Study_of_Harmonics_in_Cable_Based_Transmission_Networks.pdf

7/28/2019 Study_of_Harmonics_in_Cable_Based_Transmission_Networks.pdf

1/14

1

Study of Harmonics in Cable-based Transmission Networks

F. Faria da Silva C. L. Bak P.B. Holst

Aalborg University Aalborg University Energinet.dk

Denmark Denmark Denmark

SUMMARY

The Danish electrical transmission network is currently going through major modifications. One of the

main changes is the migration from a network based on overhead-lines (OHL) to a network based on

underground cables. High voltage cable-based networks are a new reality that presents several interest-

ing challenges for power engineers due to their differences to the more traditional OHL-based net-

works.

The electrical differences between a cable and an equivalent OHL are many, the main one being the

larger capacitance of the former. As a result, the behaviour of a cable-based network is also different

from the behaviour of an OHL-based network. Some of the main differences are: electromagnetic

transient waveforms, reactive power compensation techniques and harmonics issues.This paper focuses on harmonic studies of a cable-based transmission network, more specifically the

transmission system of West Denmark as planned to 2030. This transmission network will have the

entire 150kV level undergrounded, making it ideal for the study of harmonics in cable-based networks.

A frequency spectrum can provide information about possible harmonic excitations and the voltage

waveform during a transient. Therefore, it is necessary to assure that the model used in the digital

simulation is adequate and that the accuracy of the results is sufficient.

The paper focuses first on how the modelling depth and the modelling detail influence a simulation

outcome. It compares two models, a first where an external equivalent network is used and a second

where an extended network modelled by means of lumped-parameters models is used. The paper also

describes the effect of the modelled area length, i.e., the modelling depth, in the results.

The paper also addresses the influence of the cable bonding, both-ends bonding and cross-bonding, in

the frequency spectrum and the impact that the number of cross-bond sections has in the frequency

spectrum. The analysis is made for both positive-sequence and zero-sequence by means of simulations

in the 2030 West Denmark transmission network and a meticulous mathematical analysis that is fully

presented in appendix.

It is demonstrated that a cross-bond cable has more resonant points than a cable bonded in both-ends.

It is also proven that the magnitude of the parallel resonance points is larger for a cross-bonded cable

than for a cable bonded in both-ends, whereas the magnitude of the series resonance points is lower.

To finalise, the paper analyses how a deviation in the thickness of the different layers affects the fre-

quency spectrum. It is demonstrated that the deviations allowed by the standard are sufficiently large

to result in non-negligible changes in the frequency spectrum.

KEYWORDSHVAC Cables; Harmonics; Modelling; Cable bonding

21, rue dArtois, F-75008 PARIS C4-108 CIGRE 2012http : //www.cigre.org


2/14

2

1. Modelling Approaches

The size of the simulation model (i.e., the modelling depth) influences the simulation results. The

more detailed a model is the more accurate is the simulation. However, unless frequency scan field

measurements are available, it is not feasible to do an accurate frequency dependent (FD) networkwithout designing the entire system by means of FD-models. Thus, simplifications are necessary.

The normal approach is to divide the system into two areas, a detailed study area and an external net-

work area [4]. The common approach for estimating the required modelling size is outlined in [5]:1. Design a detailed system up to a distance of two or three busbars from the point of interest and use

an equivalent network for the rest of the grid;

2. Repeat the previous point, but increase the modelling depth of the detailed area in one busbar;

3. Compare the frequency spectrums for both systems;

4. Repeat the process until the difference between the spectrums is minimum around the frequencies

of interest;

Two different approaches can be used for the modelling of the external network:

- An equivalent network is designed: The detailed area is modelled by means of FD-models. The ex-

ternal network is modelled by means of an equivalent network, using a N-ports or 2-ports equiva-

lent (this approach will be namedDi-Eq, where i is the modelling depth of the detailed area);

- The entire system is modelled: The detailed area using FD-models and the external network by

means of lumped-parameter models for voltage levels equal or superior to 150kV, for the lower

voltage levels an N-port equivalent network is used (this approach will be named Di-L, where i is

the modelling depth of the detailed area);

Figure 1 and Figure 2 compare the frequency spectrums for the two modelling approaches previously

described; using a N-ports equivalent network for theDi-Eq modelling approach. The cables modelled

by means of FD-models are the same in both models, the difference lies in the modelling of the re-

maining network.

