Licentiate Thesis
Stefan Gustafsson 2012-11-27
Electromagnetic dispersion
modeling and analysis for
HVDC power cables
Stefan Gustafsson
Electromagnetic dispersion modeling and analysis for HVDC power cables
Licentiate Thesis Engineering Physics
2012
A thesis for the Degree of Licentiate of Philosophy in Engineering Physics
Electromagnetic dispersion modeling and analysis for HVDC power cables Stefan Gustafsson
Linnæus University
School of Computer Science, Physics and Mathematics SE-351 95 V¨axj¨o, Sweden
http://www.lnu.se
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Abstract
Derivation of an electromagnetic model, regarding the wave propagation in a very long (10 km or more) High Voltage Direct Current (HVDC) power cable, is the central part of this thesis. With an existing “perfect” electromagnetic model there are potentially a wide range of applications.
The electromagnetic model is focused on frequencies between 0 and 100 kHz since higher frequencies essentially will be attenuated. An exact dispersion relation is formulated and the propagation constant is computed numerically. The dominating mode is the first Transversal Magnetic (TM) mode of order zero, denoted TM01, which is also referred to as the quasi-TEM mode. A comparison is made with the second propagating TM mode of order zero denoted TM02.
The electromagnetic model is verified against real time data from Time Domain Reflection (TDR) measurements on a HVDC power cable. A mismatch calibration procedure is performed due to matching difficulties between the TDR measurement equipment and the power cable regarding the single-mode transmission line model.
An example of power cable length measurements is addressed, which reveals that with a “perfect” model the length of an 80 km long power cable could be estimated to an accuracy of a few centimeters. With the present model the accuracy can be estimated to approximately 100 m.
In order to understand the low-frequency wave propagation characteristics, an exact asymptotic analysis is performed. It is shown that the behavior of the prop- agation constant is governed by a square root of the complex frequency in the low- frequency domain.
This thesis also focuses on an analysis regarding the sensitivity of the propaga- tion constant with respect to some of the electric parameters in the model. Variables of interest when performing the parameter sensitivity study are the real relative per- mittivity and the conductivity.
Keywords: HVDC power cable, electromagnetic model, TDR measurement, sensi- tivity analysis, dispersion relation, propagation constant, low-frequency asymptotics
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Acknowledgements
First and foremost I would like to express my gratitude to my main-supervisor Sven Nordebo and to my co-supervisors B¨orje Nilsson and Mats Sj¨oberg. Sven and B¨orje have been very helpful and supporting in the course of the work, and Mats has been very assisting in the fieldwork when measurements were carried out on power cables.
Also, many thanks to the company ABB in Karlskrona whose cables and premises we have had access to during the measurements.
One of my fellow Ph.D students, Doctor Therese Sj¨oden, also deserves a warm thanks for all the good and interesting discussions we have had.
I would like to finish off with a word of thanks to my supporting family and friends.
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Preface
The central part of this thesis treats the electromagnetic model regarding wave propagation in very long (10 km or more) High Voltage Direct Current (HVDC) power cables. Below is a list of papers included in this thesis, where each paper is based on the electromagnetic model, but with different areas of application.
Papers included in the thesis
I. S. Nordebo, B. Nilsson, T. Biro, G. Cinar, M. Gustafsson, S. Gustafsson, A.
Karlsson, and M. Sj¨oberg. Electromagnetic dispersion modeling and measure- ments for HVDC power cables. Technical Report LUTEDX/(TEAT-7211)/1- 32/(2011), Lund University, Department of Electrical and Information Tech- nology, P.O. Box 118, S-221 00 Lund, Sweden, 2011. http://www.eit.lth.se.
II. S. Nordebo, B. Nilsson, T. Biro, G. Cinar, M. Gustafsson, S. Gustafsson, A. Karlsson, and M. Sj¨oberg. Low-frequency dispersion characteristics of the multi-layered coaxial cable. Technical Report LUTEDX/(TEAT-7212)/1- 21/(2011), Lund University, Department of Electrical and Information Tech- nology, P.O. Box 118, S-221 00 Lund, Sweden, 2011. http://www.eit.lth.se.
