liserre_lecture_6[1]

Marco Liserre [email protected]

Stability of power converters connected to the grid through LCL-filters


Marco Liserre

[email protected]



Outline Grid converters connected through an LCL-filter

LCL-filter resonance used to estimate the gid inductance

Passive damping methods current sensors on the converter side current sensors on the grid side design of the passive damping

Active damping methods multiloop methods notch filter methods design of the active damping

Influence of the conditions at PCC



101

102

103

104

105

-70

-60

-50

-40

-30

-20

-10

0

frequency (Hz)

magnitude (Db)

L1

L1+L2

LCL

swres

LC

sw

swg z

hi

hi22

2

ripple attenuation

Grid converters connected through an LCL-filter

v

i

REF M. Liserre, F. Blaabjerg e S. Hansen, “Design and Control of an LCL-filter based Three-phase Active Rectifier” IEEE Transactions on Industrial Applications, Sept./Oct. 2005, vol. 41, no.5, pagg. 1281-1291.




102

103

-50

0

50

Ma

gn

itud

e [

Db

]

102

103

-300

-200

-100

0

100

Fre que ncy [Hz]

Ph

ase

[G

rad

]

The LCL-filter challenges the system stability

There is a resonant peak associated to two resonant poles

Their position changes as the grid inductance changes



vga vgb vgc

A/D

ia ib ic

A/D

iga igb igc

A/D

vca vcb vcc

A/D


Influence on the low frequency behavior

Influence on the high frequency behavior



2 2

2 21

( ) 1( )

( )

LC

res

s zi sG s

v s L s s

2 21 2

( ) 1 1( )

( )res

i sG s

v s L L Cs s

current sensors on the converter side

current sensors on the grid side




L2 L1

VSC Cf

vc i CURRENT CONTROL

L2

Cf Zb

ZTgrid ZTconv

L1

Zb

(a)

L2 L1

VSC Cf

ig vc CURRENT CONTROL

L2

Cf Zb

ZTgrid ZTconv

L1

Zb

(b)

L2 L1

VSC Cf

ig

e CURRENT CONTROL

(c)

L2 L1

VSC Cf

i

e CURRENT CONTROL

(d)

1

1

Tgrid g C

Tconv

z j x x

z jx

1

1

Tgrid g

Tconv C

z jx

z j x x

1

1

Tgrid

Tconv g c

z

z j x x x

(c) (d)

1

1

Tgrid C

Tconv

z j x

z j x

1 % error if xc is less than 10 %


REF M. Liserre, A. Dell’Aquila, F. Blaabjerg “Step-by-step design procedure for a grid-connected three-phase PWM Voltage Source Converter” International Journal of Electronics (Taylor&Francis Ed.), Agosto 2004, vol. 91, no. 8, pagg. 446-459.




a

Cv

giVSI

Modulator

gv

1L

v

i 2L

CvfC e

*v

+

-

+

-

Current control

+

-

gL+

-

Ci

Ci

i

gi

v

1j L i

2 g gj L L i

e

b

Cvgi

VSI

Modulator

gv

1L

v

i 2L

CvfC e

*v

+

-

+

-

Current control

+

-

gL+

-

Ci

Ci

gii

v

1j L i

2 g gj L L i

e




c

Cv

Ci

gi

i

v

1j L i

2 g

g g

j L i

j L i

e

VSI

Modulator

gv1L

v

i 2L

CvfC

gL

e

*v

gi+

-

+

-

Current control

+

-gv

d

giVSI

Modulator

gv

1L

v

i 2L

CvfC e

*v

+

-

+

-

Current control

+

-

gL+

-

CiCv

Ci

i

gi

v

1j L i

2 g

g g

j L i

j L i

e

gv



Grid converters connected through an LCL-filter Design procedure

1. Ripple analysis and converter-side inductor choice

2. Harmonic attenuation of the LCL-filter and choice of the resonance frequency value

3. LCL-filter optimization and choice of grid-side inductor, capacitor and damping method and value1. Installed reactive power of the filter

2. Robustness of the filter attenuation, to the grid impedance variation

3. The influence of the damping on the LCL-filter attenuation

1

1 dcMAX

VI

n L f

22

2 21

LC

res

i z

i



Grid converters connected through an LCL-filter Influence of inductor saturation

The frequency behaviour of the non-linear inductance can be studied splitting the model in a linear part and a non-linear part in accordance with the Volterra theory.

