[PPT] - Modeling and Simulation of RLC Networks & Modal Equivalents for PowerPoint Presentation

SLIDE 1

Modeling and Simulation of RLC Networks & Modal Equivalents for Transmission Networks Containing Distributed Parameter Lines

1

Model Based Systems Engineering (MBSE) Lecture Series

N. Martins Talk - Part 2

Sergio L. Varricchio, CEPEL and multiple co-authors*

(*) Sergio Gomes Jr. (CEPEL), Nelson Martins (CEPEL), Francisco D. Freitas (University of Brasilia), Carlos M. Portela (COPPE), Leonardo Lima (Kestrel Power), Franklin C. Veliz (CEPEL)

SLIDE 2

Introduction;
Modeling electrical network components in the formulations

Descriptor System (DS) and Y(s) matrix;

Distributed parameter transmission line model for Y(s) matrix
The Sequential MIMO Dominant Pole Algorithm (SMDPA) for

computing the dominant poles and residue matrices associated with MIMO TFs of infinite systems;

Performance of a multi-bus equivalent (MIMO ROM) for a

transmission network with distributed parameter lines (poles computed by SMDPA);

Modeling infinite systems by Linear Matrix Approximations

Outline of Part II

SLIDE 3

Modal Analysis

– Involves the calculation of the system matrix, its poles & zeros and their sensitivities to system parameters; – Provides system structural information: mode shapes, participation factors, TF dominant poles, reduced order models; – Matrix models are used for the study of different power system phenomena:

Eletromechanical transients
Subsynchronous resonance
Harmonic performance;
Electromagnetic Transients.

(High-frequency network modeling, all transmission lines having distributed parameters)

Introduction to Part II (1/2)

(Lumped R-L-C dynamic network modeling); (Algebraic network modeling, R+jX);

SLIDE 4

High Frequency Modeling of Electrical Networks

– 3 formulations: State Space (SS), Descriptor Systems (DS) and Y(s) matrix; – The distributed parameter nature of transmission lines (TL) can be modeled by transcendental functions having infinite poles – Infinite systems; – Infinite systems are neatly modeled by the Y(s) matrix formulation; – Finite approximations of infinite systems can be modeled in the SS and DS formulations, where TLs are represented by cascaded RLC circuits; – Various NLA methods exist to efficiently compute ROMs for large scale DS models; – A main disadvantage of the Y(s) matrix formulation is the inexistance of robust and efficient algorithms for the computation of the poles and residue matrices of multivariable TFs.

Introduction to Part II (2/2)

SLIDE 5

Basic equations:

     

t t t u B x A x T   

     

t t t

T

u D x C y  

The components of the system are described by first-order ordinary differential

equations and algebraic equations as well;

The Kirchhoff Law of Currents for each individual node of the network is then added to

these equations, to define the connection among the various existing system components;

The DS model is a generalization of the SS model and leads to a simpler and more

efficient computer implementation.

Descriptor Systems (1/5)

SLIDE 6

   

D B A T C H   

1

s s

T

 Transfer Functions

TF SISO 

       

d s s u s y s H

T

   

 b

A T c

1

   

d j j H

T

    

 b

A T c

1

TF MIMO 

 Frequency Response

         

t t t t t t t t                        u u B x A T x A T 2 2

     

t t t t t t

T

        u D x C y

 Time Response (trapezoidal rule of integration)

Descriptor Systems (2/5)

SLIDE 7

j k kj C kj

v v i R v dt di L     

kj C

i dt dv C  R L C ikj vC iL vk vj k j vC R L C vk ikj vj k j RLC Parallel Voltage Source RLC Series

f j k f f f f

v v v i R dt di L      j Rf Lf k if vk vj vf

C L

v dt di L 

kj C L C

i v R i dt dv C     1   

C j k

v v v Kirchhoff Current Law Node k 



  m mk

i  → Nodes connected to k

→

Descriptor Systems (3/5)

SLIDE 8

L12 R12 L13 R13 C3 R3 L3 C1 L1 C2 R2 L2 v2

2

L

i barra 1 barra 2 barra 3 Parameters for 3-bus system

Input → i2 = 1 pu Output → v1 (pu)

v1

Ind. (mH)
Res. ()
Cap. (F)

