Digital Resonant Current Controllers for Voltage Source...
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Transcript of Digital Resonant Current Controllers for Voltage Source...
Digital Resonant Current Controllers for Voltage Source Converters
Digital Resonant Current Controllers for Voltage
Source Converters
Author: Alejandro Gómez Yepes
Director: Jesús Doval Gandoy
Department of Electronics Technology, University of Vigo,
14 December 2011
Dissertation submitted for the degree ofDoctor of Philosophy at the University of Vigo
�Doctor Europeus� mention
Alejandro Gómez Yepes DTE, University of Vigo 1 of 66
Digital Resonant Current Controllers for Voltage Source Converters
Outline
1 Introduction
2 E�ects of Discretization Methods on the Performance of ResonantControllers
3 High Performance Digital Resonant Current Controllers Implemented withTwo Integrators
4 Analysis and Design of Resonant Current Controllers for Voltage SourceConverters by Means of Nyquist Diagrams and Sensitivity Function
5 Conclusions
Alejandro Gómez Yepes DTE, University of Vigo 2 of 66
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Outline
1 Introduction
Plant Model for Current-Controlled VSCs
Review of Current Controllers for VSCs
Objectives
2 E�ects of Discretization Methods on the Performance of Resonant Controllers
3 High Performance Digital Resonant Current Controllers Implemented with Two
Integrators
4 Analysis and Design of Resonant Current Controllers for Voltage Source Converters
by Means of Nyquist Diagrams and Sensitivity Function
5 Conclusions
Alejandro Gómez Yepes DTE, University of Vigo 3 of 66
Pag.
3-7
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Plant Model for Current-Controlled VSCs
Plant Model for Current-Controlled VSCs I
Model suitable for active �lters, active recti�ers, adjustable speed drives(decoupled back EMF), etc.
vac: grid voltage, back EMF of electric machine, etc.
L �lter
GL(s) =I(s)
V C(s)=
1
s LF +RF
vCv
dcv
ac
iVSC L
FR
F
CONTROL
& PWM
i*
LCL �lter
G CLCL(s) =
I C(s)
V C(s); GG
LCL(s) =I G(s)
V C(s)
* *
vac
iG
VSC LC
vdc
RC L
GR
G
RD
C
iC
or iG
iC
vC
CONTROL
& PWM
vD
Alejandro Gómez Yepes DTE, University of Vigo 4 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Plant Model for Current-Controlled VSCs
Plant Model for Current-Controlled VSCs II
fres
-80
-60
-40
-20
0
20
Magnitude (dB)
100 101
102
103
-270
-180
-90
0
90
Phase (deg)
Frequency (Hz)
LCL filter [ GLCL(s)= IG (s) / VC (s) ]
L filter [ GL(s)= I (s) / VC (s) ]
LCL filter [ GLCL(s)= IC (s) / VC (s) ]G
C
LCL �lters behave as L atfrequencies lower thanapprox. fres:
G CLCL(s) ≈ G G
LCL(s) ≈
≈GL(s) =1
s LF +RF
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Plant Model for Current-Controlled VSCs
Plant Model for Current-Controlled VSCs III
vdc
GC
(s)+-
iiPWMe -sTs G
L(s)+
-
vac
Plant
*
++
vac
-1 vdc
e
Complete block diagram
≡GC (s)+
-ii
GPL(s)* e
Simpli�ed block diagram
GC: current controller
GL: L �lter
GPL: plant model
GPL(s) =
Comp.delay︷ ︸︸ ︷e−sTs
ZOH (PWM)︷ ︸︸ ︷1− e−sTs
s
GL(s)︷ ︸︸ ︷1
s LF +RF
GPL(z) = Z{L−1
[GPL(s)
]}=z−2
RF
1− ρ−1
1− z−1ρ−1where ρ = e RFTs/LF
Alejandro Gómez Yepes DTE, University of Vigo 6 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Plant Model for Current-Controlled VSCs
Plant Model for Current-Controlled VSCs III
vdc
GC
(s)+-
iiPWMe -sTs G
L(s)+
-
vac
Plant
*
++
vac
-1 vdc
e
Complete block diagram
≡GC (s)+
-ii
GPL(s)* e
Simpli�ed block diagram
GC: current controller
GL: L �lter
GPL: plant model
GPL(s) =
Comp.delay︷ ︸︸ ︷e−sTs
ZOH (PWM)︷ ︸︸ ︷1− e−sTs
s
GL(s)︷ ︸︸ ︷1
s LF +RF
GPL(z) = Z{L−1
[GPL(s)
]}=z−2
RF
1− ρ−1
1− z−1ρ−1where ρ = e RFTs/LF
Alejandro Gómez Yepes DTE, University of Vigo 6 of 66
Pag.
7-11
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Hysteresis Control
Q
vac
iLF
RF
+
-
+
-
i*
+
-
S
R
Q
SWITCH
DRIVER
i
emin
emax
e
-vdc /2
+vdc /2
Simplicity
Unconditionedstability
Very fast
response
Good accuracy
Variable switching frequency (resonances, �lters,power losses, ripple...)
Interference among phases
Inability to perform selective control
Dependence on converter topology
Analog comparators
Alejandro Gómez Yepes DTE, University of Vigo 7 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Hysteresis Control
Q
vac
iLF
RF
+
-
+
-
i*
+
-
S
R
Q
SWITCH
DRIVER
i
emin
emax
e
-vdc /2
+vdc /2
Simplicity
Unconditionedstability
Very fast
response
Good accuracy
Variable switching frequency (resonances, �lters,power losses, ripple...)
Interference among phases
Inability to perform selective control
Dependence on converter topology
Analog comparators
Alejandro Gómez Yepes DTE, University of Vigo 7 of 66
Pag.
11-12
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Deadbeat Control
GDB (z)+-
iiGPL(z)
* e
GDB(z)GPL(z)
1 +GDB(z)GPL(z)= z−2 ⇒ GDB(z) =
RF
1− ρ−1
1− z−1 ρ−1
1− z−2
Simplicity
Theoretically,fastest transientamong digitalcontrollers
Sensitiveness to deviations in plant parameters
Need for vac feedforward
Sensitiveness to measurement noise
Need for dead-times compensation
Closed-loop steady-state error
Alejandro Gómez Yepes DTE, University of Vigo 8 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
PI Control in SRF
idq*
+
-+
-
idq
Plant: GL(s)´+
+
kPhidq
GPIh(s) =kPh+kIh /s
1/(LFs+RF)
jhω1LF
1/skIh Conventional PI
controller
+
-+
++
-
idq
Plant: GL(s)´
1/(LFs+RF)
jhω1LFjhω1LF
+
+
kPh
1/skIh
idq*
idq PI Controller withcross-coupling
decoupling (PICCD)
+
-+
+
kh+
++
-
idq
Plant: GL(s)´
1/(LFs+RF)
jhω1LF
idq*
idqjhω1LF
1/sRF
LFComplex-
vector PIcontroller
Alejandro Gómez Yepes DTE, University of Vigo 9 of 66
Pag.
