Periodic Control of Power Electronic Converters. Periodic Control of Power... · 2017-05-25 ·...
Transcript of Periodic Control of Power Electronic Converters. Periodic Control of Power... · 2017-05-25 ·...
Periodic Control of Power Electronic Converters
Yongheng YANG, Yi TANGAssistant Professors
[email protected], [email protected]
Tutorial at IFEEC 2017 – ECCE AsiaJune 4, 2017 | Kaohsiung Exhibition Center, Kaohsiung
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About the Presenters
Yongheng YANG Assistant Professor at Aalborg University
He received the B.Eng. degree in electrical engineering and automation from Northwestern Polytechnical University, Shaanxi, China, in 2009 and the Ph.D. degree in electrical engineering from Aalborg University, Aalborg, Denmark, in 2014.
He was a postgraduate student with Southeast University, Jiangsu, China, from 2009 to 2011.In 2013, he was a Visiting Scholar at Texas A&M University, College Station, TX, USA. Since 2014, he has been with the Department of Energy Technology, Aalborg University, where currently he is an Assistant Professor. He has published more than 100 technical papers and coauthored a book Periodic Control of Power Electronic Converters (London, UK: IET). His research includes grid integration of renewable energies, power electronic converter design, analysis and control, and reliability in power electronics.
Dr. Yang is a Member of the IEEE Power Electronics Society (PELS) Students and Young Professionals Committee. He served as a Guest Associate Editor of IEEE J. Emerg. Sel. Top. Power Electron. (JESTPE) and a Guest Editor of Applied Sciences. He is an Associate Editor of CPSS Transactions on Power Electronics and Applications.
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About the Presenters
Yi TANG Assistant Professor at Nanyang Technological University
He received the B.Eng. Degree in electrical engineering from Wuhan University, Wuhan, China, in 2007 and the M.Sc. and Ph.D. degrees from the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore, in 2008 and 2011, respectively.
From 2011 to 2013, he was a Senior Application Engineer with Infineon Technologies Asia Pacific, Singapore. From 2013 to 2015, he was a Postdoctoral Research Fellow with Aalborg University, Aalborg, Denmark. Since March 2015, he has been with Nanyang Technological University, Singapore as an Assistant Professor. He is the Cluster Director in advanced power electronics research program at the Energy Research Institute @ NTU (ERI@N).
Dr. Tang serves as an Associate Editor for the IEEE J. Emerg. Sel. Top. Power Electron. (JESTPE). He Received the Infineon Top Inventor Award in 2012.
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About the Tutorial
Introduction (15 mins)
Fundamentals in Periodic Control (45 mins)
Advanced Periodic Control Schemes (30 mins)
Frequency-Adaptive Periodic Control Strategies (30 mins)
Continuing Developments (30 mins)
Summary and Discussions (10 mins)
Periodic Control of Power Electronic Converters
Coffee Break (10 mins)
Coffee Break (10 mins)
Fundamentals in Periodic Control
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Aalborg University
Adapted from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:European_Union_(orthographic_projection).svghttps://upload.wikimedia.org/wikipedia/commons/c/c1/Denmark_regions.svg
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Aalborg University
Adapted from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:European_Union_(orthographic_projection).svghttps://upload.wikimedia.org/wikipedia/commons/c/c1/Denmark_regions.svg
PBL-Aalborg Model (Problem-based learning)
Inaugurated in 197422,000 students
2,300 faculty
Aalborg
Esbjerg Copenhagen
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Aalborg University Campus
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Energy Production | Distribution | Consumption | Control
Power Electronics Centered
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Focuses at E.T.
E.T. Facts40+ Faculty members100+ Ph.D. students30+ RA and post-docs30+ Visiting scholars and
students30+ Technical and
administrative staff2 In-house company
divisions
60%+ of the above manpowerare in power electronicsand its applications
2 in-house company divisions heavily involve in power electronics
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Power Electronics in today’s power systems:
Today’s Power Systems
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Power Electronics Dominated power systems:
Future
Danish Energy Agency, “Overview map of the Danish power infrastructure in 1985 and 2015”. https://ens.dk/sites/ens.dk/files/Statistik/foer_efter_uk.pdf, last accessed Mar. 6, 2017.
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Revisit or Reinvent the way that electrical energy is processed:
Why Power Electronics
Generation…
Consumption…
InterfacesIntegration to electric gridPower transmission, distribution, conversion, control
Power Electronics enable
efficient, reliable, flexible conversion and control of electrical energy
40% Energy Consumption is in electrical energy
60% by 2040
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What is the Power Electronic technology:
Power Electronic Scopes
Refers to efficient control and conversion of electrical power by power semiconductor devices
William E. Newell, “Power Electronics-Emerging from Limbo,” IEEE Trans. Ind. Appl., vol. IA-10, no. 1, pp. 7-11, Jan./Feb.. 1974.
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Side Effect
Power electronic Systems:
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Power electronic conversion brings Harmonics:
Side Effect
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Power electronic conversion brings Harmonics:
Side Effect
Line notching Motor vibration Overheating Triggering resonance Equipment dysfunctional Nuisance tripping …
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π approximation – Liu Hui’s algorithm: ( )
One Harmonic Origin
Wikimedia: https://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithm
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Harmonics are related to Power Converter Topologies:
One Harmonic Origin
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Harmonics are related to Power Converter Topologies:
One Harmonic Origin
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Harmonics are related to Power Converter Topologies:
One Harmonic Origin
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Harmonics are related to Power Converter Topologies:
One Harmonic Origin
n-pulse converters produce dominant nk±1 (k = 0, 1, …) order harmonics due to n-pulse commutation
6k±1
12k±1
24k±1
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Switching Grid Distortions
Harmonics due to Switching and Background Distortions:
What’s More
1
ab g2 2
gab1 1 1
h hg c g gi i
n nh
g gh h
hi v v dt i vL L
v dtL
dt
According to KVL:
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Switching harmonic Injection and Compensation:
A Double-Edged Sword
Pulse Width Modulation (PWM):
1ab pwm dc pwm pwm dc dc
2
nh
hv d v d d v v
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Injection Compensation
Switching harmonic Injection and Compensation:
A Double-Edged Sword
Pulse Width Modulation (PWM):
1 1ab pwm dc pwm dc dc pwm dc pwm
2 2
n nh h
h hv d v d v v d v d
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Switching harmonic Injection and Compensation:
A Double-Edged Sword
Well-designed converter controller (dpwm) can remove certain harmonics
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Feedback control for Zero-Error Tracking:
Zero-Error Feedback Control
1 11 1c p c p
E s R s Y s R s D sG s G s G s G s
If Gc(s) ∞, Y(s) R(s), but system should be stable;
For periodic signals, Gc(s) ∞ only at desired frequencies is necessary.
