A Free Tool in Laplace Domain for Assessing the Extra High Voltage Cable Energization

Roberto Benato, Sebastian Dambone Sessa Department of Industrial Engineering

University of Padova, Padova, Italy

roberto.benato@unipd.it, sebastian.dambonesessa@unipd.it

Davide Pietribiasi Prysmian Power Link

Milan, Italy davide.pietribiasi@prysmiangroup.com

Abstract—The purpose of this paper is to present a free tool easily implementable in commercial mathematical software (e.g. Matlab) which allows computing the switching overvoltages due to the cable (in single or double circuit) energization. Also the use of shunt reactive compensation can be accounted for. In order to achieve this goal, the authors have made an extensive use of Lapalce domain. The paper shows both a numerical inverse Laplace transform and a novel method which does not need a numerical inverse transform. In fact, by representing the transmission line as a cascade of n cells constituted of lumped L-C ladder, it is straightforward to make the inverse Laplace transform of rational functions as polynomials. Index Terms--Switching overvoltages, Insulated cables, Inverse Laplace Transform, ladder circuit.

I. INTRODUCTION When facing electro-magnetic transients, engineer minds prepare themselves to different types of commercial software. One of the most reliable and used is ATP/EMTP-RV. Differently, Academia, even if using the same commercial software, aims at finding complementary tools including self implementation in mathematical software environment. As it is well known, a powerful tool for transient analysis is the Laplace domain [1-3] where the time domain response is obtained by Inverse Laplace Transform (ILT). By assuming a symmetric structure of a transmission lines, the A, B, C and D parameters can be expressed as a function of the Laplace variable and the switching overvoltage due to energization holds transcendental terms (i.e. hyperbolic trigonometric functions). Their inversion can be cumbersome since the function to be inverted is not rational (i.e. it is not a ratio of polynomials). In these cases, the use of a numerical inverse Laplace transform (NILT) becomes unavoidable. In technical literature some attempts have been proposed in order to overcome NILT by means of a modelling constituted of one PI lumped circuit [4]. Differently, this paper presents a novel representation of the transmission line as a cascade of n cells constituted of lumped L-C ladder (with acronym LLCL). Using R-C ladder circuit [5] has been proposed in circuit theory end chiefly applied to microwave contest [6] but in this

specific application a L-C ladder representation gives a straightforward and powerful tool to easily compute the switching overvoltages in shunt-compensated long extra-high voltage (EHV) cables.

II. THE COMPLETE SOLUTION OF THE SYSTEM WITH UNIFORMLY DISTRIBUTED PARAMETER CABLE LINE

Let us consider the energization at no-load of an EHV insulated cable line (with acronym ICL) with length d which is shunt-compensated at both ends. The energizing grid is represented by its equivalent circuit i.e. a cosine generator u(t) with a series inductance lsc which can be computed by knowing the three-phase short-circuit current at the sending-end port. The transmission matrix of the ideal ICL (in EHV ICL, typical values of the ratio r/(ωl) and g/(ωc) are 0.07 and 0.0007 respectively) can be written in Laplace domain,

( )( ) ( )( ) ( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⋅⋅

⋅⋅= kdcosh

Zkdsinh

kdsinhZkdcoshs,d

c

cM (1)

where c

Zcl

= is the characteristic impedance and cs)s(k l⋅=

is the propagation constant. By posing cd ⋅= lτ , (1) can be re-written as in (2):

( )( ) ( )( ) ( ) ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡= ττ

ττ

scoshZ

ssinhssinhZscosh

s,dc

cM (2)

Figure 1 EHV shunt compensated cable line

ℓsc

ℓsr ℓsr

R

u(t)

d

2

By means of (2), it is possible to compute the whole transmission system pertaining to the cascade of four two-port networks (TPNs) i.e. the elementary TPN of series impedance Zsc=slsc with matrix Msc, the elementary TPN of shunt admittance Ysr=1/(slsr) with matrix Msr, the TPN of cable line with matrix in (2) and the elementary TPN of shunt admittance Ysr=1/(slsr) at receiving-end with matrix Msr. Once the equivalent transmission matrix (which is given by the ordered product of each transmission matrix) has been computed, the no-load voltage at receiving-end UR(s) is immediately given by setting IR(s)=0 so that:

( ) ( )

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⋅

=

src

sr2csc

2sr

2sc

2c

sr

scR

sZZsZssinh12scosh

sUsU

l

llll

l

l ττ

(3)

where U(s) is the Laplace transform of the supplying generator. If u(t)=cosωt, by considering an unitary peak value, it is immediate to obtain: ( ) ( )22 sω

ssU+

=

The switching overvoltages due to the shunt compensated cable energization are given by the ILT of (3). Unfortunately, it is not analytically possible since it has transcendental terms. Consequently, a NILT is necessary and has been ideated by the authors and implemented in Matlab environment. The proposed NILT is based on the residue theorem of the complex analysis.