The simulated network is the West Denmark transmission network as planned to 2030. Appendix A

shows the network single-line diagram and more information about the network is available in [1].The frequency spectrum is obtained for a point in the 150kV network, where a 22.1km cross-bonded

cable is connected to a phase shift-transformer (LEM node), resulting in a low frequency resonant

point. The cable is connected between the LEM and STSV nodes and it is open in the receiving end

(STSV node), so that the frequency scan indicates the frequencies excited during the energisation tran-

sient [1].

100 150 200 250 300 350 4000

1000

2000

3000

4000

5000

6000

agntue

m

Frequency [Hz] 100 150 200 250 300 350 4000

500

1000

1500

2000

2500

3000

3500

agntue

m

Frequency [Hz] (a) (b)

Figure 1 - Frequency spectrums seen from the LEM node with the cable open in the STSV end.

a) Blue: Model D1-L; Magenta: Model D1-Eq b) Black: Model D2-L; Magenta: Model D2-Eq


3/14

3

100 150 200 250 300 350 4000

500

1000

1500

2000

agntu

e

m

Frequency [Hz] 100 150 200 250 300 350 4000

500

1000

1500

2000

2500

M

agnitude[Ohm]

Frequency [Hz] (a) (b)

Figure 2 - Frequency spectrums seen from the LEM node with the cable open in the STSV end.

a) Orange: Model D4-L; Magenta: Model D4-Eq b) Red: Model D5-L; Magenta: Model D5-Eq

The observation of the frequency spectrums indicates that:

1. The number of resonance frequencies of the L-Net is equal or superior to the number of resonance

frequencies of the equivalent Eq-Net;

2. The magnitude of the impedance at the main resonance frequency (~250 Hz) is larger in the Eq-

Net than in the equivalent L-Net;

3. The modelling depth has more influence on the Eq-Net frequency spectrum than on the L-Net fre-

quency spectrum;

4. Only the model D5 presents similar results for both modelling approaches.

An Eq-Net is always less complex than the corresponding L-Net, with the notable exception of the

limit case when the entire network is modelled. As the modelling depth increases, the number of lines

and other elements in the model also increase, followed by an increase in the size of the N-port exter-

nal equivalent network. As a result, an increasing number of resonance points are expected.

The L-Net contains all the network generators, transformers and loads, and the only difference when

the modelling depth increases lies in the modelling of some of the cables, which changes fromlumped-parameters models to FD-models. Thus, an increase in the modelling depth does not affect the

frequency spectrum of the L-Net as much as in the Eq-Net. A corollary of the previous paragraph is

that the Eq-Net model may require the modelling of a large area of the network in order to present ac-

curate results.

The simulation is expected to be slower for the L-Net approach than for the equivalent Eq-Net model.

However, the Eq-Net requires more cables modelled by means of FD-models in order to provide accu-

rate results. The simulation of a FD-modelled cable is approximately 10 times slower than the simula-

tion of the equivalent lumped-parameters cable [1]; Moreover, if the cable is cross-bonded, the simula-

tion of the FD-modelled cable is 10*x times slower, wherex is the number of minor sections. Conse-

quently, contrary to what common sense would initially indicate, an accurate simulation can very often

be faster for a L-Net than for an Eq-Net.

2. Bonding Influence

Cables can be installed in different bonding configurations, being the most common for long HVAC

cables, cross-bonding and both-ends bonding [2], [3]. Preference is normally given to the former over

the last due to its lower losses. There are, however, cable installations that require the use of both-ends

bonding, more notably submarine cables due to financial cost associated to the cross-bonding of cables

sections in the sea.

The influence of the cross-bond sections on the frequency spectrum should also be ascertained as the

bonding can shift the resonance frequencies. For the first examples, the 2030 West Denmark network

is used. The conclusions are later validated through a mathematical development.

The objective of the analysis is to determine the differences between having and not having a cross-

bond cable. Four different bonding configurations of the LEM-STSV cable, the same from the previ-

ous section, have been prepared:


4/14

4

- Both-Ends Bonding

- Cross-bond cable with

- 1 Major-section

- 6 Major-sections

- 12 Major-sections

Frequency spectrums used in the following examples are estimated for the same reference points of

the previous section. This way, there is a lower frequency resonance point present, resultant of the ca-

ble-transformer interaction, and several higher frequency resonance points, which are solely associated

to the cable.