To achieve zero-error tracking (i.e., E(s) 0):
cG s
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What is a Periodic Signal:
Periodic Signals
https://en.wikibooks.org/wiki/Signals_and_Systems/Periodic_Signals
A signal is a periodic signal if it completes a pattern within a measurable time frame, called a period and repeats that pattern over identical subsequent periods.
“”
Decomposed into its Fourier Series
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Harmonic signal generators (Internal Models):
Internal Models of Periodic Signals
0
2 2
00
cos 0
c c s j
c c s jω
kk u t G s G ss
ksk ωt G s G ss ω
Sinusoidal signal:
DC signal:
It is clear that if the harmonic signal generators (internal models) are included in the controller Gc(s), Gc(s) at the interested harmonic
frequencies. Consequently, Y(s) R(s), i.e., zero-error tracking is achieved.
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Internal Model Principle
In the early 1970s, Francis, Wonham et al. laid the foundation of
regulation theory with the Internal Model Principle which states that perfect asymptotic rejection/tracking of persistent inputs can only be attained by replicating the signal generator in a stable feedback loop.
Wonham summarized the internal model principle: “Every good regulator must incorporate a model of the outside world”.
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IMP Periodic Control
Internal Model Principle based PID control:
Internal model for DC signals
Control accuracy
Stability and dynamics
Stability and dynamics
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IMP Periodic Control
Internal Model Principle based periodic control:
Internal model for periodic signals (Resonant and repetitive control)
Control accuracy
Stability and dynamicsIntroducing Periodic Control for Power Electronic Converters
Questions?
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Periodic signal generator (Internal Models of all harmonics):
Internal Model of Any Periodic Signal
0
0
Impulse Step Harmonics
rc 2210 0 0
1 1 1 2ˆ21
sT
sTn
e sG sT s Te s nω
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Periodic signal generator (Internal Models of all harmonics):
Internal Model of Any Periodic Signal
T0 = 0.02 s
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Development of Conventional Repetitive Control (CRC):
Conventional Repetitive Control
0
0
( ) 1rc
rc rc 2210 0 0
( ) 1 1 1 2 21 ( )
c c
sT Q ssT sT
sTn
k Q s e sG s e k eT s TQ s e s nω
Control gain krc/T0 for all frequencies: identical convergence rate Time lead Tc at all harmonics: increase stability Q(s) is usually a low pass filter: increase stability
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Digital periodic signal generator (Internal Models of all harmonics):
Digital Conventional Repetitive Control
0
0
/
rc /ˆ
1 1
s
s
T TN
N T T
z zG sz z
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Digital Conventional Repetitive Control
Conventional RC Scheme in the discrete-time domain:
rcrc
( )1 ( )
N
fN
k Q z zG z G z
Q z z
ω ±iω0, i = 0, 1, …, N/2, or (N-1)/2 , Grc(z) ∞ Identical gain at all harmonics: krc×2/N
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General “PID” System (digital RC + feedback control):
Plug-in Digital CRC System
Stability Conditions:
1
c p
c p
G z G zH z
G z G z The feedback control system is stable
rc1 1fQ z k G z H z
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Achievable Zero-Phase Compensation:
Plug-in Digital CRC System
Assuming
dB z z B z B zH z
A z A z
If
1d
f
z A z B zG z
B z bwith
2max jωb B e , and 1Q z
Then,
21
0 1jω
f
B eB z B zG z H z
b bZero-Phase Compensation is achieved.
2
rc rc11 1 1f
B zQ z k G z H z k
b Q z
Stability range of the control gain:
rc0 2k
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Linear Phase Compensation Design for the CRC system:
Plug-in Digital CRC System
In practice, it is impossible to obtain an accurate transfer function of H(z),
1
1 fhjθ ωjωf fH
B z B zG z H z z G e e
bwith z ε
0fHθ ω
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Linear Phase Compensation Design for the CRC system:
Plug-in Digital CRC System
To simplify the design, a linear phase-lead compensator Gf(z) is introduced:
pfG z z
rc rc 1
N p
N
Q z zG z k
Q z z
rc
2cos0 H
jω
θ pωk
H e
Linear phase-lead compensator: simplest but effective At all harmonics, identical lead steps: not zero-phase compensation
and reduced stability range of krc
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Linear Phase Compensation – an example:
Plug-in Digital CRC System
If we have a feedback control system 2
0.5 0.4320.487 0.429
zH zz z
with 10 kHzsf
rc2 2 0 1.12 2Hπ πkπ θ pω kπ k
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Internal Model for a specific harmonic of interest:
Internal Model of Selected Harmonics
1 2 2
2 2 2
1 1 1ˆ2
cos
sin 1ˆ2
hh hh
hh
h h
h
hh
sG ss jω s jωs ω
ω j jG ss jω s jω
ω
ω tω
t
s
Any periodic signal can be decomposed into the sum of a set of harmonics (i.e.,
cosines and sines) and its DC component. The internal model of a periodic signal is equivalent to the sum of the internal models of its harmonics and DC component.
