grid converters for photovoltaic and wind power systems · • the dead‐beat controller belongs...

Grid Converters for Photovoltaic and Wind Power Systems

by R. Teodorescu, M. Liserre and P. RodriguezISBN: 978‐0‐470‐05751‐3Copyright Wiley 2011

Chapter 12p

Grid Current Control

Outline

• Introduction

• Harmonic requirementsHarmonic requirements

• Current control overview

• Linear current control with separated modulation Use of averaging

PI‐based control

Dead‐beat controlDead beat control

Resonant control

Harmonic compensation

• Modulation techniques Bipolar and unipolar modulation

Three‐phase modulation (continuous and discontinuous)Three phase modulation (continuous and discontinuous)

Multilevel modulation

Interleaved converters

• Operating limits of the grid converter

2

Introduction1Li 2L gLgigv

b)

20

-10

0

v fC Cv edcvC

mag

nitu

de (D

b

-60

-50

-40

-30

-20

L1+L2+Lg

LCL

v

frequency (Hz)10

110

210

310

410 5 -70

• The influence of the capacitor of the filter will be neglected since it is only dealing withthe switching ripple frequencies. In fact at frequencies lower than half of the resonancefrequency the LCL‐filter inverter model and the L‐filter inverter models have the samef h t i tifrequency characteristic

• PI‐based current control implemented in a synchronous frame is commonly used inthree‐phase converters

• In single‐phase converters the PI controller capability to track a sinusoidal reference islimited and Proportional Resonant (PR) can offer better performances

• Modulation has an influence on the design of the converter (dc voltage value), losses andEMC problems including leakage current

3

Harmonic requirements: PV‐systems

• In Europe there is the standard IEC 61727

• In US there is the recommendation IEEE 929

• The recommendation IEEE 1547 is valid for all distributed resourcestechnologies with aggregate capacity of 10 MVA or less at the point of commong gg g p y pcoupling interconnected with electrical power systems at typical primaryand/or secondary distribution voltages

• All of them impose the following conditions regarding grid current harmonic• All of them impose the following conditions regarding grid current harmoniccontent

• The total THD of the grid current should not be higher than 5%g g

4

Harmonic requirements: WT‐systems• In Europe the standard 61400‐21 recommends to apply the standard 61000‐3‐6

valid for polluting loads requiring the current THD smaller than 6‐8% depending on the type of network

harmonic limit 5th 5 6 %5 5-6 %7th 3-4 %

11th 1.5-3 % 13th 1 2 5 %

• In case of several WT systems

13th 1-2.5 %

1

Nhi

hII

• In WT systems asynchronous and synchronous generators directly connected to h d h l h

1i i

the grid have no limitations respect to current harmonics

5

Current Control overview

PWM current control methods

ON/OFF controllers with pulse width modulator

linear Non - linearlinear

fuzzypassivity

PI predictiveD d b t

resonanttiti

hysteresis predictiveoptimized

Dead - beat repetitive

p

6

Linear current control with separated

• Continuous switching vector which components are the duty cycle of each

modulation: use of averaging

22( ) ( ) ( ) ( )3 a b cd t d t d t d t

converter leg

• Average model

1dq d d d

di ti t Ri t e t v t

dt L

• Average model

1qd q q q

dt Ldi t

i t Ri t e t v tdt L

d d dc

q q dc

v t d t v tv t d t v t

• Linearized model v (t)=V

10 0dc

d d d d

VRi t i t d t e td L L L

• Linearized model vdc(t)=Vdc

100q q q qdci t i t d t e tR VdtL LL

7


• Typically PI controllers are used for the current loop in grid inverters

modulation: PI current controlyp y p g

• Technical optimum design (damping 0.707 overshoot 5%)

e ( ) IPI P

kG s k

s

v

v i( )dG s ( )fG s( )PIG si

1( )1 1.5d

s

G sT s

( ) 1i sgv ( ) 1( )( )f

i sG sv s R Ls

1 0

b)0.2

0.4

0.6

0.8

10-1 100 101 102 103 104-20

-15

-10

-5

Mag

nitu

de (D

b

-0.6

-0.4

-0.2

0 10 10 10 10 10 10

-200

-100

0

e (D

egre

e)