The Volterra-series expansion of the flux is 5

1i

i

t t

v e

L1 ii1

1 1

1

,...,n nn

i ii

L

2 1

21

ii

L

3 1 23

1

,i ii

L

non-linear inductance



2 1 2 2

1 1 1

2resres f g gC L L L L

Considering the closed loop current control where only a proportional controller is considered for the sake of simplicity

2 2

2 2 2 2 2( )

1.5

g f

g f

P L C

c

s P res P L C res

k s zG s

LT s Ls k s k z

• Different grid impedances lead to different resonance frequencies that can be detected

103

-25

-20

-15

-10

-5

0

5

10

15

20

25From: Input Point To: Output Point

Mag

nitu

de (

dB)

resonance frequency 3230 Hzin case Lg=3.2 mH

resonance frequency 3530 Hz in case Lg=1.2 mH

Use of the LCL-filter resonance to estimate the grid inductance

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.05/T

0.50/T

0.45/T

0.40/T

0.05/T

0.10/T

0.15/T

0.20/T0.25/T

0.30/T

0.35/T

0.50/T

0.35/T

0.30/T0.25/T

0.10/T

0.15/T

0.20/T

0.80.9

0.45/T

0.40/T

0.3

0.10.2

0.7

0.40.50.6





0 0.05 0.1 0.15 0.2-20

-10

0

10

20

time [s]

conv

erte

r cu

rren

t [A

]

0 10 20 30 40 50 60 70 800

2

4

6

harmonic order

ampl

itude

of

the

harm

onic

s [A

] 49th harmonic corresponding to 2450 Hz resonance frequency

Use of the LCL-filter resonance to estimate the grid inductance

Different methods can be used to excite the system resonance, such as: increase the proportional gain of the current control; add other zeros and poles in the controller in order to push the LCL-filter

poles out of the stability region;

both methods change the resonance frequency of the closed loop system saturate the ac voltage command for the PWM modulator

it does not change the resonance frequency

0 0.05 0.1 0.15 0.2

-1

-0.5

0

0.5

1

time [s]

mod

ulat

ing

sign

al



Evaluation of the proposed algorithm

0 20 40 60 80 100

0

0.02

0.04

0.06

-1 -0.5 0 0.5

-1

-0.5

0

0.5

1

0.9

1.6e3

0.80.70.60.50.40.3

3.6e3

4e3

0.10.2

400

1.6e32e32.4e3

2.8e3

3.2e3

1.2e3

400

1.2e3

3.6e3

4e3

800

800

2e32.4e3

2.8e3

3.2e3

Imag A

xis

Real Axis

Root Locus Editor (C)

Experimental spectrum of the grid currentSimulated

Root locus

Test in case the grid inductance is equal to 0 mH




Experimental spectrum of the grid currentSimulated

Root locus

Test in case the grid inductance is equal to 1.5 mH

0 20 40 60 80 100

0

0.01

0.02

0.03

0.04

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.9

1.6e3

0.80.70.60.50.40.3

3.6e3

4e3

0.10.2

400

1.6e32e32.4e3

2.8e3

3.2e3

1.2e3

400

1.2e3

3.6e3

4e3

800

800

2e32.4e3

2.8e3

3.2e3

Imag A

xis

Real Axis





0 20 40 60 80 1000

0.02

0.04

0.06

-1 -0.5 0 0.5

-1

-0.5

0

0.5

1

0.9

1.6e3

0.80.70.60.50.40.3

3.6e3

4e3

0.10.2

400

1.6e32e32.4e3

2.8e3

3.2e3

1.2e3

400

1.2e3

3.6e3

4e3

800

800

2e32.4e3

2.8e3

3.2e3

Imag A

xis

Real Axis


0 20 40 60 80 1000

0.01

0.02

0.03

0.04

-1 -0.5 0 0.5 1

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0.9

1.6e3

0.80.70.60.50.40.3

3.6e3

4e3

0.10.2

400

1.6e32e32.4e3

2.8e3

3.2e3

1.2e3

400

1.2e3

3.6e3

4e3

800

800

2e32.4e3

2.8e3

3.2e3

Imag A

xis

Real Axis


0 mH

1.5 mH

REF M. Liserre, R. Teodorescu, F. Blaabjerg, “Grid impedance estimation via excitation of LCL-filter resonance” to be published on IEEE Transactions on Industry Applications



Increasing the switching/sampling frequency, the losses decrease but at the same time the damping becomes less effective

Passive damping: current sensors on the converter side

h

gdd hihiRP 2)(3losses

L1L2

Cf

Rd

vvce

iig ic

2

21

2

2

2

2

1

1

)(

)(

resdT

LCd

sLL

RLs

zsL

Rs

sLsv

si

main terms of the sum are for the index h near to the multiples of the switching frequency order.

