≥⇒≤ 1
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
59
DB-117
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling
• Average model of boost rectifier• Average model of VSI• Average models in rotating coordinates
4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
DB-118
aps bps cps
ans bns cns
L
C
p
n
dcvCAvBCv
ABvav
bvcv
R
LLdcllll v
Lvs
Ldtid
−−− −⋅=
rrr
31
31
llT
lldc isi −− ⋅=rr
LLllLL v
RCi
Cdtvd
−−− −=
rrr11
Averaging
LLdcllll v
Lvd
Ldtid
−−− −⋅=
rrr
31
31
llT
lldc idi −− ⋅=rr
LLllLL v
RCi
Cdtvd
−−− −=
rrr11
Development of VSI Average Model
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
60
DB-119
L3
L3
L3
RC
• Fourth order system due to degeneration
RC
RC
dcab vd ⋅
dcbc vd ⋅
dcca vd ⋅
abab id ⋅ bcbc id ⋅ caca id ⋅
dcv
ABv
BCv
CAv
abi
bci
cai
dci
Equivalent Circuit forVSI Average Model
DB-120
L3
L3
L3
RC
• Steady-state ac voltages and currents are sinusoidal if• Difficult to define small-signal model. Operating point ?
RC
RC
dcab vd ⋅
dcbc vd ⋅
dcca vd ⋅
abab id ⋅ bcbc id ⋅ caca id ⋅
dcv ABv
BCv
CAv
abi
bci
cai
dci
For:
⇒⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
π+π−=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)3/2cos()3/2cos(
)cos(
ωtDωtD
ωtD
ddd
m
m
m
ca
bc
ab
const.=dcV
Steady-State Operation under DC Input and Balanced Sinusoidal Duty-Cycles
dcABdcabAB VV
ωCRRωL
ωCRR
≤⇒⋅⋅
++
+= VDV
j13j
j1
and
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
61
DB-121
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling
• Average model of boost rectifier• Average model of VSI• Average models in rotating coordinates
4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
DB-122
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−−−
+−
=
21
21
21
)3
2sin()3
2sin(sin
)3
2cos()3
2cos(cos
32
/πωπωω
πωπωω
ttt
ttt
T abcdq0
Choose
where: ω=2πf, f is ac line frequency(source frequency for boost rectifier;desired output frequency for VSI)
abcdq0 XTX ⋅= dq0abc XTX ⋅= −1
)( / abcdq0TT =
Coordinate Transformation
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
62
DB-123
dcllLLll vd
Lv
Ldtid
⋅−= −−−
rrr
31
31
RCvid
Cdtvd dc
llT
lldc −⋅= −−
rr1
dcdq0dq0dq0 vdT
LvT
LdtiTd
⋅⋅−⋅=⋅ −−
− rrr
111
31
31)(
RCviTTd
Cdtvd dc
llT
lldc −⋅⋅= −
−−
rr11
dq0abc XTX ⋅= −1
dcdq0dq0dq0
dq0 vdL
TvL
Tdtid
Tidt
dT⋅⋅−⋅=⋅+⋅ −−−
− rrr
r
31
31 111
1
RCviTdT
Cdtvd dc
llT
lldc −⋅⋅⋅⋅= −−
rr)(1
×T
Coordinate Transformation– Boost Rectifier –
DB-124
Coordinate Transformation– Boost Rectifier –
dcdq0dq0dq0
dq0 vdL
vLdt
idi
dtdTT ⋅−=+⋅
− rrr
r
31
311
RCvid
Cdtvd dc
dq0Tdq0
dc −⋅=rr1
RCviTdT
Cdtvd dc
llT
lldc −⋅⋅⋅⋅= −−
rr)(1
dcdq0dq0dq0
dq0 vdL
vLdt
idi
dtdTT ⋅−=+⋅
− rrr
r
31
311
dq0ll ddTrr
=⋅ − dq0ll iiTrr
=⋅ −
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
63
DB-125
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−+−
−−−−
−−
⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−−−−
+−
=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−+−
−−−−
−−
⋅=
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+−+
−−−
−
⋅=⋅=⋅−
0)3
2cos()3
2sin(
0)3
2cos()3
2sin(0cossin
32
21
21
21
)3
2sin()3
2sin(sin
)3
2cos()3
2cos(cos
32
0)3
2cos()3
2sin(
0)3
2cos()3
2sin(0cossin
32
)
21)
32sin()
32cos(
21)
32sin()
32cos(
21sincos
32(
T1
πωωπωω
πωωπωωωωωω
πωπωω
πωπωω
πωωπωω
πωωπωωωωωω
πωπω
πωπω
ωω
tt
tttt
ttt
ttt
tt
tttt
T
tt
tt
tt
dtdT
dtdTT
dtdTT
Coordinate Transformation
DB-126
23)
32(cos)
32(coscos 222 =++−+
ππ xxx
Using the following trigonometric relationships
23)
32(sin)
32(sinsin 222 =++−+
ππ xxx
0)3
2cos()3
2sin()3
2cos()3
2sin(cossin =+⋅++−⋅−+⋅ππππ xxxxxx
0)3
2cos()3
2cos(cos =++−+ππ xxx
0)3
2sin()3
2sin(sin =++−+ππ xxx
Coordinate Transformation
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
64
DB-127
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −=⋅
−
0000000
1
ωω
dtdTT
dcdq0dq0dq0dq0 vd
Liv
Ldtid
⋅−⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −−=
rrrr
31
0000000
31 ω
ω
Therefore:dcdq0dq0
dq0dq0 vd
Lv
Ldtid
idt
dTT ⋅−=+⋅− rr
rr
31
311
RCvid
Cdtvd dc
dq0Tdq0
dc −⋅=rr1
RCvid
Cdtvd dc
dq0Tdq0
dc −⋅=rr1
Coordinate Transformation– Boost Rectifier –
DB-128
dc
ca
bc
ab
CA
BC
AB
ca
bc
ab
vddd
Lvvv
Liii
dtd
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
31
31
[ ]RCv
iii
dddCdt
vd dc
ca
bc
ab
cabcabdc −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=
1
(abc coordinates)
dc
0
q
d
0
q
d
0
q
d
0
q
d
vddd
Liii
vvv
Liii
dtd
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −−
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
31
0000000
31 ω
ω
[ ]RCv
iii
dddCdt
vd dc
0
q
d
0qddc −
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=
1 (dq0 coordinates)
State-Space Equations– Boost Rectifier –
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
65
DB-129
Equivalent Circuit in dq0 Coordinates– Boost Rectifier –
di L3
dvqiLω3
dcd vd ⋅
qi L3
qvdiLω3
dcq vd ⋅
0i L3
0vdc0 vd ⋅
dcv
dd id ⋅ qq id ⋅ 00 id ⋅
C R
.3j,3j,3j cabcab iωLiωLiωL ⋅⋅⋅
The cross-coupling terms,
and ,
in dc coordinates (dq0),
account for the voltage drops
across inductances at line
frequency in ac coordinates,
qiωL3 diωL3
DB-130
0≡++ CABCAB vvv
0≡++ cabcab iii
0≡++ cabcab ddd
Since
0i L3
0vdc0 vd ⋅
0≡0v
0≡0i
0≡0d
0-channel can be omitted
0-Channel
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
66
DB-131
dcv
dd id ⋅ qq id ⋅
C R
di L3
dvqiLω3
dcd vd ⋅
qi L3
qvdiLω3
dcq vd ⋅
dcq
d
q
d
q
d
q
d vdd
Lii
vv
Lii
dtd
⋅⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ −−⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
31
00
31
ωω
[ ]RCv
ii
ddCdt
vd dc
q
dqd
dc −⎥⎦
⎤⎢⎣
⎡⋅=
1
Equivalent Circuit in dq0 Coordinates– Boost Rectifier –
DB-132
LLdcllll v
Lvd
Ldtid
−−− −⋅=
rrr
31
31
llT
lldc idi −− ⋅=rr
LLllLL v
RCi
Cdtvd
−−− −=
rrr11
Transformation
(abc coordinates)
dq0dq0dcdq0dq0 v
Livd
Ldtid rrrr
31
0000000
31
−⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −−⋅= ω
ω
dq0Tdq0dc idi
rr⋅=
dq0dq0dq0dq0 v
RCvi
Cdtvd rrrr
1
0000000
1−⋅
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡ −−= ω
ω
(dq0 coordinates)
State-Space Equations– Voltage Source Inverter –
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
67
DB-133
dqdqdcdqdq v
Livd
Ldtid rrrr
31
00
31
−⋅⎥⎦
⎤⎢⎣
⎡ −−⋅=
ωω
dqTdqdc idi
rr⋅=
dqdqdqdq v
RCvi
Cdtvd rrrr
10
01−⋅⎥
⎦
⎤⎢⎣
⎡ −−=
ωω
dcv
dd id ⋅ qq id ⋅C
R
di L3dv
qiLω3
dcd vd ⋅
L3qvdiLω3
dcq vd ⋅ R
C
qi
dci
dvCω
qvCω
Equivalent Circuit in dq0 Coordinates– Voltage Source Inverter –
DB-134
Buck Rectifier / CSI dq0 Model– similar approach but different results –
R
R
Cdcd id ⋅ dvdi
dcq id ⋅ qvqi
C
qvCω
dvCω
dcd id ⋅dvdi
dcq id ⋅qvqi
RC
dd vd ⋅
qq vd ⋅
L
dcvpnv
dci
dd vd ⋅
qq vd ⋅
L
dcv pnv
dciCurrent Source Inverter (CSI)
Buck Rectifier
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
68
DB-135
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling
• Small-signal model of boost rectifier• Small-signal model of VSI• Three-phase modulator modeling