-1 -0.5 0 0.5 1

-1

-0.8

100 101 102 103 104-400

-300

Frequency (Hz)

Pha

s

8

Shortcomings of PI controller

1

0.1

0.15

0.2

0.023 0.0235 0.024 0.0245 0.025 0.0255 0.026 0.0265 0.027 0.02750.8

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

steady‐state magnitude and phase

0.019 0.0192 0.0194 0.0196 0.0198 0.02 0.0202 0.0204 0.0206 0.0208 0.021

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05magnitude and phase

error

limited disturbance rejection capability

• When the current controlled inverter is connected to the grid, the phase errorresults in a power factor decrement and the limited disturbance rejection

bili l d h d f id f d f d icapability leads to the need of grid feed‐forward compensation

• However the imperfect compensation action of the feed‐forward control due tothe background distortion results in high harmonic distortion of the current andconsequently non‐compliance with international power quality standards

9

Use of a PI controller in a rotating frame

didi

gdv

k

In order to overcome the limit of the PI in dealing with

ii

i

di

di

L

qiL

je je

abcv

v

v

IP

kks

limit of the PI in dealing with sinusoidal reference and

harmonic disturbances, the PI control is implemented in

a rotating frameL

qi

gqvgv gv

f gV

abc

IP

kks

a rotating frame

abc

gvgv

je

gdv

gqv

dcv

dcV

P

IP

kks

di

P

Q qi

P

igv 1

3

ga a gb b gc c

gab a gbc b gca c

P v i v i v i

Q v i v i v i

IP

kks

Ikk

0( )

0

IP

PI dqI

P

kk

sG sk

k

Q

IPk

s P s

10

Use of a PI controllers in a rotating frame in

di

gdv

single‐phase systems

• An independent Q control ishi d

IP

kks

i di

d

digd

v

Lv

achieved

• A phase delay block createthe virtual quadrature f

gV

ije i qi

i

je v

L2je

IP

kks

the virtual quadraturecomponent that allows toemulate a two‐phase system

h f h

gv

gv

gv

g

gdv

qi

gqv

2je

• The v component of thecommand voltage is ignoredfor the calculation of the

dc controllerV

je gqv

MPPT dcV di

dci

dcv

Ikk

duty‐cycle

MPPT

P

Q

IPk

s

qi

gdvQ

gqv

Q

11

Use of a PI controllers in two rotating frames

• Under unbalanced conditions inorder to compensate the

di

PIi v

order to compensate theharmonics generated by theinverse sequence present in theid lt b th th iti d i

PIi v

je je

grid voltage both the positive‐ andnegative‐sequence referenceframes are required

qi

di

PIi v

• Obviously using this approach,double computational effort mustbe devoted

PI

PI

i

v

je je

be devoted qi

PI

12

Linear current control with separated modulation: Dead‐beat controller

• The dead‐beat controller belongs to the family of the predictivecontrollers

• They are based on a common principle: to foresee the evolution of thecontrolled quantity (the current) and on the basis of this prediction: To choose the state of the converter (ON OFF predictive) or To choose the state of the converter (ON‐OFF predictive) or

The average voltage produced by the converter (predictive with pulse widthmodulator)

• The starting point is to calculate its derivative to predict the effect of thecontrol action

Th t ll i d l d th b i f th d l f th filt d f• The controller is developed on the basis of the model of the filter and ofthe grid, which is used to predict the system dynamic behavior: thecontroller is inherently sensitive to model and parameter mismatches