2.1 Comparison of both-ends bonding with cross-bonded cable with 12 major-sections

Figure 3 shows the impedance spectrum considering the LEM-STSV cable as being bonded in both-

ends or cross-bonded with twelve major sections.

The resonance frequency seen around 250 Hz is the cable-transformer resonance. The resonance

points seen in the second figure for frequencies higher than 600 Hz are the cable resonance frequen-

cies.

0 100 200 300 400 500 6000

500

1000

1500

2000

2500

3000

3500

Magnitude[

hm]

Frequency [Hz] 2000 4000 6000 8000 10000 12000 14000

0

100

200

300

400

500

600

700

800

Magnitude[Ohm]

Frequency [Hz] Figure 3 - Frequency spectrums seen from the LEM node with the cable open in the STSV end. Red:

Cross-bonded with 12 major-sections; Blue: Both-ends bonding

The first resonance point is at approximately 250 Hz or the 5th harmonic. At this frequency, there is a

difference between the impedances of the two bonding configurations, and the resonance frequency is

almost the same for both configurations.

As the frequency increases, the differences between the two bonding configurations start to become

evident:

- The cross-bonded cable has more resonance points than the cable bonded in both-ends;

- The magnitude of the cross-bond cable impedance is larger at the parallel resonance points and

lower at the series resonance points when compared with the cable bonded in both-ends;

The mathematical demonstration of the results is long and it is, for that reason, made in Appendix B.A summary and physical explanation of the results is done next.

The conductor positive-sequence series inductance is larger in a cross-bond cable than in an equivalent

both-end bonded cable, whereas the series resistance is larger for a both-end bonded cable. The con-

ductor positive-sequence shunt admittance is equal for both bonding configurations [1].

These differences in the series impedance result in lower resonance frequencies for a cross-bond cable.

They are also responsible for the larger impedance magnitude at the parallel resonance points and

lower impedance magnitude at the series resonance points of the cross-bond cable, when compared to

the cable bonded in both-ends.

From a physical point of view, the larger inductance in the cross-bond cable is a result of having a

lower current circulating in the screen when compared to the both-end bonded cable. As the current is

lower, the magnetic field induced by screen current is also lower, resulting in a larger inductance

value.


5/14

5

2.2 Comparison of cross-bonded cables with different number of sections

Figure 4 compares the frequency spectrum for a different number of major cross-bond sections.

0 5000 10000 150000

200

400

600

800

1000

Magnitude[Ohm]

Frequency [Hz] Figure 4 - Comparison of the frequency spectrum in PSCAD/EMTDC. Black: One major cross-bonded

sections; Green: Six major cross-bonded sections; Red: 12 major cross-bonded sections;

The higher the number of cross-bond sections, the closer the results are to the ideal cross-bonding, i.e.,

a continuous transposition and grounding of the screen. Using the 12 major cross-bond sections sce-

nario as reference, it is seen that the cable with only one major section starts to diverge after the first

parallel resonance point (~2.5 kHz) and the one with 6 major sections after the third parallel resonance

point (~7k Hz).

After these frequencies, the respective spectrums present an unexpected behaviour. They have more

resonance points, whose magnitude does not always decrease with the increase in frequency.

This behaviour is the result of the larger imbalance present when less cable sections are included in the

model. Figure 5 shows the voltage in the cable receiving end when injecting a 134.5kV peak voltage

in the sending end at different frequencies. The figure shows the results for a cable bonded in both-

ends and an equivalent cable with a major cross-bond section.

0 5000 10000 150000

100

200

300

400

500

600

700

800

Voltage[kV]

Frequency [Hz]

Figure 5 - Voltage in the cable receiving end for a 134.5 kV peak voltage in the cable sending end.Blue: Bonded in both-ends; Red: One major cross-bonded section

When observing the cross-bonded cable spectrum it can be seen that between two larger peak voltages

(red circles), two smaller overvoltages (green circles) are present. This is the result of the two crossing

of the screens.

