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

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)

• 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

• 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);

• 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)

• Basic equations:

ttt uBxAxT

ttt TuDxCy

• 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)

DBATCH 1

ss T

Transfer Functions

TF SISO

dssu

sysH T

bATc

1

djjH T

bATc1

TF MIMO

Frequency Response

tttt

ttt

t

uuBxATxAT

22

tttttt T uDxCy

Time Response (trapezoidal rule of integration)


jkkjC

kjvviRv

dt

diL

kjC i

dt

dvC

R

L

C

ikj

vC

iLvk vj

k jvC

R L C

vk ikj vj

k j

RLC Parallel Voltage Source RLC Series

fjkff

f

f vvviRdt

diL

jRf Lf

k

ifvk vjvf

CL v

dt

diL

kjCLC iv

Ri

dt

dvC

1

0 Cjk vvv

Kirchhoff Current Law

Node k 0m

mki

→ Nodes connected to k

→


L12

R12

L13

R13

C3 R3 L3

C1 L1

C2 R2 L2v2 2Li

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

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

0 2 4 6 8 10 12 14 16 18 20

Ten

são

(p

u)

Tempo (ms)

MATLAB

PSCAD

Voltage (

pu

)

Time (ms)

Descriptor Systems (5/5) Matlab script vs PSCAD Validation

130 21220 iii

3

2

3

2

1

13

12

30

20

10

13

12

3

2

1

1

1

11

11

1111

1

11

11

11

111

1

11

111

1

11

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

71

30

3

3 33

3 ivR

idt

dvC CL

C

(7)

(7)

(13)

(13)

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:

• Basic equations:

sss uBxY

sss TuDxCy

• 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

Transfer Function

SISO TF

MIMO TF DBYCH 1

ss T

dssu

sysH T

bYc

1

djjH T

bYc1

Frequency Response

Y(s) Matrix Formulation

Parallel RLC Voltage Source Series RLC

R L Ck j

R

L

C

k j k kfi

kfv

kfR

kfL

CsLsR

yseries 1

1

2

2

1

1

CsLsR

CsL

ds

dyseries

sCsLR

yparallel 11

LsC

ds

dyparallel

2

1

01

kf

n

j

jkj ivy

kkk fffk vizv

kkk fff LsRz

k

k

f

fL

ds

dz

where:

Y(s) Matrix Formulation – Basic Elements

13131212

111

11

LsRLsRCsy

L12

R12

L13

i13

R13

i3i2

i12

i20

C3 R3 L3v3

i30

C1v1

i10

3LiC2 R2 L2v2 2Li

barra 1

barra 2 barra 3

Rf

Lf

if

vf

fff

v

i

i

i

v

v

v

z

yy

yy

yyy

3

2

3

2

1

3331

2221

131211

100

010

001

000

001

00

00

1

sY

Voltage Source

1313

3113

1

LsRyy

LCK 0313212111 fivyvyvy

LTK fff vizv 1

Y(s) Matrix – 3-bus System Equations

21313

133113

LsR

L

ds

dy

ds

dy

21313

132

1212

121

11

LsR

L

LsR

LC

ds

dy

dsdz

dsdydsdy

dsdydsdy

dsdydsdydsdy

ds

sd

f000

00

00

0

3331

2221

131211

Y

L12

R12

L13

i13

R13

i3i2

i12

i20

C3 R3 L3v3

i30

C1v1

i10

3LiC2 R2 L2v2 2Li

barra 1

barra 2 barra 3

Rf

Lf

if

vf

Fonte de Tensão

f

fffL

ds

LsRd

ds

dz


L12

R12

L13

i13

R13

i3i2

i12

i20

C3 R3 L3v3

i30

C1v1

i10

3LiC2 R2 L2v2 2Li

barra 1

barra 2 barra 3

Rf

Lf

if

vf

fff

v

i

i

i

v

v

v

z

yy

yy

yyy

3

2

3

2

1

3331

2221

131211

100

010

001

000

001

00

00

1

ff

v

i

i

i

v

v

v

v

v

v

3

2

3

2

1

3

2

1

000

000

000

0100

0010

0001

A compact description

sss uDxCy

MIMO System

sss uBxY

sss uDxCy


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

0 250 500 750 1000 1250 1500

|z12| (

pu

)

Frequência (Hz)

Sistema Descritor

Matriz Y(s)

-200

-100

0

100

200

0 250 500 750 1000 1250 1500

Fas

e(z 1

2) (g

rau

s)