17-23
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Repetitive Controllers
Common characteristics
Tracking of multiple harmonics by a simple scheme
Di�cult frequency adaptation
Same parameters for all peaks
Recursive form
+
+ z-nd+ns outputinput
z -ns
F1(z)
F2(z)
Krep No selectiveness
Steady-state errorDi�cult tuning
DFT-based
+
+GDFT(z) output
input
z -ns
Krep
Selectiveness
No steady-state error
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Proportional+Resonant (PR) Controllers
Equivalent to a conventional PI in positive-sequence SRF + another one innegative-sequence SRF
Conventional PI:
GPIh(s) = kPh+kIhs
Pos.seq.SRF G+
PIh(s) = GPIh (s− jhω1) = kPh
+kIh
s−jhω1
Neg.seq.SRF G−PIh
(s) = GPIh (s+ jhω1) = kPh+
kIhs+jhω1
GPRh(s) = G+
PIh(s)+G−PIh
(s) =
KPh︷ ︸︸ ︷2 kPh
+
KIh︷ ︸︸ ︷2 kIh
s
s2 + h2ω21
= KPh+KIh
R1h(s)︷ ︸︸ ︷s
s2 + h2ω21
Delay compensation: G dPRh
(s) = KPh+KIh
Rd1h
(s)︷ ︸︸ ︷s cos(φ′h)− hω1 sin(φ′h)
s2 + h2ω21
Total controller: GC(s) =
nh∑h
G dPRh
(s) =
KPT︷ ︸︸ ︷nh∑h
KPh+
nh∑h
KIh Rd1h
(s)
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Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Proportional+Resonant (PR) Controllers
Equivalent to a conventional PI in positive-sequence SRF + another one innegative-sequence SRF
Conventional PI:
GPIh(s) = kPh+kIhs
Pos.seq.SRF G+
PIh(s) = GPIh (s− jhω1) = kPh
+kIh
s−jhω1
Neg.seq.SRF G−PIh
(s) = GPIh (s+ jhω1) = kPh+
kIhs+jhω1
GPRh(s) = G+
PIh(s)+G−PIh
(s) =
KPh︷ ︸︸ ︷2 kPh
+
KIh︷ ︸︸ ︷2 kIh
s
s2 + h2ω21
= KPh+KIh
R1h(s)︷ ︸︸ ︷s
s2 + h2ω21
Delay compensation: G dPRh
(s) = KPh+KIh
Rd1h
(s)︷ ︸︸ ︷s cos(φ′h)− hω1 sin(φ′h)
s2 + h2ω21
Total controller: GC(s) =
nh∑h
G dPRh
(s) =
KPT︷ ︸︸ ︷nh∑h
KPh+
nh∑h
KIh Rd1h
(s)
Alejandro Gómez Yepes DTE, University of Vigo 11 of 66
Pag.
24-26
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Proportional+Resonant (PR) Controllers
Equivalent to a conventional PI in positive-sequence SRF + another one innegative-sequence SRF
Conventional PI:
GPIh(s) = kPh+kIhs
Pos.seq.SRF G+
PIh(s) = GPIh (s− jhω1) = kPh
+kIh
s−jhω1
Neg.seq.SRF G−PIh
(s) = GPIh (s+ jhω1) = kPh+
kIhs+jhω1
GPRh(s) = G+
PIh(s)+G−PIh
(s) =
KPh︷ ︸︸ ︷2 kPh
+
KIh︷ ︸︸ ︷2 kIh
s
s2 + h2ω21
= KPh+KIh
R1h(s)︷ ︸︸ ︷s
s2 + h2ω21
Delay compensation: G dPRh
(s) = KPh+KIh
Rd1h
(s)︷ ︸︸ ︷s cos(φ′h)− hω1 sin(φ′h)
s2 + h2ω21
Total controller: GC(s) =
nh∑h
G dPRh
(s) =
KPT︷ ︸︸ ︷nh∑h
KPh+
nh∑h
KIh Rd1h
(s)
Alejandro Gómez Yepes DTE, University of Vigo 11 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Analysis and design of PR Controllers
Open-loop
1
0
10
20
30
40
50
Magnitude (dB)
101 102-180
-135
-90
-45
0
45
Phase (deg)
Frequency (Hz)
KPTKI KI KI3 5
fc
KI = 2000h
KI = 500h
KI = 0h
PMP
f1 3f1 5f1
Closed-loop
0
1
2
3
4
5
67
Magnitude (abs)50 150 250 350 450 550 650 750 850 950
-180
-135
-90
-45
0
45
90
Phase (deg)
Frequency (Hz)
h = 0
2 samples h
KPT → fc, PMP
KIh → bandwidth around hf1
hf1 < fc ∀h such that φ′h = 0
φ′h → anomalous peaks & stability
GC(s) = KPT+
nh∑h
KIh Rd1h
(s)
Alejandro Gómez Yepes DTE, University of Vigo 12 of 66
Pag.
26-27,53
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Vector Proportional+Integral (VPI) Controllers
Equivalent to a complex-vector PI in positive-sequence SRF + another one innegative-sequence SRF
GVPIh(s) =
G+cPIh︷ ︸︸ ︷
GcPIh (s− jhω1) +
G−cPIh︷ ︸︸ ︷GcPIh (s+ jhω1) =
= KPh
R2h︷ ︸︸ ︷s2
s2 + h2ω21
+KIh
R1h︷ ︸︸ ︷s
s2 + h2ω21
= Khs (
Plantcancellation︷ ︸︸ ︷s LF +RF )
s2 + h2ω21
Delay compensation: G dVPIh
(s) = Kh(s LF +RF)
[s cos(φ′h)− hω1 sin(φ′h)
]s2 + h2ω2
1
=
= KPh
Rd2h
(s)︷ ︸︸ ︷s2 cos(φ ′h)− s hω1 sin(φ ′h)
s2 + h2ω21
+ KIh
Rd1h
(s)︷ ︸︸ ︷s cos(φ ′h)− hω1 sin(φ ′h)
s2 + h2ω21
Alejandro Gómez Yepes DTE, University of Vigo 13 of 66
Pag.
26-27,53
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Vector Proportional+Integral (VPI) Controllers
Equivalent to a complex-vector PI in positive-sequence SRF + another one innegative-sequence SRF
GVPIh(s) =
G+cPIh︷ ︸︸ ︷
GcPIh (s− jhω1) +
G−cPIh︷ ︸︸ ︷GcPIh (s+ jhω1) =
= KPh
R2h︷ ︸︸ ︷s2
s2 + h2ω21
+KIh
R1h︷ ︸︸ ︷s
s2 + h2ω21
= Khs (
Plantcancellation︷ ︸︸ ︷s LF +RF )
s2 + h2ω21
Delay compensation: G dVPIh
(s) = Kh(s LF +RF)
[s cos(φ′h)− hω1 sin(φ′h)
]s2 + h2ω2
1
=
= KPh
Rd2h
(s)︷ ︸︸ ︷s2 cos(φ ′h)− s hω1 sin(φ ′h)
s2 + h2ω21
+ KIh
Rd1h
(s)︷ ︸︸ ︷s cos(φ ′h)− hω1 sin(φ ′h)
s2 + h2ω21
Alejandro Gómez Yepes DTE, University of Vigo 13 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Review of Current Controllers for VSCs
Analysis and design of VPI ControllersClosed-loop
0
0.5
1
1.5
2
2.5
50 150 250 350 450 550 650 750 850 950 1050 1150-200
-150
-100
-50
0
50
100
Magnitude (abs)
Phase (deg)
Frequency (Hz)
Kh = 50
Kh = 200Kh
Kh → bandwidth around hf1
φ′h → anomalous peaks & stability (only required at ↑ hf1/fs)
Alejandro Gómez Yepes DTE, University of Vigo 14 of 66
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1
Digital Resonant Current Controllers for Voltage Source Converters
Introduction
Objectives
Main Objectives of this PhD Thesis
To provide an in-depth study and comparison of the e�ects ofdiscretization strategies
To develop optimized discrete-time implementations with a good
tradeo� between
accuracy
resource-consumption & simplicity
To propose an analysis and design methodology for resonant controllersby means of Nyquist diagrams, suitable for cases with more than onecross-over frequency.