L1 8.0 R2 80.0 C1 23.9 L2 424.0 R3 133.0 C2 8.0 L3 531.0 R12 0.46 C3 11.9 L12 9.7 R13 0.55 L13 11.9

i2

Nominal Frequency: 50 Hz Nominal Voltage: 20 kV MVA base: 10 MVA

Descriptor Systems (4/5)

Matlab script vs PSCAD Validation

SLIDE 9

Voltage at Bus # 1 following a step in the current injected in Bus 2

0.3
0.2
0.1

0.0 0.1 0.2 0.3 0.4 2 4 6 8 10 12 14 16 18 20

Tensão (pu) Tempo (ms)

MATLAB PSCAD

Voltage (pu) Time (ms)

Descriptor Systems (5/5)

Matlab script vs PSCAD Validation

SLIDE 10

 

13

2 12 20

i i i    

                                                                                                                                                                

3 2 3 2 1 13 12 30 20 10 13 12 3 2

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3 3 2 2 1

i i v v v v i i i i v i i v i i v R R R R R

f f C L C L C f

                                                                                        

3 2 1 13 12 30 20 10 13 12 3 3 2 2 1

3 3 2 2 1

v v v i i i i v i i v i i v dt d L L L C L C L C

f C L C L C f

 

7 1

30 3 3

3 3 3

i v R i dt dv C

C L C

   

(7)  (7)  (13)  (13) 

SLIDE 11

                                                                                     

3 2 3 2 1 13 12 30 3 3 20 2 2 10 1 3 2 1

1 1 1 i i v v v v i i i i v i i v i i v v v v

f f C L C L C

Descriptor System Matrices (2/2)

Let us consider the nodal voltages as the output variables:

SLIDE 12

Basic equations:

     

s s s u B x Y 

     

s s s

T

u D x C y  

Elements
Diagonal yii: Summation of the all elementary admittances

connected to node i;

Off-diagonal yij: negative value of summation of all elementary

admittances connected between nodes i and j;

SS and DS formulations are particular cases of Y(s): Y(s) = (sT – A)
Voltage sources are modeled by additional equations;
The derivative of Y(s) with respect to s, for the computation of the

system poles, is automatically built by coding simple rules that are similar to those used for building Y(s).

Y(s) Matrix Formulation

SLIDE 13

 Transfer Function

SISO TF  MIMO TF 

 

D B Y C H  

1

s s

T

       

d s s u s y s H

T

  

 b

Y c

1



  

d j j H

T

   

 b

Y c

1  Frequency Response

Y(s) Matrix Formulation

SLIDE 14

Parallel RLC Voltage Source Series RLC

R L C k j R L C k j k

k

f

i

k

f

v

k

f

R

k

f

L C s L s R yseries 1 1   

2 2

1 1              C s L s R C s L ds dyseries sC sL R yparallel    1 1 L s C ds dyparallel

2

1  

1

 





k

f n j j kj

i v y

k k k

f f f k

v i z v  

k k k

f f f

L s R z  

k k

f f

L ds dz  where:

Y(s) Matrix Formulation – Basic Elements

SLIDE 15

                    

13 13 12 12 1 11

1 1 L s R L s R C s y L12 R12 L13 i13 R13 i3 i2 i12 i20 C3 R3 L3 v3 i30 C1 v1 i10

3

L

i C2 R2 L2 v2

2

L

i barra 1 barra 2 barra 3 Rf Lf if vf                                                

f f f

v i i i v v v z y y y y y y y

3 2 3 2 1 33 31 22 21 13 12 11

1 1 1 1 1

 

s Y

Voltage Source

           

13 13 31 13

1 L s R y y

LCK

3 13 2 12 1 11

   

f

i v y v y v y

LTK

f f f

v i z v  

1

Y(s) Matrix – 3-bus System Equations

SLIDE 16

  2

13 13 13 31 13

L s R L ds dy ds dy   

    

                 