13

Dead‐beat controller

• The information on the model is used to decide the switching state of theconverter with the aim to minimize the possible commutations (ON‐OFFconverter with the aim to minimize the possible commutations (ON‐OFFpredictive) or the average voltage that the converter has to produce in order tonull it

e

• In case it is imposed that the error at the

v i( )fG z( )DBG zi

• In case it is imposed that the error at theend of the next sampling period is zerothe controller is defined as “dead‐beat”.It b d t t d th t it i th

gv

It can be demonstrated that it is thefastest current controller allowingnulling the error after two sampling

i d*iperiods i

i

ONt ONt

sTk 1k 2k sT

14

Dead‐beat controller

)1()1()1()(1)1()1( kekekiakikvkv )1()1()1()()1()1( kekekib

kib

kvkv

)()1()(1)1( kekekikvkv Neglecting R! )()1()()1( kekekib

kvkv

0.8

1

0.8

1

0.2

0.4

0.6

0.2

0.4

0.6

450 460 470 480 490 500 510 520 530 540 550-0.2

0

450 460 470 480 490 500 510 520 530 540 550-0.2

0

15

Dead‐beat controller: limits

0

10

20

0

10

20

-30

-20

-10

0

curre

nt [A

]

-30

-20

-10

0

curre

nt [A

]

Due to PWM!

0.04 0.045 0.05 0.055 0.06-60

-50

-40

time [s]0.04 0.045 0.05 0.055 0.06

-60

-50

-40

time [s]time [s] time [s]

Pole-Zero Map

10.6/T

0.5/T0.4/T

10

20

y Ax

is

0

0.2

0.4

0.6

0.8

0.90.80.70.60.50.40.30.20.1

/T

0.9/T

0.8/T

0.7/T 0.3/T

0.2/T

0.1/T

-20

-10

0

10

ent [

A]

Imag

inar

y

0 8

-0.6

-0.4

-0.2

0/T

0.9/T

0.8/T

0.7/T 0.3/T

0.2/T

0.1/T

-50

-40

-30curre

Due to parameter error!

Real Axis

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-1

-0.80.6/T

0.5/T0.4/T

0.04 0.045 0.05 0.055 0.06-60

time [s]

16


R t t l i b d th f G li d I t t (GI)

modulation: Resonant controller• Resonant control is based on the use of Generalized Integrator (GI)

• A double integrator achieves infinite gain at a certain frequency, called resonancefrequency, and almost no attenuation outside this frequency

• The GI will lead to zero stationary error and improved and selective disturbance rejection

2 2

ss

GI

The GI will lead to zero stationary error and improved and selective disturbance rejectionas compared with PI controller

B o d e D ia g r a m2 0 0 2

in

- 2 0 0

- 1 0 0

0

1 0 0

Mag

nitu

de (d

B)

0

0.5

1

1.5

out

2 2

ss

- 9 0

0

9 0

1 8 0

Phas

e (d

eg)

2 0 0

1.5

-1

0.5

0

1 01

1 02

1 03

- 1 8 0

F r e q u e n c y ( H z )0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

-2

Resonant control

)()()( jsGjsGsG DCDCAC • The resonant controller can be obtained via afrequency shift

2k s

2 2

2( ) IAC

k sG ss

( ) IDC

kG ss

k

600

800

) 60

80

)

2 2

2( )2

I cAC

c

k sG ss s

( )1 ( )

IDC

c

kG ss

0

200

400

600

Mag

nitu

de (

dB)

0

20

40

60

Mag

nitu

de (

dB)

101

102

103

0

Frequency (Hz)

50

100

deg)

101

102

103

0

Frequency (Hz)

50

100

deg)

101

102

103

-100

-50

0

Pha

se (

d

Freq enc (H )10

110

210

3-100

-50

0

Pha

se (

d

Frequency (Hz) Frequency (Hz)

Bode plots of ideal and non‐ideal PR with kP = 1, kI = 20, = 314 rad/s, c = 10 rad/s18

Resonant control• The stability of the system should be taken into consideration

e

v i( )dG s ( )fG s( )P RESG s

i

• The phase margin (PM) decreases as the resonant frequency approach to thecrossover frequency

2 2P Isk k

s

B o d e D ia g r a m Bode Diag ram

q y

2 21

P Isk k

s R Ls

- 1 0 0

0

1 0 0

2 0 0

3 0 0

4 0 0

Mag

nitu

de (d

B)

0

100

200

300

400

Mag

nitu

de (d

B)

g

- 2 0 0

1 0 0

9 0

0

9 0

1 8 0

Phas

e (d

eg)