Alejandro Gómez Yepes DTE, University of Vigo 15 of 66
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Outline
1 Introduction
2 E�ects of Discretization Methods on the Performance of Resonant Controllers
Digital Implementations of Resonant Controllers
Resonant Poles Displacement
E�ects on Zeros Distribution
E�ects on Delay Compensation
Summary of Optimum Discrete-Time Implementations
Experimental Results
Conclusions
3 High Performance Digital Resonant Current Controllers Implemented with Two
Integrators
4 Analysis and Design of Resonant Current Controllers for Voltage Source Converters
by Means of Nyquist Diagrams and Sensitivity Function
5 Conclusions
Alejandro Gómez Yepes DTE, University of Vigo 16 of 66
Pag.
43-45
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Digital Implementations of Resonant Controllers
Digital Implementations
1 Discretization of continuous transfer function (by Tustin, zero-polematching, etc.)
2 Two discrete integrators
Direct int.: Forward Euler Feedback int.: Backward Euler (f&b)Direct int.: Backward Euler Feedback int.: Backward Euler + z−1 (b&b)Direct int.: Tustin Feedback int.: Tustin (t&t)
PR
1
s
1
s
+
++
-
outputinput KIh
KPh
X2 2
1h ω
VPI
1
sX
1
s
2 2
1h ω
+
++
-
outputinputKIh
KPh
Alejandro Gómez Yepes DTE, University of Vigo 17 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Resonant Poles Displacement
Resonant Poles Displacement (fs = 10 kHz)
0.8 0.85 0.9 0.95 1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
850 Hz
Real Axis
Imaginary Axis
750 Hz
650 Hz
550 Hz
450 Hz
350 Hz
150 Hz
50 Hz
250 Hz
351 Hz
350 Hz
349 Hz
zoom
A (f)B (b)C (t, t&t)D (f&b, b&b)E (zoh, foh,tp, zpm, imp)
Alejandro Gómez Yepes DTE, University of Vigo 18 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Resonant Poles Displacement
Resonant Poles Displacement (fs = 10 kHz)
0.8 0.85 0.9 0.95 1
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
850 Hz
Real Axis
Imaginary Axis
750 Hz
650 Hz
550 Hz
450 Hz
350 Hz
150 Hz
50 Hz
250 Hz
351 Hz
350 Hz
349 Hz
zoom
A (f)B (b)C (t, t&t)D (f&b, b&b)E (zoh, foh,tp, zpm, imp)
Alejandro Gómez Yepes DTE, University of Vigo 18 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Resonant Poles Displacement
Resonant Poles Displacement (variable fs)
Alejandro Gómez Yepes DTE, University of Vigo 19 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (E methods)
-100
-80
-60
-40
-20
0
20
40
60
Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
Phase (deg)
Frequency (Hz)
R1h (s)h
Alejandro Gómez Yepes DTE, University of Vigo 20 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (E methods)
-100
-80
-60
-40
-20
0
20
40
60
Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
Phase (deg)
Frequency (Hz)
zoom
R1h (s)h
impR1h (z)h
Better stability at
high frequencies
Alejandro Gómez Yepes DTE, University of Vigo 20 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (E methods)
-100
-80
-60
-40
-20
0
20
40
60
Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
Phase (deg)
Frequency (Hz)
zoom
R1h (z)=R1h (z)hh
R1h (s)h
impR1h (z)zoh zpmh
6,3º Worse stability at
high frequencies
Better stability at
high frequencies
Alejandro Gómez Yepes DTE, University of Vigo 20 of 66
Pag.
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Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (E methods)
-100
-80
-60
-40
-20
0
20
40
60
Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
Phase (deg)
Frequency (Hz)
zoom
R1h (z)=R1h (z)hh
R1h (z)≈R1h (z)tp foh
hh
R1h (s)h
impR1h (z)
zoh zpmh
6,3º Worse stability at
high frequencies
Better stability at
high frequencies
Same phase
response as R1h (s)h
Alejandro Gómez Yepes DTE, University of Vigo 20 of 66
Pag.
50-51
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (D methods)
-100
-80
-60
-40
-20
0
20
40
60Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
135
Phase (deg)
Frequency (Hz)
R1h (s)h
Alejandro Gómez Yepes DTE, University of Vigo 21 of 66
Pag.
50-51
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (D methods)
-100
-80
-60
-40
-20
0
20
40
60Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
135
Phase (deg)
Frequency (Hz)
zoom
R1h (s)
R1h (z)f&b
h
h
6,4º
Alejandro Gómez Yepes DTE, University of Vigo 21 of 66
Pag.
50-51
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R1h(s) (D methods)
-100
-80
-60
-40
-20
0
20
40
60Magnitude (dB)
101
102
103
-180
-135
-90
-45
0
45
90
135
Phase (deg)
Frequency (Hz)
zoom
R1h (s)
R1h (z)f&b
h
h
R1h (z)h
b&b
6,4º6,4º
Alejandro Gómez Yepes DTE, University of Vigo 21 of 66
Pag.
49-50
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R2h(s) (E methods)
-50
0
50
100
Magnitude (dB)
101
102
103
-45
0
45
90
135
180
Phase (deg)
Frequency (Hz)
R2(s)R2h (s)h
Alejandro Gómez Yepes DTE, University of Vigo 22 of 66
Pag.
49-50
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R2h(s) (E methods)
-50
0
50
100
Magnitude (dB)
101
102
103
-45
0
45
90
135
180
Phase (deg)
Frequency (Hz)
R2zoh(z)
R2imp(z)
R2(s)
zoom
R2h (s)h
R2h (z)imph
Low gain at high
frequencies
Alejandro Gómez Yepes DTE, University of Vigo 22 of 66
Pag.
49-50
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R2h(s) (E methods)
-50
0
50
100
Magnitude (dB)
101
102
103
-45
0
45
90
135
180
Phase (deg)
Frequency (Hz)
R2zoh(z)
R2imp(z)
R2(s)
zoom
R2h (s)h
R2h (z)imph
R2h (z)zoh
h
6,2º
Low gain at high
frequencies
Alejandro Gómez Yepes DTE, University of Vigo 22 of 66
Pag.
49-50
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of R2h(s) (E methods)
-50
0
50
100
Magnitude (dB)
101
102
103
-45
0
45
90
135
180
Phase (deg)
Frequency (Hz)
R2zoh(z)
R2imp(z)
R2(s)
zoom
R2h (s)h
R2h (z)imph
R2h (z)zoh
h
R2h (z)≈R2h (z)≈R2h (z)tp zpm
h
foh
h
6,2º
Low gain at high
frequencies
h
Alejandro Gómez Yepes DTE, University of Vigo 22 of 66
Pag.
