Jun 12, 2018 - Sigma-Delta Based Modulation Method for. Matrix Converters. Simone Orcioni, Giorgio Biagetti, Paolo Crippa, Laura Falaschetti, Claudio.
Sigma-Delta Based Modulation Method for Matrix Converters
Simone Orcioni, Giorgio Biagetti, Paolo Crippa, Laura Falaschetti, Claudio Turchetti June 12, 2018 DII, Dipartimento di Ingegneria dell’Informazione Universit` a Politecnica delle Marche, Italy
Table of contents
Introduction Σ∆ matrix converter modulator Σ∆ modulator
Objective Function Input output filters Simulation Results Conclusions
1
Introduction
Matrix Converters
• Matrix converters are forced commutated converters based on a matrix of bidirectional switches. Their main feature is the absence of a DC link with the function of energy reserve. • Although not so widespread in industrial applications, matrix converters have the following desirable properties: • • • •
simple and compact circuitry arbitrary amplitude and frequency of load voltage sinusoidal input and output current arbitrary input power factor
J. Rodriguez, M. Rivera, J. W. Kolar, and P. W. Wheeler, “A review of control and modulation methods for matrix converters,” IEEE Transactions on Industrial Electronics, vol. 59, no. 1, pp. 58–70, Jan 2012.
2
Matrix converter control schemes
• Direct transfer function approach • introduced low-frequency modulation matrix concept • output is obtained by multiplying this modulation index with the input voltages M. Venturini and A. Alesina, “The generalised transformer: A new bidirectional, sinusoidal waveform frequency converter with continuously adjustable input power factor,” in 1980 IEEE Power Electronics Specialists Conference, June 1980, pp. 242–252. A. Alesina and M. Venturini, “Analysis and design of optimum-amplitude nine-switch direct ac-ac converters,” Power Electronics, IEEE Transactions on, vol. 4, no. 1, pp. 101–112, Jan 1989.
3
Matrix converter control schemes • Space Vector Modulation (SVM)
4
Matrix converter control schemes • Space Vector Modulation (SVM) • first SVM method was based on the concept of fictitious DC link, so aimed to control indirect matrices.
P. D. Ziogas, Y. G. Kang, and V. R. Stefanovic, “Rectifier-inverter frequency changers with suppressed dc link components,” IEEE Transactions on Industry Applications, vol. IA-22, no. 6, pp. 1027–1036, Nov 1986.
4
Matrix converter control schemes • Space Vector Modulation (SVM) • first SVM method was based on the concept of fictitious DC link, so aimed to control indirect matrices. • this approach was extended to control load voltage, input power factor and to improve the modulation performance
L. Huber and D. Borojevic, “Space vector modulator for forced commutated cycloconverters,” in Industry Applications Society Annual Meeting, 1989. Conference Record of the 1989 IEEE, Oct 1989, pp. 871–876 vol.1. ——, “Space vector modulated three-phase to three-phase matrix converter with input power factor correction,” Industry Applications, IEEE Transactions on, vol. 31, no. 6, pp. 1234–1246, Nov 1995. 4
Matrix converter control schemes • Space Vector Modulation (SVM) • first SVM method was based on the concept of fictitious DC link, so aimed to control indirect matrices. • this approach was extended to control load voltage, input power factor and to improve the modulation performance • recent SVM method is not limited to indirect matrix converters and it is based on the instantaneous space vector representation of input and output voltages and currents
D. Casadei, G. Serra, A. Tani, and L. Zarri, “Optimal use of zero vectors for minimizing the output current distortion in matrix converters,” Industrial Electronics, IEEE Transactions on, vol. 56, no. 2, pp. 326–336, Feb 2009.
4
Σ∆ modulation • Σ∆ modulation is based on oversampling and noise shaping.
5
Σ∆ modulation • Σ∆ modulation is based on oversampling and noise shaping. • Oversampling spreads quantization noise in a wider band than the signal band where the information resides (base-band).