The higher the number of major cable cross-bonded sections, the more balanced the cable and the

coupling. As a result, the entire cable behaves like a uniform single section; if only one major cross-

bond section is present, the cable behaves almost like three different cables.


6/14

6

2.3 Zero-Sequence frequency spectrum

Most of the zero-sequence current components of a buried cable return in the screen of the cable [6].

As a result, the current in the screen is roughly equal to the current in the conductor for any random

given point of the cable.

Consequently, there should be almost no current flowing into the ground at the grounding points, and

the frequency spectrums should be the same for all types of bonding.

Figure 6 shows the frequency spectrums for several bonding configurations.

0 5000 10000 150000

50

100

150

200

Magnitude[Ohm]

Frequency [Hz] Figure 6 - Frequency spectrum for the D2-eq system. Blue: Both-ends bondings; Black: One major

cross-bonded section; Green: Six major cross-bonded sections; Red: 12 major cross-bonded sec-

tions

The resonance frequencies are the same for all the examples because of the reasons explained in the

previous two paragraphs. The both-end bonded cable and the cable with only one major cross-bond

section have the same magnitude at all times, but the same is not the case for a cable with more cross-

bond sections that have a different magnitude for some of the resonance frequencies.

Figure 7 shows the line reactance as a function of the length, using the line wavelength as reference. It

is seen that the reactance of an open line is not the same at all points of the lineA cable with multiple cross-bond sections also has multiple grounding points. If one of those ground-

ing points corresponds to a point of the line where the reactance has a very high value, the current in

the screen flows to the ground at that point, changing the magnitude of the impedance.

0 L/4 L/2 3L/4 L

-600

-400

-200

0

200

400

600

Wavelength

Magnitude[]

Figure 7 - Reactance of an open line as a function of the wavelength

Example

The example is given for the cable with six major cross-bond sections. For a parallel resonance situa-

tion, the maximum current in the cable occurs for the point(s) that are equal to or a multiple of the ca-

ble wavelength, i.e., /2; 3/2; 5/2 .

Figure 8 shows the relative current along the cable for the first two parallel resonance frequencies. For

the first resonance frequency, the maximum current is in the middle of the cable line. For the second

resonance frequency, the peak currents are in the first and last quarter of the cable line.


7/14

7

In the first case, the screen is grounded in a point corresponding to maximum current, which means a

very high reactance in that point. Thus, the current flows to the ground in that point, but this does not

happen in the second case, which has the peak currents in the middle point between two grounds.

As a consequence of this, and using the both-end bonded cable as reference, the magnitude of the im-

pedance is smaller for the first resonance point (~4 kHz) and virtually equal for the second resonance

point (~8 kHz).

I

I

(a) (b)

Figure 8 - Relative current along a cable with six major cross-bonded sections. a) 1st parallel reso-

nance point (~4 kHz); 2nd parallel resonance point (~8 kHz)

It should be noted that the presence of other conductors near the cable can change the results presented

in this section as these conductors present a possible path for the zero-sequence components.

3. Sensitivity analysis of the cable layers thicknesses

A frequency spectrum is very sensitive to the cable parameters. The existing IEC standards [7] and [8]

allow a deviation in the thickness of some of the cable layers of up to 10%. It is therefore important to

assess how the spectrum is affected by these deviations.

The thicknesses of the cable conductor and insulation are changed and the frequency spectrum esti-

mated:

- Conductor thickness -> Changes the conductor resistivity as well;

- Insulation thickness -> Changes the insulation permittivity as well;

Deviations in the screen and outer insulation thicknesses were also simulated. However, the simula-tions did not show any visible changes in the frequency spectrum and the results were not analysed

further.

The layers are analysed separately, meaning that for a deviation in the conductor thickness, only the

conductor thickness is changed. However, an increase in the conductor thickness also represents a

change in conductor resistivity, whereas in the insulation results in a change in the permittivity [9].

The standard limits the thickness deviation to a maximum of 10%, but in this analysis a maximum of

20% is considered, allowing the gathering of more data. The simulated deviations are of 1%, 5%,

10%, 15% and 20%. The frequency spectrums range between 1 Hz and 1 kHz (20th harmonic),

with a step of 1 Hz, and the cable data is shown in Table 1. Only the LEM-STSV cable is changed, the

rest of the network remains unaltered.