50-51
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of GVPIh(s) (D methods)
-60
-40
-20
0
20
40
60
80Magnitude (dB)
101
102
103
0
45
90
135
180
Phase (deg)
Frequency (Hz)
GVPIh(s)h
Alejandro Gómez Yepes DTE, University of Vigo 23 of 66
Pag.
50-51
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of GVPIh(s) (D methods)
-60
-40
-20
0
20
40
60
80Magnitude (dB)
101
102
103
0
45
90
135
180
Phase (deg)
Frequency (Hz)
zoom
f&bGVPIh(z)h
GVPIh(s)h
Alejandro Gómez Yepes DTE, University of Vigo 23 of 66
Pag.
50-51
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Zeros Distribution
E�ects on Zeros of GVPIh(s) (D methods)
-60
-40
-20
0
20
40
60
80Magnitude (dB)
101
102
103
0
45
90
135
180
Phase (deg)
Frequency (Hz)
zoom
f&bGVPIh(z)h
b&bGVPIh(z)h
GVPIh(s)h
Alejandro Gómez Yepes DTE, University of Vigo 23 of 66
Pag.
54-57
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Delay Compensation
E�ects on Delay Compensation (E methods)
Alejandro Gómez Yepes DTE, University of Vigo 24 of 66
Pag.
57-58
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
E�ects on Delay Compensation
E�ects on Delay Compensation (D methods)
Alejandro Gómez Yepes DTE, University of Vigo 25 of 66
Pag.
59-60
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Summary of Optimum Discrete-Time Implementations
Optimum Discrete-Time Implementations
Alejandro Gómez Yepes DTE, University of Vigo 26 of 66
Pag.
60-62
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Experimental Results
Experimental Setup (APF)
Alejandro Gómez Yepes DTE, University of Vigo 27 of 66
Pag.
60-62
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Experimental Results
Experimental Setup (APF)
Alejandro Gómez Yepes DTE, University of Vigo 27 of 66
Pag.
65-67
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Experimental Results
Resonant Poles Displacement I
Alejandro Gómez Yepes DTE, University of Vigo 28 of 66
Pag.
65-67
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Experimental Results
Resonant Poles Displacement II
Alejandro Gómez Yepes DTE, University of Vigo 29 of 66
Pag.
67,69-70
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Experimental Results
Discrete-Time Delay Compensation
Alejandro Gómez Yepes DTE, University of Vigo 30 of 66
Pag.
72,74
Digital Resonant Current Controllers for Voltage Source Converters
E�ects of Discretization Methods on the Performance of Resonant Controllers
Conclusions
Conclusions of Chapter II
An exhaustive study and comparison of discretization techniques applied
to resonant controllers has been presented, in terms of their e�ects on
steady-state error
stability
It has been shown that the choice of discretization strategy is a crucial
aspect for resonant controllers
The optimum discrete-time implementations have been assessed
Alejandro Gómez Yepes DTE, University of Vigo 31 of 66
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Outline
1 Introduction
2 E�ects of Discretization Methods on the Performance of Resonant Controllers
3 High Performance Digital Resonant Current Controllers Implemented with Two
Integrators
Previous Schemes based on Two Integrators
Correction of Poles
Correction of Zeros
Experimental Results
Conclusions
4 Analysis and Design of Resonant Current Controllers for Voltage Source Converters
by Means of Nyquist Diagrams and Sensitivity Function
5 Conclusions
Alejandro Gómez Yepes DTE, University of Vigo 32 of 66
Pag.
75-77
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Previous Schemes based on Two Integrators
Previous Schemes based on Two Integrators
GPRh(s)
1
s
1
s
+
++
-
outputinput KIh
KPh
X2 2
1h ω
GVPIh(s)
1
s
X
1
s
2 2
1h ω
+
++
-
outputinputKIh
KPh
G dPRh
(s)
+
+
-
+
-
X
cos( )h
φ ′
X
X
outputinput
1 sin( )h
hω φ ′
KIh
KPh
2 2
1h ω
s
1
s
1
Simple frequency adaptation
Resonant poles deviation→steady-state error
Leading angle φh deviation→anomalous peaks
instability
Alejandro Gómez Yepes DTE, University of Vigo 33 of 66
Pag.
79-82
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Poles
Correction of Poles I
Accurate resonant poles: 1− 2z−1 cos (hω1Ts)+z−2
Schemes based on 2 int.: 1− 2z−1(1−h2ω21T
2s /2) +z−2
2nd order
Taylor
approx.
Proposed correction:
h2ω21 → Ch = 2
nT/2∑n=1
(−1)n+1h2nω2n1 T 2n−2
s
(2n)!=
nT = 4︷ ︸︸ ︷h2ω2
1−h4ω41T
2s
12+h6ω
61T
4s
360︸ ︷︷ ︸nT = 6
+. . .
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1
2
3
4
5
6
7
8
9
10
hf1/fs
Resonant frequency error (Hz)
nT=8nT=6nT=4nT=2Original
nT = 4: valid formost cases
Alejandro Gómez Yepes DTE, University of Vigo 34 of 66
Pag.
79-82
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Poles
Correction of Poles I
Accurate resonant poles: 1− 2z−1 cos (hω1Ts)+z−2
Schemes based on 2 int.: 1− 2z−1(1−h2ω21T
2s /2) +z−2
2nd order
Taylor
approx.
Proposed correction:
h2ω21 → Ch = 2
nT/2∑n=1
(−1)n+1h2nω2n1 T 2n−2
s
(2n)!=
nT = 4︷ ︸︸ ︷h2ω2
1−h4ω41T
2s
12+h6ω
61T
4s
360︸ ︷︷ ︸nT = 6
+. . .
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50
1
2
3
4
5
6
7
8
9
10
hf1/fs
Resonant frequency error (Hz)
nT=8nT=6nT=4nT=2Original
nT = 4: valid formost cases
Alejandro Gómez Yepes DTE, University of Vigo 34 of 66
Pag.
79-82
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Poles
Correction of Poles II
GPRh(z)
s
11
T
z−
−
+
-
X
outputinput 1
s
11
T z
z
−
−
−
KPh
KIh+
+
2
1ω +
-
4 2
1 s
12
Tω
Resonant poles
correction
h2
h4
Ch
GVPIh(z)
s
11
T
z−
−
+
-
X
outputinput 1
s
11
T z
z
−
−
−
KPh
2
1ω +
-
4 2
1 s
12
Tω
Resonant poles
correction
h2
h4
KIh
+
+
Ch
G dPRh
(z)
s
11
T
z−
−
+
+
+
+
-
X
outputinput 1
s
11
T z
z
−
−
−
KPh
cos( )h
φ ′
X
X
1sin( )
hhω φ ′
KIh
2
1ω +
-
4 2
1 s
12
Tω
Resonant poles
correction
h2
h4
Ch
Alejandro Gómez Yepes DTE, University of Vigo 35 of 66
Pag.
78,82-83
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of Zeros
GdPRh
(s) = KPh+KIh ·
·s cos(φ′h)− hω1 sin(φ′h)
s2 + h2ω21
φ′h: target leading angle
φh: actual leading angle
Ideally:
φh = φ′h == −∠∠∠GPL(e
jhω1Ts)
0
0
20
40
60
80
100
120
700 710 720 730 740 750 760 770 780 790
-90
0
90
180
800
Phase (deg)
Frequency (Hz)
Magnitude (dB)
h= (no delay comp.)