Submitted to the Journal of Engineering Mathematics.
III. S. Gustafsson, S. Nordebo, and B. Nilsson. Electromagnetic dispersion mod- eling and sensitivity analysis for HVDC power cables. Technical report, pub- lished in Diva as URI: urn:nbn:se:lnu:diva-22296, Linnæus University, School of Computer Science, Physics and Mathematics, 351 95 V¨axj¨o, Sweden, 2012.
IV. S. Gustafsson, T. Biro, G. Cinar, M. Gustafsson, A. Karlsson, B. Nilsson, S.
Nordebo, and M. Sj¨oberg. Electromagnetic dispersion modeling and measure- ments for HVDC power cables. Submitted to IEEE Transactions on Power Delivery, October 15, 2012.
Contributions to the papers
In paper I and II the author of this thesis has been involved in the work concerning the electromagnetic model, measurements and most of the numerical computations.
In paper III and IV (which is a synthesis of paper I and III) the author has had the main responsibility regarding analytic results, numerical computations and writing.
Related papers, not included in the thesis
• S. Gustafsson, S. Nordebo, and B. Nilsson. An electromagnetic dispersion model for HVDC power cables. The Sixth International Conference on “In- verse Problems: Modeling and Simulation”, May 21-26, 2012, Antalya, Turkey.
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• S. Nordebo, S. Gustafsson, and B. Nilsson. Wave modeling for HVDC power cables. RadioVetenskap och Kommunikation (RVK), Poster Session, March 6, 2012, Stockholm, Sweden.
• S. Nordebo, B. Nilsson, T. Biro, G. Cinar, M. Gustafsson, S. Gustafsson, A.
Karlsson and M. Sj¨oberg. Wave modeling and fault localization for underwater power cables. The eleventh International Conference on Electromagnetics in Advanced Applications (ICEAA), September 12-16, 2011, Torino, Italy.
• S. Nordebo, S. Gustafsson, and B. Nilsson. Fault localization for HVDC power cables. The Sixth International Conference on “Inverse Problems: Modeling and Simulation”, May 21-26, 2012, Antalya, Turkey.
• S. Nordebo, S. Gustafsson, A. Ioannidis, and B. Nilsson. System identifica- tion for wave guides with complicated structure and complex media. Mod- ern Mathematical Methods in Science and Technology (M3ST), August 26-28, 2012, Kalamata, Greece
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Contents
1 Introduction 1
2 The HVDC power cable 2
3 Measurements on the HVDC power cable 3
4 The electromagnetic model 4
5 Included papers 6
6 Conclusions 7
I Electromagnetic dispersion modeling and measurements for HVDC power cables
II Low-frequency dispersion characteristics of the multi-layered coaxial cable
III Electromagnetic dispersion modeling and sensitivity analysis for HVDC power cables
IV Electromagnetic dispersion modeling and measurements for HVDC power cables
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1 Introduction
Never before has the demand for electricity been as great as it is now, and with more well-developed countries and more power consuming products, that is a trend expected to increase rather than decrease. With a society so dependent on elec- tricity the power grid needs to have high reliability and the electric power has to be distributed in a safe and environmental friendly manner. The growing need for electricity is placing new demands on the power grid regarding higher and higher power transmission levels, and consequently this requires new power products, such as new power cables. To cost-effective transport power over long distances, feeding oil/gas platforms with power, and to deliver power from off-shore wind power plants are all examples of important power cable applications [1, 4]. With an increased need for power cables follows more attention and focus on power cable mainte- nance and monitoring, e.g. Partial Discharge (PD) surveillance [11, 14, 15], Power Line Communication (PLC) techniques [7, 8], fault localization problems and length measurements. In the light of this, measurements (e.g. Time-Domain Reflection, TDR) and electromagnetic modeling of the power cable are of significance to better understand its complex behavior and structure.
Although there could be built in fiber optic cables used for e.g. PLC techniques, the main purpose of a power cable is to transport large amounts of electric power over long distances, and to be able to transmit these high voltages and currents the cable needs to be designed in a very special way. More details regarding the cable and its design can be found in section 2.