Internal models of the selected harmonics approach to infinity at harmonic frequencies ±ωh. Therefore, zero-error tracking of
periodic signals can be achieved at frequencies of ±ωh.
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Development of Resonant Control (RSC):
Resonant Control
2 2
cos sincos h h h
h h h h hh
s θ ω θG s k ω t θ k
s ω
Control gain kh for the harmonic: convergence rate tuning Phase-lead compensation θh: system stability No need for the low pass filter Q(s) as in the repetitive control
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Parallel Resonant Control (MRSC) for multiple harmonics:
Multiple Resonant Control
2 2
cos sin
h h
h h hM h h
h N h N h
s θ ω θG s G s k
s ω
Digital Implementation:
2 1 2 2
1 2
1 1 cos sin 1 2 sin sin2 2
( )1 2 cosh
h sh h s h
M hh N h h s
ω Tz θ ω T z z θG z k
ω z ω T z
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Plug-in MRSC enabling selective harmonic cancellation:
Plug-in Digital MRSC System
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable Roots of 1+GM(z)H(z) = 0 are inside the unit circle
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Periodic Control of CVCF single-phase PWM inverters:
Application Case
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Periodic Control of CVCF single-phase PWM inverters:
Application Case
Parameters Nominal value Unit
DC-link voltage vdc 250 V
Inductor filter Lf 3.3 mH
Capacitor filter Cf 100 µF
Resistive load R 60 Ω
Rectifier inductor Lr 3.3 mH
Rectifier capacitor Cr 1000 µF
Rectifier resistor Rr 60 Ω
Switching frequency 10 kHz
Sampling frequency 10 kHz
Reference voltage vc 155.6sin(100πt) V*
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Periodic Control of CVCF single-phase PWM inverters:
Application Case
*
3
dc dc dc
27.76 4.15 10 28.76c c cv k v k v ku k
v v v
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
with a fundamental-frequency RSC
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
with the repetitive control (i.e., RC)
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
with multiple resonant controllers (i.e., MRSC)
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DFT-based Repetitive Control
DFT-based Band-Pass Filter of selected harmonics:
Discrete Fourier Transform
dh1
2 2cos z aN
Nia dh
i
πF z h i N z Q zN N
N = 100, Na = 0
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DFT-based Repetitive Control
DFT-based Band-Pass Filter of selected harmonics:
Discrete Fourier Transform
dh1
2 2cos z aN
Nia dh
i
πF z h i N z Q zN N
DFT dh1
2 2cosh
h h
NNi
ah N i h N
πF z F z h i N zN N
DFT1
a aN
N Nih D
iF z b i z z Q z z
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DFT-based Repetitive Control
DFT-based Band-Pass Filter of selected harmonics:
Discrete Fourier Transform
DFT1
a aN
N Nih D
iF z b i z z Q z z
A Comb Filter is developed.
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DFT-based Repetitive Control
DFT-based Internal Model of selected harmonics:
Discrete Fourier Transform
DFT1
a aN
N Nih D
iF z b i z z Q z z
DFTDFT
DFT
ˆ1
F zG z
F z
For example, if N = 100, Na = 0, and h = 0, 1, 2, ..., 49 (all pass), then
100 49
100
1 0
2 2cos iDFT
i h
πiF z h z zN N
100
DFT 100ˆ
1zG z
z
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DFT-based Repetitive Control
Control gain 2kF/N for all frequencies: identical convergence rate Phase lead step Na at all harmonics: increase stability
DFT-based Repetitive Control scheme:
rcDFT 11
a
a
NDFT DF FN
DDFT
u z F z Q zG z k k z
e z Q zF z z
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Plug-in DFT-based RC System compatible periodic control:
Plug-in DFT-based Repetitive Control
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable Roots of 1+GDFT(z)H(z) = 0 are inside the unit circle
Design of the plug-in DFT-based RC system is similar to other plug-in periodic control systems.
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Modified DFT-based Repetitive Control
Modified DFT-based Repetitive Control scheme:
Discrete Fourier Transform
dh1
2 2cos z aN
Nih a dh
i
πF z a h i N z Q zN N
DFT dh1
2 2cosh
a
h h
N NNi
h a Dh N h N i
πF z F z a h i N z Q z zN N
rcDFT 11
a
a
NDFT DF FN
DDFT
u z F z Q zG z k k z
e z Q zF z z
Control gain 2kFah/N for the h-order harmonic: tune for proper convergence rate
Phase lead step Na at all harmonics is still identical
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MRSC scheme ≈ DFT-based RC scheme:
Unifying Periodic Control Schemes
Since MRSC GM(z) offers more degrees of freedom in adopting both independent gain and independent phase-lead compensation for each harmonic, when compared with the modified DFT-based RC G’DFT(z).
That’s to say, the modified DFT-based RC is actually a special case of the MRSC. Hence, GM(z) can be roughly approximated by G’DFT(z).
1
a
h
NDM h F
h N D
Q zG z G z k z
Q z
2 hh F
ak k
N
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RSC Scheme ≡ I scheme in the synchronous rotating frame:
Unifying Periodic Control Schemes
Zero-error tracking can be achieved using PI controllers in the stationary reference frame, and also using PR controllers in the synchronous rotating frame.
0( ) ( ) ( )
0
i
dq dq dqi
ksG s G s G s
ks
( ) ( )
2 2 2 2 2 2 2 2 2 2
2 2 2 2 2 2 2 2 2 2
20
( )2
0
αβ αβG s G s
i i i i i
αβi i i i i
k s k ω k s k ω k ss ω s ω s ω s ω s ωG s
k ω k s k ω k s k ss ω s ω s ω s ω s ω
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RSC Scheme ≡ I scheme in the synchronous rotating frame:
Unifying Periodic Control Schemes
Zero-error tracking can be achieved using PI controllers in the stationary reference frame, and also using PR controllers in the synchronous rotating frame.