2 13 13 13 2 12 12 12 1 11

L s R L L s R L C ds dy

 

             ds dz ds dy ds dy ds dy ds dy ds dy ds dy ds dy ds s d

f 33 31 22 21 13 12 11

Y L12 R12 L13 i13 R13 i3 i2 i12 i20 C3 R3 L3 v3 i30 C1 v1 i10

3

L

i C2 R2 L2 v2

2

L

i barra 1 barra 2 barra 3 Rf Lf if vf

Fonte de Tensão

 

f f f f

L ds L s R d ds dz   

Y(s) Matrix – 3-bus System Equations

SLIDE 17

L12 R12 L13 i13 R13 i3 i2 i12 i20 C3 R3 L3 v3 i30 C1 v1 i10

3

L

i C2 R2 L2 v2

2

L

i barra 1 barra 2 barra 3 Rf Lf if vf

                                                   

f f f

v i i i v v v z y y y y y y y

3 2 3 2 1 33 31 22 21 13 12 11

1 1 1 1 1                                                        

f f

v i i i v v v v v v

3 2 3 2 1 3 2 1

1 1 1

A compact description      

s s s u D x C y  

MIMO System      

s s s u B x Y 

     

s s s u D x C y  

Y(s) Matrix – 3-bus System Equations

SLIDE 18

DS compared to Y(s) Matrix:

Transfer Impedance between buses 1 & 2

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 250 500 750 1000 1250 1500

|z12| (pu) Frequência (Hz)

Sistema Descritor Matriz Y(s)

200
100

100 200 250 500 750 1000 1250 1500

Fase(z12) (graus) Frequência (Hz)

Sistema Descritor Matriz Y(s)

Y(s) matrix- Freq. response results for 3-bus system

SLIDE 19

Matriz Y(s) – Modelagem de Redes

z y y k vk j vj ik ij

 

l z z

c

  sinh

 

l y z

c

  csch 1

     

l l yc     csch coth

   

s Z s Y yc 

c c

y z 1 

   

s Y s Z                                                    

j k s m m s j k j k

v v y y y y v v z y z z z y i i 1 1 1 1

       

l y l l y z y y

c c s

        csch csch coth 1

 

l y y

c s

  coth

 

l y y

c m

  h csc         2 tanh l y y

c

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 20

Matriz Y(s) – Modelagem de Redes  

l y y

c s

  coth

 

l y y

c m

  h csc

   

s Z s Y yc 

   

s Y s Z  

   

l l ds d y l ds dy ds dy

c c s

csch coth     

     

l l l ds d y l ds dy ds dy

c c m

coth csch csch                ds dZ y ds dY ds dy

c c 2

2 1           2 1 ds dZ Y ds dY Z ds d

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 21

Matriz Y(s) – Modelagem de Redes

1

C s Y 

1

C ds dY 

     

g i e

Z Z Z Z   

     

ds dZ ds dZ ds dZ ds dZ

g i e

  

   

e e

L s Z

1



   

e e

L ds dZ

1



Positive sequence capacitance & indutance, computed by matrix reduction considering ideal conductor and soil.

  

e

L C

1 1,

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 22

Matriz Y(s) – Modelagem de Redes

 

     

s m s n n s k Z

s i 

x y Condutor Aço ri re

 

e

r s s k     2 1

 

       

1 1 1 1

      K I K I s n

 

       

1 1 1 1 1 1

      K I K I s m     s r

i

    s r

e 1

I0, I1 → Modified Bessel functions of first kind for integer orders 0 & 1, respectively. K0, K1 → Modified Bessel functions of first kind for integer orders 0 & 1, respectively.

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 23

Matriz Y(s) – Modelagem de Redes

 

                 

            

2

1 s m ds s dm s n ds s dn s m s k s m s n ds s dk n ds dZ

s i

   

s s k ds s dk 2 

                 

ds d d dK I K ds d d dI ds d d dK I K ds d d dI ds s dn

1 1 1 1 1 1 1 1 1 1 1 1

                   

                 

s d d d dK I K ds d d dI s d d d dK I K s d d d dI ds s dm

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

                   

s ds d 2    s ds d 2

1 1

  

   

   

1

I d dI

     

2

2 1

      I I d dI

   

    

1

K d dK

     

2

2 1

       K K d dK

 

     

s m s n n s k Z

s i 

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 24

Matriz Y(s) – Modelagem de Redes

x y D12 H1 1 2 3  

   

                                       

 

    3 1 3 1 2 2 2 2 3 1

2 ln ln 6

i i j j ij j i ij j i i i i g

D H H D p H H H p H s Z

 

   

                                     

 

    3 1 3 1 2 2 3 1

ˆ ˆ ˆ ln ln 6

i i j j ij ij ij j i ij i i i i g

ds p H d p H s D H H H ds dp p H s H p H ds dZ

 

     s s p 1

 

2 2

2 ˆ

ij j i ij

D p H H H    

 

2 2

3

p s ds dp      

 