- 100

0

0

90

180

Phas

e (d

eg)

1 01

1 02

1 03

- 1 8 0

- 9 0

Fr e q u e n c y ( H z ) Frequenc y (Hz )10

110

210

3-180

-90

P

PM

19

Tuning of resonant control• The gain kP is founded by ensuring the desired bandwidth using either rlocus Matlab

function or SISOTOOL

Th i t l t t k t t li i t th t d t t h• The integral constant kI acts to eliminate the steady‐state phase error

6

2 2( )c P Io

sG s k ks

6

2 2( )c P Io

sG s k ks

0

2

4o

0

2

4 o

-4

-2

-4

-2

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-6

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05-6

kI = 100 kI = 500

• A higher kI will "catch" the reference faster but with higher overshoot

• Another aspect is that kI determines the bandwidth centered at the resonancefrequency in this case the grid frequency where the attenuation is positive Usually thefrequency, in this case the grid frequency, where the attenuation is positive. Usually, thegrid frequency is stiff and is only allowed to vary in a narrow range, typically ± 1%

20

Discretization of generalized integrators

GI integrator decomposed in two simple integrators Forward integrator for direct path and backward for

2 2

2

1

1

y s u s v sy s s su s s v s y s

ò y u

g

1 1 1

2

( )k k s k ky y T u v

Forward integrator for direct path and backward for feedback path

ys

ò 2

v21k k s kv v T y

The inverter voltage reference

*2 2( ) I

i Pk su s s k

s

Difference equations kp

Control diagram of PR implementation

1 1 1

*,

k k s I k s k

i k P k k

y y T k T v

u k y

Difference equations

ii*

ii

iiui*

ò

ò

yu

v

p

kI

2

aw

Gf(s)

,

21

1

1

i k P k k

k k s k

k k

k k

v v T y

y y

ò 2

max max,y y y yaw

y y y y

1k kv v

max max,y y y y

21

Use of P+resonant controller in stationary

frame

2 2I

Pk sk

s

ii

i

i

v

v

*v

Current controller

abc

i

i

v

2 2I

Pk sk

s

abc

Current controller

• PLL is still indispensable for reference generationPLLgv gv

gv

,f

gV

abcdv

IP

kks

dcv

dc controllerV

dcV

P

controllerP

sin

cos

i

gv

i

i

I

IP

kks

IP

kks

1

3

ga a gb b gc c

gab a gbc b gca c

P v i v i v i

Q v i v i v i

PQ

controllerP

calculationPQ scontrollerQ

calculationPQ

Q

22

From PI in a rotating‐frame to P+res for each phase

• In the hypothesis

• H11(s) = H22(s) = IP

kk H11(s)

+

+

Vd(s)Ed(s)

H11(s) H22(s)

• H12(s) = H21(s) = 0

Pks

H12(s)

H21(s)

H22(s)

+

+ Vq(s)Eq(s)

)()()()()()(

22

11

tethtvtethtv

qq

dd

)()()( 22 qq

0 02 2 2 2 2 2

0 0 0

3 32 2 ( ) 2 2 ( )

I P I I P I IP

k s k k s k k k s kks s s

k 0 0 0

( , . ) 0 02 2 2 2 2 2

0 0 0

2 2 ( ) 2 2 ( )

2 3 3( )3 2 2 ( ) 2 2 ( )

3 3

a bc P I I I P I Ic P

s s s

k k s k k s k k s kG s ks s s

k k s k k k s k k s

( , )0

0

IP

d qc

I

kksG

kk

0 0

2 2 2 2 2 20 0 0

3 32 2 ( ) 2 2 ( )P I I P I I I

Pk k s k k k s k k sk

s s s

0 Pks

23

Linear controllers: from PI in a rotating‐frame to P+res for each phase

zva

( )( ) ( )0 2 1 11 1 ( )( ) ( )0 1 1 23

( )

aa a

bc c

c

v ti t i tRd v ti t i tRdt L

v t

( ) ( ) ( ) 0a b cv t v t v t

)(01)(01)( tvtiRtid aaa

zzvbvc

)(

)(10)(

)(0)(

)(tvtiRLtidt c

a

c

a

c

a

E h t i d t i d l b it lt !