Oversampling Pe (f)
f fB=fS1/2
fS2/2
PCM conversion at Nyquist frequency PCM conversion and oversamplinq
5
Σ∆ modulation • Σ∆ modulation is based on oversampling and noise shaping. • Oversampling spreads quantization noise in a wider band than the signal band where the information resides (base-band).
Oversampling Pe (f)
f fB=fS1/2
fS2/2
PCM conversion at Nyquist frequency PCM conversion and oversamplinq
• Noise shaping further reduces the noise in the base-band, increasing it to frequencies higher than the signal band.
Oversampling and noise shaping Pe (f)
f fB=fS/2
fS/2
PCM conversion at Nyquist frequency PCM conversion oversampling and noise-shaping
5
Σ∆ modulation Quantizer
Integrator
Integrator
e[n] x[n]
+
-
u1[n]
v1[n]
+
-1
z
+
-
u2[n]
+
-1
z
v2[n]
+
y[n]
DAC
Second order Σ∆ modulator.
• Y (z) = X (z)z −1 + E (z)(1 − z −1 ), • STF(z) = z −1 and NTF(z) = (1 − z −1 )2 .
6
Σ∆ modulation Quantizer
Integrator
Integrator
e[n] x[n]
+
-
u1[n]
v1[n]
+
+
-
-1
z
u2[n]
-1
z
+
v2[n]
y[n]
+
DAC
Second order Σ∆ modulator.
• Y (z) = X (z)z −1 + E (z)(1 − z −1 ), • STF(z) = z −1 and NTF(z) = (1 − z −1 )2 . 8
7
50
1st order NTF 2nd order NTF 3rd order NTF
0
Magnitude (dB)
Magnitude
6
5
4
-50
-100
1st order NTF 2nd order NTF 3rd order NTF
3 -150
2 -200
1
0 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
-250 0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Normalized Frequency (xð rad/sample)
Normalized Frequency (xð rad/sample)
(a)
(b)
0.9
NTF of Σ∆ modulators from I to III order (a) linear scale (b) dB.
6
Goals
• Use of Σ∆ modulation as a control method of matrix converters. • Presentation of a novel filter topology aimed to be used with Σ∆ modulator.
7
Σ∆ matrix converter modulator
Σ∆ matrix converter modulator Conventional direct matrix converter
S Three phase generator
Input filter
I
O
Output filter
L
GS 3
M 3
V
Load
Three phase current AD converter
Three phase voltage AD converter
C
OS
OS
vabcS
iabcL matrix control
a2
vabcΣ∆
a1 Three phase generator
z −1
vabc des
z −1
QΣ∆
b2
b1
b2
b1
en (v )k en (Q)k
min
z −1
Q = −QO z −1
Quantizer a2
a1
Σ−∆
Scheme of the entire system.
8
Switch Matrix Conventional direct matrix converter
S Three phase generator
I
Input filter
O
Output filter
L
GS 3
M 3
V
Load
Three phase current AD converter
Three phase voltage AD converter
C
OS
OS
vabcS
iabcL matrix control
a2
vabcΣ∆
a1 Three phase generator
z −1
vabc des
z −1
QΣ∆
b2
b1
b2
b1
en (v )k en (Q)k
min
z −1
Q = −QO z −1
Quantizer a2
a1
Σ−∆
Switch Matrix.
9
Switch Matrix
s11
s12
s13
iaO
vaO
s21
s22
s23
ibO
vbO
s31
s32
s33
icO
vcO
iaI vaI
ibI vbI
icI vcI
Direct converter switch matrix: voltage and current naming conventions.
10
Switch Matrix
s11
s12
s13
iaO
s21
s22
s23
ibO
vbO
s31
s32
s33
icO
vcO
iaI vaI
ibI vbI
icI vcI
Direct converter switch matrix: voltage and current naming conventions.
vaO
vacb O (t) = Sk vabc I (t) iabc I (t) = SkT iabc O (t)
with k indexes valid incidence matrix configurations: X
sij = 1, i = 1, . . . , 3 .
j
The number of valid incidence matrices is K = 27.