5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
DB-136
Linearization
Autonomous dynamic system:
If is analytic it can be expressed as Taylor series:fr
Retaining the first 3 terms results in linear approximation of :fr
But the dynamic system is NOT linear because:
A Bx&r = xr ur+ 0≠gr+
K
rrr
rrrrrrr
rr
rrrrr
r
rrr
rrr
rrrrr
r
rrrrrrrrr
+
⎥⎦
⎤⎢⎣
⎡−
∂∂
+−−∂∂
∂+−
∂∂
+
+−⋅∂
∂+−⋅
∂∂
+=
202
002
0000
22
0200
2
000
000
00
)(),())((),()(),(!2
1
)(),()(),(),(),(
uuu
uxfuuxxux
uxfxxx
uxf
uuu
uxfxxx
uxfuxfuxf
)(),()(),(),(),( 000
000
00 uuu
uxfxxx
uxfuxfuxf rrr
rrrrr
r
rrrrrrrrr
−⋅∂
∂+−⋅
∂∂
+≅
000
000
000000 ),(),(),(),(),( u
uuxfx
xuxfuxfu
uuxfx
xuxf
dtxd r
r
rrrr
r
rrrrrrr
r
rrrr
r
rrrr
⋅∂
∂−⋅
∂∂
−+⋅∂
∂+⋅
∂∂
≅
),( uxfdtxd rrrr
=
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
69
DB-137
Linearization
)(),()(),(),(),( 000
000
00 uuu
uxfxxx
uxfuxfuxfdtxd rr
r
rrrrr
r
rrrrrrrrrr
−⋅∂
∂+−⋅
∂∂
+≅=
dtxd
dtxd
dtXd
dtxd ~~ rrrr
=+=
uUu ~rrr+=
uu
uxfxx
uxfdtxd
UXUX
~),(~),(~
),(),(
rr
rrrr
r
rrrr
rrrr⋅
∂∂
+⋅∂
∂≅
If is an equilibrium point , and is perturbation around it:),( UXrr
)~
,~
( ux rr),( 00 ux rr
0),( ≡UXfrrr
xXx ~rrr+=
0=dtXdr
),(
),(
UXu
uxfrr
r
rrr
∂∂
=B
uxx ~~~ rr&r ⋅+⋅= BA
),(
),(
UXx
uxfrr
r
rrr
∂∂
=A
DB-138
Average Large-Signal ModelBoost Rectifier
dcv
dd id ⋅ qq id ⋅
C R
di L3
dvqiLω3
dcd vd ⋅
qi L3
qvdiLω3
dcq vd ⋅
A steady-state operating point:
md VV ⋅=23
(Vm: Max line-to-line voltage)
0=qV
dc
dd V
VD =
dc
dq V
LID ω3−=
d
dcd DR
VI⋅
=
0=qI
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
70
DB-139
Linearization – Boost Rectifier
dcq
d
q
d
q
d
q
d vdd
Lii
vv
Lii
dtd
⋅⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ −−⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
31
00
31
ωω
[ ]RCv
ii
ddCdt
vd dc
q
dqd
dc −⎥⎦
⎤⎢⎣
⎡⋅=
1
Linearization
dcq
ddc
q
d
q
d
q
d
q
d vDD
LV
dd
Lii
vv
Lii
dtd ~
31
~~
31
~~
00
~~
31
~~
⋅⎥⎦
⎤⎢⎣
⎡−⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ −−⎥
⎦
⎤⎢⎣
⎡=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
ωω
[ ] [ ]RCv
ii
DDCI
Idd
Cdtvd dc
q
dqd
q
dqd
dc~
~~
1~~1~−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅+⎥
⎦
⎤⎢⎣
⎡⋅=
DB-140
Small-Signal Model – Boost Rectifier
⎥⎦
⎤⎢⎣
⎡⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−
+⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
−
−−
−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
q
d
q
d
qd
dc
dc
dc
q
d
qd
q
d
dc
q
d
vv
L
L
dd
CI
CI
LV
LV
vii
RCCD
CD
LD
LD
vii
dtd
~~
00310
031
~~
30
03
~
~~
13
0
30
~
~~
ω
ω
A Bx&r = xr ur+ D vr+
IntrinsicSystem Dynamics
ControlInput
DisturbanceInput
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
71
DB-141
Small-Signal Model – Boost Rectifier
dcv~
dd Id ⋅~
C R
di~ L3
dv~
qiL~3ω
dcd vD ~⋅
qi~ L3
qv~
diL~3ω
dcq vD ~⋅
dcd Vd ⋅~
dcq Vd ⋅~
dd iD ~⋅ qq Id ⋅~
qq iD ~⋅
DB-142
Open-Loop Transfer Functions
)()()()()(
~~
*221
21
pspspszszsK
di idddidddiddd
d
d
+⋅+⋅+
+⋅+⋅=
)()()(
)(~~
*221 pspsps
zsK
di iddqiddq
q
d
+⋅+⋅+
+⋅=
0
20
40
60
80
-300
-250
-200
-150
-100
1 10 100 1000 10000 100000
-40
-20
0
20
40
60
0
50
100
150
200
1 10 100 1000 10000 100000Frequency [rad/sec] Frequency [rad/sec]
Mag
nitu
de [d
B]
Pha
se [°
]
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
72
DB-143
Open-Loop Transfer Functions
)()()(
)()(~~
*221
21
pspsps
zszsK
d
i iqddiqddiqdd
d
q
+⋅+⋅+
+⋅+⋅=
)()()(
)()(~~
*221
*
pspsps
zszsK
d
i iqdqiqdqiqdq
q
q
+⋅+⋅+
+⋅+⋅=
-40
-20
0
20
40
60
-200
-150
-100
-50
0
50
1 10 100 1000 10000 100000
0
20
40
60
80
80
100
120
140
160
180
1 10 100 1000 10000 100000Frequency [rad/sec] Frequency [rad/sec]
Mag
nitu
de [d
B]
Pha
se [°
]
DB-144
Open-Loop Transfer Functions
)()()(
)()(~~
*221
1
pspsps
zszsK
dv RHPvdcdqvdcdq
q
dc
+⋅+⋅+
−⋅+⋅=
)()()()()()(
~~
*221
21
pspspszszszsK
dv RHPvdcddvdcddvdcdd
d
dc
+⋅+⋅+
−⋅+⋅+⋅=
Frequency [rad/sec] Frequency [rad/sec]
Mag
nitu
de [d
B]
Pha
se [°
]
0
20
40
60
80
-50
0
50
100
150
200
1 10 100 1000 10000 100000
-40
-20
0
20
40
60
-200
-100
0
100
200
1 10 100 1000 10000 100000
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
73
DB-145
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling
• Small-signal model of boost rectifier• Small-signal model of VSI• Three-phase modulator modeling
5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
DB-146
Average Large-Signal Model – VSI
A steady-state operating point:
qd
d CVR
VI ω−=
dq
q CVR
VI ω+=
dc
qdd V
LIVD
ω3−=
dc
dqq V
LIVD
ω3+=
dcv
dd id ⋅ qq id ⋅C
R
di L3dv
qiLω3
dcd vd ⋅
L3qvdiLω3
dcq vd ⋅ R
C
qi
dci
dvCω
qvCω
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
74
DB-147
Linearization – VSI
Linearization
⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ω
ω−−⋅⎥
⎦
⎤⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
q
d
q
ddc
q
d
q
dvv
Lii
vdd
Lii
dtd
31
00
31
[ ] ⎥⎦
⎤⎢⎣
⎡⋅=
q
dqddc i
iddi
⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ −−
⎥⎥⎦
⎤
⎢⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
q
d
q
d
q
d
q
d
vv
RCvv
ii
Cvv
dtd 1
001ω
ω
⎥⎦
⎤⎢⎣
⎡−
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ω
ω−−⋅⎥
⎦
⎤⎢⎣
⎡+⋅
⎥⎥⎦
⎤
⎢⎢⎣
⎡=
⎥⎥⎦
⎤
⎢⎢⎣
⎡
q
d
q
ddc
q
ddc
q
d
q
dvv
Lii
vDD
LV
dd
Lii
dtd
~~
31
~~
00~
31
~~
31
~~
[ ] [ ] ⎥⎦
⎤⎢⎣
⎡⋅+
⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅=
q
dqd
q
dqddc I
Idd
ii
DDi ~~~~
~
⎥⎦
⎤⎢⎣
⎡−⎥
⎦
⎤⎢⎣
⎡⋅⎥
⎦
⎤⎢⎣
⎡ −−
⎥⎥⎦
⎤
⎢⎢⎣
⎡=⎥
⎦
⎤⎢⎣
⎡
q
d
q
d
q
d
q
d
vv
RCvv
ii
Cvv
dtd
~~1
~~
00
~~
1~~
ωω
DB-148
Small-Signal Circuit Model – VSI
dcv~
dd Id ⋅~
C R
di~
L3
dv~qiL~3ω
qi~
L3
qv~diL~3ω
dcd Vd ⋅~
dd iD ~⋅ qq Id ⋅~
qq iD ~⋅
RC
dci~ qvC~ω
dvC~ω
dcd vD ~⋅
dcq Vd ⋅~
dcq vD ~⋅
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
75
DB-149
Small-Signal State-Space Model – VSI
dcq
d
q
ddc
dc
q
d
q
d
q
d
q
d
vL
DL
D
dd
LV
LV
vvii
RCC
RCC
L
L
vvii
dtd ~
00
3
3
~~
0000
30
03
~~
~~
110
1013100
0310
~~
~~
⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+⎥⎥⎦
⎤
⎢⎢⎣
⎡⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
+
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
⋅
⎥⎥⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢⎢⎢
⎣
⎡
−ω−
ω−
−ω−
−ω
=
⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢
⎣
⎡
A Dx&r = xr ur+ B + vr
DB-150
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling
• Small-signal model of boost rectifier• Small-signal model of VSI• Three-phase modulator modeling
5. Closed-Loop Control Design
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
76
DB-151
Natural and Uniform Sampling
+_
Modulatingsignal
Natural Sampling Uniform Sampling
TS
Ton
TS
Ton
Carrier
+_
0
1
0
1
DB-152
Trailing- and Leading-Edge Modulation
+_
Trailing-Edge Modulation Leading-Edge Modulation
+_
TS
Ton
TS
Ton
Both are the natural sampling
0
1
0
1
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
77
DB-153
Review of Modulator Modeling for DC-DC Converters
Natural sampling
• Unity gain• No delay
Uniform sampling
1. Trailing-edge modulation• Unity gain• Phase delay at a modulation frequency
