Computing Small-Signal Stability Boundaries for Large

To appear in IEEE Transactions on Power Systems Vol. 1, No. 2, May 2003

Computing Small-Signal Stability Boundaries for Large-Scale Power

Systems

Sergio Gomes Jr., CEPELNelson Martins, CEPELCarlos Portela, COPPE/UFRJ

Rio de Janeiro - Brazil

2

Acknowledgements

• Herminio Pinto, previously with CEPEL

• Paulo Eduardo M. Quintão, FPLF-PUC/RJ

• Alex de Castro, FPLF-PUC/RJ

3

System Equations Linearized at Singular Point X0

( )( )uy

ux,hx,fxT

==⋅ &

( ) 0xf 0 =

udyu

∆⋅+∆⋅=∆∆⋅+∆⋅=∆⋅

xcbxJxT &

4

Basic Equations for the Hopf Algorithm

( )[ ]1v =⋅

=⋅−⋅c

0vJT pspecλ

•λspec is the specified value for a complex pole, to be reached through an appropriate change to the system parameter “p”

•The second equation defines the norm of the eigenvector “v”

•Solution by Newton-Raphson

5

Small-Signal Stability and Security Boundaries

( ) 0, =ωσB5%15%

25%

σ

jω

Where ζ = 35%

( ) ωζ

ζσωσ ⋅−

+=21

,B

ωσλ j±=

6

Non-Linear Equations for Security Boundary (Modified Hopf) Algorithm

( ) 0=p,xf 0

( )[ ] 0vxJT 0 =⋅−⋅λ p,

1=⋅ vc

( ) 0=ωσ,B

7

Matrix Problem to be Solved at Every Iteration

JT −⋅σ

c

newrev

newimv

σ∆

ω∆

p∆

0

0

1

0

JT −⋅σ

T⋅ω−

T⋅ω

c

σ∂∂B

ω∂∂B

B−

revT ⋅ imvT ⋅−

revT ⋅imvT ⋅

revJp∂∂

−

imvJp∂∂

−

=•

8

Compact Form for Hopf Equations Involving the Prior Factorization of the

System Jacobian Equations

[ ] [ ] [ ]

[ ] [ ] [ ]

−

=

∆

ω∆

σ∆

⋅

ω∂∂

σ∂∂

⋅ℑ⋅ℜ⋅ℑ

⋅ℜ⋅ℑ−⋅ℜ

BpBB

0

1

0

211

211

vcvcvc

vcvcvc

9

Hopf Bifurcations

• Compute parameter values that cause critical eigenvalues to cross small-signal stability boundary

• Hopf bifurcations are computed for:

– Single-parameter changes

– Multiple-parameter changes (minimum distance in the

parameter space)

10

Experience with Inter-Area Oscillation Problems (1/2)

1984Taiwan1982-1983Western Australia

1978Scotland-England1975South East Australia

1971-1974Italy-Yugoslavia-Austria1971-1970Mid-continent Area Power Pool (MAPP)Late 1960’sSweden-Finland-Norway-Denmark1964-1978Western USA (WSCC)1962-1965Saskatchewan-Manitoba-Ontario West

1959Michigan-Ontario-Quebec

11

Experience with Inter-Area Oscillation Problems (2/2)

2000Brazilian North-South1998South-African Cone1998Argentinean Interconnected System (SADI)1992Venezuela-Colombia Interconnection1986Brazilian North-Northeast1985Ontario Hydro

1985-1987Southern Brazil

1985Ghana-Ivory Coast

Original slide from Dr. Prabha Kundur, with thelast five additions by the authors

12

North-Northeast Interconnection (1988)