10
Switch Matrix The three phase power supply voltages: s11
s12
s13
iaO
vaO
s21
s22
s23
ibO
vbO
s31
s32
s33
icO
vcO
iaI vaI
ibI vbI
icI vcI
Direct converter switch matrix: voltage and current naming conventions.
√ 2VS sin(2πfS ) √ vb S (t) = 2VS sin(2πfS − 2π/3) √ vc S (t) = 2VS sin(2πfS + 2π/3) va S (t) =
The desired, star connected, output voltages are √ 2Vdes sin(2πfdes + φ) √ vb des (t) = 2Vdes sin(2πfdes + φ − 2π/3) √ vc des (t) = 2Vdes sin(2πfdes + φ + 2π/3) va des (t) =
where φ is the initial phase difference between input and output voltages, fdes is the desired frequency and and Vdes the RMS voltage.
10
Σ∆ modulator Conventional direct matrix converter
S Three phase generator
I
Input filter
O
Output filter
L
GS 3
M 3
V
Load
Three phase current AD converter
Three phase voltage AD converter
C
OS
OS
vabcS
iabcL matrix control
a2
vabcΣ∆
a1 Three phase generator
z −1
vabc des
z −1
QΣ∆
b2
b1
b2
b1
en (v )k en (Q)k
min
z −1
Q = −QO z −1
Quantizer a2
a1
Σ−∆
Σ∆ modulator.
11
Σ∆ modulator
STF(z) = NTF(z) =
z z 2 − z(2 + b1 + a1 + b2 + a2 ) + 1 + b2 + a2 z2
z 2 − z(2 + b1 + b2 ) + 1 + b2 − z(2 + b1 + a1 + b2 + a2 ) + 1 + b2 + a2
12
Σ∆ modulator z z 2 − z(2 + b1 + a1 + b2 + a2 ) + 1 + b2 + a2
STF(z) = NTF(z) =
z2
z 2 − z(2 + b1 + b2 ) + 1 + b2 − z(2 + b1 + a1 + b2 + a2 ) + 1 + b2 + a2
Magnitude Response (dB) 1
10 0
Imaginary Part
Magnitude (dB)
0.5 -10 -20 -30
2
0
-0.5
-40 -50
-1 0
0.2
0.4
0.6
Normalized Frequency (
0.8
rad/sample)
-1
-0.5
0
0.5
1
Real Part
NTF magnitude, poles and zeros (fZ = 692 Hz) 12
Load Voltage and Input Power Factor Control • Minimization of: • Error on Load Voltage: ek (v )[n + 1] = |vabc des [n + 1] − vabc L [n + 1]| • Error on Power factor: ek (Q)[n + 1] = Qdes [n + 1] − QIk [n + 1]
13
Load Voltage and Input Power Factor Control • Minimization of: • Error on Load Voltage: ek (v )[n + 1] = |vabc des [n + 1] − vabc L [n + 1]| • Error on Power factor: ek (Q)[n + 1] = Qdes [n + 1] − QIk [n + 1]
• After some calculations and approximations we obtained • ek (v )[n + 1] = |v abc des [n + 1] − Sk [n + 1]vabc S [n]| • ek (Q)[n + 1] = 3VS2 2πfS C − √13 (v∆,S [n] · SkT [n + 1]iabc L [n])
where QS = 0 =⇒ QI = −Qfilter and v∆,S = [vbc,S , vca,S , vab,S ]
13
Load Voltage and Input Power Factor Control • Minimization of: • Error on Load Voltage: ek (v )[n + 1] = |vabc des [n + 1] − vabc L [n + 1]| • Error on Power factor: ek (Q)[n + 1] = Qdes [n + 1] − QIk [n + 1]
• After some calculations and approximations we obtained • ek (v )[n + 1] = |v abc des [n + 1] − Sk [n + 1]vabc S [n]| • ek (Q)[n + 1] = 3VS2 2πfS C − √13 (v∆,S [n] · SkT [n + 1]iabc L [n])
where QS = 0 =⇒ QI = −Qfilter and v∆,S = [vbc,S , vca,S , vab,S ]
13
Load Voltage and Input Power Factor Control • Minimization of: • Error on Load Voltage: ek (v )[n + 1] = |vabc des [n + 1] − vabc L [n + 1]| • Error on Power factor: ek (Q)[n + 1] = Qdes [n + 1] − QIk [n + 1]
• After some calculations and approximations we obtained • ek (v )[n + 1] = |v abc des [n + 1] − Sk [n + 1]vabc S [n]| • ek (Q)[n + 1] = 3VS2 2πfS C − √13 (v∆,S [n] · SkT [n + 1]iabc L [n])