In previous literature:
1 wrong target (2 samples)→ φ′h 6= −∠∠∠GPL(ejhω1Ts)
2 discretization→ inaccuracy: φh 6= φ′h
Alejandro Gómez Yepes DTE, University of Vigo 36 of 66
Pag.
78,82-83
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of Zeros
GdPRh
(s) = KPh+KIh ·
·s cos(φ′h)− hω1 sin(φ′h)
s2 + h2ω21
φ′h: target leading angle
φh: actual leading angle
Ideally:
φh = φ′h == −∠∠∠GPL(e
jhω1Ts)
0
0
20
40
60
80
100
120
700 710 720 730 740 750 760 770 780 790
-90
0
90
180
800
Phase (deg)
Frequency (Hz)
Magnitude (dB)
h= /8 h= /4
h= /4h= /8
h=3 /8
h= (no delay comp.)
h=3 /8
In previous literature:
1 wrong target (2 samples)→ φ′h 6= −∠∠∠GPL(ejhω1Ts)
2 discretization→ inaccuracy: φh 6= φ′h
Alejandro Gómez Yepes DTE, University of Vigo 36 of 66
Pag.
78,82-83
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of Zeros
GdPRh
(s) = KPh+KIh ·
·s cos(φ′h)− hω1 sin(φ′h)
s2 + h2ω21
φ′h: target leading angle
φh: actual leading angle
Ideally:
φh = φ′h == −∠∠∠GPL(e
jhω1Ts)
0
0
20
40
60
80
100
120
700 710 720 730 740 750 760 770 780 790
-90
0
90
180
800
Phase (deg)
Frequency (Hz)
Magnitude (dB)
h= /8 h= /4
h= /4h= /8
h=3 /8
h= (no delay comp.)
h=3 /8
In previous literature:
1 wrong target (2 samples)→ φ′h 6= −∠∠∠GPL(ejhω1Ts)
2 discretization→ inaccuracy: φh 6= φ′h
Alejandro Gómez Yepes DTE, University of Vigo 36 of 66
Pag.
83-85
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression I
Objective:
φ′h ≈∣∣∠GPL(e
jhω1Ts)∣∣
2 samples: inaccurate
Proposed:
φ′h =π
2︸︷︷︸φ′
o
+3
2︸︷︷︸λ
TsGPL(z)
f90 << fs
Alejandro Gómez Yepes DTE, University of Vigo 37 of 66
Pag.
83-85
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression I
Objective:
φ′h ≈∣∣∠GPL(e
jhω1Ts)∣∣
2 samples: inaccurate
Proposed:
φ′h =π
2︸︷︷︸φ′
o
+3
2︸︷︷︸λ
TsGPL(z)
f90 << fs
h
Alejandro Gómez Yepes DTE, University of Vigo 37 of 66
Pag.
83-85
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression I
Objective:
φ′h ≈∣∣∠GPL(e
jhω1Ts)∣∣
2 samples: inaccurate
Proposed:
φ′h =π
2︸︷︷︸φ′
o
+3
2︸︷︷︸λ
TsGPL(z)
f90 << fs
h
≈ •Ts
o≈
h
Alejandro Gómez Yepes DTE, University of Vigo 37 of 66
Pag.
85-87
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression II
1000 1200 1600 1800 2000 2200-180
-90
0
90
180-40
-20020
40
60
80
1400
Phase (deg)
Magnitude (dB)
1400 1420 1440 1460 1480 1500-180
-90
0
90
180-40
0
40
80
Magnitude (dB)
Phase (deg)
Frequency (Hz)
PM
PM0
1
2
3
1000 1200 1400 1600 1800 2000 2200-180
-90
0
90
180
Phase (deg)
Frequency (Hz)
Magnitude (abs)
´Proposed h
2 samples h
´
Greater phase
margin
Open-loop
Closed-loop Less anomalous
peaks
-180
Frequency (Hz)
Alejandro Gómez Yepes DTE, University of Vigo 38 of 66
Pag.
85-87
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression II
1000 1200 1600 1800 2000 2200-180
-90
0
90
180-40
-20020
40
60
80
1400
Phase (deg)
Magnitude (dB)
1400 1420 1440 1460 1480 1500-180
-90
0
90
180-40
0
40
80
Magnitude (dB)
Phase (deg)
Frequency (Hz)
PM
PM0
1
2
3
1000 1200 1400 1600 1800 2000 2200-180
-90
0
90
180
Phase (deg)
Frequency (Hz)
Magnitude (abs)
´Proposed h
2 samples h
´
Greater phase
margin
Open-loop
Closed-loop Less anomalous
peaks
-180
Frequency (Hz)
Alejandro Gómez Yepes DTE, University of Vigo 38 of 66
Pag.
85-87
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression II
1000 1200 1600 1800 2000 2200-180
-90
0
90
180-40
-20020
40
60
80
1400
Phase (deg)
Magnitude (dB)
1400 1420 1440 1460 1480 1500-180
-90
0
90
180-40
0
40
80
Magnitude (dB)
Phase (deg)
Frequency (Hz)
PM
PM0
1
2
3
1000 1200 1400 1600 1800 2000 2200-180
-90
0
90
180
Phase (deg)
Frequency (Hz)
Magnitude (abs)
´Proposed h
2 samples h
´
Greater phase
margin
Open-loop
Closed-loop Less anomalous
peaks
-180
Frequency (Hz)
Alejandro Gómez Yepes DTE, University of Vigo 38 of 66
Pag.
85-87
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Improvement in φ′h Expression II
1000 1200 1600 1800 2000 2200-180
-90
0
90
180-40
-20020
40
60
80
1400
Phase (deg)
Frequency (Hz)
Magnitude (dB)
1400 1420 1440 1460 1480 1500-180
-90
0
90
180-40
0
40
80
Magnitude (dB)
Phase (deg)
Frequency (Hz)
PM
PM0
1
2
3
1000 1200 1400 1600 1800 2000 2200-180
-90
0
90
180
Phase (deg)
Frequency (Hz)
Magnitude (abs)
´Proposed h
2 samples h
´
Greater phase
margin
Open-loop
Closed-loop Less anomalous
peaks
-180
Alejandro Gómez Yepes DTE, University of Vigo 38 of 66
Pag.
87-90
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of φh Error Due to Discretization I1st order Taylor approximation centered at 0
z−1 [ cos (φ′h)−hω1Ts sin (φ′
h) ]− z−2 cos(φ′h)
1storder Taylor
approx. centered at 0
z−1 cos (hω1Ts+φ′h) −z−2 cos(φ′h)
Original delay compensation
s
11
T
z−
−
+
+
+
+
-
X
outputinput 1
s
11
T z
z
−
−
−
KPh
cos( )h
φ ′
X
X
1sin( )
hhω φ ′
KIh
2
1ω +
-
4 2
1 s
12
Tω
h2
h4
Ch
Accurate
s
11
T
z−
−
+
+
+
+
-
X
output
input 1
s
11
T z
z
−
−
−
KPh
+
-
cos( )h
φ ′
X
X
1 scos( )
hh Tω φ ′+
KIh
2
1ω +
+
4 2
1 s
12
Tω
h2
h4
Ch
Ts-1
Optimized
s
11
T
z−
−
+
-
X
1ω∆
X outputinput 1
s
11
T z
z
−
−
−
KPh
+
-
ah
bh
ch
dh
+
+
2
1ω +
+
4 2
1 s
12
Tω
h2
h4
Ch
Ts-1
+
-
+
+
1storder Taylor
approx. centered at 0
1st order Taylor
approximation
centered at hω1n
ah-dh: pre-calculatedconstants
Alejandro Gómez Yepes DTE, University of Vigo 39 of 66
Pag.
