The document studies the transient stability of parallel connected inverters in a microgrid system operating in stand-alone mode. It develops a mathematical model to describe the behavior of each inverter and the electrical network. Lyapunov stability theory is then applied to analyze the stability of the overall microgrid system. Simulations are performed using Power Electronics tools to validate the theoretical results.

1. Study of transient stability for parallel connected
inverters in Microgrid system works in stand-alone
F. Andrade, J. Cusido, L. Romeral, J. J. Cárdenas
1

2. 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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3. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Microgrid concept
the Microgrid is defined as an aggregation of
loads and microsources operating like single
system providing both power and heat.
 Inverters can be classified in current
source inverters (CSI) or voltage-source
inverters (VSI).
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4. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Studies Stability
[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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5. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Studies Stability
[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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6. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Outline
Non-linear
model
Lyapunov
Function
Small
Result
Signal
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7. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• A Mathematical model
 
I1 I2
Z3
Renewable
Renewable
Source 1
Source 2
 
V1 Z1 Z2 V2
Inverter 1 Filter 1 Filter 1 Inverter 2
f
Ps Pi s
Pi s Vd I d Vq I q s f
Qi s Vd I q Vq I d
f
Qs Qi s
s f
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8. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• A Mathematical model
 
I1 I2
Z3
Renewable
Renewable
Source 1
Source 2
 
V1 Z1 Z2 V2
Inverter 1 Filter 1 Filter 1 Inverter 2
0 kpP
V V0 kv Q
droop curves (f vs P and V vs Q)
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9. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• working in the time domain
 f kp f Pi
V V kv Qi
f f
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10. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• working in the time domain
 f kp f Pi
V V kv Qi
f f

V Vd jVq
Vq V sin Vq
arctan
Vd V cos Vd
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11. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• working in the time domain
 f kp f Pi
V V kv Qi
f f
 
VqVd Vd Vq
 
2 2
Vd Vq V Vd jVq
2 2 Vq V sin Vq
V Vd Vq arctan
Vd V cos Vd
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12. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• A Generator
 f kp f P
 kp f Vd
Vd f Vd Vq Q
V
 kv f Vq
Vq Vd f Vq Q
V
We have a set of equations which describe the behavior of each inverter
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13. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• The whole system
1 f1 1 k p1 f1 1P
 k p1 V
f 1 d1
Vd 1 V
f 1 d1 V
1 q1 Q1
V1
 k v1 V
f 1 q1
Vq1 V
1 d1 V
f 1 q1 Q1
V1
2 f2 2 k p2 f2 2P
 k p2 V
f 2 d2
Vd 2 V
f 2 d2 V
2 q2 Q2
V2
 kv 2 V
f 2 q2
Vq 2 V
2 d2 V
f 2 q2 Q2
V2
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14. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• The network
 
I1 I2
Z3
Renewable
Renewable
Source 1
Source 2
 
V1 Z1 Z2 V2
Inverter 1 Filter 1 Filter 1 Inverter 2
I1d G11 B11 G12 B12 Vd 1
 
I1 Y1 Y3 Y3 E1 I1q B11 G11 B12 G12 Vq1
 
I2 Y3 Y2 Y3 E2 I 2d G12 B12 G22 B22 Vd 2
I 2q B12 G12 B22 G22 Vq 2
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15. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• The network
 
I1 I2
Z3
Renewable
Renewable
Source 1
Source 2
 
V1 Z1 Z2 V2
Inverter 1 Filter 1 Filter 1 Inverter 2
P1
Vd 1 Vq1 0 0 G11 B11 G12 B12 Vd 1
Q1 Vq1 Vd 1 0 0 B11 G11 B12 G12 Vq1
P2 0 0 Vd 2 Vq 2 G12 B12 G22 B22 Vd 2
Q2 0 0 Vq 2 Vd 2 B12 G12 B22 G22 Vq 2
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16. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• The whole system
 2 X2X5 X3X6 X3X5 X2X6
X1 1 X1 2 X2 X 32 3 4
2
X2 X 32 2
X2 X 32
 2 X2X5 X3X6 X3X5 X2X6
X2 1X 2 X1 X 3 5X2 X2 X 32 6X2 7X2
2
X2 X 32 2
X2 X 32
 2 X2X5 X3X6 X3X5 X2X6
X3 1X3 X1 X 2 5X3 X2 X 32 6X3 7 X3
2
X2 X 32 2
X2 X 32
 X2X5 X3X6 X3X5 X2X6
X4 1X 4 2 X 52 2
X6 3 4
X 52 2
X6 X 52 2
X6
 X2X5 X3X6 X3X5 X2X6
X5 1X5 X4X6 5X5 X 52 2
X6 6X5 7 X5
X 52 2
X6 X 52 2
X6
 X2X5 X3X6 X3X5 X2X6
X6 1 X6 X4X5 5 X 6 X 52 2
X6 6 X6 7 X6
X 52 2
X6 X 52 2
X6
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17. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Study of Stability for the Microgrid by means of Lyapunov’s
method
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18. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• System
 