0 02 2 2 2 2 2

3 32 2 ( ) 2 2 ( )

I P I I P I IP

k s k k s k k k s kks s s

22

00sKK ip

Each current is determined only by its voltage!

0 0 0

( , . ) 0 02 2 2 2 2 2

0 0 0

0 0

2 2 ( ) 2 2 ( )

2 3 3( )3 2 2 ( ) 2 2 ( )

3 3

a bc P I I I P I Ic P

P I I P I I I

s s s

k k s k k s k k s kG s ks s s

k k s k k k s k k sk

22

20

2

02

).,(1

00

00)(

sKK

ssKK

s

sG

ip

ip

p

cbac

2 2 2 2 2 20 0 02 2 ( ) 2 2 ( ) Pk

s s s

2

02 sp

24

Linear controllers: results (ideal grid conditions)

current errorharmonic spectrum

PI controller in a rotating frame

current error

P+resonant controller for each phase

current errorharmonic spectrum

P+resonant controller for each phase

25

Linear controllers: results (equivalence of PI in dq and P+res in )

PI controller in a rotating frame triggering LCL‐filter resonance

P+resonant in stationary frame triggering LCL filter resonancey triggering LCL‐filter resonance

26

Comparison under fault

Single‐phase fault:

With dead beat controllerWith dead‐beat controller the current overshoot is

limited

27


• The decomposition of signals into harmonics with the aim ofmonitor and control them is a matter of interest for various electricand electronic systemsand electronic systems

• There have been many efforts to scientifically approach typicalproblems (e.g. faults, unbalance, low frequency EMI) in powerp ( g , , q y ) psystems (power generation, conversion and transmission) throughthe harmonic analysis

• The use of Multiple Synchronous Reference Frames (MSRFs), earlyproposed for the study of induction machines, allowscompensating selected harmonic components in case of two phasecompensating selected harmonic components in case of two‐phasemotors, unbalance machines or in grid connected systems

28


• The harmonic components of power signals can be represented in stationaryor synchronous frames using phasors

• In case of synchronous reference frames each harmonic component istransformed into a dc component (frequency shifting)p ( q y g)

i5 Signal

Signal

7q

i5je

7

Time Time

5q5d

5

7

i

i7je

7

Time Time

SignalSignal

7d

• If other harmonics are contained in the input signal, the dc output will bedi t b d b i l th t b il filt d t

disturbed by a ripple that can be easily filtered out

29

Harmonic compensation by means of synchronous dq‐frames

• Two controllers should beimplemented in two frames

5 0di

1 si v

5

implemented in two framesrotating at -5 and 7

• Or nested frames can be used i.e.i l ti i th i

i v

5je 5je

1 s

implementing in the mainsynchronous frame twocontrollers in two frames rotatingt 6 d 6

5 0qi

7 0di 7

at 6 and -6• Both solutions are equivalent also

in terms of implementation

7di

i

v

7je 7je

1 s

burden because in both the casestwo controllers are needed

7 0qi

i v

1 s

7q

30


gdv

IP

kks

i

je

i

i

di

qi

g

je

v

v

L

i

L

i i

i i

gv

gv

je

di

qi

gqv

IP

kks

v

i

je

gdv

gqv 5Iks

je

i

ije

5v

5v

55Ik

s

k

5

5

7Iks

je

i

ije

7v

7v

7Ik7 7s

731


vi

IP

kks

i

di

qi

gdv

L

i

L

di

di

je

v

v

PLLgv

gv

je

di

qi

gqv

IP

kks

v

ii

qi

qi

je

gdv

gqv 5Iks

je

di

qije

5dv

5qv

65Ik

s

k

6

6

7Iks

je

di

qi

6

je

7dv

7qv

7Ik6 6s

632

Harmonic compensation by means of stationary ‐frame

Besides single frequency compensation (obtained with the generalizedintegrator tuned at the grid frequency), selective harmonic compensation canalso be achieved by cascading several resonant blocks tuned to resonate at thedesired low‐order harmonic frequencies to be compensated.