RI O

XINGU

RI O

DAS

A LMA

S

RIO

AR

AGUAIA

RIO

TOCANT

I NS

R IOPA

RA NA

O ANT I NZI NH

RI O

S. F

R

ANCI S

CO

R IO

F OR M

OSO

R I OCOR R E N T E

R I O P AR AGU A C U

OJE Q

U I T I N H ONH

A

RI O

A RA

A I

R I OP R E T O

RIO

TO

CANT

INS

AGUA

S.SIMAOITUMBIARA

S.GOTARDO 2

JAGUARAPORTOMARIMBONDO

IRAPE

PIRAPORA

FSA

MOCAMBINHO

JEQUITINHONHA

ALMEN

S. ANTONIO DE JESUS

ITABAIANINHA

JARDIM

PENEDO

SOBRADINHO

P.BR

BOA ESPERANCA

EUNAPOLIS

ELISEU

PICOS

BOM NOME

COREMAS

ACU

TAP/XIN1

F.3

M.A.GRANDE

MIRADOR

BELEM

S.DIVISA

SALVADOR

SME/TMA

GOV

MASCARENHAS

ALTAMIRA

BELO MONTE

TUCURUI

REP

V. CONDE

IPI/MAR

ESTREITO

MARABA

SERRA QUEBRADA

IMPERATRIZ

FRAGOSO

LAJ/SOB

LAJ/IRE

BARREIRAS

IRECE SOB/SLV

IRE/SLV

MUSSURE

S. LUIS

FORTALEZA

PIRIPIRI

CRATEUSTERESINA

QUIXADA

MILAGRES

RUSSAS

MOSSORO

SOBRAL PENTECOSTES

B. ESP/MIL

SJP/MIL

MIRANDA

PRES. DUTRA

SEC.T/P

SERRA da

S. ROMAO

SME/BJLBOM JESUS

GOV

FUNIL

TRES MARIAS

XAVANTES

BANDEIRANTES

CORUMBA

NOVA PONTE

CAPIM BRANCO

CAMACARI

NIQUELANDIA

LAJ/SJP

SAM/BJL

IR

V.GRANDE

VALADARES

MANGABEIRA

VERMELHA

COLOMBIA

MARTINS

MESA

S.G.PARA TAQUARIL

VARZEADA PALMA

IPATINGA

T.OESTES.LUZIA

MACEIOJAGUARI

NATAL

NATAL II

DA LAPA

S. JOAO DO PIAUI

ICO

BJL/GMB

PERISES

C. PENA

BANABUIU

QUI/REC

RECIFE

MIL/REC

XINGO

ANGELIM

AFONSO

ITABAIANACICERODANTAS

OLINDINA

SENHOR DOBONFIM

JUAZEIRO

ANHANGUERA

EMBORCACAO

M.CLAROS

NEVES MESQUITA

DOURADACACHOEIRA

ITAPEBI

SAMAMBAIA

POMPEU

GU

PERITORO

ITAPARICA

PAULOMOXOTO MESSIAS

RIBEIRAO

BRASILIA GERAL

BARRO ALTO

CATU

DERIVACAO

S. ISABEL

RT.

C

C. GRANDE

TACAIMBOGOIANINHA

PAU FERRO

PAPAGAIO

S.3BLAJEADO

IPUEIRAS

PEIXE

CANA BRAVA

TUPIRATINS

500 kV AC TRANSMISSION REINFORCEMENT ALTERNATIVENORTH-SOUTH INTERCONNECTION

ALTERNATIVES

DCIMPERATRIZ

ACIMPERATRIZ

S. DA MESA S. DA MESA

PEIXE

TUPIRATINS

LEGEND

HYDROELECTRIC PLANT

SUBSTATION

LT 230 kV

LT 500 kV

EletrobrásGRUPO COORDENADOR DE PLANEJAMENTO

DO S S ISTEM AS ELÉTR ICO S - GCPSLT 345 kV

SOUTHEAST

NORTHEAST

NORTH

CENTRAL WEST

Northeast Generation

TucuruíGenerator

13

Brazilian N/NE InterconnectionSpontaneous oscillations controlled by operator through MW reduction

Tucurui's Power Plant Recordings (N/NE System)

0 1 2 3 4 5 6 7 8 9 10

Vt=4%

Pe=190 MW

1 s

Efd = 200 V

Peo= 1550 MW

Vto=104%

Time in seconds (s)

Oscillation problem later solved via PSS retuning

Section 7.9 of CIGRE TF 38.02.16 Document“Impact of the Interaction Among P. System Controls”

Field Tests to Determine the Effectiveness of the TCSC Controllers in Damping the

Brazilian North-South Intertie Oscillations

C. Gama L. Ängquist G. Ingeström M. NoroozianEletronorte ABB

15

The North-South Brazilian Interconnection

16

TCSCs Located at the Two Ends of the North-South Intertie

17

System Staged Tests – no PODs on 2 TCSCsBrazilian North-South System goes Unstable!