   

ds dp D p H H p H H ds p H d

ij j i j i ij 2 2

2 2 2 ˆ      

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 25

vf 1 2 Rf Z Y Y R3 L3 C3 3 Lf L12

 

               ds dz ds dy ds dy ds dy ds dy ds dy ds dy ds dy ds s d

f 33 32 23 22 21 12 11

Y

25

 

Z Y C s L s R s y 1 1

3 3 3 33

    

         

l s s y ds d C L s R L ds dy

c

      coth

3 2 12 3 3 33

     

l s s yc  coth

 

               

f

z y y y y y y y s 1 1

33 32 23 22 21 12 11

Y

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 26

vf 1 2 Rf Z Y Y R3 L3 C3 3 Lf L12

26

f f f

v i v v v z y y y y y y y                                           1 1 1

3 2 1 33 32 23 22 21 12 11

 

1 

t

c

 

              

f

i v v v v

3 2 1 3

1

 

1 

t

b

 

f

v s b x Y  x ct v 

3

 

f t

v s v b Y c

1 3 



 

  b

Y c

1 1 3 31 

   s v v v s G

t f

Y(s) Matrix – Distrib. Param. Transmission Line

SLIDE 27

Sequential MIMO Dominant Pole Algorithm

(SMDPA)

SLIDE 28

   

D B Y C H  

1

s s

T

   

e T

s s D B Y C H  

1

 Direct Matrix

i

D

e i

D D D  

 

ds s d

s

H K

 

 lim

 

e s i

s s D K H D   

 

lim

 

e i i i i

s s s D D K R H      

 1

 Partial Fraction Expansion

SMDPA – Fundamental Concepts (1/3)

SLIDE 29

   

D K H H    s s s ˆ

 



  

       

k i i i k i k k k

j j

1

ˆ R R H

 Strictly Proper part of H(s)

 

D K R H     

 

s s s

i i i 1

The pole k will be dominant in H(s) if the magnitude of ( ||Rk||2 / |k| ) is sufficiently large so as to cause a peak in the plot of max[Ĥ(j)] in the close neighborhood of the frequency k. Let k = k + j k be a pole with an associated residue matrix Rk, then:

 Dominant Pole

SMDPA – Fundamental Concepts (2/3)

SLIDE 30

 Reduced Order Model (ROM)

  Set of N dominant poles and associated residue matrices.

   

D K R H H      



   

s s s s

N i i i N

i

1

       



   

       

N i i i N

i

s s s s s s

1

R D K H H H H

   

 

     j j

MOR

H

max

 MIMO ROM Deviation TF  Norm of MIMO ROM Deviation TF

SMDPA – Fundamental Concepts (3/3)

SLIDE 31

 The set of dominant poles of H(s) may be efficiently computed only when eliminating

from H(s):

The N previously computed poles (deflation);
Matrices K and D.

 The Newton method should therefore be applied to the MIMO ROM deviation TF:

   



   

     

N i i i

i

s s s s

1

R D K H H

SMDPA – Newton Method (1/3)

SLIDE 32

 

 

lim

1 min

 

  

s

H

   

min

   s s f

 Newton equationing      

 

   

 

k k k k k

s ds s d s s s

min * min min

) ( 1 v H w

     

  

     

ds s d ds s d ds s d

N

H H H  

   



   

    

N i i i N

i

s ds s d

1 2

K R H

   

B X Y

B

 s s

   

C X Y

C

 s s T

       

s ds s d s ds s d

T B C

X Y X H  

   

s s s

k k

  

1

vmin, wmin  eigenvectors associated with min.

The sparse solution of these two matrix equations sistemas require a single LU factorization and various solves.

mim  minimum eigenvalue of

  1



s H

 Pole of

 

s H

mim, vmin, wmin) function eig of Matlab.

SMDPA – Newton Method (2/3)

SLIDE 33

0.1 1.0 10.0 1000 1500

max[Ĥ(j)] (pu) Frequência (Hz)

0.1 1.0 10.0 1000 1500

max[Ĥ(j)] (pu) Frequência (Hz)

 Determining the initial pole estimates

f1 f2 f3 f4 f5 f6 f7

 

 

8 1

2 f f j    s f8

SMDPA – Newton Method (3/3)

SLIDE 34

Computation of the Residue Matrix for a Pole

 Definition of the Integration Curve



j Real(s)

   

 

  







4 1

1

k P P C

ds s ds s

k k

H H 

P2 P1 P3 P4 j Real(s) ≡ P5 C

 ds

s j

C



  H R 2 1

SMDPA – Pole Residue Matrix (1/2)

SLIDE 35

       

k k k

P l P P l        1 5 . , , H J

Legendre-Gauss Method with Error Control

k

P

k k

P P  

k k

P P   2

 

k k

P l P    1

k k

P l P  

 

k k k

P m P    1

k k k

P m P  

1 1  k

P

   

k k

P l P s       1 5 .