where QS = 0 =⇒ QI = −Qfilter and v∆,S = [vbc,S , vca,S , vab,S ] Conventional direct matrix converter
S Three phase generator
Input filter
I
O
Output filter
L
GS 3
M 3
V OS
Three phase voltage AD converter
Load
Three phase current AD converter
C OS
13
Multi-objective optimization
The solution of the multi-objective optimization can be found by normalizing the two errors to unity and choosing the point of the Pareto frontiers (in the space of the two normalized errors) closer to the origin. This can be accomplished by doing n o min ek N (v )2 + ek N (Q)2 k
where ek N (v ) =
ek (v )[n+1] Vdes +VS
and ek N =
ek (Q)[n+1] |Qdes [n+1]|
.
The matrix configuration that minimizes the above functional is chosen in the quantizer of the second order Σ∆.
14
Input output filters Conventional direct matrix converter
S Three phase generator
Input filter
I
O
Output filter
L
GS 3
M 3
V
Load
Three phase current AD converter
Three phase voltage AD converter
C
OS
OS
vabcS
iabcL matrix control
a2
vabcΣ∆
a1 Three phase generator
z −1
vabc des
z −1
QΣ∆
b2
b1
b2
b1
en (v )k en (Q)k
min
z −1
Q = −QO z −1
Quantizer a2
a1
Σ−∆
Input output filters.
15
Input output filters R L
L C
C
Traditional filters: (left) simple LC filter, (right) filter with losses
16
Input output filters R
L
C
L C
Proposed filter with losses only at resonance.
• Losses act only at the resonance frequency, increasing the efficiency of the converter.
16
Input output filters R
L
C
L C
Proposed filter with losses only at resonance.
• Losses act only at the resonance frequency, increasing the efficiency of the converter. • Important for Σ∆ modulation where, unlike the PWM, noise power is present also at frequencies lower than the switching frequency.
16
Input output filters R
L
C
L C
Proposed filter with losses only at resonance.
• Losses act only at the resonance frequency, increasing the efficiency of the converter. • Important for Σ∆ modulation where, unlike the PWM, noise power is present also at frequencies lower than the switching frequency. • Additional reactive components, though must be of the same value as the main ones, do not store a significant amount of power nor are subject to high voltages/currents, so can be much smaller and cheaper. 16
Simulation Results
Results: Source and Load
Source and load simulation results. All x-axes report time in second. From top to bottom: VL [V], IL [A], VS [V], IS [A] (VS = 230 V, fS = 50 Hz, Vdes = 70.7 V, fdes = 150 Hz.) 17
Results: Source power
Instantaneous source power. From top to bottom: Active Power [W], Reactive Power [var], Power Factor. (PF[n] = √ P[n] . PAv S = 1.994 kW, 2 2 P[n] +Q[n]
QAv S = −18.7 var, and PFAv S = 0.997.) 18
Results: Load power
Instantaneous Load Power. All x-axes report time in second. (top) Active Power [W], (bottom) Reactive Power [var]. PAv L = 1.97 kW, and QAv L = 743 var. 19
Results: Input ans output filter resistance currents
Input filter IR [A] vs time [s].