Table 1 Cable dataCross-

section

Conductor

radius

Insulation

radius

Screen Ra-

dius

Outer

radius

Conductor

resistivity

Insulation rel.

Permittivity

1200mm2 0.022mm 0.042mm 0.042358mm 0.047mm 3.55e-08m 3.02

Figure 9 shows the frequency spectrum for different insulation thicknesses when using model D1-L.

The increase in the insulation thickness results in a decrease in the capacitance. Thus, the resonance

frequency increases when the insulation thickness increases and decreases when the thickness de-

creases.

A change in the thickness of the cable insulation will result in larger changes at the ~250 Hz resonance

frequency than at the other resonance frequencies. The ~250 Hz resonance frequency is the direct re-

sult of the cable-transformer interaction. Consequently, a change in the cable is more noticeable at this

resonance frequency.


8/14

8

200 250 300 3500

500

1000

1500

agn

u

e

m

Frequency [Hz] Figure 9 - Frequency spectrum for different insulation thicknesses. Red: Reference; Blue: 5%; Green:

10%; Black: 15%; Magenta: 20%. Solid lines: Thickness increases; Dashed Lines: Thickness de-

creases

The behaviour of the impedance magnitude is not so linear. It increases, for example, when the thick-

ness decreases for the first resonance point (~210 Hz), but it has the opposite behaviour for the main

resonance point (~250 Hz).

The increase of the capacitance is normally followed by a decrease in the impedance at resonance fre-quency, e.g., a parallel LCcircuit. The opposite behaviour at ~210 Hz is explained by a high capaci-

tance and inductance of the network behind the transformer, including the transformer at that specific

frequency.

Figure 10 shows the frequency deviation in function of the thickness deviation. The deviation is com-

mon to the different modelling approaches and to increase the model complexity does not mean a re-

duction of the deviation.

-20 -15 -10 -5 0 5 10 15 20-10

-5

0

5

10

15

FrequencyDevia

tion[Hz]

Thickness Deviation [%] -20 -15 -10 -5 0 5 10 15 20

-15

-10

-5

0

5

10

15

20

FrequencyDevia

tion[Hz]

Thickness Deviation [%] Figure 10 - Difference between the resonance frequencies for deviations in the insulation thickness

when using a model with a 3-busbars depth (figure on the left) and model with a 5-busbars depth

(figure on the right). Green: f~210Hz; Red: f~250Hz; Blue: f~310Hz;

4. Example

A final example joining the theories explained in the previous chapters is prepared. Firstly, the fre-

quency spectrum for different bonding configurations is estimated and it is later seen how a deviation

in the thickness of the cable insulation affects the spectrum.

Figure 11 shows the frequency spectrum for different bonding configurations. The spectrums are esti-

mated for a 150kV node of the 2030 West Denmark Network. The node is connected to three cables

with lengths of 23.7km, 29.7km and 47.49km. The nodes up to two busbars of distance are modelled

by means of FD-models, whereas the rest of the network is modelled by means of lumped-parameters

models. All the network transformers, generators and loads are included in the model.

Appendix A shows the networks single-line diagram and the cables modelled by means of FD-

models.


9/14

9

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

140

ImpedanceM

agnitude[Ohm]

Frequency [Hz] Figure 11 Frequency spectrum for different bonding configurations. Black: All cables bonded in

both-ends; Blue: All cables with one cross-bond major section; Green: The three cables attached

to the reference node with three cross-bond major sections and the remaining cables with one

cross-bond major section; Red: All cables with six cross-bond major sections

The figure shows that the bonding arrangement influences the resonance frequencies after the 400Hz(as demonstrated in chapter 2, the bonding does not influence the lower frequencies). The best exam-

ple is around the 830Hz, where the magnitude of the impedance is as shown in Table II. It is also seen

that the frequency spectrums is smoother when the cable has more major sections. This behaviour is

expected, as the more major sections that cable has, the closer it is to an ideal cross-bonding, as dem-

onstrated in chapter 2.

Table II Peak magnitude of the impedance for the different bonding configurations

Type of Bonding Magnitude of the impedance [Ohm]

Red curve 137

Green curve 125*

Blue curve 40*Black curve 11

*The values are not for 830Hz, but for the nearest peak

points, 823Hz and 805Hz respectively

Figure 12 shows the frequency spectrum for a bonding configuration where the three cables attached

to the reference node have three cross-bond major sections and the remaining cables have one cross-

bond major section. One of the spectrums is the original spectrum (green curve), whereas the other

spectrum is considering a deviation of +10% in the thickness of the insulation for the three cables at-

tached to the reference node (black curve).