   

 

  

      



4 1 1 1 k m l P l P P l P C

k k k k k

ds s ds s H H

 

 

  

  

4 1 1 1

, , 2

k m l M i k i i k

k

P l w P J

 

 

  

   

4 1 1 1

, , 2 2 1

k m l M i k i i k

k

P l w P j J R

 ,

, , 2 2 1

1 1



 

   

k

m l M i k i i k k

P l w P j J R

4 , , 1   k

 , 2 , 1 , , 2   q m

q k

     

2 1 2 1   

 

q k q k q k k

R R R

max 4 1

4      

 k k

Determining the Error

SMDPA – Pole Residue Matrix (2/2)

SLIDE 36

The 34-bus test system with 25 distributed parameter TLs

24 13 12 26 27 34 11 28 7 8 29 9 32 25 21 20 23 33 22 31 30 10 6 1 2 3 4 5 18 19 17 14 15 16

Element Quant. Buses 34 TLs 25 Branches 12 Trafos 16 Loads 16 Generators 10

SMDPA – 34-bus Test System Results (1/4)

SLIDE 37

5 10 15 20 25 30

90
80
70
60
50
40
30
20
10

Imaginário (rad/s) 1000 Real (1/s)

5 Iterações 6 Iterações 7 Iterações 11 Iterações 12 Iterações

Most dominant pole spectrum for 34-bus system (2x2) TF. Color code identifies iterations required for SMDPA convergence from the initial set of 37 estimates

SMDPA – 34-bus Test System Results (2/4)

SLIDE 38

0.0 1.0 2.0 3.0 4.0 500 1000 1500 2000 2500 3000 3500 4000

max (pu) Frequência (Hz)

SMDPA – 34-bus Test System Results (3/4)

0.0 1.0 2.0 3.0 4.0 500 1000 1500 2000 2500 3000 3500 4000

max (pu) Frequência (Hz)

Infinite ROM  151

SLIDE 39

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1000 2000 3000 4000

MOR (pu) Frequência (Hz)

SMDPA – 34-bus Test System Results (4/4)

SLIDE 40

ROM and LMA Errors

     

s s s

N

H H H   D K R    



   

s s

N i i i

i

1

 

e T

s D B Y C 

1

     

s s s

L L

H H H  

ROM
LMA – Linear Model (finite) Aproximation

 

e L T L s

D B A T C  

1

 

 

   

% 100

max max

        

 

 

f f

d j d j

MOR

H H

Error Measures for ROM and LMA

 

 

   

% 100

max max

        

 

 

f f

d j d j

L MLA

H H

SMDPA - Infinite × Finite Systems (1/5)

SLIDE 41

How many  circuits to use in the TL models of the 34-bus system?
The tables below compare the performances of finer LMAs ROM-151

n nD nL nL  n 300 15063 15182 345.05 400 20063 20182 458.68 500 25063 25182 572.32 600 30063 30182 685.95

Statistics for LMA models n  Number of  circuits per TL nD  Number of differencial equations nL  Dimension of matrices A and T n  Dimension of matrix Y(s) ( no of buses + no of voltage sources  44)

n

MLA



(%)

MOR

 (%) 300 4.59 × 101 400 2.58 × 101 1.94 × 101 500 1.65 × 101 600 1.15 × 101

Error Measures for LMA and ROM

SMDPA - Infinite × Finite Systems (2/5)

SLIDE 42

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 2000 4000 6000 8000

MOR (pu) Frequência (Hz)

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1.0E-01 1.0E+00 2000 4000 6000 8000

MOR (pu) Frequência (Hz)

Simple procedure for improving ROM Fidelity of infinite systems
ROM – 151 for MIMO TF of 34-bus system

These eight spikes are due to the 8 pairs of poles that are dominant in the adjacent 4 - 8 KHz region, and cause the error curve within the 0 – 4 KHz window to rise. Frequency Range of Interest