Frequência (Hz)

Sistema Descritor

Matriz Y(s)

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

Matriz Y(s) – Modelagem de Redes

z

yyk

vk

j

vj

ik ij

lzz c sinh

lyz

c csch1

llyc cschcoth

sZ

sYyc

c

cy

z1

sYsZ

j

k

sm

ms

j

k

j

k

v

v

yy

yy

v

v

zy

z

zzy

i

i

11

11

lylly

zyy ccs cschcschcoth

1

lyy cs coth lyy cm hcsc

2tanh

lyy c

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


lyy cs coth

lyy cm hcsc

sZ

sYyc

sYsZ

llds

dyl

ds

dy

ds

dyc

cs csch coth

lllds

dyl

ds

dy

ds

dyc

cm coth csch csch

ds

dZy

ds

dY

ds

dyc

c 2

2

1

2

1

ds

dZY

ds

dYZ

ds

d



1CsY 1Cds

dY

gie ZZZZ

ds

dZ

ds

dZ

ds

dZ

ds

dZ gie

ee LsZ 1

ee

Lds

dZ1

Positive sequence capacitance & indutance, computed by

matrix reduction considering ideal conductor and soil.

eLC 11,



sm

sn

n

skZ

s

i

x

y

Condutor

Aço

ri re

er

ssk

2

1

10010110 KIKIsn

11010111 KIKIsm

sri0 sre1

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.



2

1

sm

ds

sdmsn

ds

sdnsm

sksm

sn

ds

sdk

nds

dZ

s

i

s

sk

ds

sdk

2

ds

d

d

dKIK

ds

d

d

dI

ds

d

d

dKIK

ds

d

d

dI

ds

sdn 1

1

100110

0

0

010

0

011001

1

1

10

sd

d

d

dKIK

ds

d

d

dI

sd

d

d

dKIK

sd

d

d

dI

ds

sdm 1

1

110111

0

0

010

0

011101

1

1

11

sds

d

2

00

sds

d

2

11

1

0 Id

dI

2

201

II

d

dI

1

0 Kd

dK

2

201

KK

d

dK

sm

sn

n

skZ

s

i



x

y

D12

H1

1

2

3

3

1

3

122

223

1

02

lnln6

iij

j ijji

ijji

i i

ig

DHH

DpHH

H

pHsZ

3

1

3

122

3

1

0ˆ

ˆ

ˆlnln

6i

ijj

ij

ijijji

ij

i ii

ig

ds

pHd

pH

s

DHH

H

ds

dp

pH

s

H

pH

ds

dZ

ssp

1

222ˆ

ijjiij DpHHH

2

2 3ps

ds

dp

ds

dp

DpHH

pHH

ds

pHd

ijji

jiij

222

22ˆ


vf

1 2Rf Z

Y Y

R3

L3

C3

3Lf L12

dsdz

dsdydsdy

dsdydsdydsdy

dsdydsdy

ds

sd

f000

00

0

00

3332

232221

1211

Y

25

Z

YCsLsR

sy11

3

33

33

lssy

ds

dC

LsR

L

ds

dyc

coth32

123

333

lssyc coth

fz

yy

yyy

yy

s

001

00

0

10

3332

232221

1211

Y


vf

1 2Rf Z

Y Y

R3

L3

C3

3Lf L12

26

f

ff

v

i

v

v

v

z

yy

yyy

yy

1

0

0

0

001

00

0

10

3

2

1

3332

232221

1211

0100tc

fi

v

v

v

v3

2

1

3 0100

1000tb

fvs bxY xctv 3

ft vsv bYc

13

bYc

1

1

331

s

vv

vsG t

f


Sequential MIMO Dominant Pole Algorithm

(SMDPA)

DBYCH 1

ss T eT ss DBYCH

1

Direct Matrix

iD

ei DDD

ds

sd

s

HK

lim

es

i ss DKHD

lim

ei

i i

i ss

s DDKR

H

1

Partial Fraction Expansion

SMDPA – Fundamental Concepts (1/3)

DKHH sssˆ

kii ik

i

k

kk

jj

1

ˆ RRH

Strictly Proper part of H(s)

DKR

H

ss

si 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


Reduced Order Model (ROM)

Set of N dominant poles and

associated residue matrices.