As an example the transfer kAs an example, the transferfunctions of a non‐idealharmonic compensator (HC)designed to compensate for

Pk

u

i

i

designed to compensate forthe 3rd, 5th and 7th harmonicsis reported. 2 2

Ik ss

22

2( )

2Ih c

hk s

G sh

2 2

3,5,7... ( )I h

h

k ss h

23,5,7 2h cs s h , , ( )

33

Harmonic compensation by means of

Open‐loop PR current control system with and Closed loop PR current control system with and

stationary ‐frameOpen loop PR current control system with and

without harmonic compensator Closed loop PR current control system with and

without harmonic compensator

• Having added the harmonic compensator, the open‐loop and closed‐loop bodegraphs changes as it can be observed with dashed line. The change consists inthe appearance of gain peaks at the harmonic frequencies, but what isinteresting to notice is that the dynamics of the controller, in terms ofbandwidth and stability margin remains unaltered

34

Disturbance rejection comparison

Disturbance rejection (current error ratio disturbance) of the PR+HC, PR and P

• Around the 5th and 7th harmonics the PR attenuation being around 125 dB and the PI attenuationonly 8 dB. The PI rejection capability at 5th and 7th harmonic is comparable with that one of asimple proportional controller the integral action being irrelevantsimple proportional controller, the integral action being irrelevant

• PR +HC exhibits high performance harmonic rejections leading to very low current THD!

35

Results: grid voltage background distortion

Effect of the grid voltage background distortion on the

currents

Use of harmonic compensators

36

Results: grid voltage background distortion

Effect of the grid voltage background distortion on the

currents

Use of harmonic compensators

37

Modulation techniques

• Characteristic parameters of these strategies are:Characteristic parameters of these strategies are: The ratio between amplitudes of modulating and carrier waves (called

modulation index M)

The ratio between frequencies of the same signals (called carrier index m)

• These techniques differ for the modulating wave chosen with the goal toobtainobtain A lower harmonic distortion

To shape the harmonic spectrum

To guarantee a linear relation between fundamental output voltage andmodulation index in a wider range

• The space ector mod lations are de eloped on the basis of the space• The space vector modulations are developed on the basis of the spacevector representation of the converter ac side voltage

38

Modulation techniques

• Analogic or digital

• Natural sampled or regular sampled

1

2

Ao

d

VV

• Natural sampled or regular sampled

• Symmetric or asymmetric 4

.1 278

.1 0

Linear

Over-modulation

Square-wave

Optimization both for the linearity and harmonic

t t0

0 .1 0 M for 15fm

.3 24

content

39

Bipolar and unipolar modulations

2dcV

2C

V v t

P

V v t

PP

2dcV

1C

dcV v t

P

dcV v t

PP

00 1

0cos2

sin2

14)(

mm

nn

cndc tntmnmMqJ

qVtv • Bipolar

10120 122cos1cos

218cos2)(

m ncn

dcdc tntmnmMmJ

mVtMVtv

• Unipolar 1m n

40

Bipolar and unipolar modulations

-0 50

0.51

1.5

v ref, v

tri

00.5

11.5

ef, v

tri

0 10 20 30 40-1.5

-10.5

t [ms]

vref vtri

50

0

50

100

v Ao [V

]

0 10 20 30 40-1.5

-1-0.5

t [ms]

v re

vref vtri

0

50

100

v Ao [V

]

0 10 20 30 40-100

-50

t [ms]

-50

0

50

100

v Bo [V

]

0 10 20 30 40-100

-50

v

t [ms]

0

50

100

v Bo [V

]

0 10 20 30 40-100

t [ms]

-100-50

050

100150

v AB [V

]

0 10 20 30 40-100

-50

t [ms]

-500

50100150

v AB [V

]

0 20.40.60.8

1

v AB/V

d

0 20.40.60.8

1

v Ao/V

d

0 10 20 30 40-150100

t [ms] 0 10 20 30 40-150-100

t [ms]

