A Practical Approach for Faster Power System Transient Simulation · 2015. 6. 16. · V K sT Ts V...

A Practical Approach for Faster Power System Transient Simulation

34th Power Affiliates Program Annual Review

May 3, 2013

Soobae Kim, Thomas Overbye

Outline • Motivation

• Numerical Integration Method – Explicit method : Second Order RK method – Multi-rate method

• Practical problem definition

• Proposed Method

• Case study

• Conclusion

2

Motivation

• Power system dynamic simulation involves a wide range of different time constants [stiff]

• In order to avoid numerical instability, explicit integration methods require very small time steps for stiff power systems

• Only a small fraction of system states shows fast dynamics

• Therefore, simulation of an entire power system with small integration time steps is not efficient

• This work aims to reduce the transient simulation time by removing fast modes from a system.

3

Numerical Integration

Explicit method : Second Order RK method

1 1 21 1*( )2 2n nx x h k k-= + +

Region of Stability

where 1 1 1( , )n nk f x t- -=2 1 1 1( , )n nk f x k t h- -= + +

2( ) 11( ) 2zh z zz

rls

= = + +

1z <

2 0hl- < <Stable Region

if 2400l = -0.05h cycle<

Real hλ

Imag

hλ

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

-2

-1

0

1

2

3

Stable Region

4

5 Figure from Jingjia Chen and M. L. Crow, "A Variable Partitioning Strategy for the Multirate Method in Power Systems," Power Systems, IEEE Transactions on, vol. 23, pp. 259-266, 2008.

Multi-rate Method

1. At each macro step(H), the slow variables(ys) are integrated.

2. Interpolate the slow variables to provide approximation for each micro step(h)

3. Fast variables can be found using the interpolated slow variables as needed

Practical Problem : BIG negative eigenvalues • WECC case (17,710 buses) SMIB result

Bus # 42711 Min. eigenvalue

-2601.8

• SMIB Participation factor

EXST1 exciter model

NEED Very Small Time Step

6

7

EXST1 exciter model

<EXST1 Exciter model> Figure from PowerWorld Manual

12

2

(1 ) (1 )( )( ) 1 ( )( )

A F A F

A F A F A F A F

V K sT K sTT sV T T s T T K K s T T s p s p+ -

+ += = =+ + + + - -

2( ) ( ) 42

A F A F A F A F A F

A F

T T K K T T K K T Tp

T T± - + + ± + + -

=where

When two lead-lag compensators are neglected,

V2

V1

T(s)

when

10-4

10-3

10-2

10-1

100

101

102

103

104

-40

-20

0

20

40

60

Mag

nitu

de [d

B]

Bode Diagram

10-4

10-3

10-2

10-1

100

101

102

103

104

-100

-80

-60

-40

-20

0

Frequency [Hz]

Pha

se [d

eg]

Original EXST1Reduced EXST1

8

1

2

(1 )( )( )( )

A F

A F

V K sTT sV T T s p s p+ -

+= =- -

1

2

(1 ) (1 )( )( )( ) 1 /

A F A Freduction

A F

V K sT K sTT sV T T p s p s p- + +

+ += = =- - -

EXST1 and its reduction of small eigenvalue

200, 0.01,0.5, 0.4

A A

F F

K TK T

= == =

0.01, 2512p p+ -= - = -

When the exciter input voltage shows only low frequency range,

the reduced model can be used

Only abrupt voltage change introduces high freq. component

Voltage change is dependent on fault given

9

Fault Type Dependency

Generator or Load fault Bus or Line to ground fault

When Load 1 open at bus 1

When Bus 1 to ground fault

109876543210

1

0.995

0.99

0.985

0.98

0.975

0.97

0.965

0.96

0.955

0.95

0.945

0.94

109876543210

1.014

1.013

1.012

1.011

1.01

1.009

1.008

1.007

1.006

1.005

1.004

1.003

1.002

1.001

1

0.999

0.998When Gen ID 1 open at bus 1

Bus 1 Voltage Magnitude

9876543210

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

When Line 1-2 to ground fault

9876543210

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

10

Fault Type Dependency

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09Single-Sided Amplitude Spectrum