2. Leading-edge modulation• Unity gain• Phase delay at a modulation frequency
ms ωTD ⋅⋅
( ) ms ωTD ⋅⋅−1
DB-154
Three-Phase Modulator Modeling
All naturally sampled modulators can be modeled by the constantgain term
• Modulators associated with analog controllers
Small-signal models have to be derived for uniformly sampledthree-phase modulators
• Modulators associated with digital controllers
carm V
F 1=
refav vcar
Vcar
Vcar
t
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
78
DB-155
Example – for Boost Rectifier and VSI
Trailing-edge modulation (2Φ-RA)
Two out of three switching functions always have pulsessynchronized to the beginning of the switching period
TS
Sa
Sc
Sb
pnn ppn ppp
Sab
Sca
Sbc
msab ωTD ⋅⋅
( ) msca ωTD ⋅⋅−1
Phase delays:
DB-156
Approximate Average Model of Digital Modulator
refdd dd
refqd qd
dsTe−
qsTe−
DigitalModulator
refdd~
dd
refqd~
qd
refdd
refqd
dd~
qd~
refd
d
dd
~~
refd
q
d
d~
~
refq
q
d
d~
~
refq
d
dd
~~ +
+
+
+
≈
sqd TTT ≈≈
desired model ofactual
actual
CoordinateTransform.
InverseCoordinateTransform.
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
79
DB-157
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
• Control approach• Current-loop design• Voltage-loop design• Limiting
6. More Complex Converters
PECon2008
DB-158
Classical Linear Control Design Approach
Power stageControl1
Control2
Output1
Output2
Compensator
Power stageControl1
Control2
Output1
Output2
Compensator2
Compensator1
Multi-Input-Multi-Output
Single-Input-Single-Output
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
80
DB-159
Closed-Loop Control Design
• Based on small-signal models
Advantage• Classical control design methods can be used
(Bode plots, root loci, etc.)
Disadvantages• Design is valid only at a certain operating point• The approach does not guarantee large-signal stability
DB-160
Control Structure: Cascade Control
1. Inner current controla. Bang-bang current control (and
PWM) in stationary coordinatesb. Current control in stationary
coordinates with two independent or three dependent current controllers (P, PI, or resonant regulators)
c. Current control in rotating coordinates with two independent current controllers (P or PI regulators)
2. Outer output voltage or torque / flux / speed / position control (PI regulators)
Modulator
CurrentControl
OutputControl
Three-phasePWM
Converter
CurrentFeedback
OutputFeedback
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
81
DB-161
Control Realization
1. Completely analog controlComplete analog control is rarely used because ofhardware complexity
2. Combination of analog and digital controlAnalog current control in stationary and digital outputvoltage or speed and flux control
3. Completely digital controlUsed in most new designs
DB-162
3 - PhaseSwitching Network
Phase-to-linecurrenttransf.
Coordinatetransf.
Inversecoordinate
transf.
SVM
6
Phase-locked
loop (PLL)
vA
vB
vCia ib
cos θ
sin θ
iab ibc
id iq dd dq
dα dβcos θ
sin θ
Open-loop Boost Rectifier Control
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
82
DB-163
COORDINATETRANSFORM.
INVERSE COORD.TRANSFORM.
SVM
A/D A/D
A/D
A/D
A/D
A/D
D/A
D/A
6
DSP
GENERATIONPLL & sin θ, cos θ
Open-loop Boost Rectifier ControlDigital Signal Processor (DSP) Implementation
refdd ′ref
qd ′
ABv
BCv
abi bci
di′
qi′
di
qi
αd βdrefdd
refqd
DB-164
refdd ′ ref
qd ′refdD ref
qDref
dd~′ refqd~′
3 - PhaseSwitching Network
A/D
6
DSPA/D D/A(ZOH)
ImpedanceAnalyzer
Test signalOutput signal
di′ qi′
di
qirefdd ref
qd
Transfer Function Measurement Set-up
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
83
DB-165
di~′
qi~′
di~
qi~
refdd~
refqd~
BOOSTRECTIFIERSMALL-SIG.
MODEL
s s (k+2) TsComputational
delay PWM delay
Sampling instant
k T (k+1) T
Modified small-signal model
dd~
qd~
dcv~
dsTe−
qsTe−
dcv ′~
“Digital Control Interface”
cT
csTe−
csTe−
qd TT ,
csTe−
Digital Delay
DB-166
Simplified approximations in continuous time domain:
di~′
qi~′
di~
qi~
refdd~
refqd~
BOOSTRECTIFIERSMALL-SIG.
MODEL
dd~
qd~
dcv~
dsTe−
qsTe−
dcv ′~
“Digital Control Interface”
( )
( )122
1
1221
2
2
deldel
deldelsT
sTsT
sTsT
e del
++
+−≈−
Often choose: switchingsamplingqdcdel TTTTTT =≈≈≈≈
csTe−
csTe−
csTe−
Digital Delay
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
84
DB-167
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
• Control approach• Current-loop design• Voltage-loop design• Limiting
6. More Complex Converters
PECon2008
DB-168
Current Loop Design
dcvdd id ⋅ qq id ⋅
C R
di L3dv
qiLω3dcd vd ⋅
qi L3qv
diLω3dcq vd ⋅
Hv Hid
Hiq
vref
v’dc
idref
iqref
di′
qi′
refqd
dd qd
refdd
di qiDigital Control Interface
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
85
DB-169
Control-to-Current Transfer Function
)()()()()(
~~
*221
21
pspspszszsK
di idddidddiddd
d
d
+⋅+⋅+
+⋅+⋅=
sK
KH ipid +=
Mag
nitu
de [d
B]
Pha
se [°
]
10
20
30
40
50
60
70
-300
-250
-200
-150
-100
0.1 1 10 100 1000 10000 100000Frequency [Hz]
DB-170
Control-to-Current Transfer Function
)()()(
)()(~~
*221
*
pspsps
zszsK
d
i iqdqiqdqiqdq
q
q
+⋅+⋅+
+⋅+⋅=
sK
KH ipiq +=
Mag
nitu
de [d
B]
Pha
se [°
]
10
20
30
40
50
60
70
80
100
120
140
160
180
0.1 1 10 100 1000 10000 100000Frequency [Hz]
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
86
DB-171
Current Loop-Gain
• D channel loop-gain Td• Bandwidth is limited by delay (fsw=20kHz)
Mag
nitu
de [d
B]
Pha
se [°
]
-20
0
20
40
60
80
-350
-300
-250
-200
-150
-100
-50
0
0.1 1 10 100 1000 10000 100000Frequency [Hz]
DB-172
Current Loop-Gain
• Q channel loop-gain Tq• Bandwidth is limited by delay (fsw=20kHz)
Mag
nitu
de [d
B]
Pha
se [°
]
-20
0
20
40
60
80
100
-350
-300
-250
-200
-150
-100
-50
0.1 1 10 100 1000 10000 100000Frequency [Hz]
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
87
DB-173
Current Regulation
• Peak is more pronounced when gain increases
-20
-15
-10
-5
0
5
-350
-300
-250
-200
-150
-100
-50
0
0.1 1 10 100 1000 10000 100000
-20
-15
-10
-5
0
5
0.1 1 10 100 1000 10000 100000
-400
-300
-200
-100
0
0.1 1 10 100 1000 10000 100000Frequency [Hz] Frequency [Hz]
Mag
nitu
de [d
B]
Pha
se [°
]
dref
di
i~
~′
qref
q
ii
~~′
DB-174
Current Loop with D and Q Decoupling
Hv Hid
Hiq
vref
v’dc
idref
i’d
iqref
i’q
3ωL/v’dc
3ωL/v’dc
qsT
dcq
sTdcq
refd
sTrefd ie
vωLdeviωLdedd ddd
d′⋅⋅
′+=⋅′′+=⋅= −−− 13)/3(
))
refqd
refdd
dsT
dcq
sTdcd
refq
sTrefq ie
vωLdeviωLdedd qqq
q′⋅⋅
′−=⋅′′−=⋅= −−− 13)/3(
))
refqd)
refdd)
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
88
DB-175
Decoupled D and Q Channels
di L3
dvdcd vd ⋅
)
qi L3
qvdcq vd ⋅
)
dcv
dd id ⋅ qq id ⋅
C R
• Similar to two parallel dc-dc boost converters after d and q decoupled
qqsT
dc
dcdcd
qdcd
iωLievvLωvd
iωLvd
d 33
3
−′′
+=
=−
−)))
ddsT
dc
dcdcq
ddcq
iωLievvLωvd
iωLvd
q 33
3
+′′
−=
=+
−)))
0≈
0≈
DB-176
Cross-Coupling Effect After Decoupling
qref
di
i~
~′
dref
q
ii
~~′
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