0 5 10 15 20 25 30 35 40 45 50 55 600

0.2

h0 5 10 15 20 25 30 35 40 45 50 55 60

00.2

h

Due to the unipolar PWM the odd carrier and associated sideband harmonics arel t l ll d l i l dd id b d h i (2 1) t dcompletely cancelled leaving only odd sideband harmonics (2n‐1) terms and

even (2m) carrier groups.41

Three‐phase modulation techniques

-0.50

0.51

1.5

v ref, v

tri

50

0

50

100

v An [V

]

0 10 20 30 40-1.5

-1

t [ms]

vref vtri

0

50

100

o [V]

0 10 20 30 40-100

-50

t [ms]

050

100150

[V]

0 10 20 30 40-100

-50

v Ao

t [ms]

50

100

V]

0 10 20 30 40-150-100-50

E

t [ms]

10

20

0 10 20 30 40-100

-50

0

v on [V

t [ms]100150

0 10 20 30 40-20

-10

0

i o [A]

t [ms]

10

20

0 10 20 30 40-150-100-50

050

v AB [V

]

t [ms]0 10 20 30 40

-20

-10

0

10

i d [A]

t [ms][ ] [ ]

The basic three‐phase modulation is obtained applying a bipolar modulation to each of thethree legs of the converter. The modulating signals have to be 120 deg displaced. The phase‐to‐phase voltages are three levels PWM signals that do not contain triple harmonics If theto phase voltages are three levels PWM signals that do not contain triple harmonics. If thecarrier frequency is chosen as multiple of three, the harmonics at the carrier frequency andat its multiples are absent.

42

Extending the linear range (m=1,1)

SPWM

SVM

43

Three‐phase continuous modulation techniques

Continuous modulationsContinuous modulations• Sinusoidal PWM with Third Harmonic Injected – THIPWM. If the third harmonic

has amplitude 25% of the fundamental the minimum current harmonic contentis achieved; if the third harmonic is 17% of the fundamental the maximal linearrange is obtained

• Suboptimum modulation (subopt). A triangular signal is added to theSuboptimum modulation (subopt). A triangular signal is added to themodulating signal. In case the amplitude of the triangular signal is 25% of thefundamental the modulation corresponds to the Space Vector Modulation(SVPWM) with symmetrical placement of zero vectors in sampling time

vdc/2 vdc/2 vdc/2

ACRv ACRv ACRvTHIPWM1/6 THIPWM1/4 SVPWM

(SVPWM) with symmetrical placement of zero vectors in sampling time

0

-vdc/2

0

-vdc/2

0

-vdc/2

onv

DCRv

onv

DCRv

onv

DCRv

0 /2 2/3 2 0 /2 2/3 2 0 /2 2/3 2

Maximum linear range Minimum current harmonics Maximum linear range and almost optimal current harmonics

Triple harmonic injection 1/6 Triple harmonic injection 1/4 Space vector - Triangular 1/4

44

Three‐phase discontinuous modulation techniques

The discontinuous modulations formed by unmodulated 60 degsegments in order to decrease the switching lossessegments in order to decrease the switching losses• Symmetrical flat top modulation, also called DPWM1

• Asymmetrical shifted right flat top modulation, also called DPWM2

• Asymmetrical shifted left flat top modulation, also called DPWM0

vdc/2

0

vdc/2

0

ACRv DPWM0=-/6

DCRvACRv

DPWM1=0

DCRvvdc/2

0

ACRvDPWM2

=/6DCRv

0 /2 2/3 2

-vdc/2

0 /2 2/3 2

-vdc/2onv onv

0 /2 2/3 2

-vdc/2

onv

45

Multilevel converters and modulation techniques

• Wind turbine systems: high power ‐> 52dcV

y g pMW Alstom converter

• Photovoltaic systems: many dc‐links for atransformerless solution

0V 0V0V

V

transformerless solution2dcV

Different possibilities:• Alternative phase opposition (APOD) where carriers in adjacent bands are

A B C

phase shifted by 180 deg

• Phase opposition disposition (POD), where the carriers above the referencezero point are out of phase with those below zero by 180 degzero point are out of phase with those below zero by 180 deg