DKR

HH

ss

ssN

i i

iN

i

1

N

i i

iN

i

ssssss

1

RDKHHHH

jjMOR Hmax

MIMO ROM Deviation TF

Norm of MIMO ROM Deviation TF


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

ssss

1

RDKHH

SMDPA – Newton Method (1/3)

0lim1

min

s

sH

0min ssf

Newton equationing

kk

kk

k

sds

sdss

s

min

*

minmin

)(

1

vH

w

ds

sd

ds

sd

ds

sd NHHH

N

i i

iN

i

sds

sd

12

KRH

BXY B ss

CXY C ssT

s

ds

sds

ds

sd TBC X

YX

H

sss kk 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 1sH

Pole of

sH

mim, vmin, wmin)

function eig of Matlab.


0.1

1.0

10.0

1000 1500

m

ax[Ĥ

(j

)] (

pu

)

Frequência (Hz)

0.1

1.0

10.0

1000 1500

m

ax[Ĥ

(j

)] (

pu

)

Frequência (Hz)

Determining the initial pole estimates

f1 f2 f3 f4 f5 f6 f7

810 2 ffj s

f8


Computation of the Residue Matrix for a Pole

Definition of the Integration Curve

j

Real(s)

4

1

1

k

P

PC

dssdss

k

k

HH

P2 P1

P3 P4

j

Real(s)

≡ P5

C

dssj C

HR2

1

SMDPA – Pole Residue Matrix (1/2)

kkk PlPPl 15.0,, HJ

Legendre-Gauss Method with Error Control

kP kk PP kk PP 2 kk PlP 1 kk PlP kkk PmP 1 kkk PmP 1 1kP

kk PlPs 15.0

4

1 1 1k

m

l

PlP

PlPC

k kk

kk

dssdss HH

4

1 1 1

,,2

k

m

l

M

i

kiik

k

PlwP

J

4

1 1 1

,,22

1

k

m

l

M

i

kiik

k

PlwP

jJR ,,,

22

1

1 1

km

l

M

i

kiik

k PlwP

jJR 4,,1 k

,2,1,0,2 qm qk

2

1

2

1 qk

qk

qkk RRR max

4

1

4 k

k

• Determining the Error

SMDPA – Pole Residue Matrix (2/2)

The 34-bus test system with 25 distributed parameter TLs

241312

26

27 34 11

287

8

29

9

32

25

21

20

23 33

22 31

30106

1 2 34 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)

0

5

10

15

20

25

30

-90 -80 -70 -60 -50 -40 -30 -20 -10 0

Imag

inár

io (

rad/s

)

10

00

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


0.0

1.0

2.0

3.0

4.0

0 500 1000 1500 2000 2500 3000 3500 4000

m

ax(p

u)

Frequência (Hz)


0.0

1.0

2.0

3.0

4.0

0 500 1000 1500 2000 2500 3000 3500 4000

m

ax(p

u)

Frequência (Hz)

Infinite

ROM 151

1.0E-05

1.0E-04

1.0E-03

1.0E-02

0 1000 2000 3000 4000

MO

R(p

u)

Frequência (Hz)


ROM and LMA Errors

sss NHHH

DKR

ss

N

i i

i

i

1

eT s DBYC

1

sss LL HHH

• ROM

• LMA – Linear Model (finite) Aproximation

eLTL s DBATC

1

%100

0

max

0

max

f

f

dj

dj

MOR

H

H

• Error Measures for ROM and LMA

%100

0

max

0

max

f

f

dj

djL

MLA

H

H

SMDPA - Infinite × Finite Systems (1/5)

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 × 101

400 2.58 × 101

1.94 × 10

1

500 1.65 × 101

600 1.15 × 101

Error Measures for LMA and ROM


1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 2000 4000 6000 8000

MO

R(p

u)

Frequência (Hz)

1.0E-05

1.0E-04

1.0E-03

1.0E-02

1.0E-01

1.0E+00

0 2000 4000 6000 8000

MO

R(p

u)

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


1.0E-05

1.0E-04

1.0E-03

1.0E-02

0 1000 2000 3000 4000

MO

R(p

u)

Frequência (Hz)

1.0E-05

1.0E-04

1.0E-03

1.0E-02

0 1000 2000 3000 4000

MO

R(p

u)

Frequência (Hz)

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

• ROM - 151 ROM - 167

ROM 151

ROM 167


(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

Err

o (

%)

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

Err

o (

%)

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

Err

o (

%)

Número de s

Comparing the Performances of LMAs ROMs

LMAs ROMs


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.

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.

• 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

Thank you!

Nelson Martins

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