A PR controller is equivalent to the combination of two PI controllers.
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Comparison
Conventional Repetitive ControlW/O consideration of the harmonic distribution in power converters
Accurate: compensate any known periodic signal
Recursive: compact form, light computation, easy-implementation
Slow: limited gain. It’s impossible to optimize its transient response by tuning gains independently at selected harmonic frequencies.
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Comparison
Multiple Resonant ControlConsidering the harmonic distribution, multiple RSC components with independent gain kh and phase lead compensation θh at each harmonic frequency
Paralleled connection: multiple RSC components can yield high control accuracy. However, too many RSC components will yield heavy parallel computation burden and tuning difficulty in implementation.
Independent gain (and much larger) kh and phase lead compensation θh enable MRSC to optimize its transient response and stability.
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Comparison
DFT-based Repetitive ControlConsidering the harmonic distribution in power converters, multiple selective harmonics with identical or independent gains and identical phase lead step Na
Compatible phase delay compensation: equivalent to linear phase-lead compensation RC scheme.
Dynamic optimization: modified DFT-based RC allows users to optimize its dynamics by tuning coefficients (i.e. gains) at selected harmonics.
Flexible harmonic compensation: a large amount of parallel computation for implementation, which is proportional to the fundamental period N. It may be suitable for high performance fixed-point DSP implementation.
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Summary
Fundamentals in Periodic Control:
CRC, MRSC, and DFT-based RC are the fundamental periodic control schemes.
Compatible stability criteria are achieved for the three plug-in fundamental periodic control systems.
General “PID” control scheme is formed by combing the feedback control and fundamental periodic control.
Optimal periodic control is needed to achieve fast dynamics, high accuracy, good compatibility, and easy-for-implementation.
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Summary
Fundamentals in Periodic Control:
CRC, MRSC, and DFT-based RC are the fundamental periodic control schemes.
Compatible stability criteria are achieved for the three plug-in fundamental periodic control systems.
General “PID” control scheme is formed by combing the feedback control and fundamental periodic control.
Optimal periodic control is needed to achieve fast dynamics, high accuracy, good compatibility, and easy-for-implementation.
Introducing Advanced Periodic Control
Questions?
10 Minutes
Advanced Periodic Control
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Harmonics are Unevenly Distributed in Power Converters:
Unevenly Distributed Harmonics
n-pulse converters produce dominant nk±1 (k = 0, 1, …) order harmonics due to n-pulse commutation
6k±1
12k±1
24k±1
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A comparison of RC and MRSC schemes:
What Inspired It
Repetitive Control
• Recursive form• Internal models of all harmonics
• Identical gain for all harmonics
• Accurate but relatively slow dynamic response
Multiple Resonant Control
• Parallel structure• Only internal models of the
selected harmonics
• Can optimize gains for the selected harmonics
• Fast but heavy parallel computation burden
How to optimize periodic controllers for selective harmonic mitigation for high accuracy, fast dynamics, cost-effective and easy implementation?
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Complex Internal Model of selected harmonics:
Generic Harmonic Generator
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Complex Internal Model of selected harmonics:
Generic Harmonic Generator
0
0
2 / /
m 2 / /
02 2 2 210 0 0 0 0
ˆ1
1 1 22
π s nω j m n
π s nω j m n
k
u s eG se s e
s jmωn nT s jmω T s jmω n k ω
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T0 = 0.02 s 6k+1
Complex Internal Model of selected harmonics:
Generic Harmonic Generator
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Take the advantages of RC and MRSC schemes:
Parallel Structure Repetitive Control
0
0
2 / /1 1
psrc pm m pm 2 / /0 0
ˆ1
π s nω j m nn n
π s nω j m nm m
eG s k G s ke
If kpm = krc/n, then
0
0
1rc
psrc m0
1rc 0
2 2 2 20 10 0 0 0 0
rc rc rc2210 0 0
ˆ
1 1 22
1 1 22 1
n
m
n
m k
sT
sTk
kG s G s
n
k s jmωn nn T s jmω T s jmω n k ω
s ek k G sT s T es nω
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Take the advantages of RC and MRSC schemes:
Parallel Structure Repetitive Control
0
0
2 / /1 1
psrc pm m pm 2 / /0 0
ˆ1
π s nω j m nn n
π s nω j m nm m
eG s k G s ke
In practice, a low-pass or band-pass filter Qm(s) and a phase-lead compensator Gf(s) are adopted,
0
0
1
psrc pm m0
2 / /1
pm 2 / /0
ˆ
1
n
fm
π s nω j m nnm
fπ s nω j m nm
m
G s k G s G s
e Q sk G s
e Q s
Further, let kpm = krc/n and Qm(s) = Q(s),
0 0
00
2 / /1rc
psrc rc2 / /0 11
π s nω j m n sT nn
f fsT nπ s nω j m nm
e Q s e Q skG s G s k G s
n e Q se Q s
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Take the advantages of RC and MRSC schemes:
Parallel Structure Repetitive Control
The parallel structure repetitive control Gpsrc(s) is equivalent to the conventional repetitive control Grc(s) when kpm = krc/n and Qm(s) = Q(s).