18

System Staged Tests – with 2 PODsBrazilian North-South System is now stable

19

Large-Scale, Decentralized Oscillation Damping Control Problem

20

Hopf Bifurcations – Test System Utilized

• System Data:– 2400 buses, 3400 lines/transformers, 2520 loads– 120 generators, 120 AVRs, 46 PSSs, 100 speed-governors– 1 HVDC link (6,000 MW)– 4 SVCs– 2 TCSCs

• Jacobian Matrix:– 13.062 lines– 48.521 non-zeros– 1676 states

21

Hopf Bifurcations – Test System Problem

• The TCSCs are located one at each end of the North-South intertieand are equipped with PODs to damp the 0.17 Hz mode

• The Hopf bifurcation algorithms were applied to compute eigenvalue crossings of the security boundary (5% damping ratio)for simultaneous gain changes in the two PODs

22

Eigenvalue (Pole) Spectrum for Brazilian North-South System (1,676 eigenvalues)

Poles in window near the origin are shown enlarged in next slides

-16

-12

-8

-4

0

4

12

16

-10 -8 -6 -4 -2 0Real Part (1/s)

Eigenvalue Spectrum

8

23

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

5%

North-Southmode

Adverse controlInteraction mode

Root Locus (focusing on the critical poles within small window) when Reducing the Gains

of the 2 TCSCs

24

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


of the 2 TCSCs

25

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


of the 2 TCSCs

26

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


of the 2 TCSCs

27

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


of the 2 TCSCs

28

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

K=0

K=0

North-Southmode


5%


of the 2 TCSCs

29

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%

Root Locus (focusing on the critical poles within small window) when Increasing the Gains of the

2 TCSCs

30

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

31

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

32

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

33

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

34

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

35

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

36

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

37

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

38

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

39

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

40

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

41

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%


2 TCSCs

42

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

K=2.16

K=2.16

North-Southmode


5%


2 TCSCs

43

North-South System with no TCSCs

Interarea Mode: λ = −0,0335 + j 1,0787 (f = 0,17 Hz, ζ = 3,11 %)

Step Response (N-S Tie line Power)

-1.0-0.8-0.6-0.4-0.20.00.2

0 4 8 12 16 20Tempo (s)

NNE

SSE

Mode Shape for Generator Rotor Speeds

44

North-South System with 2 TCSCs

Inter-area mode without TCSCs: λ = −0,0335 + j 1,0787 (ζ = 3,11%)

Inter-area mode with TCSCs: λ = −0,318 + j 1,044 (ζ = 29,14 %)

-1.0-0.8-0.6-0.4-0.20.00.2

0 2 4 6 8 10 12 14 16 18 20Tempo (s)

Step Response (N-S Tie line Power)

45

Computation of S. Signal Stability Boundaries

• Different algorithms, heuristics, parameter limits and boundaries– Change of a single parameter (Newton)– Change of multiple parameters (Lagrange)– Step-length control– Maximum and minimum limits for parameter values– Small-signal security boundaries

• Cases investigated by the authors– Simultaneous change in the gains of the 2 TCSCs– Independent changes in the gains of the 2 TCSCs– Varying other system parameters

46

Hopf Bifurcation Results

0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%Determining Kmin for 5% Security Boundary

47


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode



48


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode



49


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode



50


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%

K=0.0647

Determining Kmin for 5% Security Boundary

51


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode


5%Determining Kmax for 5% Security Boundary

52


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6

K=0.6

North-Southmode



53


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6North-South

mode



K=0.6

54


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6North-South

mode



K=0.6

55


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6North-South

mode



K=0.6

56


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6North-South

mode



K=0.6

57


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6North-South

mode



K=0.6

58


0.0

0.4

0.8

1.2

1.6

2.0

-0.8 -0.6 -0.4 -0.2 0.