87-90
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of φh Error Due to Discretization I1st order Taylor approximation centered at 0
z−1 [ cos (φ′h)−hω1Ts sin (φ′
h) ]− z−2 cos(φ′h)
1storder Taylor
approx. centered at 0
z−1 cos (hω1Ts+φ′h) −z−2 cos(φ′h)
Original delay compensation
s
11
T
z−
−
+
+
+
+
-
X
outputinput 1
s
11
T z
z
−
−
−
KPh
cos( )h
φ ′
X
X
1sin( )
hhω φ ′
KIh
2
1ω +
-
4 2
1 s
12
Tω
h2
h4
Ch
Accurate
s
11
T
z−
−
+
+
+
+
-
X
output
input 1
s
11
T z
z
−
−
−
KPh
+
-
cos( )h
φ ′
X
X
1 scos( )
hh Tω φ ′+
KIh
2
1ω +
-
4 2
1 s
12
Tω
h2
h4
Ch
Ts-1
Optimized
s
11
T
z−
−
+
-
X
1ω∆
X outputinput 1
s
11
T z
z
−
−
−
KPh
+
-
ah
bh
ch
dh
+
+
2
1ω +
+
4 2
1 s
12
Tω
h2
h4
Ch
Ts-1
+
-
+
+
1storder Taylor
approx. centered at 0
1st order Taylor
approximation
centered at hω1n
ah-dh: pre-calculatedconstants
Alejandro Gómez Yepes DTE, University of Vigo 39 of 66
Pag.
87-90
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of φh Error Due to Discretization I1st order Taylor approximation centered at 0
z−1 [ cos (φ′h)−hω1Ts sin (φ′
h) ]− z−2 cos(φ′h)
1storder Taylor
approx. centered at 0
z−1 cos (hω1Ts+φ′h) −z−2 cos(φ′h)
Original delay compensation
s
11
T
z−
−
+
+
+
+
-
X
outputinput 1
s
11
T z
z
−
−
−
KPh
cos( )h
φ ′
X
X
1sin( )
hhω φ ′
KIh
2
1ω +
-
4 2
1 s
12
Tω
h2
h4
Ch
Accurate
s
11
T
z−
−
+
+
+
+
-
X
output
input 1
s
11
T z
z
−
−
−
KPh
+
-
cos( )h
φ ′
X
X
1 scos( )
hh Tω φ ′+
KIh
2
1ω +
-
4 2
1 s
12
Tω
h2
h4
Ch
Ts-1
Optimized
s
11
T
z−
−
+
-
X
1ω∆
X outputinput 1
s
11
T z
z
−
−
−
KPh
+
-
ah
bh
ch
dh
+
+
2
1ω +
-
4 2
1 s
12
Tω
h2
h4
Ch
Ts-1
+
-
+
+
1storder Taylor
approx. centered at 0
1st order Taylor
approximation
centered at hω1n
ah-dh: pre-calculatedconstants
Alejandro Gómez Yepes DTE, University of Vigo 39 of 66
Pag.
90-92
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Correction of Zeros
Correction of φh Error Due to Discretization II
0 10 20 30 40 60 70 80 90 100-60
-50
-40
-30
-20
-10
0
10
50
f1
h
Fundamental frequency (Hz)
Phase lead error (deg)
0 10 20 30 40 50 60 70 80 90 100-70
-60
-50
-40
-30
-20
-10
0
10
20
30
Fundamental frequency (Hz)
Phase lead error (deg)
h
f1h = 45
h = 43
h = 41
zoom
0
48 50 52
0 10 20 30 40 50 60 70 80 90 100-5
-4
-3
-2
-1
0
1
2
Fundamental frequency (Hz)
Phase lead error (deg)
h
f1
A
3
4
5
1st order Taylor approx.
centered at 0
1st order Taylor
approx. centered at
h 1n
0.15
-0.2
f1 = 90 Hzf1 = 25 Hz
Original Accurate
Optimized
Alejandro Gómez Yepes DTE, University of Vigo 40 of 66
Pag.
92-96
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Experimental Results
Correction of Poles I
Alejandro Gómez Yepes DTE, University of Vigo 41 of 66
Pag.
92-96
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Experimental Results
Correction of Poles II
Alejandro Gómez Yepes DTE, University of Vigo 42 of 66
Pag.
97-98
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Experimental Results
Correction of Zeros
Alejandro Gómez Yepes DTE, University of Vigo 43 of 66
Pag.
98-99
Digital Resonant Current Controllers for Voltage Source Converters
High Performance Digital Resonant Current Controllers Implemented with Two Integrators
Conclusions
Conclusions of Chapter III
Enhanced digital implementations of resonant controllers based on two
integrators are contributed, with an optimized trade-o� between accuracy
and simplicity.
Correction of poles: improvement in steady-state error
Correction of zeros: improvement in stability margins
The advantages of the originals are maintained: low computational
burden and easy frequency adaptation
The steady-state error as a function of the order of Taylor seriesapproximation of the poles has been studied. A fourth order issatisfactory for most cases
An expression is proposed for the target leading angle, in order toachieve a better compensation of the plant phase lag
Alejandro Gómez Yepes DTE, University of Vigo 44 of 66
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Outline
1 Introduction
2 E�ects of Discretization Methods on the Performance of Resonant Controllers
3 High Performance Digital Resonant Current Controllers Implemented with Two
Integrators
4 Analysis and Design of Resonant Current Controllers for Voltage Source Converters
by Means of Nyquist Diagrams and Sensitivity Function
Limitations of Previous Approaches
Analysis of Stability Margins by Means of Nyquist Diagrams
Relation Between Closed-Loop Anomalous Peaks and Sensitivity Function
Minimization of Sensitivity Function
Experimental Results
Conclusions
5 Conclusions
Alejandro Gómez Yepes DTE, University of Vigo 45 of 66
Pag.
101
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Limitations of Previous Approaches
Limitations of Previous Approaches
PR controllers
PMP is usually employed as indicator of stability. However, when thereare more 0 dB crossings (e.g., high-power, selective control...), PMP isno longer valid
Usually, phase margins are optimized. However:
the phase margin is a less reliable indicator than the
sensitivity peak 1/ηclosed-loop anomalous peaks are directly related to η,not to phase margin
VPI controllers
No methods have been proposed to measure and optimize proximity to
instability and closed-loop anomalous peaks
Ambiguity in φ′h selection: leading angles of 1 & 2 samples have beenproposed
Alejandro Gómez Yepes DTE, University of Vigo 46 of 66
Pag.
104-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers I
-30 -20 -10 0 10 20 40
-30
-20
-10
0
10
20
30
40
Real Axis
Imaginary Axis
0 rad/s
-40-40 30 -1.5 -1 -0.5 0 0.5 1 1.5
-1.5
-1
-0.5
0
0.5
1
1.5
fs
Real Axis
Imaginary Axis
zoom
c
GPL(z) (only proportional gain)KPT
KPT de�nes ωc, PMP, GMP and ηP
PMh ≤ PMP, GMh ≤ GMP and ηh≤ ηP
Alejandro Gómez Yepes DTE, University of Vigo 47 of 66
Pag.