0 0
00
2 / /1rc
psrc rc2 / /0 11
π s nω j m n sT nn
f fsT nπ s nω j m nm
e Q s e Q skG s G s k G s
n e Q se Q s
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Take the advantages of RC and MRSC schemes:
Parallel Structure Repetitive Control
2 / /1 1
psrc pm pm 2 / /0 0
ˆ1
j πm n N nn nm
m f fj πm n N nm m m
e z Q zG z k G z G z k G z
e z Q z
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Plug-in Digital PSRC System
Take the advantages of RC and MRSC schemes:
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable Roots of 1+Gpsrc(z)H(z) = 0 are inside the unit circle
1
00 2
n
pmm
k
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Real Internal Model of selected harmonics:
Selective Harmonic Generator
0 0
0 0
0
0 0
2 / / 2 / /
sm m m 2 / / 2 / /
2 /
4 / 2 /
1 1ˆ ˆ ˆ2 2 1 1
cos(2 / ) 12cos(2 / ) 1
π s nω j m n π s nω j m n
π s nω j m n π s nω j m n
πs nω
πs nω πs nω
e eG s G s G se e
πm n ee πm n e
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Real Internal Model of selected harmonics:
Selective Harmonic Generator
Internal models for 6k±1 order harmonics
Internal models for 4k±1 order harmonics
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Real Internal Model of selected harmonics:
Selective Harmonic Generator
T0 = 0.02 s 6k±1
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Using real internal models for Selective Harmonic Control:
Selective Harmonic Control
0 0
0 0
0
0 0
2 / / 2 / /
msm 2 / / 2 / /
2 / 2
m 4 / 2 / 2
2 1 1
cos(2 / )
2cos(2 / )
π s nω j m n π s nω j m n
fπ s nω j m n π s nω j m n
πs nω
fπs nω πs nω
e Q s e Q skG s G s
e Q s e Q s
πm n e Q s Q sk G s
e πm n e Q s Q s
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Using real internal models for Selective Harmonic Control:
Selective Harmonic Control
/ 2
sm m 2 / / 2
cos(2 / )2cos(2 / )
N n
fN n N n
πm n Q z z Q zG z k G z
z πm n Q z z Q z
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Plug-in Digital SHC System
Enabling Fast Dynamics – Selective harmonic control scheme:
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable Roots of 1+Gsm(z)H(z) = 0 are inside the unit circle, i.e.,
|Q2(z)(1-kmGf(z)H(z))|<1
0 2mk
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Plug-in Digital SHC System
Enabling Fast Dynamics – Selective harmonic control scheme:
In practice, a linear phase-lead compensator Gf(z) = zc is adopted.
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Selective Harmonic Control Repetitive Control
Stability range
Equivalent gain
Comparison
A comparison of SHC and RC schemes:
0 2mk rc0 2k
0
12
mnkT 0
1rck
T
If km = krc
At the selected nk±m order harmonics,
Convergence rate of SHC is n/2 times faster than that of RC.
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Odd Order Harmonic RC – an SHC for single-phase converters:
Odd Order Harmonic Repetitive Control
Internal models for 4k±1 order harmonics
At 4k±1 order harmonics, the convergence rate of SHC is 2 times faster than that of RC. This scheme is especially suitable for single-phase (4-pulse) converters.
/2 2
orc or /2 21
N c
N
z Q zG z k
z Q z
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6k±1 Order Harmonic RC – an SHC for three-phase converters:
6k±1 Order Harmonic Repetitive Control
Internal models for 6k±1 order harmonics
At 6k±1 order harmonics, the convergence rate of SHC is 3 times faster than that of RC. This scheme is especially suitable for three-phase (6-pulse) converters.
/6 /3 2
rc rc /6 /3 2
/ 21
N c N c
N N
z Q z z Q zG z k
z Q z z Q z
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/ 2
sm m 2 / / 2
cos(2 / )2cos(2 / )
N n
fN n N n
πm n Q z z Q zG z k G z
z πm n Q z z Q z
Optimized Gain for each selective harmonic control module:
Optimal Harmonic Control
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Optimized Gain for each selective harmonic control module:
Optimal Harmonic Control
/ 2
OHC sm m 2 / / 2
cos(2 / )2cos(2 / )
m m
N n
fN n N nm N m N
πm n Q z z Q zG z G z k G z
z πm n Q z z Q z
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Optimally Weighted Gain leads to fast dynamics:
Plug-in Digital OHC System
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable
and km ≥ 00 2m
mm N
k
Control gain for each selective harmonic control module Gsm(z) can be optimally weighted (e.g., according to the harmonic distribution) fast dynamics
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Dual-Module RC Scheme – an OHC for single-phase converters:
Dual-Module Repetitive Control
Internal models for 4k±1 and 4k±2 order harmonics
/2
orc or /2
( )( )
1 ( )
N co
No
z Q zG z k
z Q z
/2
erc er /2
( )( )
1 ( )
N ce
Ne
z Q zG z k
z Q z
/2 /2
DMRC or er/2 /2
Odd-order Harmonics Even-order Harmonics
1 1
N No e c
N No e
z Q z z Q zG z k k z
z Q z z Q z
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/2 /2
DMRC or er/2 /21 1
N c N co e
N No e
z Q z z Q zG z k k
z Q z z Q z
Dual-Module RC Scheme – an OHC for single-phase converters:
Dual-Module Repetitive Control
Convergence rate of Dual-Module RC is up to 2 times faster than that of RC, a universal PC scheme for single-phase converters
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Advanced Periodic Control of CVCF single-phase PWM inverters:
Application Case
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Advanced Periodic Control of CVCF single-phase PWM inverters:
Application Case
Parameters Nominal value Unit
DC-link voltage vdc 80 V
Inductor filter Lf 20 mH
Capacitor filter Cf 45 µF
Resistive load R 15 Ω
Rectifier inductor Lr 1 mH
Rectifier capacitor Cr 500 µF
Rectifier resistor Rr 22 Ω
Switching frequency 10 kHz
Sampling frequency 10 kHz
Reference voltage vc 50sin(100πt) V*
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Advanced Periodic Control of CVCF single-phase PWM inverters:
Application Case
*
3
dc dc dc
90 8.4 10 90c c cv k v k v ku k
v v v
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
without any advanced periodic control
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
without any advanced periodic control
1r 199
1
100%j
ii
ii
Mh j
M
10.25 0.5 0.25Q z z z
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
with various advanced periodic control
RC, krc = 1.2 ORC, kor = 1.2
DMRC, kor = 0.4, ker = 0.8 DMRC, kor = 0.8, ker = 0.4
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Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
with various advanced periodic control
RC, krc = 1.2 ORC, kor = 1.2
DMRC, kor = 0.4, ker = 0.8 DMRC, kor = 0.8, ker = 0.4
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Periodic Control THD, % Speed, s Even harmonics?