K=0.6North-South

mode


5%

K=2.1172

Determining Kmax for 5% Security Boundary

K=0.6

59

Hopf Bifurcations Results

• Two crossings of the security boundary were found, for POD gains far away from the nominal values (Knom = 0.6 pu):

2.1172 > K > 0.0647

• Computational cost of Hopf bifurcation algorithm

– Single-parameter changes : 0.16 s (per iteration)

– Multiple-parameter changes : 0.35 s (per iteration)

60

Step Responses for Different TCSC Gains: Knom = 0.6 and Kmin = 0.06471

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.

0.1

0. 5. 10. 15. 20.Time (s)

Nominal Gain Minimum Gain

ξ = 5 %

Freq = 0.171 Hz

61

Step Responses for Different TCSC Gains:Knom = 0.6 and Kmax = 2.1172

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.

0.1

0. 5. 10. 15. 20.Time (s)

Nominal Gain Maximum Gain

ξ = 5 %

Freq = 0.0788 Hz

62

Jacobian Matrix for Brazilian System Model (13, 062 equations with 48,626 non-zeros )

63

Eigenvalue (Pole) Spectrum for Brazilian North-South System (1,676 eigenvalues)

-16

-12

-8

-4

0

4

8

12

16

-10 -8 -6 -4 -2 0Real Part (1/s)

Eigenvalue Spectrum

64

Hopf Matrix and its LU Factors (26,127 lines)

256,915 non-zeros108,288 non-zeros

65

Convergence Performance of the Hopf Algorithm (Kmin Solution)

0.0647119-0.053749 + j 1.073660.0647119-0.053749 + j 1.073650.0646849-0.053750 + j 1.073640.0596167-0.053862 + j 1.07593

0.00911156-0.055743 + j 1.113520.264483-0.049219 + j 0.9831510.600000-0.31793 + j 1.04370

Gains of the two PODsEigenvalue EstimateIteration

66

Convergence Performance of the Hopf Algorithm (Kmax Solution)

52.11725-0.024789 + j 0.49516752.11759-0.024790 + j 0.49518652.04841-0.024886 + j 0.497105

11.301.59247-0.059773 + j 0.52549421.301.14716-0.12364 + j 0.56710331.300.864511-0.19803 + j 0.60087241.300.727598-0.28045 + j 0.61841151.300.600000-0.34641 + j 0.579610

ζ (%)POD GainsEigenvalue EstimateIteration

67

Pole Spectrum of 1,676-State System

Small window near the origin is shown enlarged in next 3 slides

-16

-12

-8

-4

0

4

12

16

-10 -8 -6 -4 -2 0Real Part (1/s)

Eigenvalue Spectrum

8

68

Enlarged View of Small Window in the 1676-Pole Spectrum, for Knom = 0.6

0.00

0.40

0.80

1.20

1.60

2.00

-0.80 -0.60 -0.40 -0.20 0.00Real Part (1/s)

-0.31793 + 1.04375j

69

Enlarged View of Small Window in the 1676-Pole Spectrum, for Kmin = 0.06471

0.00

0.40

0.80

1.20

1.60

2.00

-0.8 -0.6 -0.4 -0.2 0Real Part (1/s)

-0.053777 + 1.0736j

70

Enlarged View of Small Window in the 1676-Pole Spectrum, for Kmax = 2.1172

0.00

0.40

0.80

1.20

1.60

2.00

-0.8 -0.6 -0.4 -0.2 0Real Part (1/s)

-0.024777 + 0.495155j

71

Conclusions

• Developed Hopf algorithm showed robust performance for large-scale systems

• Ensuring that all crossings of the security border have been determined is not an easy task for large-scale problems

• Finding the closest bifurcation point in the multi-parameter space requires optimization techniques (Minimum Distance to Hopf)

• Hopf algorithms and Minimum Distance to Hopf algorithms may be useful in the design of decentralized, non-linear controllers in power systems

• The authors have also been applying these algorithms to other problem areas.

Computing Small-Signal Stability Boundaries for Large

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