III. A NILT FOR COMPUTING SWITCHING OVERVOLTAGES DUE TO ICL ENERGIZATION

In complex analysis, a residue of a function f(z) in a point z0 is the first Laurent series coefficient C-1 calculated in z0. If z0 is a first order singularity of the function f(z) it can also be written: [ ])z(f)zz(limC)z(R 0

zz10

0

−==→

− (4)

If f(z) has a number n of singularities { }nzzz ,,, 21 K contained in a set Γ , then the following theorem is valid:

∫Γ∂

+++= )z(R)z(R)z(R)z(fi2

1n21 K

π (5)

and it takes the name of residue theorem. It can be demonstrated that choosing a proper set, an equivalence exists between the Laplace inverse transform and the residues:

[ ] )()()()(21)( 21

1n

ix

ix

st sRsRsRsFei

sFL +++== ∫∞+

∞−

− Kπ

(6)

Without demonstrating this theorem, it can be seen that the definition of analytical Laplace inverse transform presented in (6) is clearly similar to the first term of (5). The terms

{ }nsss ,,, 21 K are commonly known as the poles of F(s). The first point of this analysis is to calculate the poles of (3). This function can be seen, for the sake of clearness, as the ratio of two functions:

)s(D)s(U)s(UR = (7)

where U(s) is, as already mentioned, the Laplace transform of the applied voltage u(t) and D(s) is the denominator of (3). Now, it is possible to group the poles of UR(s) in two sets:

• poles of the input U(s) • poles of the ICL 1/D(s)

Since the Laplace transform of a cosine impressed voltage is known, the poles of U(s) can be found immediately: ωisIN ±= (8) where ω is the angular frequency of the supply voltage. The poles of 1/D(s) are not analytically computable, since the denominator has transcendental terms. An algorithm, by using the Newton Raphson method, has been built in Matlab to solve the equation D(s)=0. The number n and precision ε of roots for this equation can be set manually to achieve the desired final algorithm performance. From this calculation a set of n poles is obtained. If we consider an ideal ICL, without dissipation effects due to the resistance and the conductance, these poles are purely imaginary: { } { }nn iiisss μμμ ,,,,,, 2121 KK ±= with nμμμ <<< K21 (9) The second step is the computation of the residues of estUR(s). The poles { }nsss ,,, 21 K are by definition those terms which zero D(s). It is clear how the application of (4) to UR(s) would result in an undetermined form. The easiest way to solve the limit is to expand UR(s) using Taylor. Equation (3) can be rewritten explaining the term U(s), assumed for convenience with unitary peak value:

( )( ) ( ) ( )

( ) ( ) ( )22

22

src

sc2sr

2srsc

2c

sr

scR

sωsCsBsτsinhAsτcosh

s

sωsZ

s)(Zsτsinh12sτcosh

ssU

+⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ +⋅+⋅

=

=

+⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛ ++⋅+⎟⎟

⎠

⎞⎜⎜⎝

⎛+⋅

=

l

llll

l

l

where:

c

scsr

sr

srscc

sr

sc

ZC

)(ZBA

ll

l

ll

l

l=

+=+= 12

The Taylor expansion has to be calculated on a specific point, in our case the poles of UR(s). A residue expression, function of s, can be derived from each group of poles. Since they are purely imaginary, it can be demonstrated that the residues will be purely real. The Taylor expansion allows solving the limit and then calculating the residues. The one derived from the input pole is:

3

( ) ( ) ⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ −⋅−

=

ωωωω

ω

BCτsinτcosA2

e)t(Rti

input (10)

whereas those for a generic ICL pole ii is μ±= are:

( ) ( )τKτKetR

iiii

tii

iICL

i

μμμ μ

sincos)(

sin,cos,, −

= (11)

where:

)3(2)(

)(2

22

22sin,

22cos,

ωμμ

τμω

μμ

μωτμ

−−+⎟⎟⎠

⎞⎜⎜⎝

⎛+−=

⎟⎟⎠

⎞⎜⎜⎝

⎛−−+−=

ii

ii

ii

iii

CBBAK

CBAK

By arranging the results according to (5), it is possible to obtain the expression of the transient response of the ICL after energization with a cosine voltage generator, as the sum of the residues:

∑=

+=n

iiICLinput tRtRtu

1, )()()( (12)

It is worth mentioning that

• the complex exponential can be rewritten by the Euler formula as the sum of a cosine and a sine: )sin()cos( tite ti ωωω += (13) thus it is a term varying in time domain with a pulsation ω and unitary magnitude;

• since the poles are always complex and conjugate, for every μi the imaginary components of (13) are zeroed and only the real part remains, confirming that u(t) is in the time domain, as expected;

• the second term of (12) is the transient component: it determines the peak value of the switching overvoltage. If the hypothesis of ideal line was not assumed, it would yield 0)t(Rlim

ii,ICL

t=∑∞→

. It can

also be noted that the amplitude of )(, tR iICL decreases for i increasing, so that the contribute of higher frequencies to the total peak value is negligible.

Equation (12) has been implemented in Matlab in order to have a unique tool to analyse ICL for comparison or quick check purposes.

IV. A NOVEL LUMPED L-C LADDER METHOD Even if the abovementioned NILT already represents a good tool to have ICL energization switching overvoltages, the authors have tried to find a more direct and simpler method. This novel approach consists in representing the ICL as a cascade of n LLCLs each representing a d/n length of the cable line so achieving the model of fig. 2.

Figure 2 EHV shunt compensated ICL represented by a cascade of lumped L-

C ladder

The number n can be chosen arbitrarily but precise results can be obtained even with n=2. As it has been already mentioned, the ICL is considered to be ideal i.e. the series resistance and the shunt conductance are neglected. For EHV ICL, with typical values of the ratio r/(ωl) and g/(ωc) equal to 0.07 and 0.0007 [7], respectively, this seems to be a licit approximation. This hypothesis could be not verified for MV and LV cables. The transmission matrix TΔ of one two-port LLCL is simply expressed by:

( )⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ ⋅+

=⎥⎦

⎤⎢⎣

⎡ +=Δ

1ndsc

nds

ncds1

1YZZY1

s,d 2

22l

l

T (14)

The cascade of n two-port LLCLs which are all equal is given by:

( )

n

2

22

1ndsc

nds

ncds1

s,d

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡ ⋅+

== Δ

ll

nTT (15)

The term A(s) of T can be expressed in a general form as in (16) which has been blocked at the third addend:

( )( )

( )( ) ( )

( ) ...dkn

1nn2n241

dkn

nn21

1sA 44

222

2

2

+−⋅+⋅

++⋅

+= (16)

The equivalent matrix of the system shown in fig. 2 is given by the product of the involved transmission matrices i.e.

srn

srsceq MTMMT Δ= (17)

At no-load the receiving-end voltage is linked to the term Aeq(s) of this matrix and it holds only polynomials i.e. it is a rational functions whose ILT is immediate. In order to have a view of Acasc(s) for n=2 and without reactors, first it is necessary to construct the matrix Mcasc

ℓsc

ℓsr ℓsr

R

u(t) cd/ncd/n

ℓd/n ℓd/n

4

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

+

++⋅⎥

⎦

⎤⎢⎣

⎡=

.......n

dcsndsc2

.......nds)cds1(

10s1

323

2

2

sc2222

sc

l

lllcascM

and then its (1,1) element is given by:

)dc(ns)cd2

ncd(

ns)scd1()s(A 32

sc

4

sc

22222

casc llll

l ++++== (1,1)Mcasc

This novel LLCL method can be applied also to real ICL. Since the complexity of this issue, it is not here considered but it has been presented in [8]. In this case, it is worth noting that LLCL cannot accommodate frequency dependent parameters. Since the paper focus is the switching overvoltages which are characterized by a not wide frequency spectrum, this limitation is acceptable and the method gives a very good precision in the peak evaluation. Also the case of switching condition with t>0 can be considered with LLCL method [8].

V. COMPARISON OF LLCL WITH NILT AND EMTP_RV MODEL

By considering a 2500 mm2 400 kV EHV double circuit cable line, the no-load receiving-end voltage was calculated by means of both the LLCL and the NILT methods. The voltage computation has been performed by taking into account both the line with the shunt compensation reactors and without them. The case study layout is represented in fig.3. Each circuit of the line is composed of three single-core Milliken-type (M-type) cables (see fig.4) and their geometrical characteristics are reported in Tab.1. An EMTP-RV model (see fig. 5) has been implemented since it has been chosen as reference and the NILT and LLCL have been compared with it.