Repetitive control, krc = 1.2 0.8 0.2 YES
Odd-harmonic RC, kor = 1.2 1.2 0.1 NO
Dual-module RC, kor = 0.4, ker = 0.8 0.8 0.32 YES
Dual-module RC, kor = 0.8, ker = 0.4 0.5 0.16 YES
Application Case – Results
State Feedback Control of the CVCF single-phase PWM inverter:
with various advanced periodic control
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Summary
Selective Harmonic Control:
Stability criterion of SHC is compatible to that of RC. When selected harmonics dominate the tracking errors,
SHC can be used to achieve much faster convergence rate than RC
SHC occupies less computation resources than RC Tracking accuracy of SHC is a little less than that of RC
Recursive form for easy-implementation
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Summary
Optimal Harmonic Control:
Take the advantages of the RC and MRSC schemes, and it allows optimizing the control gains,
OHC can achieve high control accuracy due to the removal of selected clusters of harmonics (up to all)
OHC offers fast dynamics due to parallel combination of optimally weighted SHC modules
Cost-effective and easy real-time implementation due to the universal recursive SHC modules
Design is compatible with other periodic control schemes
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Summary
Optimal Harmonic Control:
Take the advantages of the RC and MRSC schemes, and it allows optimizing the control gains,
OHC can achieve high control accuracy due to the removal of selected clusters of harmonics (up to all)
OHC offers fast dynamics due to parallel combination of optimally weighted SHC modules
Cost-effective and easy real-time implementation due to the universal recursive SHC modules
Design is compatible with other periodic control schemes
Introducing Frequency Adaptive Periodic Control
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Frequency Dependency
Periodic Control for grid-connected power converters:
Implemented in low-cost digital control units;
Control in various reference frames (abc, dq, and αβ)
Currents should synchronize with the grid voltages;
Grid frequency is not constant.How will the controllers behave? What are the solutions?
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Frequency Dependency
Periodic Control for grid-connected power converters:
Implemented in low-cost digital control units;
Control in various reference frames (abc, dq, and αβ)
Currents should synchronize with the grid voltages;
Grid frequency is not constant.How will the controllers behave? What are the solutions?
50
Time (1 hour)
50.4
49.6
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Frequency Dependency
Performance of the periodic control is Frequency-Dependent:
Implemented in low-cost digital controllers
ω0 treated as a constant
Fixed sampling frequency fs (also Ts) for simplicity
Grid frequency is time-varying
o i.e., ωpll is not strictly constant
o N = fs/f0 = 2πfs/ω0 will be a fractional
1 2
2 2 2 1 201 2
hi
hs
k z zG z
h T z z
RC 1
Nrc
N
k zG z
z
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Frequency Adaptability
Performance of the periodic control is Frequency-Dependent:
0 0 0ˆ Δ Δ Δg pllω ω ω ω ω ω
Actual grid frequency can be expressed as
Grid frequency changes
Frequency estimator errors
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Frequency Adaptability
Performance of the periodic control is Frequency-Dependent:
Actual grid frequency can be expressed as
00 2 2 2
00 0
ˆ 1ˆˆ
h hh
jk hω k δG jhωhω δ δhω hω
0 0
0 0
0
ˆ2 /0 rc
RC 0 rc ˆ2 /0 ˆ 1
ˆˆ
ˆ1 2 2cos 2
π jhω ω
π jhω ω
Q jhω
Q jhω e kG jhω k
Q jhω e πhδ
Frequency sensitivity of RSC and RC schemes can be obtained
0 0 0ˆ Δ Δ Δg pllω ω ω ω ω ω
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Frequency Adaptability
Performance of the periodic control is Frequency-Dependent:
It calls for
Frequency Adaptive Periodic Control
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Frequency Adaptive Resonant Control
Directly Feeding the Frequency to the resonant control:
rsc2 2ˆah h
h
u s sG s ke s s ω
Considering phase compensation, the frequency adaptive multiple resonant control is obtained
2 2
ˆcos sinˆ
h h
h h haM ah h
h N h N h
s θ ω θG s G s ks ω
Frequency adaptive resonant control
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Frequency Adaptive Repetitive Control
Impossible to implement z –N if N is fractional:
arc rc 1
N
fN
Q z zG z k G z
Q z z
Frequency adaptive repetitive control
N = T0/Ts
Solution 1 – Variable Sampling Rate (VSR)Ensuring N is always a constant integer if frequency changes. VSR approach enables RC to compensate harmonics due to frequency variations.
Increased the real-time implementation complexity, such as online controller redesign.
Cannot deal with multiple signals with coprime frequencies simultaneously (It is impossible to ensure all Ni to be integers).
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Solution 2 – Fractional Delay (FD) at fixed sampling rateFixed sampling rate significantly simplifies the design of the frequency adaptive RC scheme. Fractional delay (FD) filters can approximate the real delay.
Simple in real-time implementation – minor software modifications.
Tolerate large frequency variations (good portability).
Can deal with multiple signals with coprime frequencies simultaneously (It is possible to approximate all Ni at the same time).
Frequency Adaptive Repetitive Control
Impossible to implement z –N if N is fractional:
arc rc 1
N
fN
Q z zG z k G z
Q z z
Frequency adaptive repetitive control
N = T0/Ts
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Frequency Adaptive Repetitive Control
Lagrange Interpolation fractional delay filter:
i iN F NN Fz z z z
Integer part Easy to implement
Fractional part Polynomial approximation
0
nF k
kk
z A z
with0
n
kii k
F iAk i
0
in
NN kk
kz z A z
Integer part Easy to implement
Approximated partEasy to implement
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Frequency Adaptive Repetitive Control
Lagrange Interpolation fractional delay filter:
FD filter with n = 3 gives an excellent approximation of z-F within bandwidth of 75% the Nyquist frequency; while 50% the Nyquist frequency, if n = 1.