Frequency (Hz)

|Y(f)

|

Bus to Ground faultLine to Ground faultGenerator OutageLoad Outage

FFT results of voltage magnitude

0 10 20 30 40 50 60 70 80 90 1000

0.01

0.02

0.03

0.04

0.05

0.06Single-Sided Amplitude Spectrum

Frequency (Hz)

|Y(f)

|

Bus 5Bus 4Bus 3Bus 2Bus 1

11

Bus Fault Location Dependency

21.91.81.71.61.51.41.31.21.110.90.80.70.60.5

1

0.95

0.9

0.85

0.8

0.75

0.7

0.65

0.6

0.55

0.5

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

When Bus to ground fault at Bus 5,

Electrically Close Electrically Far

Bus 5 Bus 1

Bus 1

Bus 5

Voltage Magnitude at different bus

location FFT results

of voltage magnitude

Bus 1

Bus 5

12

Proposed Approach I : Fault Dependent Model Reduction

13

Case Study : GSO 37 bus Case

JO345 EXST1 exciter min. eigenvalue : -2102

1. JO345 Gen ID1 OPEN

2. DEMAR69 Solid Bus to Ground fault

3. JO345 Solid Bus to Ground fault

0 1 2 3 4 5 6 7 8 9 10-10

-8

-6

-4

-2

0

2

4

6

8

Time [sec]

Gen

erat

or M

var T

erm

inal

[Mva

r]

Generator JO345 ID=2 Terminal Output when JO345 ID=1 Gen Outage

Original ModelReduced Model

0 1 2 3 4 5 6 7 8 9 10150

160

170

180

190

200

210

Time [sec]

Gen

erat

or M

W T

erm

inal

[Mva

r]

Generator JO345 ID2 Terminal Output when JO345 ID=1 Gen Outage


14

GSO 37 bus Case : Simulation results

Generator Outage

Bus Voltage Magnitude at JO345

JO345 Gen ID1 OPEN

JO345 ID2 Gen MW output

JO345 ID2 Gen Mvar output

0 1 2 3 4 5 6 7 8 9 101.02

1.025

1.03

1.035

Time [sec]

Vol

tage

Mag

nitu

de [p

u]

Bus Voltage when JO345 ID=1 Generator Outage


0 1 2 3 4 5 6 7 8 9 10-10

0

10

20

30

40

50

60

70

80

90

Time [sec]

Gen

erat

or M

var T

erm

inal

[Mva

r]

Generator JO345 ID1 Terminal Output when DEMAR69 Bus Fault


0 1 2 3 4 5 6 7 8 9 10140

142

144

146

148

150

152

154

156

158

160

Time [sec]

Gen

erat

or M

W T

erm

inal

[Mva

r]

Generator JO345 ID1 Terminal Output when DEMAR69 Bus Fault


15


Bus Fault - FAR


DEMAR69 Solid Bus to Ground fault



0 1 2 3 4 5 6 7 8 9 10

0.92

0.94

0.96

0.98

1

1.02

1.04

1.06

Time [sec]

Vol

tage

Mag

nitu

de [p

u]

Bus Voltage when DEMAR69 Bus Fault


17

GSO 37 bus Case : Computation Time Benefits

Exciter Model

Numerical Integration

Max. Time Step

(cycle)

Time to Solve (sec)

Max. Angle Difference

(deg.)

Newton Solution Results

Total Newton

Solutions

Total Newton

Iterations

Number of Jacobian

Factorizations

Number of Forward

/Backward Substitutions

Original Single Rate 0.05 47.99 26.854 13975 12773 40 12773

Multi Rate 2.4 1.08 27.011 443 433 40 433

Reduced Single Rate 2.4 1.01 26.901 431 404 36 404

Bus Fault - FAR

DEMAR69 Solid Bus to Ground fault

Simulation Time : 10 sec

0 1 2 3 4 5 6 7 8 9 10-60

-40

-20

0

20

40

60

80

Time [sec]

Mva

r Ter

min

al O

utpu

t [M

var]