89
DB-177
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
• Control approach• Current-loop design• Voltage-loop design• Limiting
6. More Complex Converters
PECon2008
DB-178
Output Voltage Loop Design
)()()(
~~
HL
RHP
dref
dcc psps
zsKiv
G+⋅+
−⋅==
-200
-150
-100
-50
0
0.1 1 10 100 1000 10000 100000Frequency [Hz]
Pha
se [°
]M
agni
tude
[dB
]
-20
-15
-10
-5
0
5
10
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
90
DB-179
COMPENSATOR DESIGN
Voltage compensator
Place z as high as possible for required phase margin.Place p for loop-gain
attenuation.Attainable voltage-loop bandwidth:
ωc
Outer loop-gain
Gc
Hv
zv pv v
v
ω < 1/4 zc RHP
K (1 + s / z )vvH =v s (1 + s / p )v
DB-180
Attainable Voltage Loop Bandwidthwith Delay
-10
0
10
20
30
10100
1,00010,000
-350-300-250-200-150-100
-50
K
Frequency [kHz]
Gai
n [d
B]Ph
ase
[deg
]
Voltage compensator
Zero is placed close to the crossover to improve the phase margin
K (1 + s / z )vvH =v s (1 + s / p )v
Kv
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
91
DB-181
Time Domain Simulation Results
Phase voltages and currents (PFC operation)
Output voltage
DB-182
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
• Control approach• Current-loop design• Voltage-loop design• Limiting
6. More Complex Converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
92
DB-183
LIMITING IN THREE-PHASE CONVERTERS
Modulator
CurrentControl
OutputControl
Three-phasePWM
Converter
CurrentFeedback
OutputFeedback
Limiter
Limiter
DB-184
Boost Rectifier/VSI Switching State Hexagon
ab
bc
ca
v ref
Maximal attainable voltage vector lies on the hexagonFor sinusoidal average output voltages the voltage vector must be inside the inscribed circle
dcrefref Vdv ⋅⋅=
23rr
1123 22 ≤+⇒≤⇒⋅≤ qd
refdc
ref dddVvrr
Duty Cycle Limiting
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
93
DB-185
d lim q lim
Vector angle is kept constant( d ) and ( d ) should be fed back to the anti-windup current controllersIf output of each current controller is limited separatelyit can result in unattainable voltage vector
dd
dq
limd q
limd d
dlimref
d ref
1If 22 >+ qd dd
22limqd
dd dd
dd+
=
22limqd
qq dd
dd
+=
Duty Cycle Limiting
DB-186
idref
id
iqref
iq
3ωL/vdc
3ωL/vdc
refqd
refdd
Ki /s
Kp
μ
Ki /s
Kp
μ
ddlim
ddlim
dqlim
dqlim
Variable Limit PI (“Smart Anti-windup”) Current Controllers
μ → ∞
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
94
DB-187
Maximal current magnitude is limited
(i ) + ( i )ref ≤ Imaxd2 ref
q2√
Limiting of each reference current component is determined by the control algorithm
EXAMPLEBoost rectifier control
Output voltage
regulator
Imaxidref ( i )d
reflim
( i )qref
limiqref
(I ) - ( i )max2 ref
d2√ lim
CurrentControl dq
dd
Input displacement
factor controller
ref
ref
Current Limiting
DB-188
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters• Three-phase four-wire (four-phase) converter• Multilevel converters• Parallel converters
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
95
DB-189
Limitations of Three-Leg Converters
2RR
R
Output Voltages
-2.4
-1.8
-1.2
-0.6
0
0.6
1.2
1.8
2.4
0 30 60 90 120 150 180
• There is no path for the neutral current.• Output voltages are unbalanced.
Conventional inverter is not good for highly unbalanced load.
AvBvCv
aibici
AvCvBv
Nv
dcv
DB-190
• Provide three-phase four-wire• Deal with unbalanced and nonlinear load
R
R 2R
dcv
av bvcv
xvAv Bv Cv
Nv
aibici
ni nL
L
C
Load
Applications
Advantages• The fourth leg provides a neutral current path• Compared to generating vn with capacitive divider:
• Much smaller dc-link capacitance is needed • Full utilization of dc-link voltage (15% higher) with SVM
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
96
DB-191
Three-Dimensional Space Vector forFour-Leg Converter
• Vγ is the zero-sequence component• Vγ is related to the neutral current
α
−β
γ
V→
Vα
Vβ
Vγ
0=++= cnbnan VVVVγ
a-b-c coordinates α−β−γ coordinates
dcv
av bvcv
xvAv Bv Cv
Nv
aibici
ni nL
L
C RA
RB RC
DB-192
A Nonlinear Load
Vector Trajectory
in α-β-γCoordinates
Trajectory Projection on
α-β Plane
Trajectory Projection on
γ -Axis
Desired voltage
for sinusoidal
Load current
[ ]TCBA iii
-600-400
-2000
200400
600
-600-400
-2000
200400
600
-500
0
500
AlfaBeta
Gam
ma
0.3 0.302 0.304 0.306 0.308 0.31 0.312 0.314 0.316-500
-400
-300
-200
-100
0
100
200
300
400
500
Vga
mm
a
time0.2950.295 0.3 0.305 0.31 0.315 0.32
-150
-100
-50
0
50
100
150
Il gam
ma
time
-300-200
-1000
100200
300
-300-200
-1000
100200
300-300
-200
-100
0
100
200
300
AlfaBeta
Gam
ma
-500 0 500-600
-400
-200
0
200
400
600
Alfa
Bet
a
-300 -200 -100 0 100 200 300-300
-200
-100
0
100
200
300
Alfa
Bet
a
[ ]Tcxbxax vvv[ ]TCNBNAN vvv
Ai
Bi
Ci
Ni
C
N
B
A
Three Single-Phase Diode Bridge Rectifiers with C Filter
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
97
DB-193
Switching Model
dcv
avbv
cvxv
AvBv
Cv
Nv
aibici
Ni
L
C
LAi LBi LCi
dci
nL
[ ]⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=
c
b
a
cbadc
iii
sssi dc
c
b
a
cx
bx
ax
vsss
vvv
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
xps aps bps cps
ans bns cnsxns
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
xpcp
xpbp
xpap
c
b
a
ssssss
sss
cbaN iiii −−−=
DB-194
Average Model in Stationary Coordinates
• Controlled voltage sources are balanced for balanced load• Controlled voltage sources are unbalanced for unbalanced load• Difficult to describe the circuit operation in abc coordinates
dcvdci
axvbxvcxv
xv
L
C
aibici
NinL
LAi LBi LCi
AvBv
Cv
Nv
dc
c
b
a
cx
bx
ax
vddd
vvv
⋅⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡[ ]
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡⋅=
c
b
a
cbadc
iii
dddi⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
π+ωπ−ω
ω=
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
)3/2cos()3/2cos(
)cos(
tVtV
tV
vvv
m
m
m
refCN
BN
AN
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
98
DB-195
Average Model in Rotating Coordinates
+
+
+
+
++
++
d
q
0
dv
qv
ov
dωCv
qωCv
qωLi
dωLi
di
qi
oi
Ldi
Lqi
Loi
C
L
nLL 3+
dcd vd ⋅
dcq vd ⋅
dco vd ⋅
• 2 decoupled subsystems with reduced system orders• Transfer functions can be derived based on the dc operating point
dcv dci
ooqqdddc idididi ⋅+⋅+⋅=
L
C
C
DB-196
Transfer Functions
102 103
40
60
Gai
n (d
B)
102 103-180-90
090
180
Phas
e (d
egre
e)
Frequency (Hz)
+
+
+
+
102 103
1020304050
Gai
n (d
B)
102 103-180-90
090
180
Phas
e (d
egre
e)
Frequency (Hz)
• Sampling delay and PWM delay are incorporated in the model• The derived model agrees with the measurement very well
dd
qd
od
d
d
dv
q
d
dv
d
q
dv
q
q
dv
o
o
dv
dv
qv
ov
d
d
dv
q
d
dv
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
99
DB-197
Impact of Load Power Factor on Control-to-Output Voltage TFs
Resistive, capacitive (PF=-0.8), and inductive (PF=0.8) are all at 150 kW
Graph4
25.0
30.0
35.0
40.0
45.0
50.0
55.0
60.0
65.0
70.0