104-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers IGPL(z) (only proportional gain)KPT
-30 -20 -10 0 10 20 40
-30
-20
-10
0
10
20
30
40
Real Axis
Imag
ina
ry A
xis
0 rad/s
-40-40 30
KP |GPL(z)|T
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
fs
Real Axis
Imagin
ary
Axis
KP |GPL(z)|
c
T
GPL(z)
zoom
== GPL(z)
KPT de�nes ωc, PMP, GMP and ηP
PMh ≤ PMP, GMh ≤ GMP and ηh≤ ηP
Alejandro Gómez Yepes DTE, University of Vigo 47 of 66
Pag.
104-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers I
-30 -20 -10 0 10 20 40
-30
-20
-10
0
10
20
30
40
Real Axis
Imaginary Axis
0 rad/s
-40-40 30
KP |GPL(z)|T
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
fs
Real Axis
Imaginary Axis
KP |GPL(z)|
cPMP
P
GMP
T
GPL(z)
zoom
== GPL(z)
GPL(z) (only proportional gain)KPT
KPT de�nes ωc, PMP, GMP and ηP
PMh ≤ PMP, GMh ≤ GMP and ηh≤ ηP
Alejandro Gómez Yepes DTE, University of Vigo 47 of 66
Pag.
104-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers IGPL(z) (only proportional gain)KP
[KP +KI R1 (z)]GPL(z) (proportional+resonant term)d
h
T
T h
-30 -20 -10 0 10 20 40
-30
-20
-10
0
10
20
30
40
Real Axis
Imaginary Axis
0 rad/s
-40
(∞)-40 30
KP |GPL(z)|T
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
fs
Real Axis
Imaginary Axis
KP |GPL(z)|
(∞)
cPMP
P
GMP
T
GPL(z)
zoom
== GPL(z)
KPT de�nes ωc, PMP, GMP and ηP
PMh ≤ PMP, GMh ≤ GMP and ηh≤ ηP
Alejandro Gómez Yepes DTE, University of Vigo 47 of 66
Pag.
104-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers IGPL(z) (only proportional gain)KP
[KP +KI R1 (z)]GPL(z) (proportional+resonant term)d
h
T
T h
-30 -20 -10 0 10 20 40
-30
-20
-10
0
10
20
30
40
Real Axis
Imaginary Axis
0 rad/s
-40
h 1
(∞)-40 30
KP |GPL(z)|T
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
fs
Real Axis
Imaginary Axis
h= /2+ h+
h 1
KP |GPL(z)|
(∞)
cPMP
P
GMP
T
+ GPL(e )
GPL(z)
zoom
KIh
=jh Ts
= GPL(z)
KPT de�nes ωc, PMP, GMP and ηP
PMh ≤ PMP, GMh ≤ GMP and ηh≤ ηP
Alejandro Gómez Yepes DTE, University of Vigo 47 of 66
Pag.
104-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers IGPL(z) (only proportional gain)KP
[KP +KI R1 (z)]GPL(z) (proportional+resonant term)d
h
T
T h
-30 -20 -10 0 10 20 40
-30
-20
-10
0
10
20
30
40
Real Axis
Imaginary Axis
0 rad/s
-40
h 1
(∞)-40 30
KP |GPL(z)|T
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
fs
Real Axis
Imaginary Axis
h= /2+ h+
h 1
KP |GPL(z)|
(∞)
PMh
h
cPMP
P
GMh
GMP
T
+ GPL(e )
GPL(z)
zoom
KIh
=jh Ts
= GPL(z)
KPT de�nes ωc, PMP, GMP and ηP
PMh ≤ PMP, GMh ≤ GMP and ηh≤ ηP
Alejandro Gómez Yepes DTE, University of Vigo 47 of 66
Pag.
106-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers II
GMh > 0 and PMP > 0, but system is unstable
GMh < 0, but system is stable
φ′h = 0 and hω1 > ωc, but system is stable
-1.5 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Axis
Imaginary Axis
h
(∞)
15
20
25
30
KP
-1
GM <0h
PM >0P
GM >0h
T
Alejandro Gómez Yepes DTE, University of Vigo 48 of 66
Pag.
106-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers II
GMh > 0 and PMP > 0, but system is unstable
GMh < 0, but system is stable
φ′h = 0 and hω1 > ωc, but system is stable
-1.5 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Axis
Imaginary Axis
h
(∞)
15
20
25
30
KP
-1
GM <0h
PM >0P
GM >0h
T
Alejandro Gómez Yepes DTE, University of Vigo 48 of 66
Pag.
106-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers II
GMh > 0 and PMP > 0, but system is unstable
GMh < 0, but system is stable
φ′h = 0 and hω1 > ωc, but system is stable
-1.5 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Axis
Imaginary Axis
h
(∞)
15
20
25
30
KP
-1
GM <0h
PM >0P
GM >0h
T
Alejandro Gómez Yepes DTE, University of Vigo 48 of 66
Pag.
106-107
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers II
GMh > 0 and PMP > 0, but system is unstable
GMh < 0, but system is stable
φ′h = 0 and hω1 > ωc, but system is stable
-1.5 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Axis
Imaginary Axis
h
(∞)
15
20
25
30
KP
-1
GM <0h
PM >0P
GM >0h
T
Alejandro Gómez Yepes DTE, University of Vigo 48 of 66
Pag.
106,108
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers III
ηh provides more information about the actual proximity to
instability than PMh
-1.5 -1 -0.5 0 0.5-1.5
-1
-0.5
0
0.5
Real Axis
Imaginary Axis
PMh=PMh=22.3ºh=0.24
h=0.40
(∞)
h=0.8727 rad´
h=0 rad´
Alejandro Gómez Yepes DTE, University of Vigo 49 of 66
Pag.
106-108
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers IV
φ′h = −∠GPL(ejhω1Ts) is usually pursued, but it is not the optimum
Objective: φ′h such that ηh becomes maximum, i.e., ηh =ηP ∀h
Real Axis
Imaginary Axis
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-0.5
0
0.5
1
1.5
h=0.32
h= P=0.49
PMh=46.7ºPMh=43º
h=0.8874 radh= GPL(e )
jh Ts1
´
(∞)
-1
´ -
Alejandro Gómez Yepes DTE, University of Vigo 50 of 66
Pag.
106-108
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of PR Controllers V
η de�nes:
Proximity to instability
Oscillations during transients (damping)
Maximum steady-state error
max{|S(z)|}= 1/η⇐
{|D(z)| ≤ηS(z) = E(z)/I∗(z) = 1/D(z)
ηP is set by KPT :
KPT = F1 (ηP, Ts, RF, LF )
Alejandro Gómez Yepes DTE, University of Vigo 51 of 66
Pag.
110-112
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Analysis of Stability Margins by Means of Nyquist Diagrams
Nyquist Diagrams of VPI Controllers
|GPL(z)| is cancelledφh = φ′h + arctan(hω1LF/RF)
}Greater stability margins and easier
design than PR
0
-1-1.5 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0.5
1
1.5
Real Axis
Imagin
ary
Axis
GM >>0h
(∞)
h
PM ≈h h
h= /2+ h++ GPL(e )-jh Ts
-0.15 -0.1 -0.05 0.05 0.1 0.15-0.1
0.05
0.1
0.15
50
Kh
100
150
200
Real Axis
Imag
ina
ry A
xis
(∞)-0.05
0
0
zoom
Stable ⇔ γh > 0 ∀h
Alejandro Gómez Yepes DTE, University of Vigo 52 of 66
Pag.