SMDPA - Infinite × Finite Systems (3/5)

SLIDE 43

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1000 2000 3000 4000

MOR (pu) Frequência (Hz)

1.0E-05 1.0E-04 1.0E-03 1.0E-02 1000 2000 3000 4000

MOR (pu) Frequência (Hz)

Assessing ROM Fidelity for the MIMO TF of 34-bus System
ROM - 151 ROM - 167

ROM  151 ROM  167

SMDPA - Infinite × Finite Systems (4/5)

SLIDE 44

(300, 0.459) (400, 0.258) (500, 0.165) (600, 0.115) 0.0 0.1 0.2 0.3 0.4 0.5 100 200 300 400 500 600 700

Erro (%) Número de s

(300, 0.459) (400, 0.258) (500, 0.165) (600, 0.115) 140 150 160 170 180 0.0 0.1 0.2 0.3 0.4 0.5 100 200 300 400 500 600 700

Ordem Erro (%) Número de s

(300, 0.459) (400, 0.258) (500, 0.165) (600, 0.115) (151, 0.194) (167, 0.093) 140 150 160 170 180 0.0 0.1 0.2 0.3 0.4 0.5 100 200 300 400 500 600 700

Ordem Erro (%) Número de s

Comparing the Performances of LMAs ROMs

LMAs ROMs

SMDPA - Infinite × Finite Systems (5/5)

SLIDE 45

Conclusions for Part II (1/2)

Electrical network modeling with its RLC series and paralell components,

current and voltage sources, in the DS and Y(s) matrix formulations;

Development of the first reliable Newton algorithm for computing the

dominant poles of SISO and MIMO TFs of infinite systems (SMDPA). The method’s reliability comes from the very effective pole deflation procedure and the accurate computation of the pole residue matrices; – The residue is numerically computed as the path integral around the pole which was here obtained by the Legendre-Gauss quadrature method.

The modeling accuracy of the DS and Y(s) formulations was verified by

the close matching between their simulation results and those obtained with ATP or PSCAD for various test systems.

Y(s) allows the exact modeling of linear systems incorporating time delay.

SLIDE 46

Conclusions for Part II (2/2)

SMDPA yields high fidelity ROMs over a specified frequency window,

for use as equivalents in transmission network electromagnetic transient studies.

Multi-bus ROMs produced by SMDPA directly from infinite system

models are a more practical option than LMA (Linear Matrix Approximation) models;

In attempting to obtain accuracy over a wider frequency range, LMA

models may soon reach uncomfortably large dimensions and present severe numerical stiffness.

Application of SMDPA to other areas of engineering, physics and

mathematics is yet to be explored.

SLIDE 47

MARTINS, N., BOSSA, T.H. S. A Modal Stabilizer for the Independent Damping Control of

Aggregate Generator and Intraplant Modes in Multigenerator Power Plants. IEEE Transactions on Power Systems, USA, Vol. 29, Issue: 6, p. 2646 - 2661, November 2014.

DE MARCO, F.J., MARTINS, N., FERRAZ, J.C., An Automatic Method for Power System

Stabilizers Phase Compensation Design. IEEE Transactions on Power Systems, USA, Vol. 28, Issue: 2, p. 997-1007, May 2013.

VARRICCHIO, S.L., FREITAS, F.D., MARTINS, N., VELIZ, F.C., Computation of Dominant Poles

and Residue Matrices for Multivariable Transfer Functions of Infinite Power System

Models. IEEE Transactions on Power Systems, USA, Vol. 30, Issue:3, p. 1131 - 1142, May

2015.

GOMES Jr, S.; MARTINS, N.; PORTELA, C. Sequential Computation of Transfer Function

Dominant Poles of s-Domain System Models. IEEE Transactions on Power Systems, USA,

Vol. 24, No. 2, p. 776-784, May 2009.
GOMES Jr, S.; PORTELA, C.; MARTINS, N., Detailed Model of Long Transmission Lines for

Modal Analysis of ac Networks. Proc. International Conference on Power System Transients, Rio de Janeiro, Brazil, June 2001.

ROMMES, J. and MARTINS, N., Efficient Computation of Multivariable Transfer Function

Dominant Poles Using Subspace Acceleration. IEEE Transactions on Power Systems , USA,

Vol. 21, No. 4, p. 1471-1483, November 2006.

References

SLIDE 48

Thank you!

Nelson Martins

48