• Phase disposition (PD), where all the carriers are in phase across all bands46

Multilevel converters and modulation

Cascaded inverter

techniques

abdcV

i L

Cascaded inverter

dcV i L

Neutral point clamped

c

e

v

V

a

be

v

dcV

n

cd

dcV

012

0

2122cos1cos14

cos)(N

icndc

dc

mtntmnmMmJV

tMNVtv

1 1

012 2122cos1cos2m n i

icn mtntmnmMmJm

( 1)i

iN

, 1, 2,3...m kN k • Carrier shifting i Ng

47

Carrier shifting

-1

0

1

v ref, V

tri

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

100

0

100v ab

[V]

t [s]

0

1

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

-100vef

, Vtri

t [s]

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

-1

0

100

[V]

v re

t [s]

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

-100

0

200

v cd [

V]

t [s]

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

-200

0

1

pu]

v ad [V

t [s]

0 10 20 30 40 50 600

0.5

h

A [p

48

PD Modulation for NPC

0

1

Vtri

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

-1

01

V]

Vre

f, V

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06-1

t [s]

Van

[V

tri

1

01

Vre

f, [V

]

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06

-1

0

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06-10

t [s]

0200

Vbn

[V

]

0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06-200

0

t [s]1

pu]

Vab

[

Best WTHD !0 10 20 30 40 50 60

0

0.5

h

A [p

49

Inter‐leaved modulation

Shifting the carrier signals (interleaving modulation)reduces the harmonic content in the current withreduces the harmonic content in the current withthe same principle that leads cascaded convertersto have a reduced harmonic content.

H h it i th t t h d d

Parallel inverters

Inter-leaved connection

Without interleaved With interleaved

However here it is the current to have a reducedharmonic content.a

bdcV

i

'L

Inter-leaving

0.04 0.05 0.06 0.07 0.08-101

10.04 0.05 0.06 0.07 0.08-101

1

Without interleaved With interleaved

c d

e

v

dcV 'L

ginductances

0.04 0.05 0.06 0.07 0.08-10

0.04 0.05 0.06 0.07 0.08-100

0100

100

0.04 0.05 0.06 0.07 0.08-10

0.04 0.05 0.06 0.07 0.08-100

0100

100

0.04 0.05 0.06 0.07 0.08-100

0100

0.04 0.05 0.06 0.07 0.08-200

0200

20

0.04 0.05 0.06 0.07 0.08-100

0100

0.04 0.05 0.06 0.07 0.08-200

0200

20

0.04 0.05 0.06 0.07 0.08-20

020

0.04 0.05 0.06 0.07 0.08-20

020

50

Operating limits of the grid converter

Power processed by the converter

2 2 283

dcM V EP E

L

This means that the higher the dc voltage and the smaller the inductance, thehigher the power rating of the converter.

Current sharing ratio transistor/diodes2M

The transistors are conducting about 93% of the current while the load on the

cos3 boost

Mk

gdiodes is very low. In this situation the load on the transistors is slightly higher thanfor normal inverter operation on an induction motor. This is due to the smalldisplacement angle of the grid inverter compared to a typical phase angle of thep g g p yp p gstator current of an induction motor.

51

Conclusions

• The PR uses Generalized Integrators (GI) that are double integrators achievingvery high gain in a narrow frequency band centered on the resonant frequencyand almost null outside

• This makes the PR controller to act as a notch filter at the resonance frequencyand thus it can track a sinusoidal reference without having to increase theswitching frequency or adopting a high gain as it is the case for the classical PIswitching frequency or adopting a high gain, as it is the case for the classical PIcontroller

• PI adopted in a rotating frame achieves similar results, it is equivalent to the usef th PR’ f h hof three PR’s one for each phase

• Also single phase use of PI in a dq frame is feasible

• Dead‐beat controller can compensate current error in two samples but it isead beat co t o e ca co pe sate cu e t e o t o sa p es but t saffected by PWM limits and parameters mismatches

• Dead‐beat controller is faster in limiting overcurrent during faults

f d l h h b• A review of modulation techniques has been given

52

grid converters for photovoltaic and wind power systems · • the dead‐beat controller belongs...

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