study of transient stability for parallel connected inverters in microgrid system works in stand...
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
F. Andrade, J. Cusido, L. Romeral, J. J. Cárdenas
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Introduction
[1] Networks have been undergoing significant changes in the last few years due to the growth of distributed generation (DG)
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Inverters can be classified in current source inverters (CSI) or voltage-source inverters (VSI).
Microgrid conceptthe Microgrid is defined as an aggregation of loads and microsources operating like single system providing both power and heat.
• Studies Stability
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
[Vásquez et al., 2009], [Guerrero et al, 2004] , [Guerrero et al, 2006] , [Coelho et al, 2002], Pogaku et al, 2007], [Nikkhajoei y Iravani, 2007]
• Studies Stability
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
[Vásquez et al., 2009], [Guerrero et al, 2004] , [Guerrero et al, 2006] , [Coelho et al, 2002], Pogaku et al, 2007], [Nikkhajoei y Iravani, 2007]
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• OutlineNon-linear
model
Lyapunov Function
Small Signal Result
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• A Mathematical model
Ren
ewab
leSo
urce
1
Filter 1Inverter 1 Inverter 2
Ren
ewab
leSo
urce
2
Filter 1
1V
1I
2V
2I
1Z 2Z
3Z
qqddi IVIVsP
dqqdi IVIVsQ
sPs
sP if
f
sQs
sQ if
f
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• A Mathematical model
Ren
ewab
leSo
urce
1
Filter 1Inverter 1 Inverter 2
Ren
ewab
leSo
urce
2
Filter 1
1V
1I
2V
2I
1Z 2Z
3Z
Pk p 0
QkVV v 0
droop curves (f vs P and V vs Q)
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• working in the time domain
ifpf Pk
ifvf QkVV
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• working in the time domain
ifpf Pk
ifvf QkVV
qd jVVV
cosVVd
sinVVq
d
q
V
Varctan
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• working in the time domain
ifpf Pk
ifvf QkVV
qd jVVV
cosVVd
sinVVq
d
q
V
Varctan
22qd
qddq
VV
VVVV
22qd VVV
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• A Generator
QV
VkVVV
QV
VkVVV
Pk
qfvqfdq
dfpqdfd
fpf
We have a set of equations which describe the behavior of each inverter
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• The whole system
22
22222222
22
22222222
222222
11
11111111
11
11111111
111111
QV
VkVVV
QV
VkVVV
Pk
QV
VkVVV
QV
VkVVV
Pk
qfvqfdq
dfpqdfd
fpf
qfvqfdq
dfpqdfd
fpf
• The network
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Ren
ewab
leSo
urce
1
Filter 1Inverter 1 Inverter 2
Ren
ewab
leSo
urce
2
Filter 1
1V
1I
2V
2I
1Z 2Z
3Z
2
2
1
1
22221212
22221212
12121111
12121111
2
2
1
1
2
1
323
331
2
1
q
d
q
d
q
d
q
d
V
V
V
V
GBGB
BGBG
GBGB
BGBG
I
I
I
I
E
E
YYY
YYY
I
I
• The network
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Ren
ewab
leSo
urce
1
Filter 1Inverter 1 Inverter 2
Ren
ewab
leSo
urce
2
Filter 1
1V
1I
2V
2I
1Z 2Z
3Z
2
2
1
1
22221212
22221212
12121111
12121111
22
22
11
11
2
2
1
1
00
00
00
00
q
d
q
d
dq
qd
dq
qd
V
V
V
V
GBGB
BGBG
GBGB
BGBG
VV
VV
VV
VV
Q
P
Q
P
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• The whole system
26
25
6253672
625
635266
26
256554616
26
25
6253572
625
635256
26
255564515
26
25
625342
625
63523
26
252414
23
22
6253372
322
635236
23
223521313
23
22
6253272
322
635226
23
222531212
23
22
625342
322
63523
23
222111
XX
XXXXX
XX
XXXXXXXXXXXX
XX
XXXXX
XX
XXXXXXXXXXXX
XX
XXXX
XX
XXXXXXXX
XX
XXXXX
XX
XXXXXXXXXXXX
XX
XXXXX
XX
XXXXXXXXXXXX
XX
XXXX
XX
XXXXXXXX
• Study of Stability for the Microgrid by means of Lyapunov’s method
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• System
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Ren
ewab
leSo
urce
1
Filter 1Inverter 1 Inverter 2
Ren
ewab
leSo
urce
2
Filter 1
1V
1I
2V
2I
1Z 2Z
3Z
Variable Value unitLine transmission (Z3) 0.5+3i Ω
Local load (Z1) 13+6i ΩLocal load (Z2) 25+13i Ω
Cut-off freq. of measuring filter (ωf) 37.7 rd/sFrequency droop coefficient (kp) 0.0005 rd/s/W
Voltage droop coefficient (kv) 0.0005 V/VARNominal frequency 377 rd/s
• Study of Stability for the Microgrid by means of Lyapunov’s method
• Lyapunov‘s Funtion
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
26
25
26
25
24
23
22
23
22
21)(
XX
XXD
CX
XX
XXB
AXXV
kk
XXV 0)(
0)0( V
0)0( f
• Lyapunov‘s Funtion
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
26
25
26
25
24
23
22
23
22
21)(
XX
XXD
CX
XX
XXB
AXXV
kk
XXV 0)(
0)0( V
0)()()()()()(
66
55
44
33
22
11
XX
XVX
X
XVX
X
XVX
X
XVX
X
XVX
X
XV
0)( XVdt
d
• Using Matlab for find the values
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
066
2/126
2555
2/126
25
4433
2/123
2222
2/123
2211
XXXXFXXXXE
XCXXDXXXCXXXXBXAX
• Eigenvalues of lineal system
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
XAX~~
15.032.29.154006.0061.00
42.25.3686.9097.0954.00
97.034.07.37956.0097.00
000005.154
157.094.00217.057.360
944.0157.00936.0466.07.37
A
;5.37;7.377.35;9.26;8.10;0 654321
• Simulation results of a DG Microgrid system by means of Power Electronics tools
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
0.5 1 1.5 21500
1600
1700
1800
1900
2000
2100
2200
t(s)
kW
P Power Elect. model
0.5 1 1.5 2-500
0
500
1000
t(s)
VAR
Q Power Elect. model
L=8mHL=0.5mHL=5mH
L=8mHL=5mHL=0.5mH
• Simulation results of a DG Microgrid system by means of Nonlineal model.
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
0.5 1 1.5 2-500
0
500
1000
t(s)
VA
R
Q Nonlineal Model
0.5 1 1.5 21600
1700
1800
1900
2000
2100
2200
t(s)
kW
P Nonlineal Model
L=8mHL=5mHL=0.5mH
L=8mHL=5mHL=0.5mH
• Study of stability by means of Lyapunov function and root locus plot.
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
0 0.2 0.4 0.6-20
-18
-16
-14
-12
-10
-8
-6
-4
-2
0Lyapunov function Plot
t(s)-40 -30 -20 -10 0 10
-30
-20
-10
0
10
20
30Root locus Plot
0
L=8mH
L=0.5mHL=8mH
L=0.5mH
L=8mH
L=0.5mH
L=8mH
L=0.5mH
L=8mH
L=0.5mH
0)( XV
• Conclusions
• A nonlinear state-space model of a Microgrid.
• It has analyzed the model by means of both stability studies: Lyapunov function and root locus plot.
• Using that Lyapunov function, we can determined the region of asymptotic stability.
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Thank you for your Attention
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Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
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