f(Hz)5.0 10.0 20.0 50.0 100.0 0.2k 0.5k 1.0k 2.0k 5.0k
(dB) : f(Hz)
db((vd/dd))
db((vd/dd))
db((vd/dd))
Resistive LoadPF = 1
Capacitive LoadPF = -0.8
Inductive LoadPF = 0.8
• Capacitive load shift the resonant frequency to lower frequency• Inductive load leads to a higher system order and higher resonant peaking• Both capacitive and inductive loads are worse loads than resistive load
d
ddv
DB-198
Control Block Diagram
VA
VB
VC
VN
4-leg Inverter Unbalanced and Nonlinear LoadFilter
Modulator Rotating Transformation
abc/dqo
VAN,BN,CNIABC
VdqoIdqo
Vdqo_ref-
+Current
CompensatorVoltage
Compensator
ddqo
Rotating Transformation
dqo/abc
dan dbn dcn
Vdc
+-
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
100
DB-199
Double-Loop Design
CONVERTERCONVERTER
+
+
+
+
–
–
–
drefv
qrefv
orefv
dv
ov
dd
qd
od
vdH
vqH
voH
d
d
di
q
d
di
d
q
di
q
q
di
o
o
di
–
–
–
idH
iqH
di
qi
oiioH
• Decoupling is not included
drefi
qrefi
orefi
d
d
iv
q
q
iv
o
oiv
+
+
q
d
iv
d
q
iv L
O
A
Dqv
DB-200
Switching State Vectorsin Three-Dimensional Space
a b
βα
γ
c
axvbxv
cxvdcv
xa
bc
n
p
dcv
n pnn
dcv
n npnn nnpn ppnn pppn abc
• 8 Switching State Vectors with non-negative γ-component
p abc
• 8 Switching State Vectors with non-positive γ-component
• 2×6 vectors with non-zero α & β• 2 “zero vectors” with non-zero γ• 2 zero vectors
β
γ
α
γ
a b
c
axvbxv
cxvdcv
xa
bc
n
p
• 16 Switching State Vectors
pnnn
pnnp
abcx
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
101
DB-201
Three-Dimensional Switching Vectors
pnnn
ppnn
npnn
nppn
nnpn
pnpn
pppn
pnnp
nnpp
npnp pnpp
ppnp
nppp
nnnp
ppppnnnn
α
−β
γ
pnnx
ppnxnpnx
nppx
nnpx pnpx
pppx
nnnx
V23 g
β
α
where x={p,n}
Projection on α−β plane
α
γ
dcvv3
1=γ
dcvv3
2=γ
dcvv3
3=γ
0=γv
dcvv3
1−=γ
dcvv3
2−=γ
dcvv3
3−=γ
dcv32
DB-202
Three-Dimensional SVM Step 1: Prism Identification
Prism II Prism III
Prism IV Prism V Prism VI
Prism I
pnnn
pnnp
ppnn
pppn
nnnp
nnnnpppp
Vref ppnp
ppnn
ppnp
npnn
pppn
nnnp
nnnnpppp
Vref
npnp
npnn
npnp
nppn
pppn
nnnp
nnnnpppp
Vrefnppp
nppn
nppp
nnpn
pppn
nnnp
nnnnpppp
Vref
nnpp
nnpn
nnpp
pnpn
pppn
nnnp
nnnnpppp
Vref pnpp
pnpn
pnpp
pnnn
pppn
nnnp
nnnnpppp
Vref
pnnp
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
102
DB-203
Three-Dimensional SVM Step 2: Tetrahedron Identification
pnnn
pnnpppnp
ppnn
pppn
nnnp
nnnnpppp
Vref
Tetrahedron 1
[+ 0 0]
[0 - -][0 0 -]
pnnn
pnnpppnp
ppnn
pppn
nnnp
nnnnpppp
Vref
Tetrahedron 2
[+ 0 0]
[+ + 0]
[0 0 -]
pnnn
pnnp
ppnp
ppnn
nnnp
nnnnpppp
Vref
pppn
Tetrahedron 4
[0 0 -]
[0 - -] [- - -]
pnnn
pnnpppnp
ppnn
pppn
nnnp
Vref
nnnnpppp
Tetrahedron 3
[+ 0 0]
[+ + 0]
[+ + +]
DB-204
Three-Dimensional SVM:Step 3: Duty-Cycle Calculation
332211 VdVdVdVref
rrrr++=
3210 1 dddd −−−=
1Vr
2Vr
0Vr
3Vr
refVr
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
103
DB-205
Sa
Sb
Sc
Sf
V1V2VZ1 V3
Ts(k) Ts(k+1)
1d2d
(a) Rising-Edge Aligned
3d
Sa
Sb
Sc
Sf
(b) Falling-Edge AlignedTs(k) Ts(k+1)
V3V2 VZ1V1
1d 2d 3d
Sa
Sb
Sc
Sf
(c) Symmetric AlignedTs(k) Ts(k+1)
V3V2 VZ1V1 V1V3 V2
2d1
2d2
2d2
2d1
2d3
2d3
Sa
Sb
Sc
Sf
V2V1 VZ1 V2 V1
zd1d 2d zd1d2d
(d) Alternative SequenceTs(k) Ts(k+1)
V3 V3
3d 3d
zd
zd
zd
VZ1
Three-Dimensional SVM: Step 4: Switching Sequencing (Minimum Loss)
DB-206
Experimental Results with Only Voltage Loops Closed
ILA
ILB
ILC
In
VAG
VBG
VCG
In
Output voltage and neutral currentLoad current and neutral current
Unbalanced Load
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
104
DB-207
A Four-Leg Inverter with Common-Mode Filter Function
• Common-Mode Noise
• A 4-Leg Inverter for Common-Mode NoiseElimination
• A New Modulation and Control Scheme for InverterPower Supplies with Unbalanced/Nonlinear Load
DB-208
A Four-Leg Inverter with Common-Mode Filter Function
4-leg inverter for unbalanced / nonlinear load
In order to have both functions, we need
A new switching modulation strategy
A new control design because of the series capacitor in neutral leg
4-leg inverter for common-mode reduction
If , there is NO common-mode noise!
0≡+++ xcba vvvv
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
105
DB-209
Use Only Switching Vectors with2 p’s and 2 n’s
6 available switching vectors out of 24 vectors
α
β
γ
pnnn
ppnn
npnn
nppn
nnpn
pnpn
pppn
pnnp
nnpp
npnppnpp
ppnp
nppp
nnnp
ppppnnnn
γ =1
γ =2/3
γ =1/3
γ =0
γ =-1/3
γ =-2/3
γ =-1
ref
α
β
pnnp
ppnnnpnp
nppn
nnpp pnpn
ref
npnp4ppnn3pnnp2pnpn1ref VVVVV ⋅+⋅+⋅+⋅= dddd
14321 =+++ dddd
DB-210
Common-Mode Noise Reductionand Trade-Offs
250V/div 25V/div
Conventional 4-leg inverter 4-leg inverter with CM reduction
Common-mode voltage Common-mode voltage
Three-phase inductor current Three-phase inductor current
100A/div 100A/div
• DM ripple is increased because of reduced vector choice• 15% smaller maximum modulation index
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
106
DB-211
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters• Three-phase four-wire (four-phase) converter• Multilevel converters• Parallel converters
PECon2008
DB-212
Major Multilevel Three-Phase Topologies
C1
0
VdSc1
Sc2
Sc3
vb vc
va
Sc’1
Sc’2
Sc’3
Sb1
Sb2
Sb3
Sb’1
Sb’2
Sb’3
Sa1
Sa2
Sa3
Sa’1
Sa’2
Sa’3
C2
C3
C1
0
VdSc1
Sc2
Sc3
vb vc
Cf Cf
va
Cf Cf
Cf
Cf
Cf
Cf
CfSc’1
Sc’2
Sc’3
Sb1
Sb2
Sb3
Sb’1
Sb’2
Sb’3
Sa1
Sa2
Sa3
Sa’1
Sa’2
Sa’3
C2
C3
C2a
vaC1a
C3a
C2b
C1b
C3b
C2c
C1c
C3c
va1
va2
va3
vbvb1
vb2
vb3
vn
vcvc1
vc2
vc3
Nabae, 1980; Choe, 1991; Carpita, 1991Diode clamped Flying capacitor
Meynard, 1992
Cascaded H-bridge
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
107
DB-213
Merits of Multilevel Converter Technologies
Advantages• Easy voltage sharing among devices• Improved spectral performance of output waveforms
• Reduced dv / dt resulting in reduced reflections and damage to insulation• Reduced switching and conduction losses
Disadvantages• Increased number of devices• Increased control complexity• Depending on topology:
– Issues with capacitors balancing – Issues with bi-directional power flow
Vab/Vdc
3-levlel 4-levlel 5-levlel
DB-214
Multilevel Converter Control
Comp. Comp. dqabc SVM Power Stage
& Load
Standard ac current and voltagecontrol in dq0 coordinates
• Average and small-signal models are the same as for two-level VSIor boost rectifier, except for multiple dc voltages
• Closed-loop control of ac quantities in rotating dq coordinates is the same as for two-level VSI or boost rectifier
• Additional controller for voltage balancing on dc-link capacitors• Modulator is significantly more complex
refdqvr refdqir
dqabc
dqabc
abcir
abcvr
DC CapacitorBalancing
dcvr
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
108
DB-215
-2-1
01
2
-2-1
01
2
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
020
vab/Vdcvbc/Vdc
v ca/V
dc
vab
vca
vbc
200 220
022002
202
210
120
021
012
102
201
121010
101212
122011
112001
211100
221110
REFV→
⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡
−−−
⋅=ikkjji
Vv dcijkr
..
