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IEEE 2000 Summer Meeting Seattle, Washington, USA, 16th - 20th July, 2000

The single-phase RLC branch depicted in Fig. 1 will be the

basic network element in this paper.

R L Cvk vj

vCikj

Fig. 1: RLC Branch

The electrical behavior of this element can be described by

a set of two ordinary differential equations of first order:

Ckj

kjjk vdt

diLiRvv ++= (1)

kjC i

dt

dvC = (2)

Equation (1) is general and holds for the particular cases

whereL orR are zero. However, when there is no capacitor in

the branch, (2) must be replaced by:

0=Cv (3)

4 Descriptor System for Single-phase RLC NetworksA given network can be represented for harmonic studies by

the interconnection of single-phase RLC elements. For each

element, (1) and (2) can be written in matrix form:

jkC

kj

C

kjvv

v

iR

v

i

dt

d

C

L

0

1

0

1

01

1

0

0

+

+

=

(4)

where the current kji through the inductor and the voltage Cv

across the capacitor are the chosen state variables. Symbols

kv and jv denote the voltages at nodes kandj, respectively.

If there is no capacitor, (4) needs to be modified:

jkC

kj

C

kjvv

v

iR

v

i

dt

dL

0

1

0

1

10

1

00

0

+

+

=

(5)

The electric network model contains two differentialequations for each existing RLC branch and one algebraic

equation (the KCL) per system node. After interconnecting

the equations for all RLC branches, the following descriptor

system equation is obtained [1], [2], [5]:

uBxAxT += (6)xCy = (7)

where y and u are the nodal voltage and current vectors,respectively. The dimensions of the square matrices A and T

are ml nn +2 , where ln is the number of RLC branches and

nn is the number of network nodes.

A simple RLC circuit with one redundant state is used in [2]

to describe the network modeling by means of the descriptorsystem technique.

5 Harmonic Impedance seen from a System NodeApplying the Laplace Transform to (6) and (7):

( ) ( ) ( )sss uBATx 1= (8)

( ) ( )ss xCy = (9)

where ( )sx , ( )su and ( )sy are the Laplace transforms ofx, uand y, respectively.

From the above two equations:

( ) ( ) ( )sss uBATCy 1= (10)

The impedance matrix ( )sZ can be defined from (10):

( ) ( ) BATCZ 1 = ss (11)

The kkz element of ( )sZ is equal to the ( )knl +2 diagonal

element of ( ) 1 ATs [1], [2], [5]:

( ) ( ) ( )knkk lssz += 2

1diag AT (12)

Let kT and kA be the matrices obtained by canceling the

knl +2 row and column of the matrices T and A,respectively. Thus, the knl +2 diagonal element of

( )

1

ATs is given by [1], [2], [5]:

( ) ( )[ ]( )( )( )AT

ATAT

== +

det

detdiag 2

1

s

sssz kkknkk l (13)

Equation (13) is a generalization for descriptor systems of

its counterpart [3], [4] developed for conventional state space

systems. It shows that:

The system poles are the generalized eigenvalues [16] ofthe matrix pair { }TA , :

( ) iiis vTvAAT 0det == (14)

The zeros, associated with the self-impedance of node k,are the generalized eigenvalues of the matrix pair

{ }kk TA , : ( ) ikiikkks vTvAAT == 0det (15)

where i and i are the generalized eigenvalues associatedwith the pairs { }TA , and { }kk TA , and iv and iv are theirassociated generalized eigenvectors.

6 Test SystemThe results described in this paper relate to the 3-bus testsystem studied in [1], [2], [4] and [5]. This system can be

modeled by the interconnection of several RLC branches [4],

[5], as shown in Fig. 2.

bus 1

bus 2 bus 3

Lcc

L12

R12

R2

L2

Ih2 Ih3 L3 R3

R13

L13

Ih1

C2

C3

C1

Fig. 2: Test System

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The system frequency is 50Hzand the numerical values of

its elements are given in Table 1.

Table 1: System parameter values

Inductance (mH) Resistance () Capacitance (F)

ccL 8.0 2R 80.0 1C 23.9

2L 424.0 3R 133.0 2C 8.0

3L 531.0 12R 0.46 3C 11.9

12L 9.7 13R 0.55

13L 11.9

The impedance moduli as a function of frequency, seen

from each system bus (self-impedance), are shown in figures3, 4 and 5.

