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Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Stability of power converters connected to the grid through LCL-filters
Marco Liserre
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Grid converters connected through an LCL-filter
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
vga vgb vgc
A/D
ia ib ic
A/D
iga igb igc
A/D
vca vcb vcc
A/D
Grid converters connected through an LCL-filter
Influence on the low frequency behavior
Influence on the high frequency behavior
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Grid converters connected through an LCL-filter
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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 %
Grid converters connected through an LCL-filter
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Grid converters connected through an LCL-filter
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Grid converters connected through an LCL-filter
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
resonance frequency 3530 Hz in case Lg=1.2 mH
resonance frequency 3230 Hz in case Lg=3.2 mH
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Evaluation of the proposed algorithm
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
Root Locus Editor (C)
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Evaluation of the proposed algorithm
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
Root Locus Editor (C)
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
Root Locus Editor (C)
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
-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
Passive damping: current sensors on the converter side
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
-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
Passive damping: current sensors on the converter side
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Passive damping: current sensors on the converter side
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Passive damping: current sensors on the converter side
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
-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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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 .
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Dynamic test
Start of rectifying mode at full load (a); no load (b)
(a) (b)
settling time of 30 ms
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Active damping
multiloop (use of more sensors)
notch filter in cascade
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
active damping plug-in
Active damping
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Active damping: lead network
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
)(
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Active damping: lead network
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Active damping: lead network
-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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Active damping: lead network
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Active damping: lead network
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
undamped active damped2 2
2 2( ) o
ADo
z zG z
z p
Active damping: use of a notch filter
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Root locus of the undamped system with one delay
converter side current sensors
grid side current sensors
-2 0 2-2
0
2
-2 0 2-2
0
2
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Root locus of the passive damped system
converter side current sensors
grid side current sensors
-2 0 2-2
0
2
-2 0 2-2
0
2
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Use of genetic algorithm for active damping optimisation
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Use of genetic algorithm for active damping optimisation
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Use of genetic algorithm for active damping optimisation
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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 the PCC
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
Influence of the conditions at PCC
introduction of 100 F capacitance
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
use active damping !
use passive damping !
strong grid
intermediate grid
weak grid
use active damping !
Grid stiffness influence: LCL-filter
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
-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.
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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
Marco Liserre [email protected]
Stability of power converters connected to the grid through LCL-filters
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