113,115-117
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Relation Between Closed-Loop Anomalous Peaks and Sensitivity Function
Relation in PR Controllers
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-0.5
0
0.5
1
1.5
Imaginary Axis
Real Axis
B
A (∞)
-1
D(z)
D(z)
c
A: hω1
B: D(e jωBTs) = ηh
01
2
3
4
5
67
8
Magnitude (abs)
600 610 620 630 640 650 660 670 680 6900
45
90
135
180
225
Phase (deg)
Frequency (Hz)
B
A
700
S(z)=E(z)/I*(z)
012
3
4
5
678
Magnitude (abs)
600 610 620 630 640 650 660 670 680 690-135
-90
-45
0
45
Phase (deg)
Frequency (Hz)700
B
A
CL(z)=I(z)/I*(z)
Alejandro Gómez Yepes DTE, University of Vigo 53 of 66
Pag.
113,115-117
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Relation Between Closed-Loop Anomalous Peaks and Sensitivity Function
Relation in PR Controllers
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-0.5
0
0.5
1
1.5
Imaginary Axis
Real Axis
h
B
A (∞)
-1
D(z)
D(z)
c
A: hω1
B: D(e jωBTs) = ηh
01
2
3
4
5
67
8
Magnitude (abs)
600 610 620 630 640 650 660 670 680 6900
45
90
135
180
225
Phase (deg)
Frequency (Hz)
B
A
700
|S(e )|j BTs
= 1/ h
S(z)=E(z)/I*(z)
012
3
4
5
678
Magnitude (abs)
600 610 620 630 640 650 660 670 680 690-135
-90
-45
0
45
Phase (deg)
Frequency (Hz)700
B
A
|CL(e )|j BTs
1/ h
CL(z)=I(z)/I*(z)
Alejandro Gómez Yepes DTE, University of Vigo 53 of 66
Pag.
113,115-117
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Relation Between Closed-Loop Anomalous Peaks and Sensitivity Function
Relation in VPI Controllers
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Imaginary Axis
Real AxisA (∞)
B
D(z)
D(z)h
A: hω1
B: D(e jωBTs) = ηh
00.5
1
1.5
2
2.5
3
3.54
Magnitude (abs)
1300 1310 1320 1330 1340 1350 1360 1370 1380 1390-45
0
45
90
135
180
Phase (deg)
Frequency (Hz)1400
B
A
|S(e )|j BTs
= 1/ h
S(z)=E(z)/I*(z)
0
0.5
11.5
2
2.5
3
3.54
1300 1310 1320 1330 1340 1350 1360 1370 1380 1390-180
-135
-90
-45
0
45
Magnitude (abs)
Phase (deg)
Frequency (Hz)1400
B
A
|CL(e )|j BTs
1/ h
CL(z)=I(z)/I*(z)
Alejandro Gómez Yepes DTE, University of Vigo 54 of 66
Pag.
117-118
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Minimization of Sensitivity Function
Minimization of S(z) in PR Controllers
Objective: φ′h such that ηh becomes maximum, i.e., ηh =ηP ∀h
Solution: φ′h = −∠GPL(ejhω1Ts) + ∠D(ejhω1Ts)
-1.5 -1 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0
0.5
1
1.5
Real Axis
Imaginary Axis
GPL(z)KPDesired asymptote at h 1
D(e )jh 1Ts
GPL(e )jh 1Ts
KP GPL(e )jh 1Ts
D(e )jh 1Ts
h= D(e )+jh 1Ts
T
T
Alejandro Gómez Yepes DTE, University of Vigo 55 of 66
Pag.
118-119
Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Minimization of Sensitivity Function
Minimization of S(z) in VPI Controllers
Optimum: φ′h = 32Ts ⇐
{max. ηh ⇒ γh = π
2⇒ φh = −∠GPL(e
jhω1Ts)φh = φ′h + arctan(hω1LF/RF)
0
-1-1.5 -0.5 0 0.5 1 1.5-1.5
-1
-0.5
0.5
1
1.5
Real Axis
Imagin
ary
Axis
h
PM ≈ /2h
h≈ /2
(∞)
≈1
Alejandro Gómez Yepes DTE, University of Vigo 56 of 66
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Experimental Results
Test I
Objective of Test I: prove the sensitivity minimization around hω1 achieved bythe proposed φ′h expressions
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Experimental Results
Test I (PR Controller)
650 Hz 656 Hz
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Experimental Results
Test I (VPI Controller)
1300 Hz 1303 Hz
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Experimental Results
Test II (Transient)
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Experimental Results
Test II (Transient)
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Experimental Results
Test II (Steady-State)
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Digital Resonant Current Controllers for Voltage Source Converters
Analysis and Design of Resonant Current Controllers for VSCs by Nyquist Diagrams and Sensitivity Function
Conclusions
Conclusions of Chapter IV
PR and VPI controllers, including delay compensation, are analyzed mymeans of Nyquist diagrams. The e�ect of each freedom degree on thetrajectories is studied, and their relation with the sensitivity function andits peak value is assessed
Optimization of the sensitivity peak permits to achieve a betterperformance and stability in resonant controllers than optimization of thegain or phase margins
A systematic method, supported by closed-form analytical expressions, is
proposed to optimize
stability
avoidance of closed-loop anomalous peaks
transient response (greater damping of frequencies at which the
trajectory is closer to the critical point)
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Digital Resonant Current Controllers for Voltage Source Converters
Conclusions
Outline
1 Introduction
2 E�ects of Discretization Methods on the Performance of Resonant Controllers
3 High Performance Digital Resonant Current Controllers Implemented with Two
Integrators
4 Analysis and Design of Resonant Current Controllers for Voltage Source Converters
by Means of Nyquist Diagrams and Sensitivity Function
5 Conclusions
Conclusions
Future Research
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Digital Resonant Current Controllers for Voltage Source Converters
Conclusions
Conclusions
Conclusions
An exhaustive study and comparison of discretization techniques applied
to resonant controllers has been presented, in terms of
steady-state error
stability
Implementations based on two integrators, that overcome the issues of
the original ones, have been proposed. They achieve an optimized
tradeo� between
accuracy
simplicity
It is proved that to minimize the sensitivity peak permits a better
performance and stability in resonant controllers than to maximize the
gain or phase margins. A systematic method is proposed to obtain
high stability
reduced closed-loop anomalous peaks
reduced oscillations in transients
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Digital Resonant Current Controllers for Voltage Source Converters
Conclusions
Future Research
Future Research
Optimization of transient response for distributed power generationsystems
Torque ripple minimization
Injection of current harmonics in multi-phase drives for fault-toleranceoperation and increase of average torque
Active compensation of undesired current components caused bydead-time in multi-phase converters
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Digital Resonant Current Controllers for Voltage Source Converters
End
Digital Resonant Current Controllers for Voltage
Source Converters
Author: Alejandro Gómez Yepes
Director: Jesús Doval Gandoy
Department of Electronics Technology, University of Vigo,
14 December 2011
Dissertation submitted for the degree ofDoctor of Philosophy at the University of Vigo
�Doctor Europeus� mention
Alejandro Gómez Yepes DTE, University of Vigo 66 of 66
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