..
...
0
1
n-2
n-1
a
b
c
a
b
c
1...0
n-1n-2..
1...0
n-1n-2..
1...0
n-1n-2..+
+
+
+Vdc
Vdc
Vdc
Vdc
• Voltage space-vectors can be represented in a three dimensional line-to-line coordinate system
Space-Vector Representation of Three-Level Converters
• Switching model of n-level converter:Three single-pole n-throw switches
DB-216
Voltage Space-Vectors of Three-Level Converters
200022
120
102
210
012
002 202
220020
021 221110
211100
101212
112001
122011
121010
201
REFM→
0 1 2
1
2
α
β
0=++ cabcab vvv:KVL
REFV→
• All the voltage space-vectors of a three-phase converter are located on a plane
va
vc
vbsa
sb
scvdc
vdcva
vc
vbsa
sb
scvdc
vdc
va
vc
vbsa
sb
scvdc
vdcva
vc
vbsa
sb
scvdc
vdc
• Every small vector has two corresponding switching states—redundancy.
Large Vector LV→
Medium Vector MV→
Small Vector SV→
Small Vector SV→
• Voltage space-vectors have different length: small, medium and large.
1
2
0
1
2
0
1
2
0
1
2
0
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
109
DB-217
SVM for Multilevel Three-Phase Converters?
• Switching states: n3 =125
• Switching vectors: 3n(n-1)+1 = 61
• Triangular areas: 6(n-1)2 = 96
Five-level converter
Complexity?Speed?
How to do SVM? 0 0 1
0 0 2
0 0 3
0 0 4
0 1 0
0 1 1
0 1 2
0 1 3
0 1 4
0 2 0
0 2 1
0 2 2
0 2 3
0 2 4
0 3 0
0 3 1
0 3 2
0 3 3
0 3 4
0 4 0
0 4 1
0 4 2
0 4 3
0 4 4 1 0 0
1 0 1
1 0 2
1 0 3
1 0 4
1 1 0
1 1 2
1 1 3
1 1 4
1 2 0
1 2 1
1 2 2
1 2 3
1 2 4
1 3 0
1 3 1
1 3 2
1 3 3
1 3 4
1 4 0
1 4 1
1 4 2
1 4 3
1 4 42 0 0
2 0 1
2 0 2
2 0 3
2 0 4
2 1 0
2 1 1
2 1 2
2 1 3
2 1 4
2 2 0
2 2 1
2 2 3
2 2 4
2 3 0
2 3 1
2 3 2
2 3 3
2 3 4
2 4 0
2 4 1
2 4 2
2 4 3
2 4 4
3 0 0
3 0 1
3 0 2
3 0 3
3 0 4
3 1 0
3 1 1
3 1 2
3 1 3
3 1 4
3 2 0
3 2 1
3 2 2
3 2 3
3 2 4
3 3 0
3 3 1
3 3 2
3 3 4
3 4 0
3 4 1
3 4 2
3 4 3
3 4 4
4 0 0
4 0 1
4 0 2
4 0 3
4 0 4
4 1 0
4 1 1
4 1 2
4 1 3
4 1 4
4 2 0
4 2 1
4 2 2
4 2 3
4 2 4
4 3 0
4 3 1
4 3 2
4 3 3
4 3 4
4 4 0
4 4 1
4 4 2
4 4 3
0 0 1
0 0 2
0 0 3
0 0 4
0 1 0
0 1 1
0 1 2
0 1 3
0 1 4
0 2 0
0 2 1
0 2 2
0 2 3
0 2 4
0 3 0
0 3 1
0 3 2
0 3 3
0 3 4
0 4 0
0 4 1
0 4 2
0 4 3
0 4 4 1 0 0
1 0 1
1 0 2
1 0 3
1 0 4
1 1 0
1 1 2
1 1 3
1 1 4
1 2 0
1 2 1
1 2 2
1 2 3
1 2 4
1 3 0
1 3 1
1 3 2
1 3 3
1 3 4
1 4 0
1 4 1
1 4 2
1 4 3
1 4 42 0 0
2 0 1
2 0 2
2 0 3
2 0 4
2 1 0
2 1 1
2 1 2
2 1 3
2 1 4
2 2 0
2 2 1
2 2 3
2 2 4
2 3 0
2 3 1
2 3 2
2 3 3
2 3 4
2 4 0
2 4 1
2 4 2
2 4 3
2 4 4
3 0 0
3 0 1
3 0 2
3 0 3
3 0 4
3 1 0
3 1 1
3 1 2
3 1 3
3 1 4
3 2 0
3 2 1
3 2 2
3 2 3
3 2 4
3 3 0
3 3 1
3 3 2
3 3 4
3 4 0
3 4 1
3 4 2
3 4 3
3 4 4
4 0 0
4 0 1
4 0 2
4 0 3
4 0 4
4 1 0
4 1 1
4 1 2
4 1 3
4 1 4
4 2 0
4 2 1
4 2 2
4 2 3
4 2 4
4 3 0
4 3 1
4 3 2
4 3 3
4 3 4
4 4 0
4 4 1
4 4 2
4 4 3
DB-218
Coordinates and Transformation
⎥⎦
⎤⎢⎣
⎡
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
+−
−−
+
−
−
=⎥⎥⎥
⎦
⎤
⎢⎢⎢
⎣
⎡=
refq
refd
dcca
bc
ab
ref vv
Vddd
d
)3
2sin(
)3
2sin(
)3
2cos(
)3
2cos(sincos
321
πθ
πθ
πθ
πθθθ
r
0,-1,1
0,-2,2
-1,0,1
-1,-1,2
-2,2,0
-2,1,1
-2,0,2
1,-1,0
1,-2,1
0,1,-1
-1,2,-1
-1,1,0
2, 0,-2
2,-1,-1
2,-2,0
1,1,-2
1,0,-1
0,2,-2
ab
bc
ca
0 0 1
0 0 2
0 1 0
0 1 1
0 1 2
0 2 0
0 2 1
0 2 2 1 0 0
1 0 1
1 0 2
1 1 0
1 1 2
1 2 0
1 2 1
1 2 2
2 0 0
2 0 1
2 0 2
2 1 0
2 1 1
2 1 2
2 2 0
2 2 1
• Transform
from dq to abc:dcref VV
r
• Choose line-to-line coordinates: ab, bc and caq
d
refVr
dref→
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
110
DB-219
falls in a triangle as follows:
Identify the Vectors and Calculate their Duty Cycles
0,-1,1
0,-2,2
-1,0,1
-1,-1,2
-2,2,0
-2,1,1
-2,0,2
1,-1,0
1,-2,1
0,1,-1
-1,2,-1
-1,1,0
2, 0,-2
2,-1,-1
2,-2,0
1,1,-2
1,0,-1
0,2,-2
ab
bc
ca
dref→
• Calculate the floor and ceiling of the projections: cacaca
bcbcbc
ababab
cdfcdfcdf
≤≤≤≤≤≤
, ,
refcabcab dfffr
then2 If −=++
dref→
cabcab fcf , ,)( bcbc fd −
cabcab cff , ,)( caca fd −
cabcab ffc , ,)( abab fd −
dref→
cabcab ccf , ,)( abab dc −
cabcab fcc , ,)( caca dc −
cabcab cfc , ,)( bcbc dc −
falls in a triangle as follows:refcabcab dfffr
then1 If −=++•
•
DB-220
1. The projections are
2. The floors and ceilings are
An Example
202.1=abd 440.0=bcd 642.1−=cad
1=abf2=abc
0=bcf1=bcc
2−=caf1−=cac
358.01,0,1440.02,1,1202.02,0,2
1,0,1
2,1,1
2,0,2
=−=−
=−=−
=−=−
−
−
−
caca
bcbc
abab
fddfddfdd
3. Since , the nearest three vectors and their duty cycles are:
1−=++ cabcab fff
0,-1,1
0,-2,2
-1,0,1
-1,-1,2
-2,2,0
-2,1,1
-2,0,2
1,-1,0
1,-2,1
0,1,-1
-1,2,-1
-1,1,0
2, 0,-2
2,-1,-1
2,-2,0
1,1,-2
1,0,-1
0,2,-2
ab
bc
ca
dref→
→dref with a length of 1.7 and an angle of 45º from ab axis
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
111
DB-221
p n n
NPINP
CDCSa
Sc
Sb
CDC
iaib
ic
Large vectors do not affect NP balance
Vab= VdcVbc= 0Vca= -Vdc
Ampitude=
Phase =
dcV32
⋅
o0
VabVca
Vbc
p n nn p p
p p nn p n
n n p p n p
Vca= Vdc
Vab= Vdc
Vbc= 0
Capacitor Balancing Problem in Three-LevelNeutral-Point (NP) Clamped Converter
INP = 0
DB-222
Full load current is connected to NPfor the duration of the medium vectorduty cycle
Vab= VdcVbc= - Vdc/2Vca= -Vdc/2
Ampitude = Vdc
Phase = o30-
VabVca
Vbc
o n p
p o n
o p n
n p o
n o p p n o
Vca= Vdc/2
Vab= VdcVbc= Vdc/2
p n o
NPINP
CDCSa
Sc
Sb
CDC
iaibic
Medium vectors affect the charge balance in the neutral point causing the low frequency ripple
INP = ic
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
112
DB-223
p o o
NPINP
CDCSa
Sc
Sb
CDC
iaib
ic
Small vectors come in pairs
Freedom to select either vector in a pair helps balance NP
Vab= Vdc/2Vbc= 0Vca= -Vdc/2
Ampitude =
Phase =
dcV31
⋅
o0
NPINP
CDCSa
Sc
Sb
CDC
iaib
ic
o n n
VabVca
Vbc
p o oo n n
o p on o n
o p pn o o
o o pn n o
p o po n o
p p oo o n
Vca= Vdc
Vab= Vdc
Vbc= 0
Small vectors also affect the capacitor charge balance and their effect can be controlled
INP = ia
INP = ib + ic = – ia
DB-224
Modified SVM algorithm can effectively compensatethe voltage ripple in the neutral point
• Based on the commanded balancing effort, virtual small vector• can be computed (need to use both vectors every cycle)
)evev(v32V j2
caj
bcabγγ +⋅+=
32πγ =
• Compute the location of voltage space vectors in every switching cycle
• Determine the sector location of the current VREF
• Compute the duty cycles
0VSV→
LV→
MV→
→'0SV0SV
→
S0'
S0VS0 Vc)(1VcV→→→