Output filter IR [A] vs time [s].
IR I = 0.11 A, IR O = 0.98 A =⇒ PAv,losses = 7.92 W =⇒ η = 98.8%
corresponding to a 80 Plus Titanium level
20
Results: System Parameters Summary Param.
Description
Value
RMS supply phase voltage supply frequency
230 V
filter inductance wye connected filter capacitance filter resistance cutt-off frequency
4 mH
filter inductance wye connected filter capacitance filter resistance cutt-off frequency
2 mH
Input VS fS
50 Hz
Input Filter L C R fH
26.4 µF 20 Ω 692 Hz
Output Filter L C R fH
13.2 µF 8Ω 1.38 kHz
21
Results: System Parameters Summary System Parameters.
Param.
Description
Value
load inductance load resistance RMS output phase voltage output frequency
2 mH
Load L R Vdes fdes
5Ω 70.7 V 150 Hz
Σ−∆ fs fΣ∆ fZ
sampling frequency Σ∆ clock frequency NTF zeros frequency
9 kHz 100 kHz 692 Hz
22
Conclusions
Conclusions
• A Σ∆ modulation method for the control of matrix converters. • Advantages • with respect to PWM based method, of taking into accounts past errors, with its integrating stages • with respect to an SVM implementation, of requiring a simpler implementation
23
Conclusions
• A Σ∆ modulation method for the control of matrix converters. • Advantages • with respect to PWM based method, of taking into accounts past errors, with its integrating stages • with respect to an SVM implementation, of requiring a simpler implementation
• A novel filter topology is also presented in order to take into account the particular frequency distribution of the Σ∆ quantization error. • Advantages • passive active only at resonance frequency
23
Thanks for your attention!
23
Output Voltage Error
• Objective: vabc O [n + 1] = f (Sk [n + 1], vabc S [n]) and use it in Σ∆ modulator to have vabc O = vabc des (t). • Since vabc L is a low-pass filtered version of vabc O then vabc L (t) = vabc O (t) = vabc des (t) The output current and the input voltage are sampled at low frequency (fS = 9 kHz) while the Σ∆ operates at a higher frequency (fΣ∆ = 100 kHz). • Matrix converter: vabc O [n + 1] = Sk [n + 1]vabc I [n + 1] • Furthermore, vabc I ' vabc S , and vabc S [n + 1] ' vabc S [n] • =⇒ vabc O [n + 1] ' Sk [n + 1]vabc S [n] • =⇒ ek (v )[n + 1] = |vabc des [n + 1] − vabc L [n + 1]| = |vabc des [n + 1] − Sk [n + 1]vabc S [n]| .
24
Matrix Input Reactive Power Error • Error on matrix input reactive power: ek (Q)[n + 1] = |Qdes [n + 1] − Qk [n + 1]| . • QS = 0 =⇒ QI = −Qfilter Qfilter is essentially due to the filter capacitors experimenting the three-phase supply voltage =⇒ • Qdes = 3VS2 2πfS C . • By posing v∆,S = [vbc,S , vca,S , vab,S ] 1 3 1 = √ (v∆,S [n + 1] · SkT [n + 1]iabc O [n + 1]) 3 1 ' √ (v∆,S [n] · SkT [n + 1]iabc L [n]) 3
Qk [n + 1] = √ (v∆,S [n + 1] · iabc,I [n + 1])
(1) (2) (3)
and iabc L ' iabc O 25
Multi-objective optimization
The solution of the multi-objective optimization can be found by normalizing the two errors to unity and choosing the point of the Pareto frontiers (in the space of the two normalized errors) closer to the origin. This can be accomplished by doing n o min ek N (v )2 + ek N (Q)2 k
where ek N (v ) =
ek (v )[n+1] Vdes +VS
and ek N =
ek (Q)[n+1] |Qdes [n+1]|
.
The quantizer of a second order Σ∆ choose the matrix configuration that minimizes the above functional.
26