Figure 3 Case study layout

Figure 4 Milliken-type (M-type) cable

Figure 5 EMTP-RV cable line model

It is worth nothing that the EMTP-RV model is a three-phase one, differently from those of NILT and LLCL which are single-phase ones with the hypothesis of perfectly symmetrical structure of the cable line. In the comparison between different methods, for EMTP-RV the central phase of each cable circuit has been chosen. The ICL electrical characteristics are reported in Tab.2. The consideration of the double-circuit for the computation of the equivalent Aeq, Beq, Ceq and Deq is immediate from the parameters of the single-circuit so that: Deq=Aeq=A=D; Beq=B/2; Ceq=2 C .

Tab.1 XLPE-insulated cable geometrical data

Cross sectional area/Material mm2 2500/Cu M-type

Central duct mm 10

Conductor diameter mm 64.3

Conductor screen diameter mm 69.5

Insulation diameter/material mm 119.9/XLPE

Insulation screen diameter mm 127.7

Metallic screen diameter/material mm 130.1/Al

Overall diameter mm 141.7

2500 mm2 400 kV double circuit ICL 30 km length; frequency = 50 Hz

5000 mm

R

S T

350 350

T S R

350

350

1450

1 2 3 7 8 9

4 5 6 10 11 12 phases screens

Milliken conductor Semi-conductive screen

Insulation

Semi-conductive screen

Metallic screen

Outer sheath

5

Tab.2 Cable electrical characteristics

Nominal Voltage U kV 400 Per unit length series

inductance of single-circuit, l mH/km 0.55

Per unit length shunt capacitance of single-circuit, c μF/km 0.234

Line length, d km 30 Shunt compensation reactors

at both ends H 2.62

Generator series inductance lsc mH 76.8 For the shunt compensated ICL the shunt compensation degree ξsh is equal to 0.55 [7, 9], so that it is possible to compute ℓsr as:

62.2dc

1

sh2sr =

⋅⋅⋅=

ξωl [H]

with ω=2 π f and f=50 Hz. The value of the generator series inductance ℓsc depends upon the chosen fault level at the supply node. The three-phase short circuit has been chosen equal to 10.55 kA, so that, for the single-phase modelling of the ICL:

8.76I3U

ph3,ccsc =

⋅⋅=

ωl [mH]

For the three phase EMTP-RV model, the supplying network has been modelling by means of the Fortescue theory [10, 11], by choosing also a single-phase short circuit current equal to 11.16 kA. The transient overvoltage at the no load-end of the case study line without the compensation reactors is shown in fig. 6, in which the three above mentioned methods have been compared. Fig. 7 shows a detail of the first 5 ms. Fig. 8 shows the transient voltage behaviour by considering the compensation reactors. A detail of the first 5 ms of the transient is shown in fig. 9. Figures 6-9 highlight the extremely good agreement of LLCL with NILT and with EMTP-RV.

Figure 6 Comparison of the transient overvoltages in not-compensated ICL

Figure 7 Comparison of the transient overvoltages in not-compensated ICL in the first 5 ms

As it is shown in the figures above, in the evaluation of the peak overvoltage the LLCL method gives, in case of compensated line, an underestimate of 0.53% (1.665 p.u. versus 1.674 of EMTP-RV) and an overestimate of 1% in case of not-compensated line (1.753 p.u. versus 1.734 of EMTP-RV). It is worth noting the great LLCL precision in the evaluation of the peak value and that there is a higher error in the third crest equal to 6,58 % in case of fig. 8. This is due to the ideal line assumption. Moreover, it is demonstrated that the presence of shunt compensation reactors does not sensibly affect the transient switching overvoltages. Notwithstanding they play a key role in limiting the steady state overvoltages due to the Ferranti's effect.

Figure 8 Comparison of the transient overvoltages in shunt compensated ICL

6

Figure 9 Comparison of the transient overvoltages in shunt compensated ICL in the first 5 ms Moreover, both the LLCL and NILT CPU times are remarkably lower than the EMTP-RV ones, as it is shown in tab.3. LLCL takes about half time of that taken by EMTP_RV.