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Frequency Adaptive Repetitive Control
Implement frequency adaptive RC using the fractional delay:
0
arc rc
01
i
i
nN k
kk
fnN k
kk
Q z z A zG z k G z
Q z z A z
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Plug-in Digital FA-RC System
Frequency Adaptive RC (FA-RC ) system:
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable Roots of 1+Garc(z)H(z) = 0 are inside the unit circle
rc rc
0
11 0 2f nk
kk
k G z H z kQ z A z
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Frequency Adaptive Periodic Control
Implement FA-PSRC scheme using the fractional delay:
2 /1
0apsrc pm
2 /0
01
i
i
Lj πm n N k
k mnk
fLj πm n N km
k mk
e z A z Q zG z k G z
e z A z Q z
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Frequency Adaptive Periodic Control
Implement FA-SHC scheme using the fractional delay:
22
0 0asm m 2
2
0 0
cos(2 / )
1 2cos(2 / )
i i
i i
L LN Nk k
k kk k
fL LN Nk k
k kk k
πm n Q z z A z Q z z A zG z k G z
πm n Q z z A z Q z z A z
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Frequency Adaptive DFT Repetitive Control
Virtual Variable Sampling Rate Unit Delay:
DFT 1 a
DFTF N
DFT
F zG z k
F z z
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Frequency Adaptive DFT Repetitive Control
Virtual Variable Sampling Rate Unit Delay:
0
1 1s sF N N
f fN N F N F F
f f
(1 )1 NFvz z
0 1(1 )1
1 2
1 1 0
0 11N
N NF Nv
NN N
F z F z Fz z
FF z F z
With the linear Lagrange interpolation method
aDFT1
aN
Nih v v
iF z b i z z
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Frequency Adaptive DFT Repetitive Control
aDFT
aDFTaDFT1 a
FN
v
k F zG z
F z z
Frequency Adaptive DFT RC using virtual unit delay zv :-1
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Fractional-Order Phase-Lead Compensator
Fractional-Order Linear Phase Compensation:
Linear phase-lead compensation
Φ 360s
fcf
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Fractional-Order Phase-Lead Compensator
Fractional-Order Linear Phase Compensation:
If ɸ is not an integer (i.e., fractional phase compensation is required) or due to
frequency variations, the phase lead compensation zc is not accurate, c can
be a fractional number. This can not be implemented in a fixed sampling rate system.
Φ 360s
fcf
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Fractional-Order Phase-Lead Compensator
Fractional-Order Linear Phase Compensation:
11i i in F n ncfG z z z F z Fz
Alternatively,
rc
2cos0 H
jω
θ cωk
H e
That is, a fractional c yields flexible phase lead compensation (θH + cω) and larger stability range for krc.
If ɸ is not an integer (i.e., fractional phase compensation is required) or due to
frequency variations, the phase lead compensation zc is not accurate, c can
be a fractional number. This can not be implemented in a fixed sampling rate system.
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Frequency Adaptive Periodic Control of power converters:
Application Case
Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 2017130/171
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Frequency Adaptive Periodic Control of power converters:
Application Case
Parameters Nominal value Unit
Grid voltage (RMS) Vgn 220 V
Grid frequency f0 50 Hz
Current reference amplitude Ig 5 A
Transformer leakage inductance Lg 2 mH
LCL filter inductor L1 and L2 3.6 mH
LCL filter capacitor Cf 2.35 µF
DC bus voltage vdc 400 V
Switching frequency 10 kHz
Sampling frequency 10 kHz
Repetitive control gain krc 1.8 -
*
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Frequency Adaptive Periodic Control of power converters:
Application Case
* *1 1 2
1inv g g g
dc
v k v k b i k b b i kv k
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Application Case – Results
Deadbeat Control of the grid-connected single-phase converter:
without any periodic control
Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 2017133/171
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Application Case – Results
1r 199
1
100%j
ii
ii
Mh j
M
10.1 0.8 0.1Q z z z
OHC sm 40 41 42( ) ( ) ( ) ( ) ( )mm N
G z G z G z G z G z
0 1 20.2, 1.4, 0.2k k k
Deadbeat Control of the grid-connected single-phase converter:
without any periodic control
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Application Case – Results
Deadbeat Control of the grid-connected single-phase converter:
with various periodic control
CRC
OHC
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Application Case – Results
Deadbeat Control of the grid-connected single-phase converter:
with various periodic control
CRC FA-CRC
CRC FA-CRC
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Application Case – Results
Deadbeat Control of the grid-connected single-phase converter:
with various periodic control
Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 2017137/171
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Application Case – Results
Deadbeat Control of the grid-connected single-phase converter:
with various periodic control
FA-CRC FA-OHC
Convergence rate of FA-OHC is up to n/2 times faster than that of FA-RC.
This is not affected by the frequency adaptive scheme.
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Summary
Frequency Adaptive Periodic Control:Lagrange interpolation FIR FD filter based Frequency Adaptive Periodic Control (FAPC) at a fixed sampling rate,
FIR FD filter is always stable It achieves fast on-line tuning of the fractional delay and fast
update of the coefficients
It offers a simple but very accurate real-time frequency adaptive control solution
Design of the FAPC is compatible with non-frequency-adaptive PC systems
Questions?