As the damping resistor increases, both stability is enforced and the losses grow but at the same time the LCL-filter effectiveness is reduced.



-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1Rd=0 Rd=16

Rd=wr*Lg

Frequency [Hz]102

103

-300

-200

-100

0

100D(z)G(z)

D(z)Gd(z)

Phase [deg]

-50

0

50

D(z)G(z)

D(z)Gd(z)

Magnitude [dB]

10210

3

root locus bode plot




-1 .5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

0 100 200 300 400 500

-10

0

10

0 50 100 150 200 250 300 350 400 450 500-15

-10

-5

0

5

10

15

converter side current




30 40 50-100

damping resistor value [ Ω]

d cu

rren

t [A

] 20 25 30 35 40 45 50159

10

11

100

200

300

0 10 20

0

2.3 % THD 4 %0.8 % THFHD 1.9 %

29 W losses 67 W

Excessive damping

4 5 6 7 8

-20

20

grid current




root locus

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

pole introduced by the delay

reduction of the bandwidth from 350 Hz to 200 Hz

Good method to reduce losses in high power applications at the price

of a slow down of the dynamic


REF M. Liserre, A. Dell’Aquila, F. Blaabjerg "Stability improvements of an LCL-filter based three-phase active rectifier”, PESC 2002, Cairns, Australia, June 2002.



-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1 kmax

k

max

koptimum

koptimum

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1 kmax

koptimum

kmax

koptimum

undamped passively damped

Stable without damping !

Passive damping: current sensors on the grid side



Passive damping + one sample delay

undamped passively damped

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

koptimum

koptimum

-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

koptimum

k

optimum k

max

kmax



Design algorithm: constraints

for the stability: ρMAX - maximum radius of poles of the current closed loop - ρMAX should be at least < 1 in order to have a stable current loop;

for the bandwidth: bw – the lowest between the frequencies at which the gain of the closed loop is reduced to 3 dB and at which the phase delay becomes larger than 45°;

for the LCL-filter switching ripple attenuation: ra;

for the damping losses: Pd .



Design algorithm: parameters

the current controller proportional gain kp;

the sampling frequency fsampling ;

the damping resistor value Rd.

ρMAX is a function of all the three parameters but especially of the last two in a non-linear way,

bw depends strongly and almost linearly on the second parameter,

Pd depends on the last two of them in a non-linear way

ra depends especially on the second of them.

thus a step-by-step algorithm can be written



Use of a non-linear least-mean-square method a non-linear least-mean-square method can be adopted in order to find the

optimal solution without linearising the relations: ρMAX (kp, fsampling, Rd), bw (kp, fsampling, Rd), ra (kp, fsampling, Rd), Pd (kp, fsampling, Rd)

It has been chosen to use the Levenberg-Marquardt method in conjunction with the linear search.

The Levenberg-Marquardt method uses a search direction which is a solution of a linear set of equations. The line search is based on the solution of a sub-problem to yield the search direction in which the solution is estimated to lie. The minimum along the line formed from this search direction is approximated by a polynomial method involving interpolation. Polynomial methods approximate a number of points with a polynomial whose minimum can be calculated easily

The method gives good results if the optimal solution is near the initial conditions.



Dynamic test

Start of rectifying mode at full load (a); no load (b)

(a) (b)

settling time of 30 ms



Dynamic test

Step load change from no load to 4 kW load (a); and from no load to nominal load (11 kW) (b)

(a) (b)

REF R. Teodorescu, F. Blaabjerg, M. Liserre, A. Dell’Aquila, “A stable three-phase LCL-filter based active rectifier without damping” IAS 2003, USA, October 2003.