Tab.3 CPU time comparison CPU characteristics Intel core i5-2400, 8 GB RAM

Method Computation time EMTP-RV Not-compensated line: 5.023 s

Compensated line: 4.836 s NILT Not-compensated line: 0.304 s

Compensated line:0.893 s LLCL Not-compensated line:2.037 s

Compensated line:3.103 s

VI. CONCLUSIONS The paper presents a novel method to compute the switching overvoltages by means of a simple Laplace inversion of rational functions deriving from cascade of lumped L-C ladders. This novel method has been compared with a NILT also proposed by the authors and an EMTP-RV model. LLCL and NILT have been both implemented in Matlab environment. By assuming EMTP-RV results as reference, the LLCL method gives negligible differences in the peak estimate and maximum error up to 5 %. The authors think that this precision degree of a complementary tool is enough for engineering purposes. The extreme simplicity and calculation speed of lumped L-C ladder method renders it an efficient, simple, fast and complementary tool for power engineers who want to implement self-procedures in any mathematical environment for validation of commercial software or for understanding the roles played by each parameter. The generalization to the three-phase case is under consideration as a further development of the research. The use of symbolic multiconductor cell analysis (MCA in [12]) could give the solution of switching overvoltages in asymmetrical lines: in order not to complicate the model, only two multiconductor cells could be considered. In any case, for

EHV cables which are always installed with cross-bonded screens and often with phase transpositions the symmetry degree is very high and the single-phase switching overvoltage assessment is wholly suitable.

VII. REFERENCES [1] F. A. Uribea, J. L. Naredoa, P. Morenoa, L. Guardadob: "Electromagnetic

transients in underground transmission systems through the numerical Laplace transform", International Journal of Electrical Power & Energy Systems, Vol. 24, Issue 3, pp. 215–221, 2002.

[2] X.m. Lopez-Fernandez, C. Alvarez-Marino, V. Malo-Machado, "Computation Method for Transients in Underground Cables With Lossy Earth Return Path", IEEE Transactions on Magnetics, Vol. 46, Issue: 8 pp. 2911 – 2914, 2010.

[3] P. Moreno and A. Ramirez: "Implementation of the Numerical Laplace Transform: A Review", IEEE Trans. on Power Delivery, vol. 23, n° 4, 2008, pp. 2599-2609.

[4] S. Skuletic, V. Radulovic: "Contribution to Calculation of Switching Overvoltages Originated by Line Energizing in the Networks with Lumped Parameters", Proc. of the 41st UPEC '06, Vol. 3, pp. 926 – 930, 2006.

[5] K.K. Pang: "Cascade synthesis of distributed networks by lumped ladder RC equivalents", IEEE Transactions on Circuit Theory, Vol. 17, Issue: 4, pp. 549 - 554, 1970.

[6] S.C. Burkhart and R.B. Wilcox: "Arbitrary pulse shape synthesis via nonuniform transmission lines", IEEE Trans. On Microwave theory and techniques, Vol. 38, N. 10, pp. 1514-1518, 1990.

[7] R. Benato, A. Paolucci: EHV AC Undergrounding Electrical Power. Performance and Planning, Springer, ISBN: 978-1-84882-866-7, 2010.

[8] R. Benato, S. Dambone Sessa, D. Pietribiasi: "A simple and precise method to assess the switching overvoltages in EHV insulated cables", International Review of Electrical Engineering (I.R.E.E.), Vol 10, No 4 (2015).

[9] R. Benato, S. Lauria, F.M. Gatta, L. Colla, M. Rebolini, F. Renaud: "Steady-State and Transient EHV AC Cable Shunt Reactive Compensation Assessment", Proc. of Cigré 2010, Paper C4-109.

[10] R. Benato, S. Dambone Sessa, F. Guglielmi: Determination of Steady-State and Faulty Regimes of Overhead Lines by Means of Multiconductor Cell Analysis (MCA). Energies 2012, 5, 2771-2793, doi: 10.3390/en5082771.

[11] R. Benato, S. Dambone Sessa, F. Guglielmi, E. Partal, N. Tleis: Zero sequence behaviour of a double-circuit overhead line, Electric Power Systems Research 116, pp. 419-426, 2014.

[12] R. Benato: "Multiconductor Analysis of Underground Power Transmission Systems: EHV AC Cables", Electric Power System Research, Vol. 79, Issue 1, 2009, pp. 27-38, doi: :10.1016/j.epsr.2008.05.016.

VIII. GLOSSARY ICL: Insulated Cables Line; ILT: Inverse Laplace Transform; NILT: Numerical Inverse Laplace Transform; LLCL: Lumped L-C ladder; ξsh: shunt compensation degree; M: transmission matrix; Msc: transmission matrix of equivalent supply; Msr: transmission matrix of shunt reactor compensator; c: kilometric shunt capacitance; l: kilometric series inductance; lsc: short-circuit inductance due to the supply; lsr: shunt reactive compensation inductance; d: cable length; Zc: characteristic impedance; k: propagation constant; A, B, C, D: elements of the transmission matrix; Aeq, Beq, Ceq, Deq: elements of the transmission matrix representing the double-circuit cable system.