0

5

10

15

20

25

30

35

40

45

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

ImpedanceModulus()

Fig. 3: Self-impedance seen from bus 1

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

ImpedanceModulus()


0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000

Frequency (Hz)

ImpedanceModulus()


The network model has a total of eight poles: three pairs of

complex conjugate poles plus other two real poles. These

poles are the generalized eigenvalues of finite modulus of the

matrix pair { }TA, calculated by the QZ eigenroutine [16].The total number of eigenvalues calculated by this routine is

actually equal to 23 (the order of the matrices A and T).Therefore, other 15 generalized eigenvalues of infinite

modulus corresponding to the algebraic equations are also

obtained.

The pole-zero spectra of the Test System are shown in Fig.

6, for the self-impedance of each one of the three buses. In

this figure, iAZ and iBZ denote the two pairs of complex

zeros seen from bus i ( )3,1=i . Note that only those poles(three complex pairs) and zeros (two complex pairs per bus)

with non-zero imaginary parts were plotted in this figure.

P2

P3

P1

Z1B

Z1A

Z2A

Z2B

Z3A

Z3B

0

100

200

300

400

500

600

700

800

-900 -800 -700 -600 -500 -400 -300 -200 -100 0

Real Part (s-1

)

ImaginaryPart(Hz)

System poles Zeros seen from bus 1

Zeros seen from bus 2 Zeros seen from bus 3

Fig. 6: Pole-Zero Spectra of the Test System

It is worth noting that the shapes of the impedance plots,

shown in figures 3, 4 and 5, may be sketched by hand from

the inspection of the associated pole-zero spectra. One can

observe, for instance, that the two zeros seen from bus 1

( AZ1 and BZ1 ) are close to pole 2P (488Hz) eliminating the

peak around this frequency value, in the plot of Fig. 3. Note

also that the second zero seen from bus 3 ( BZ3 ) is close to

the pole 3P (722Hz), causing a large reduction at this

frequency in the plot shown in Fig. 5.

7 Shifting a Pole or a Zero by Means of a SingleTuning Filter

This section describes a Newton-Raphson algorithm to

shift a pole or a zero to a specified location in the complex

plane by varying the parameters of a single tuning filter. The

method is then applied to solve a harmonic problem in the

3-bus test system through the addition of a single tuning

filter.

Letf be the frequency value inHz(imaginary part divided

by 2 ) of a chosen pole or zero of a system; rf the targetvalue forf (in other words, f should become equal to rf at

the solution) and fC and fL the values of capacitance and

inductance of a single tuning filter that are assumed to vary,

connected at a system bus as shown in Fig. 7.

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System bus

Z

Other Shunt

Elements

Lf

Cf

Rf

Fig. 7: Single tuning filter connected at a system bus

One may define:

ff ,LCff = (16)

The mismatch function of the frequency of the selected

pole or zero can be defined by:

( ) ( %100,

,

=

r

rffff

f

fLCfLCg (17)

The mismatch function of reactance can be defined by:

( )( ) ( )

( )%100

=

fc

fcfl

ffCx

CxLx,LCr (18)

where:

ftl Lx = (19)and

1

ft

cC

x

= (20)

The symbol t denotes the tuning frequency of the filter.

In order to shift the initial value of the frequency of the

selected pole or zero (f) to the target value (fr), while

keeping the filter properly tuned, the following mismatch

vector equation must be satisfied

=

0

0

r

g(21)

which implies in simultaneously solving (17) and (18).Applying the Newton-Raphson method to (21), the following

recurrence formula is obtained

( )k

kk

f

fk

f

f

r

g

L

C

L

C

=

+ 11 J (22)

where the index kdenotes the iteration number and J is the

Jacobian matrix:

=

ff

ff

L

r

C

r

L

g

C

g

J (23)

The expressions for the Jacobian matrix elements in the

first row are given by:

100

p

f

fp

g

r

=

(24)

werep denotes a filter parameter ( fC or fL ).

The values of p

f

correspond to the imaginary parts

(divided by 2 ) of the eigenvalue sensitivities, directlyobtained from (see [5]):

vTA

w 1

=

ppkp(25)

where v and w are the right (column) and the left (row)

generalized eigenvectors of { }TA , associated with the

generalized eigenvalue fj 2+= , and vTw =k .

The expressions for the elements located in the second row

of the Jacobian matrix are derived below.