As expected, the deviation leads to a change in both frequency and magnitude of the resonance points.

In this example the insulation thickness increases leading to a decrease of the total capacitance andlater resonance points.


10/14

10

0 200 400 600 800 1000 1200 1400 1600 1800 20000

20

40

60

80

100

120

140

ImpedanceM

agnitude[Ohm]

Frequency [Hz]

Figure 12 Frequency spectrums. Green: Original model; Black: Model with a 10% deviation in the

insulation thickness

5. Conclusions

The classic method used in the estimation of a network frequency spectrum is to compare the fre-

quency spectrums for increasing modelling depths and to stop when the frequency spectrum halts

changing. However, the total simulation running time of a cable-based network is mostly a function of

the number of cables modelled by means of FD-models. Consequently, the modelling of the network

outside of the area of interest by means of lumped-parameters models does not represent a substantial

increase in the total simulation time, whereas the accuracy of the frequency spectrum increases sub-

stantially. Moreover, the inclusion of the more distant network areas in the model allows the reduction

of the number of cables modelled by means of FD-models and of the computational burden.

The bonding of a cable affects the positive-sequence resonance frequencies, whereas the influence in

the zero-sequence spectrums is only on the magnitude of the resonance points. A cross-bond cable has

both more resonance frequencies and larger impedance magnitude at the parallel resonance frequen-cies than an equivalent cable bonded in both-ends.

However, the type of bonding has little influence in the event of a cable-transformer resonance, which

occurs for frequencies lower than the cable's first resonance frequency. For other phenomena, like

propagation of harmonics generated by different harmonic sources, the bonding type is relevant, espe-

cially for long cables with low resonance frequencies.

The cable capacitance changes when the thickness of the conductor or insulation changes. Conse-

quently, the frequency spectrum is affected if the datasheet values are incorrect or the permittivity and

resistivity constants are not properly corrected. Since the IEC standards allow a deviation in the thick-

ness of both the conductor and the insulation, a sensitivity study should be done if the resonance fre-

quencies are close to a harmonic frequency.

The total frequency deviation depends on the model used, but the use of a more complex model does

not mean lower frequency deviations.


11/14

11

Appendix A

Figure 13 - Single-line diagram of the 2030 Western Denmark transmission grid. Bold: 400KV;

Dashed: 150kV. Blue: Cable-line used in chapters one to three. Red: Cable modelled by means of

FD-models in chapter four. Note: Due to differences between the English and Danish names of

some cities, the STSV-LEM link is named STS-LKR in the map

Reference Cable

Reference node


12/14

12

Appendix B

The shunt admittance matrix is equal for both bonding configurations, but the series impedance matrix

presents some differences. By applying the method explained in [1], (1) and (2) for the positive-sequence series impedance of the conductors are obtained. Where ZCC is the core self-impedance with

earth return,ZCSi is mutual impedance between core and screen with earth return,ZSS is the screen self-

impedance with earth return,Zboth is the series impedance of a cable bonded in both-ends and Zcross

the series impedance of a cross-bond cable.

2

1 2

2

2

CS CS

both CC CS

CS SS

Z ZZ Z Z

Z Z

(1)

2cross CC CS Z Z Z (2)

The resonance points have higher magnitudes and lower frequencies for the cross-bond cable than for

the both-end bonded cable. As the differences in the resonances are noticeable for the first resonancepoint, the nominal pi model instead of the equivalent pi model can be used, which results in a simple

mathematical analysis.

The line impedance for the nominal pi model is given by (3).

2

2 2 3 2

1

2

LC j RLZ

C R j C C L

(3)

The shunt admittance is equal for both bonding types and the differences in the impedance are a func-

tion ofL andR.

Resonance Frequency

The resonance frequency is given by (4), which is obtained by developing (3). L is the imaginary part

of the series impedance (1)-(2). Consequently, the imaginary part ofZ+

cross>Z+

both. Subsequently, the

imaginary part of the last element of (1) should be negative, i.e. (5).

