Transient Stability Analysis of Inverter-Interfaced Distributed Generators in a Microgrid System

F. Andrade, J. Cusidó, L. RomeralMotion Control & Industrial Applications Center, MCIA

Universitat Politècnica de Catalunya. CTM Centre Tecnològic

Intelligent Microgrids integrate different energy resources, especially renewable source, to provide dependable, efficient operations, while works connected to the grid or islanding mode.

It can be ensured an uninterrupted reliable flow of power, economic and environmental benefits while minimizing energy loss through transmission over long distances.

The use of intelligent power interfaces between the renewable source and the grid is required.

MATHEMATICAL MODEL

INTRODUCTION

Each generator has a power DC renewable source, a DC/AC inverter, a low pass filter and it is managed by two control loops

The study shows:

a mathematic model of a Microgrid system in stand-alone based in parallel connected inverters

works with no-lineal tool and computer simulations, phase-plane trajectory analysis and method of Lyapunov for evaluate the limits of the small signal models.

Fig 2: Methodologies applied for bearing diagnosis.

Inverter 1

1V

1I

2V

2I

Inverter 2

PQ

LowFilter

Droop curves

V-I CONTROL

PQ

LowFilter

Droop curves

V-I CONTROL

Z

1ZRenewableSource 1

RenewableSource 2

2Z

Electric Utility

Fig 1: Connected Microgrid system based in power electronic interfaces.

Microgrid

The PQ controller: QkVVPk vipi 00

Variable Value UnitOperating Voltage range 218,5 – 241,5 VrmsOperating Frequency range 49 - 51 HzFreq. droop coefficient (Kp1 = Kp2) 1,33e-4 rad.s-1/W

Voltage droop coefficient (Kv1 = Kv2) 0.0015 V/VAR

Cutoff frequency filter 5 Hz

The power angle between both generators: 2112

1221

22

22

42

122121

21

41

122122

42

122121

41

cos63.063.01011

cos63.063.0104.1

sin63.01082

sin63.01067

VVVVQ

VVVVQ

VVVP

VVVP

)cos(0307.00307.04.31102.7

)cos(0307.00307.04.31102.7

)sin(0027.0104.34.31108.9

)sin(0027.0108.24.31108.9

)(

154255

35

154244

34

15425

53

33

15424

52

32

321

XXXXXX

XXXXXX

XXXXXX

XXXXXX

XXX

Xf

iii

iii

QVV

P

0482.04.31102.7

0042.04.31108.93

3

Working in the time domain:

The PQ power:

The whole model:

Equilibrium points

Fig 3: the equilibrium point X0 the range of the variable X5

The equilibrium point X0=[0.0019 320.1 320.1 239.7 239.6]

)cos(0307.00307.04.31102.70

)cos(0307.00307.04.31102.70

)sin(0027.0104.34.31108.90

)sin(0027.0108.24.31108.90

0

0)(

154255

3

154244

3

15425

53

3

15424

52

3

32

XXXXX

XXXXX

XXXXX

XXXXX

XX

Xf

220 230 240 250 260-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15Set of state equation in function of X5

X5

f1(X)f2(X)f3(X)f4(X)f5(X)

X0

STABILITY OF THE SYSTEM

CONCLUSIONS

In this paper, a nonlinear state-space model of a Microgrid is presented. The model includes the most important dynamics. The no-linear model can find the equilibrium points. The model has been analyzed by means of both studies, first, a study of small signal stability by mean of linearization and root locus plot and transient stability by mean of Lyapunov function.

The studies of small signal could be done for adjustment the controller and improve the transient response and the steady-state error. Using that Lyapunov function the region of asymptotic stability, the size of the disturbed and his duration time could be determined. These tools will allow the design of Microgrid systems with loads, generators and storage systems assuring the global stability of the system.

Using the Jacobian matrix of f(X) at the equilibrium point.

Fig 4: Root locus for 1.3e-4< Kp <7.8e-4

An analysis of the equilibrium point and small-signal stability

0

)()(

)()(

5

5

5

5

5

1

1

1

XXXf

XXf

XXf

XXf

A

407.393599.70000.00000.0363.2

3597.7407.390000.00000.0363.2

0162.00009.004.320000.090.153

0009.00145.00000.004.329.153

0000.00000.0000.10000.10000.0

A

05.328.46;05.32;1.702.16;1.702.16 54321 ii

Transient Stability Analysis of Inverter-Interfaced Distributed Generators in a Microgrid System

F. Andrade, J. Cusidó, L. RomeralMotion Control & Industrial Applications Center, MCIA

Universitat Politècnica de Catalunya. CTM Centre Tecnològic

-50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0-40

-30

-20

-10

0

10

20

30

40

Real

Imag

Root Locus Plot

Stability of Lyapunov

Considering only the first generator; it has been considerate variations in the angle, active power and frequency. It was negligence variation in the voltage. The second generator is an infinite bus with fixed frequency and voltage

)(4.31 122

21

ZfZZ

ZZ

)cos(47.1)sin(14047.1)( 111 ZZZf WhereWhere

-4 -3 -2 -1 0 1 2 3 4

-5

-4

-3

-2

-1

0

1

2

3

4

5

x

y

Fig 6: phase portrait in Z1 –Z2. (x=Z1 y=Z2)

0 0.2 0.4 0.6 0.8 1-200

0

200

400

600

800

1000

1200

1400

Time (s)

Act

ive

Po

we

r (W

)

Active Power DG1

0 0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

Active Power DG2

Time (s)

Act

ive

Po

we

r (W

)

Kp incresesKp increses

0 1 2 3 446

47

48

49

50

51

52

53

Seg

Fre

q (

Hz)

Frequencies DG1 and DG2

0 1 2 3 4-7

-6

-5

-4

-3

-2

-1

0

1Angle between V1 and V2

Seg

Ra

d

Freq1Freq2

ang1

1

0

22121 )(4.31

2

1),(

Z

duufZZZZV

-3 -2 -1 0 1 2 3-1000

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

Z1

f(Z1)

Simulations result

Fig 5: Function f(Z1)

1121 )(4,31),( ZZfZZV

the region of asymptotic stability is obtained as

)(),( 0121 ZVZZV

Fig 7: Active Power dispatch by each DGs when it is increase the Kp Fig 8: two large disturbances in the Microgrid.