10 Minutes
Further Exploration
Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 2017142/171
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Beyond periodic signal control:
Periodic Signal Processing
We are not just controlling periodic signals
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Digital Multi-Period RC system:
Multi-Period Signal Control
2
1 , 1 1
pp p
R j j k jj j k j
j k
G z R z R z R z H z R z H z
1
j
j
N
j j fN
Q z zR z k G z
Q z z
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Digital Multi-Period RC system:
Multi-Period Signal Control
2
1 , 1 1
pp p
R j j k jj j k j
j k
G z R z R z R z H z R z H z
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Digital Multi-Period RC system:
Multi-Period Signal Control
Stability Conditions:
Roots of 1+Gc(z)Gp(z) = 0 are inside the unit circle, i.e., H(z) is stable Roots of 1-(1-kjGf(z)H(z))Q(z)z -Nj = 0 are inside the unit circle
1 1 j fk G z H z Q z
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Multi-Period Resonant Control:
Multi-Period Signal Control
1 1
j
j
Np p
R j j fNj j
Q z zG z R z k G z
Q z z
Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 2017147/171
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Multi-Period Resonant Control:
Multi-Period Signal Control
1 jh
p p
MR Mj jhj j h N
G s R s R s
2 2
cos sin
jh jh
hj hj hjMj jh hj
h N h N hj
s θ ω θR s R s k
s ω
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Enhancing the Control by filtering periodic harmonics:
Periodic Signal Filtering
If the feedback controller Gc(z) ∞, then y(z) r(z), even in the presence of
disturbances d(z) in the system.
However, the reference r(z) may suffer from unexpected harmonics and leads to harmonics and distort output signals, which feedback controller Gc(z)
cannot handle it .
It calls for Periodic Signal Filtering.
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Links between notch filters and resonant controllers:
Notch Filters
A periodic signal filter should be able to attenuate the harmonic at a
specific frequency to a very low level, meaning that its magnitude response should be low enough.
rsc 2 2h h
h
k sG ss ω
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Links between notch filters and resonant controllers:
Notch Filters
1rsc 22 100
sG ss π
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Links between notch filters and resonant controllers:
Notch Filters
2 2
notch 2 2rsc
11
h hh
h h
s ωG sG s s k s ω
0
notch 0 h
s jhωG s
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Links between notch filters and resonant controllers:
Notch Filters
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Links between notch filters and resonant controllers:
Notch Filters
psf
rsc
1
1H
h
h
G sG s
When considering multiple resonant controllers, a selective periodic signal filter (i.e., with multiple notch frequencies) can be obtained in the same manner.
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Links between comb filters and periodic controllers:
Comb Filters
Furthermore, as the conventional RC can compensate all harmonics, a
full comb filter can be obtained by including the RC scheme. This should enable filtering out all signals in the frequency range.
0
0
0
2 /psf 2 /
rc2 /
1 1 11
11
πs ωπs ω
πs ω
G s eG s e
e
0
psf 0 s jhω
G s
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Links between comb filters and periodic controllers:
Comb Filters
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Links between comb filters and periodic controllers:
Comb Filters
One more step further, what if we consider the selective harmonic control scheme, a unified periodic signal filter for selective periodic signals is obtained.
0 0
0
2 / /
sc /sm
2cos 2 / 111 cos 2 / 1
sT n sT n
sT n
e πm n eG s
G s πm n e
0
sm ( )0
s j nk m ωG s
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Links between comb filters and periodic controllers:
Comb Filters
One more step further, what if we consider the selective harmonic control scheme, a unified periodic signal filter for selective periodic signals is obtained.
2 2 / /
sc /
2 cos 2 / 1cos 2 / 1
N n N n
N n
α z α πm n zG z
α πm n z
/21 NescG z αz
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Links between comb filters and periodic controllers:
Comb Filters
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Lagrange Polynomial or Virtual Unit Delayenhancing the frequency adaptability:
Frequency Adaptive Periodic Signal Filters
F i iN N F N Fz z z z
Integer part Easy to implement
Fractional part Polynomial approximation
0
nF k
kk
z A z
with0
n
kii k
F iAk i
0
F in
N N kk
kz z A z
Integer part Easy to implement
Approximated partEasy to implement
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Lagrange Polynomial or Virtual Unit Delayenhancing the frequency adaptability:
Frequency Adaptive Periodic Signal Filters
(1 )F NN N FN F N Fz z z z z
Integer part Easy to implement
Fractional part
(1 )1 NFvz z
FN Nvz z
Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 2017161/171
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Lagrange Polynomial or Virtual Unit Delayenhancing the frequency adaptability:
Frequency Adaptive Periodic Signal Filters
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Periodic Signal Filter to enhance grid synchronization:
Application Case
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Periodic Signal Filter to enhance grid synchronization:
Application Case
Parameters Nominal value Unit
Grid voltage (RMS) Vgn 230 V
Grid frequency f0 50 Hz
LCL filter inverter-side inductor L1 3.6 mH
LCL filter grid-side inductor L2 4 mH
LCL filter capacitor Cf 2.35 µF
DC bus voltage vdc 400 V
Switching frequency 10 kHz
Sampling frequency 10 kHz
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f = 52 Hz THDv = 14.5%
Application Case – Results
Periodic Signal Filter to enhance grid synchronization:
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Application Case – Results
Periodic Signal Filter to enhance grid synchronization:
THDv = 3.3%Conventional
Enhanced
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Periodic Signal Filter to enhance the current control:
Application Case
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Application Case – Results
Periodic Signal Filter to enhance the current control:
W/O Notch Filter
With Notch Filter
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Application Case – Results
Periodic Signal Filter to enhance the current control:
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Conclusion and Discussion
A Generalized P’I’D Controlcombines feedback control and Periodic Control
It will provide a simple but effective general optimal (accuracy, fast, robust,
and easy implementation) control solution to periodic signal compensation in extensive engineering applications.
Questions?
Periodic Control of Power Electronic ConvertersKeliang Zhou, Danwei Wang, Yongheng Yang, Frede BlaabjergIET 2017http://www.theiet.org/resources/books/pow-en/pelconv.cfm
Thank you!Y. Yang, Y. Tang@ Kaohsiung