2 2

LC (4)

2

1 2

2

0CS CS

CS SS

Z Zimag

Z Z

(5)

Both the real and imaginary parts ofZSS are always larger than the equivalents in ZCS2. Thus, the de-nominator of (5) has always a negative real and imaginary part (6).

2CS SS Z Z a jb (6)

The development of the numerator of (5) results in (7).

22 2 2 2 2

1 22 2 2 2 2 2

CS CS Z Z c jd e jf c d e f ce df j cd ef cf de (7)

The magnetic field is stronger between the core and screen of the same cable than between the cores or

core-screen of two different cables. Thus, d>fwhile ce, and so (7) can be simplified to (8).


13/14

13

2 2 22 2 2

1 2 1 2

2 22 2 2 2 2

1 2 1 2

2 0 2

2 2 2

CS CS CS CS

CS CS CS CS

Z Z d f df j Z Z d d g d d g

Z Z d d g dg d dg Z Z g

(8)

It is concluded that the numerator of (5) has only a negative real part. Therefore, (5) can be written as

(9).

2 21 2 2

2

2

1 2 2 2

2

' '

' '

CS CS

CS SS

CS CS

CS SS

Z Z gg a jb

Z Z a jb

Z Za g j b g

Z Z

(9)

The imaginary part of (5) is always negative and the resonance frequency of a cross-bond cable is al-

ways lower than the resonance frequency if bonded at both-ends.

Magnitude

Parallel resonance

From (3) the magnitude is given by

2 22 2 2 2 2 2

2

2 32 3

2 2 3 2

1 1

2 2

1 1

4 2 4 2

LC j RC LC j RC L LZ Z Z j

RC RC RC RCC

LC

L L L LZ Z

C R C C R C

(10)

The two variables that depend on the bonding are R andL, which are, respectively, the real and imagi-

nary parts of the series impedance matrix (to be precise, the imaginary part isXL, but for this analysis,

that is not very relevant).

From the analysis of the resonance frequency and (1), (2) and (9) it is known that the both-end bonded

cable has a higher resistance and a lower inductance. Doing the substitutions in (10), it is concluded

that the magnitude of the parallel resonance points is lower in the both-end bonded cable.

Series resonance

For a series resonance, the impedance magnitude is given by (11). The value ofL is lower for a both-

end bonded cable, resulting in lower magnitude at the series resonance points for this type of bonding.

2 2 2 22

j RL L L LCZ Z Z

C R C C

(11)


14/14

14

BIBLIOGRAPHY

[1] F. Faria da Silva, Analysis and simulation of electromagnetic transients in HVAC cabletransmission grids, Ph.D. Dissertation, Dept. Energy Technology, Univ. Aalborg, 2011

[2]

CIGRE WG B1.19, General Guidelines for the Integration of a New Underground Cable Sys-tem in the Network, CIGRE Technical Brochure 250, 2004[3] CIGRE WG B1.07, Statistics of AC Underground Cables in Power Networks, CIGRE

Technical Brochure 338, 2007[4] Mohamed Abdel-Rahman, "Frequency Dependent Hybrid Equivalents of Large Networks",

PhD Thesis, University of Toronto, 2001[5] PSCAD Application Notes, "Converting a Solved PSS/E Case to PSCAD for Transient Simu-

lations", 2006[6] IEEE Guide for the Application of Sheath-Bonding Methods for Single-Conductor Cables and

Calculation of Induced Voltages and Currents in Cable Sheaths, IEEE Std. 575-1988[7] IEC 62067, "Power cables with extruded insulation and their accessories for rated voltages

above 30 kV (Um=36kV) up to 150 kV (Um=170 kV) - Test methods and requirements", Edi-

tion 3.0, 2004[8] IEC 60840, " Power cables with extruded insulation and their accessories for rated voltages

above 150 kV (Um=170kV) up to 500 kV (Um=550 kV) - Test methods and requirements",Edition 1.0, 2001

[9] B. Gustavsen, J. A. Martinez and D. Durbak, Parameter Determination for Modeling SystemTransients Part II: Insulated Cables, IEEE Transactions on power Delivery, Vol. 20, No. 3,July 2005

Documents

Study_of_Harmonics_in_Cable_Based_Transmission_Networks.pdf