Active damping

Obtain stability without additional losses

Modify the control algorithm

Various techniques based also on the use of more sensors

Two main possibilities:

Multiloop

Notch filter



Active damping

multiloop (use of more sensors)

notch filter in cascade



active damping plug-in

Active damping



Active damping: lead network

d

MAXT

f1

1

1arcsinMAX

dd

dd

ksLT

fksLT

f 1011

1

1)(

sT

sTksL

d

dd

lead network

-40

-20

0

20

0

50

100

kdz

Frequency [Hz]102 10

3

Phase [deg]

Magnitude [dB]

10210

3

principle of operation




The increase of the lead ratio increases the phase lead but it produces higher amplifications at higher frequencies

1

adopting a low-pass filter, it is possible to select a high phase margin (80°) around the resonance frequency (2.5 kHz)

Td = 5.6*10-4 = 1.2*10-2

kd has to be chosen both on damping and dynamic considerations

discretization o

odz pz

zzkzL

)(




Instead of a low-pass filter it is enough to select carefully the lead network position

)()(1

1)(

1 zEzLzzDd

22

11

resf sLCsE

i*

_+ D(z) G(z)1zi

++

1z E(z)

D d(z)

L(z)vc

=

1 for f < 1.8 kHz because of the lead network

1 for f > 4 kHz because of

introduce phase lead for f [1.8 ÷ 4] kHz




-1 -0.5 0 0.5 1

-1

-0.5

0

0.5

1

kdz=0

kdz=1 kdz=1

kdz=0.6

kdz=0.6

kdz=0

kdz = 0.6

• best damping

• good dynamic




102

103

-50

0

50

Mag

nitu

de [

Db]

102

103

-300

-200

-100

0

100

Frequency [Hz]

Pha

se [

Gra

d]

D(z)Dd(z)G(z) D(z)G(z)

Dd(z)

Reduction of the unstable peak under 0 dB




0 0.2 0.4 0.6 0.8 1-300

-200

-100

0

100

200

300

400

lead network gain kdz

d cu

rren

t [A

]

THD = 1.6%

THFHD = 0.6%

0.3 0.32 0.34 0.36 0.38 0.4

-20

-10

0

10

20

grid

cur

rent

[A

]

lead network gain kdz

optimum steady-state performance



Dynamic performances

0.15 0.16 0.17 0.18-10

-5

0

5

10

15

20

25

time (s)

d cu

rren

t [A

]

0.15 0.16 0.17 0.18-10

-5

0

5

10

15

20

25

time (s)

d cu

rren

t [A

]

0.15 0.16 0.17 0.18-10

-5

0

5

10

15

20

25

d cu

rren

t [A

]

time (s)

0.15 0.16 0.17 0.18-10

-5

0

5

10

15

20

25

d cu

rren

t [A

]

time (s)

Passive damping 16 Ω

Passive damping 8 Ω

+one delay

Active damping

Active damping

(capacitor voltage used also for dq-

frame orientation)

REF M. Liserre, A. Dell’Aquila, F. Blaabjerg "Stability improvements of an LCL-filter based three-phase active rectifier”, PESC 2002, Cairns, Australia, June 2002.



Active damping: use of a notch filter

Current controller

id

*

id

iq

i =0q *

PI

L

PI

L

-

+

+ed

vd,av

vq,av

ud

uq

eq

+

+ +

-

--

-

dq

abcAC

TIV

ED

AM

PIN

G

no more sensors

difficultto tune



undamped active damped2 2

2 2( ) o

ADo

z zG z

z p

Active damping: use of a notch filter



Root locus of the undamped system

-2 0 2-2

0

2

-2 0 2-2

0

2

converter side current sensors

grid side current sensors



Root locus of the undamped system with one delay



-2 0 2-2

0

2

-2 0 2-2

0

2



Root locus of the passive damped system



-2 0 2-2

0

2

-2 0 2-2

0

2



Genetic algortihms

The Genetic Algorithm (GA) simulates Darwin’s theory on natural selection and Mendel’s work in genetics on inheritance: the stronger individuals are likely to survive in a competing environment.

In short the GA finds the optimum solution combining a set of randomly chosen solutions. In the following the term “individual” indicates the possible solution, the terms “gene” indicates one of the parameters of the solution and the “fitness value” indicates the degree of goodness of the individual.



Genetic algortihms

The GA process is performed through the following iterative steps:

1. selection the individuals are selected on the basis of their fitness value to reproduce in the mating pool;

2. crossover each new individual is generated by two that are reproducing. This process is performed using part of the genes characterising each individual;

3. mutation is the way to randomly produce new characters in the new individual of the population, by changing one of its genes.



Use of genetic algorithm for active damping optimisation

4 3 24 3 2 1 0

4 3 24 3 2 1 0

d i

a i z a i z a i z a i z a iD z

b i z b i z b i z b i z b i

(1 )

1

sp p

I

i

Tk i z k i

TD z

z

Stability: active damping filter

Dynamic: PI current controller




the aim is to find the best individual (i.e. the best set of coefficients for the active damping filter and the best proportional constant of the PI current controller) in order to have:

the desired damping of the high frequency poles

the desired bandwidth of the current loop.