Generator JO345 ID=1 Terminal Output when JO345 Bus Fault


0 1 2 3 4 5 6 7 8 9 100

20

40

60

80

100

120

140

160

180

Time [sec]

Term

inal

Out

put [

MW

]

JO345 ID1 MW output when JO345 Bus Fault



16


Bus Fault - CLOSE

JO345 Solid Bus to Ground fault



0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Vol

tage

Mag

nitu

de [p

u]

Bus Voltage when JO345 Bus Fault


EXST1 eigenvalues are changed

21

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Vol

tage

Mag

nitu

de [p

u]



EXST1 eigenvalue = -2102

JO345 Solid Bus to Ground fault

0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Vol

tage

Mag

nitu

de [p

u]



EXST1 eigenvalue = -1102 EXST1 eigenvalue = -501 0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Vol

tage

Mag

nitu

de [p

u]



0 1 2 3 4 5 6 7 8 9 100

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Vol

tage

Mag

nitu

de [p

u]



EXST1 eigenvalue = -261 0 1 2 3 4 5 6 7 8 9 10

0

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [sec]

Vol

tage

Mag

nitu

de [p

u]



EXST1 eigenvalue = -27

Right eigenvector

22

1 1

( ) (0)i i

n nt t

i i i ii i

x t c e x el lf f y= =

D = = Då å

State \ Eigenvalue -2102 -1102 -501 -261 -27

Machine Angle 3.19E-10 -2.20E-09 -2.30E-08 -1.58E-07 0.000558929

Machine Speed w -1.78E-09 6.43E-09 3.06E-08 1.09E-07 -3.89E-05

Machine Eqp -6.81E-05 0.00013 0.000287 0.000554 -0.004242761

Machine PsiDp 8.22E-07 -3.04E-06 -1.53E-05 -6.02E-05 -0.022657701

Machine PsiQpp 1.11E-08 -4.10E-08 -2.04E-07 -7.88E-07 0.000444479

Machine Edp 0 0 0 0 0

Exciter Va 0.999999991 -1 -1 -1 0.999728538

Exciter VF 0.000119065 -0.00023 -0.00016 -0.00031 0.003303785

Governor Filter Output -1.71E-11 1.19E-10 1.28E-09 9.17E-09 -0.000218188

Governor Desired Gate 5.42E-15 -7.20E-14 -1.70E-12 -2.34E-11 5.55E-06

Governor Gate 5.42E-14 -7.22E-13 -1.71E-11 -2.36E-10 5.96E-05

Governor Turbine Flow -6.94E-17 1.76E-15 9.19E-14 2.46E-12 -6.81E-06

Revised Approach

23

Conclusion

24

• Practical approach is proposed in order to reduce a computational burden for power system transient simulation

• The method dynamically switches to original EXST1 or to reduced one depending on SMIB analysis and right eigenvector information

• The approach shows quite good performance in simulation time and accuracy. While maintaining the accuracy, the method reduces the simulation time about 98%

Questions ?

My mistake was

19

<EXST1 Exciter model>

V2

V1

T(s)

<SEXS Exciter model> Figure from PowerWorld Manual

Limiter block function is neglected

Tred(s)

The Reduced model

20

EXST1 model is used to define the reduced model transfer function by changing model parameters.

<EXST1 Exciter model>

V2

V1

Tred(s)

Questions ?

4

QSS(Quasi-Steady-State) approximation • Replaces short-term dynamics with their equilibrium

conditions • Thus transform the differential equations to algebraic ones

that are solved together with network equations

4

Numerical Integration • Approximate computation of an integral using numerical

techniques ( , , )

0 ( , , )

x f x y t

g x y t

·=

=

Explicit Method Implicit Method

1 1 1 1* ( , , )n n n n nx x h f x y t- - - -= + 1 * ( , , )n n n n nx x h f x y t-= +[Forward Euler] [Backward Euler]

For solving a stiff ODE,

Need very small time step

Larger time step is okay, But computational burden

A Practical Approach for Faster Power System Transient Simulation · 2015. 6. 16. · V K sT Ts V...

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