⋅−+⋅= [ ]0,1c ∈
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
113
DB-225
Ripple free area
power factor angle (deg)
mod
ulat
ion
inde
x m
VabVca
Vbc
Va
Vb
Vc
ILOAD
VREFφ
φ
Neutral point can be balanced in every switching cycle only in the part of the operating region
DB-226
Ripple free areadt
I(t)iQ
T
0 a
NP∫=
0
1
2
Vpos
Vneg
0
Vdc
Cdc
Cdc
+
+
iNPNP
Finding the normalized charge ripple for all operating conditions helps design dc-link capacitors
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
114
DB-227
0
2000
4000
6000
8000
10000
12000
14000
1 0.8 0
μFSystem Parameters• 1800 V DC bus• 1% NP ripple• 200 A Peak current
max_DC
maxDC U
IKC2Δ
⋅=⋅ K - normalized amplitude of
NP charge
m=0.9
cos(φ)
With NP controlWithout NP control
As expected, the effect of NP charge control diminishes for the decreasing power factor angle
DB-228
Balancing of clamping capacitor for flying capacitorconverter is load-independent
ica1= ia ica1= - ia
• The inner capacitor handles only switching frequency ripple current• There is no clamping diode reverse recovery problem• The modulation and higher level control remains the same
• Flying capacitor converters with any number of levels can be balanced.• Diode-clamped converters with number of levels above three CANNOT!
Van= Vpn/2
CdcVpn a
Sa1
Sa2
Sa3
Sa4
Cdcia
ica1
n 0
p 2
+ CdcVpn a
Sa1
Sa2
Sa3
Sa4
Cdcia
ica1
n 0
p 2
+
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
115
DB-229
Outline
1. Introduction2. Switching Modeling and PWM3. Average Modeling4. Small-Signal Modeling5. Closed-Loop Control Design
6. More Complex Converters• Three-phase four-wire (four-phase) converter• Multilevel converters• Parallel converters
PECon2008
DB-230
Motivation
Gen
DC Distribution Bus
Filter Motor
DC/AC
AC/DC
Filter
DC/DC
AC/DC
Filter
DC/ACFilter
R
A Distributed Power System • Modular design• N+1 redundancy (reliability)• No limit for current level• Maintainability• Availability• Reduced cost, size and weight• Improved performance
Parallelability
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
116
DB-231
Possible Solutions
• Separate power supplies• Transformer isolation• Inter-phase reactors• Six-leg converter
A Six-Leg Converter
PowerSupply
#2
PowerSupply
#1
Controller
Transformer
Inter-PhaseReactors
DB-232
Issues – ModelingChallenges
• High-order system• Coupling
Converter #1
C2
Converter #2
dV
qV
1dI
2dI
2qI
1qI
13L
23L
13L
23L
113 qILω
113 dILω
223 qILω
223 dILω
11 dd Id 11 qq Id
22 dd Id 22 qq Id
1dcI
2dcI
dcV
1C
dcd Vd 1
dcq Vd 1
dcd Vd 2
dcq Vd 2
RExisting results
• High-order multi-input-multi-output modeling1
• Reduced order modeling2
2Mao, 1994
1Matakas, 1993
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
117
DB-233
Issues – Load Sharing
Challenge
• Load sharing• Voltage regulation• Modular design
Existing results
• Master/Slave1
• Droop2
Converter #1
Converter #2
Control #2
Control #1 Load-sharingcontrol
Input Output
Converter #1
Converter #2
Control #2
Control #1
Input Output
Master/Slave
DroopPoor Voltage Regulation
P
V
GoodLoad
Sharing
PoorLoad
Sharing
Good Voltage Regulation
1Siri, 19922Jamerson, 1994
DB-234
Issues – Zero-Sequence Current
source/load
load/sourceI0
a b
ab
source/load
load/sourceI0
a b c
a b c
source/load
load/sourceI0
a b c n
a b c n
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
118
DB-235
Phase-Leg Averaging
dcVdcIaI
aa Id ⋅dca Vd ⋅0
aVai
aps
ans
av
dcidcv
0
aps
ans
Tad
avdcV
dci
aI
T
t
t
ab
c
DB-236
An Averaged Three-Phase Converter Model
phase-leg model
converter model
cI
dcc Vd ⋅
cV
dcVdcI
0
bI
dcb Vd ⋅
bV
dca Vd ⋅
aVaI
aa Id ⋅ bb Id ⋅ cc Id ⋅
dcVdcIaI
aa Id ⋅dca Vd ⋅0
aV
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
119
DB-237
Extraction of Zero-Sequence Components
Define:
Where:
cbaz dddd ++=
0333 =−+−+− )d(d)d(d)d(d zc
zb
za
0''' =++ cba ddd
3' z
aaddd −=
3' zaa
ddd +=
A zero-sequence component ≡ the sum of all phases components
cba IIII ++=0 e.g.
3' z
bbddd −= 3
' zcc
ddd −=
3' zbb
ddd += 3' zcc
ddd +=
DB-238
The Model with Zero-Sequence Components
• For a single converter, there is a zero-sequence voltagebut no zero-sequence current because
cIcV
dcVdcI
0
bI bVaVaI
3dcz Vd ⋅
dca Vd ⋅'dcb Vd ⋅'
dcc Vd ⋅'
aa Id ⋅'bb Id ⋅'
cc Id ⋅'
3)/II(Id cbaz ++⋅
0≡++ cba III
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
120
DB-239
Zero-Sequence Current in Parallel Converters
2cI2cV
2dcI
2bI 2bV2aV2aI
32 dcz Vd ⋅ 32222 )/II(Id cbaz ++⋅
1aI1aV
dcV
1dcI01bI 1bV
1cV1cI
31 dcz Vd ⋅
dcc Vd ⋅'1dcb Vd ⋅'
1dca Vd ⋅'1
31111 )/II(Id cbaz ++⋅
I0
01I
02I
dcc Vd ⋅'2dcb Vd ⋅'
2dca Vd ⋅'2
1'1 cc Id ⋅1
'1 bb Id ⋅1
'1 aa Id ⋅
2'2 cc Id ⋅2
'2 bb Id ⋅2
'2 aa Id ⋅
DB-240
An Averaged Model of Zero-Sequence Dynamic
222
2'
22
111
1'1
133 a
adca
dcza
adca
dcz IRdt
dILVdVdIRdt
dILVdVd ⋅−−⋅+⋅=⋅−−⋅+⋅
0210
2121 I)R(Rdt
dI)L(L)d(dV zzdc ⋅+++=−⋅
21 LL + 21 RR +
)d(dV zzdc 21 −⋅
In AC side, there are three-loops forming three equations:
Equivalent circuitI0
222
2'2
211
11
'1
133 b
bdcb
dczb
bdcb
dcz IRdt
dILVdVdIRdt
dILVdVd ⋅−−⋅+⋅=⋅−−⋅+⋅
222
2'2
211
11
'1
133 c
cdcc
dczc
cdcc
dcz IRdt
dILVdVdIRdt
dILVdVd ⋅−−⋅+⋅=⋅−−⋅+⋅
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
121
DB-241
Zero-Sequence Duty Cycle
For example:
1d
2dpnn
npn
nnp
ppn
npp
pnp
as
bs
csppp ppn pnn nnnnnn pnn ppn
21d
40d
22d
22d
21d
021
002021
5.125.0)5.0()5.0(
ddddddddddddd cbaz
++=+++++=++=
40d
20d
DB-242
A New Zero-Sequence Control Variable k
pppdk ≡ i.e. the duration of zero-vector ppp
021
002021
32)()(
kdddkdkddkddddddd cbaz
++=+++++=++=
Therefore:
as
bs
csppp ppn pnn nnnnnn pnn ppn
0kd21d
2)1( 0dk−
2)1( 0dk−
22d
22d
21d
• k is not definable in minimum lossSVM schemes
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
122
DB-243
• I0 can be controlled by controlling k1 dynamically
• High control bandwidth can be achieved becauseit is a first-order system
0I21 LL + 21 RR +
)d(dV zzdc 21 −⋅
0I21 LL + 21 RR +
01 503 )d.(kVdc −⋅
01
0221012121
5033232
)d.-(k)dkd(ddkdddd zz
=++−++=−
where k2 is set to 0.5
A New Zero-Sequence Modelwith the New Control Variable k
DB-244
Implementation
• This part is added ontoa regular controller for asingle converter.