This is expressed through the so called fitness value f(i) of each individual i

1 1 2 2f i w d i w d i




if w1>w2 the main aim is to obtain the desired damping of the high frequency poles

if w2>w1 the main aim is to have the desired

dynamic.

RANDOM GENERATION OFTHE FIRST POPULATION

START

M<MMAX

ELITISM: SEARCH OF THEBEST INDIVIDUALS

FITNESSEVALUATION

TRUE

FALSE

1 1 2 2f i w d i w d i




MATINGPOOL

CROSSOVER

MUTATION

POPULATION AT M=M+1

END

dimension of the population: too many individuals is not a good choice

the “elitism” is used to preserve the best individuals from being eliminated



Tuning of the notch filter Comparison with the non-linear Levenberg-Marquardt optimisation

method already used for passive damping design

The non-linear least-square method finds a point characterized by 1.12 while the absolute minimum is 0.92

0.92

1.12



GA search of the optimal active damping

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

-1.5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

optimal position of the poles optimal position of the poles

final result of GA

6.5 kHz sampling frequency



Use of lead network

-1 .5 -1 -0.5 0 0.5 1 1.5-1.5

-1

-0.5

0

0.5

1

1.5

0.2 0.25 0.3 0.35 0.4-10

-5

0

5

10

time [s]

grid

cur

rent

[A

]

at 6.5 kHz sampling frequency is hard to make the system stable with active damping

REF M. Liserre, A. Dell’Aquila, F. Blaabjerg “Genetic Algorithm-Based Design of the Active Damping for an LCL-Filter Three-Phase Active Rectifier” IEEE Transactions on Power Electronics, January 2004, vol. 19, no. 1 pagg. 76-86.







Influence of the conditions at the PCC




introduction of 100 F capacitance



use active damping !

use passive damping !

strong grid

intermediate grid

weak grid

use active damping !

Grid stiffness influence: LCL-filter



-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

0.6

0.90.45/ T

0.25/ T

0.05/ T

0.05/ T

0.50.35/ T

0.40.15/ T0.3

0.2

0.10/ T

0.1

0.50/ T

0.50/ T

0.40/ T

0.45/ T

0.30/ T

0.40/ T

0.20/ T

0.35/ T

0.10/ T

0.30/ T0.25/ T0.20/ T

0.7

0.15/ T

0.8

Root Locus

Rea l Axis

Imag

inar

y A

xis

Lg rid

In each grid condition the LCL-filter converter side

impedance is adjusted such as the resonance frequency

remains unchanged (the arrow indicates the root loci branches due to higher grid

impedance)

Grid stiffness influence: LCL-filter

REF M. Liserre, F. Blaabjerg, R. Teodorescu, “Stability of Photovoltaic and Wind Turbine Grid-Connected Inverters for a Large Set of Grid Impedance Values” IEEE Transactions on Power Electronics, January 2006, vol. 21, no.1, pagg. 263-272.



Conclusions LCL-filter is used to reduce the switching ripple but it challenges the

stability of the current control loop

The LCL-filter resonance can be used to estimate the gid inductance

The different sensors position changes the 50 Hz impedance of the filter and the plant of the current control loop

Passive damping can solve stability problems but it has been proven how the excessive damping leads to low frequency ripple, reduced filter effectiveness and high losses

A reduced passive damping can be used if: the converter current is controlled with one sample delay or the grid current is controlled without delays



Conclusions Active damping is an interesting alternative to passive damping, two

approaches are possible: multiloop or notch filter

It has been clearly explained why a lead-network on the filter capacitor voltage is effective but it produces a higher overshoot

Moreover if to reduce the number of sensors the capacitor voltage is also used for dq-frame orientation a low frequency ripple is produced

The use of a notch-filter to cancel the resonance is a possible solution but it is difficult to tune



Conclusions

It is not possible to define a good design method for active damping that could be valid independently:

on the position of the current sensors,

presence of delays

tuning of the PI parameters

switching frequency

Instead the design with Genetic Algorithm of the active damping is very effective

The different grid conditions in term of grid inductance challenges both passive and active damping

liserre_lecture_6[1]

Documents

pccmarco liserre

202marco liserre

marco liserre

filtersoutlinegrid

power converters

filtersgrid

converter

active damping