Differentiating (18) with respect to fC :

1

100

=

cfl

f xCx

C

r(26)

Differentiating (18) with respect to fL :

f

l

cf L

x

xL

r

=

100

(27)

Differentiating (19) with respect to fL and the inverse of

(20) with respect to fC :

tcff

lxCL

x

= =1 (28)

Substituting (19) and (28) into (26), yields:

ftf

i LC

r100

2=

(29)

Similarly, from the manipulation of (20), (28) and (27),

one obtains:

ftf

CL

r 2100=

(30)

8 Problem ExampleAssume that current source 1hI , shown in Fig. 2, has

negligible modulus while the sources 2hI and 3hI contain

5th

and 11th

harmonic components, respectively.

The impedance value seen from bus 2 at 250 Hz

(5th

harmonic) is 44.48 and the impedance value seen frombus 3 at 550Hz(11

thharmonic) is 44.12 . Therefore, the 5th

and 11th

harmonic distortions (neglecting the transfer

impedances) are given by:

==

3Busfor12.44

2Busfor48.44

1111

55

IV

IV(31)

Assume that these distortions have exceeded their

individual limits. The next two sections describe the

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traditional and proposed solutions to reduce these harmonic

distortions.

8.1 Traditional SolutionThe traditional solution for this problem would involve the

installation of a 5th

harmonic single tuning filter at bus 2 and

a 11th harmonic single tuning filter at bus 3.

As an example, consider that a 1 F capacitor is used tobuild a 11

thharmonic single tuning filter. This implies that

the inductor value must be 83.74 mH. It was considered a

filter quality factor of 75. The frequency plots for the self-

impedance seen from bus 2 and 3, after the installation of the

filter at bus 3, are shown in Fig. 8 and Fig. 9.

Fig. 8 shows that with the installation of the 11o

harmonic

filter at bus 3, the impedance seen from bus 2 at 250Hz

(5th

harmonic) remains practically unchanged (41.67 ).Therefore, the traditional solution to this problem also callsfor the installation of a 5

thharmonic filter at bus 2.

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000

Fequency (Hz)

ImpedanceModulus()

Fig. 8: Impedance modulus seen from bus 2 - 11o harmonic filter at bus 3

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000

Fequency (Hz)

ImpedanceModulus()

Fig. 9: Impedance modulus seen from bus 3 - 11o

harmonic filter at bus 3

8.2 Proposed SolutionThe frequencies (imaginary parts divided by 2) of the

complex conjugate network zeros of the self-impedances of

the three buses (series resonance), as well as their sensitivitieswith respect to the inductances and capacitances of the Test

System, considering the 11th

harmonic filter installed at bus 3,

are presented in Table 2.

The sensitivities are normalized, being given inHz/per unit

of change of nominal parameter value.

Table 2: Zero sensitivities - 11th harmonic filter at bus 3

Node 1 Node 2 Node 3

1 2 3 1 2 3 1 2 3

f(Hz) 395 565 593 317 558 653 382 550 704

ccL 0 0 0 -42 -33 -53 -88 0 -47

2L 0 -7 0 0 0 0 -5 0 -2

3L -4 0 -1 -4 0 -1 0 0 0

12L 0 -290 0 -34 -27 -44 -37 0 -289

13L -173 0 -35 -73 -13 -158 -59 0 -32

fL -21 0 -264 -6 -206 -72 0 -275 0

1C 0 0 0 -30 -76 -173 -85 0 -189

2C 0 -269 0 0 0 0 -109 0 -145

3C -152 0 -68 -108 -2 -99 0 0 0

fC -46 0 -226 -19 -201 -51 0 -275 0

Table 2 shows that the zero 1 seen from bus 2 (317 Hz)

presents considerable sensitivities with respect to the filter

parameters ( fC , fL ). This suggests the possibility of shifting

this zero to the frequency of 250Hz, by changes in fC and

fL , minimizing the 5th

harmonic distortion at bus 2. The

series connection of these new values of fC and fL must

obviously continue to carry out its function as a 11th

harmonicfilter.

Applying the Newton-Raphson method to solve thisproblem, yielded the optimized values of FCf = 9.20 and

mHLf 9.11= in 3 iterations with absolute value of themismatch functions less than 0.1 %. The filter quality factorwas kept equal to 75.

The frequency response diagram for the self-impedance

seen from bus 2 and 3 for these new parameters of the 11th

harmonic filter, are depicted in Fig. 10 and Fig. 11. The new

frequency values of the poles and zeros are presented in

Table 3.

0

10

20

30

40

50

60

70

80

0 100 200 300 400 500 600 700 800 900 1000

Fequency (Hz)

ImpedanceModulus()

Fig. 10: Impedance modulus seen from bus 2 - optimized filter at bus 3

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