Periodic Control of Power Electronic Converters. Periodic Control of Power... · 2017-05-25 · Periodic Control of Power Electronic Converters Yongheng YANG, Yi TANG Assistant Professors

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

Periodic Control of Power Electronic Converters | Y. Yang and Y. Tang | June 4, 20172/171

Tutorial @ IFEEC 2017 – ECCE Asia, Kaohsiung

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.



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.



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



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



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



Aalborg University Campus



Energy Production | Distribution | Consumption | Control

Power Electronics Centered



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



Power Electronics in today’s power systems:

Today’s Power Systems



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.



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



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.



Side Effect

Power electronic Systems:



Power electronic conversion brings Harmonics:

Side Effect



Power electronic conversion brings Harmonics:

Side Effect

Line notching Motor vibration Overheating Triggering resonance Equipment dysfunctional Nuisance tripping …



π approximation – Liu Hui’s algorithm: ( )

One Harmonic Origin

Wikimedia: https://en.wikipedia.org/wiki/Liu_Hui%27s_%CF%80_algorithm



Harmonics are related to Power Converter Topologies:

One Harmonic Origin




One Harmonic Origin




One Harmonic Origin




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



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:



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



Injection Compensation



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





Well-designed converter controller (dpwm) can remove certain harmonics



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



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



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.



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”.



IMP Periodic Control

Internal Model Principle based PID control:

Internal model for DC signals

Control accuracy

Stability and dynamics

Stability and dynamics



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?



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



Periodic signal generator (Internal Models of all harmonics):

Internal Model of Any Periodic Signal

T0 = 0.02 s



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



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



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



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



Achievable Zero-Phase Compensation:


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



Linear Phase Compensation Design for the 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



Linear Phase Compensation Design for the 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



Linear Phase Compensation – an example:


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



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.



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



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



Plug-in MRSC enabling selective harmonic cancellation:

Plug-in Digital MRSC System


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



Periodic Control of CVCF single-phase PWM inverters:

Application Case




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*




Application Case

*

3

dc dc dc

27.76 4.15 10 28.76c c cv k v k v ku k

v v v



Application Case – Results

State Feedback Control of the CVCF single-phase PWM inverter:

with a fundamental-frequency RSC





with the repetitive control (i.e., RC)





with multiple resonant controllers (i.e., MRSC)



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






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






DFT1

a aN

N Nih D

iF z b i z z Q z z

A Comb Filter is developed.




DFT-based Internal Model of selected harmonics:


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




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



Plug-in DFT-based RC System compatible periodic control:

Plug-in DFT-based Repetitive Control


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.



Modified DFT-based Repetitive Control

Modified DFT-based Repetitive Control scheme:


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



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



RSC Scheme ≡ I scheme in the synchronous rotating frame:


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



RSC Scheme ≡ I scheme in the synchronous rotating frame:


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.



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.



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.



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.



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.



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



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



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?



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



T0 = 0.02 s 6k+1





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





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





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





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



Plug-in Digital PSRC System



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



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


πs nω

πs nω πs nω

e eG s G s G se e

πm n ee πm n e





Internal models for 6k±1 order harmonics






T0 = 0.02 s 6k±1



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 / )


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



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



Plug-in Digital SHC System

Enabling Fast Dynamics – Selective harmonic control scheme:


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



Plug-in Digital SHC System

Enabling Fast Dynamics – Selective harmonic control scheme:

In practice, a linear phase-lead compensator Gf(z) = zc is adopted.



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.



Odd Order Harmonic RC – an SHC for single-phase converters:

Odd Order Harmonic Repetitive Control


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



6k±1 Order Harmonic RC – an SHC for three-phase converters:

6k±1 Order Harmonic Repetitive Control


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



/ 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



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



Optimally Weighted Gain leads to fast dynamics:

Plug-in Digital OHC System


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



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



/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



Advanced Periodic Control of CVCF single-phase PWM inverters:

Application Case




Application Case


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



Reference voltage vc 50sin(100πt) V*




Application Case

*

3

dc dc dc

90 8.4 10 90c c cv k v k v ku k

v v v





without any advanced periodic control





without any advanced periodic control

1r 199

1

100%j

ii

ii

Mh j

M

10.25 0.5 0.25Q z z z





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






RC, krc = 1.2 ORC, kor = 1.2

DMRC, kor = 0.4, ker = 0.8 DMRC, kor = 0.8, ker = 0.4



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






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



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



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



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?




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




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



Frequency Adaptability


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





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





It calls for

Frequency Adaptive Periodic Control



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



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).



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).


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




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


Approximated partEasy to implement




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.




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



Plug-in Digital FA-RC System

Frequency Adaptive RC (FA-RC ) system:


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




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




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



Frequency Adaptive DFT Repetitive Control

Virtual Variable Sampling Rate Unit Delay:

DFT 1 a

DFTF N

DFT

F zG z k

F z z




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




aDFT

aDFTaDFT1 a

FN

v

k F zG z

F z z

Frequency Adaptive DFT RC using virtual unit delay zv :-1



Fractional-Order Phase-Lead Compensator

Fractional-Order Linear Phase Compensation:

Linear phase-lead compensation

Φ 360s

fcf





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





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.



Frequency Adaptive Periodic Control of power converters:

Application Case




Application Case


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



Repetitive control gain krc 1.8 -

*




Application Case

* *1 1 2

1inv g g g

dc

v k v k b i k b b i kv k




Deadbeat Control of the grid-connected single-phase converter:

without any periodic control




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


without any periodic control





with various periodic control

CRC

OHC






CRC FA-CRC

CRC FA-CRC











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.



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



Beyond periodic signal control:

Periodic Signal Processing

We are not just controlling periodic signals



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





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






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



Multi-Period Resonant Control:


1 1

j

j

Np p

R j j fNj j

Q z zG z R z k G z

Q z z



Multi-Period Resonant 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



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.



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




Notch Filters

1rsc 22 100

sG ss π




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




Notch Filters




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.



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




Comb Filters




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




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




Comb Filters



Lagrange Polynomial or Virtual Unit Delayenhancing the frequency adaptability:

Frequency Adaptive Periodic Signal Filters

F i iN N F N Fz z z z


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


Approximated partEasy to implement





(1 )F NN N FN F N Fz z z z z


Fractional part

(1 )1 NFvz z

FN Nvz z







Periodic Signal Filter to enhance grid synchronization:

Application Case




Application Case


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





f = 52 Hz THDv = 14.5%







THDv = 3.3%Conventional

Enhanced



Periodic Signal Filter to enhance the current control:

Application Case





W/O Notch Filter

With Notch Filter







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

Periodic Control of Power Electronic Converters. Periodic Control of Power... · 2017-05-25 · Periodic Control of Power Electronic Converters Yongheng YANG, Yi TANG Assistant Professors

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