I1 I2
Z3
Renewable
Renewable
Source 1
Source 2
 
V1 Z1 Z2 V2
Inverter 1 Filter 1 Filter 1 Inverter 2
Variable Value unit
Line transmission (Z3) 0.5+3i Ω
Local load (Z1) 13+6i Ω
Local load (Z2) 25+13i Ω
Cut-off freq. of measuring filter (ωf) 37.7 rd/s
Frequency droop coefficient (kp) 0.0005 rd/s/W
Voltage droop coefficient (kv) 0.0005 V/VAR
Nominal frequency 377 rd/s
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19. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Study of Stability for the Microgrid by means of Lyapunov’s
method
f (0) 0
• Lyapunov‘s Funtion
2 2 2 2
AX 12 X2 X3 2
CX 4 X5 X6
V (X ) B D
2 2 2 2
k X 2 X 3 k X5 X6
V (0) 0
V (X ) 0 X
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20. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Lyapunov‘s Funtion
2 2 2 2
AX 12 X2 X3 2
CX 4 X5 X6
V (X ) B D
2 2 2 2
k X 2 X 3 k X5 X6
V (0) 0
V (X ) 0 X
d
V (X ) 0
dt
V (X )  V (X )  V (X )  V (X )  V (X )  V (X ) 
X1 X2 X3 X4 X5 X6 0
X1 X2 X3 X4 X5 X6
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21. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Using Matlab for find the values

AX1 X 1 BX 2
X 2 1/ 2 
X2X2 C X 2
X 2 1/ 2 
DX 3 X 3 
CX 4 X 4
2 3 2 3
E X 2
X 2 1/ 2 
X5 X5 F X 2
X 2 1/ 2 
X6X6 0
5 6 5 6
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22. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Eigenvalues of lineal system

~ ~
X A X
37.7 0.466 0.936 0 0.157 0.944
0 36.57 0.217 0 0.94 0.157
154.5 0 0 0 0 0
A
0 0.097 0.956 37.7 0.34 0.97
0 0.954 0.097 9.86 36.5 2.42
0 0.061 0.006 154.9 2.32 0.15
1 0; 2 10.8; 3 26.9; 4 35.7 5 37.7; 6 37.5;
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23. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Simulation results of a DG Microgrid system by means of
Power Electronics tools
P Power Elect. model Q Power Elect. model
2200 1000
2100
2000
500
1900
VAR
kW
1800
0
1700
L=8mH L=8mH
1600 L=0.5mH L=5mH
L=5mH L=0.5mH
1500 -500
0.5 1 1.5 2 0.5 1 1.5 2
t(s) t(s)
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24. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Simulation results of a DG Microgrid system by means of
Nonlineal model.
P Nonlineal Model Q Nonlineal Model
2200 1000
2100
2000 500
VAR
kW
1900
1800 0
1700
L=8mH L=8mH
L=5mH L=5mH
L=0.5mH L=0.5mH
1600 -500
0.5 1 1.5 2 0.5 1 1.5 2
t(s) t(s)
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25. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• Study of stability by means of Lyapunov function and root
locus plot.
Lyapunov function Plot Root locus Plot
0 30
L=0.5mH L=0.5mH
-2
20
-4
-6 L=8mH
10 L=8mH
-8
L=0.5mH
-10 0 0
L=8mH L=0.5mH
-12
-10 L=8mH
-14
-16
-20
-18 L=8mH
L=0.5mH
-20 -30
0 0.2 0.4 0.6 -40 -30 -20 -10 0 10
t(s)

V (X ) 0
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26. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
• 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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27. Study of transient stability for parallel connected inverters in Microgrid system works in stand-alone
Thank you for your Attention
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