LoadSource
va2 vb2vc2
va1vb1
vc1
ia2ib2ic2
ic1ib1ia1
vdc
0
idc2
idc1i0
0
PI k
aps
dd qd
0kdaps
i0
bpscps
ans bps bns cps cns
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
123
DB-245
Simulation Results
Without Zero-sequence control With Zero-sequence control(Two converters with different switching frequencies operate independently)
ia1
ia2
5A/div
i01
i02
5A/div
i01 i02
5A/div
ia1 ia2
5A/div
DB-246
Clock Phase ShiftingCommon-Mode Voltage Reduction
0
2/dcv+2/dcv−
2/dcv+2/dcv−
6/dcv−
1as
1bs
1cs
2as
2bs
2cs
Common-mode voltage
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
124
DB-247
ia1 ia2
ia
20A/div
Clock Phase ShiftingDifferential-Mode Ripple Reduction
DB-248
Thank You
The work and contributions are by many CPES faculty, students, and staff.
Many global industrial and US government sponsors of CPES researchare gratefully acknowledged.
This work was supported in part by
ERC Program of the US National Science Foundation under Award Number ECC-9731677
This trip was partially sponsored by
Joint PELS / IAS / IES Chapter ofIEEE Malaysia Section
IEEE Power Electronics Society(PELS)
PECon2008
Dushan Boroyevich: Modeling and Control of Three-Phase PWM ConvertersTutorial at PECon 2008, Johor Bahru, Malaysia, 30 November 2008
Some References 1. C. T. Chen, Linear system theory and design, 3rd ed., New York, NY: Oxford University Press, 1999.
2. W. J. Rugh, Linear system theory, 2nd ed., Englewood Cliffs, NJ.: Prentice Hall, 1996.
3. John S. Bay, Fundamentals of linear state space systems, New York, NY: McGraw-Hill, 1999. Chapter 2, “Vectors and vector spaces” Chapter 3, “Linear operation on vector spaces”
4. D. G. Holmes and T. A. Lipo, Pulse Width Modulation for Power Converters: Principles and Practice, New York, NY: IEEE Press and John Wiley & Sons, 2003.
5. R. Krishnan, Electric motor drives: modeling, analysis, and control, Upper Saddle River, NJ: Prentice Hall, 2001. Chapter 5, “Polyphase induction machines” Chapter 7, “Frequency controlled induction motor drives”
6. Paul C. Krause, Analysis of electric machinery, Chapter 3, “Reference-frame theory”
7. Vincent Del Toro, Electric power systems, Chapter 8, “Parameters of electric power systems: a description in terms of symmetrical components”
8. Silva Hiti, Modeling and control of three-phase PWM converters, Ph.D. Dissertation, Virginia Tech, 1995.
9. P. T. Krein, J. Bentsman, R. M. Bass, and B. Lesieture, “On the use of averaging for the analysis of power electronic systems,” IEEE Trans. Power Electr. , vol. 5, no. 5, 1990, pp. 182-190.
10. R. D. Middlebrook, and S. Cuk, “ A general unified approach to modeling switching-converter power stages,” IEEE PESC’76, Rec., 1976, pp. 18-34.
11. G. W. Wester, and R. D. Middlebrook, “Low-frequency characterization of switched dc-dc converters,” IEEE PESC’72, Rec., 1972, pp. 9-20.
12. C. T. Rim, D. Y. Hu, and G. H. Cho, “ Transformers as equivalent circuits for switches: General proofs and D-Q transformation-based analyses,” IEEE Trans. Ind. Appl., vol. 26, no. 4, 1990, pp. 777-785.
13. J. M. Noworolski, and S. R. Sanders, “ Generalized in-place circuit averaging,” IEEE PESC’91, Rec., 1991, pp. 445-450.
14. S. R. Sanders, J. M. Noworolski, X. Z. Liu, and G. C. Verghese, “ Generalized averaging method for power conversion circuits,” IEEE Trans. Power Electr., vol. 6, no. 2, 1991, pp. 251-259.
15. K. D. T. Ngo, Topology and analysis in PWM inversion, rectification, and cycloconversion, Ph.D. Dissertation, California Institute of Technology, Pasadena, CA, 1984.
16. K. D. T. Ngo, “Low frequency characterization of PWM converters,” IEEE Trans. Power Electron.,” vol. PE-1, 1986. pp. 223-230.
17. P. Bauer, and J. B. Klaassens, “Dynamic modelling of AC power converters,” IEEE PCC 1993 Rec., pp. 502-507, 1993.
18. S. Hiti, and D. Borojevic, “Small-signal modeling and control of three-phase PWM converters,” 1994 IEEE IAS Annu. Meet., Conf. Rec., 1994.
19. R. H. Park, “Two-reaction theory of synchronous machines I and II,” AIEE Trans., vol. 48, and vol. 52, 1929 and 1933.
20. H. Mao, D. Boroyevich, and F. C. Lee, "Novel reduced-order small-signal model of three-phase PWM rectifier," PESC'96.
21. D. M. Brod, and D. W. Novotny, “Current control of VSI-PWM inverters,” IEEE Trans. Industry Appl., vol. IA-21, 1985, pp. 562-570.
22. Wu, S. B Dewan, and G. R. Slemon, “A PWM Ac-to-DC converter with fixed switching frequency, ” IEEE Trans. Industry Appl., vol. 26, 1990, pp. 880-885.
23. T. G. Habetler, “A space vector-based rectifier regulator for AC/DC/AC converters,” IEEE Trans. Power Electron., vol. 8, 1993, pp. 30-36.
24. R. Wu, S. B Dewan, and G. R. Slemon, “Analysis of an ac-to-dc voltage source converter using PWM with phase and amplitude control,” IEEE Trans. Industry Appl., vol. 27, 1991, pp. 355-363.
25. J. W. Dixon, and B T. Ooi, “Indirect current control of a unity power factor sinusoidal current boost type three-phase rectifier,” IEEE Trans. Industrial Electron., vol. 35, 1988, pp. 508-515.
26. Y. Jiang, F. C. Lee, and D. Borojevic, “Simple high performance three-phase boost rectifiers,” IEEE PESC’94, Rec., vol. 2, 1994, pp. 1158-1164.
27. H. Sugimoto, S. Morimoto, and M. Yano, “A high-performance control method of a voltage-type PWM converter,” IEEE PESC’88, Rec., 1988, pp. 360-368.
28. J. W. Kolar, H. Ertl, K. Edelmoser, F. C. Zach, “Analysis of the control behavior of a bidirectional three-phase PWM rectifier system, ” EPE '91, vol. 2, 1991, pp. 95-100.
29. T. Kawabata, T. Miyashita, and Y. Yamamoto, “Digital control of three-phase PWM inverter with L-C filter,” IEEE PESC’88, Rec., 1988, pp. 634-643.
30. S. Hiti, and D. Borojevic, “Control of front-end three-phase boost rectifier,” IEEE APEC’94, Conf. Proc., vol. 2, 1994, pp.927-933.
31. K. Biswas, M. S. Mahesh, B S. R. Iyengar, “A three-phase GTO AC to DC converter with input displacement factor and output voltage control,” 1987 IEEE IAS Annu. Meet. Conf. Rec., 1987, pp. 685-690.
32. Y. Sato, and T. Kataoka, “State feedback control of current type PWM ac-to-dc converter,” 1991 IEEE IAS Annu. Meet. Conf. Rec., 1991, pp. 840-846.
33. Busse, and J. Holtz, “Multiloop control of a unity power factor fast switching ac to dc converter,” IEEE PESC’82, Rec., 1982, pp. 171-179.
34. S. Hiti, D. Borojevic, F. C. Lee, B. Choi, and S. Lee, “Small-signal modeling of three-phase PWM buck rectifier with input filter,” 1991 VPEC Sem. Proc., 1991, pp. 229-237.
35. O. Ojo, and I. Bhat, “Analysis of three-phase PWM buck rectifier under modulation magnitude and angle control,” 1993 IEEE IAS Annu. Meet. Conf. Rec., 1993, pp. 917-925.
36. S. Hiti, V. Vlatkovic, D. Borojevic, F. C Lee, “A new control algorithm for three-phase PWM buck rectifier with input displacement factor compensation,” IEEE Trans. Power Electron., vol. 9, 1994, pp.173-180.
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