Section 3 addresses the TDR measurement that was performed on a HVDC power cable. The equipment consisted of an oscilloscope and a pulse generator, where the latter generated a rectangular pulse which was registered in the oscilloscope before and after the pulse had propagated through the power cable. The data could then be compared to and verify the electromagnetic model.
The central part of this thesis is the electromagnetic model, and depending on which paper, I-IV, we are focusing on, different applications to that model is addressed. The model concentrates on very long power cables (10 km or more) and frequencies between 0 and 100 kHz. An exact dispersion relation is derived which gives us the possibility to obtain the propagation constant related to each frequency in the range of interest. The electromagnetic model is explained further in section 4.
Section 5 summarizes the included papers, and section 6 concludes.
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2 The HVDC power cable
Power cables are an important part of the power transmission infrastructure. The power cables are constructed to transport very high voltages and currents and to be able to do that they need to be designed and manufactured in a very specific manner, see Figure 1 below. The cable modeled and measured on in this thesis is an 80 km long HVDC power cable, manufactured at ABB, HVC (High Voltage Cables) in Karlskrona, Sweden. The 500 MW and ±200 kV power cable is a part of the cable system connecting Ireland and Wales.
Figure 1: A typical example of a mass impregnated paper insulated HVDC cable to the left and four polymeric insulated HVDC cables to the right. The two in the front are sea cables whereas the two in the back are land cables.
When transporting power over long distances, low power losses are of significance, then the High Voltage Direct Current (HVDC) power cables are to prefer, compared to the High Voltage Alternating Current (HVAC) power cables. The possibility to accurately control the active power and voltages, the environmental aspects, and the ability to long distance water crossing also speak in favor of the HVDC power cable. There are several areas of application for the HVDC power cable, e.g. they can be used when powering distant islands, connecting wind farms, feeding oil and gas offshore platforms with power, or the possibility to use an underground power link instead of overhead power lines [1, 4].
HVDC power cables can be designed in different ways depending on their field of application. The insulation layer of the cable can be made of mass impregnated paper or a polymeric material. A submarine cable with an outer protecting armour can be made of one or two layers of steel wires, or no steel wires at all if the cable should be buried in ground. The cable this thesis is concerned with has a polymeric insulating material. Although the cables can be designed differently their internal structure are similar. The innermost layer is a copper or aluminum conductor which transports the power in the cable. The conductor is then surrounded by the insula- tion system, which main purpose is to confine the power to the conductor. The outer layers (lead, polymeric and steel layers) protects the cable from external damage [5].
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3 Measurements on the HVDC power cable
The 80 km long cable mentioned above was in June 2011 subjected to a Time- Domain Reflection (TDR) measurement and it was carried out at the domains of ABB in Karlskrona just after the cable had been manufactured. The purpose of doing a TDR measurement is to be able to compare and evaluate its result with an electromagnetic model described in section 4. The TDR measurement equipment consisted of an oscilloscope and a pulse generator, see Figure 2. In order to connect
The inner conductor of the coaxial cable The shield of the coaxial cable
Coaxial cable
Oscilloscope
Pulse generator
Figure 2: The measurement equipment consisted of an oscilloscope and a pulse generator, which were connected to the power cable using a coaxial cable. The inner conductor of the coaxial cable was connected to the conductor of the power cable, and the shield of the coaxial cable was connected to the lead layer of the power cable.
the coaxial cable to the power cable the armour of the power cable had to be bent back, as can be seen in Figure 3, exposing the underlying layers in the power cable.
The coaxial cable was then connected to the lead sheath and the conductor of the power cable, see Figure 2. The pulse generator was set to generate a rectangular pulse with a pulse-width of approximately 100 μs, and the initial pulse was registered in the oscilloscope. The pulse then propagated through the cable, was reflected at the other end of the cable, and propagated back through the cable, and once again registered in the oscilloscope. The initial pulse and the reflected pulse were no longer identical due to attenuation and dispersion, which is illustrated in Figure 3.
A TDR measurement equipment can be used to find and locate cable failures, but it is rarely a simple task to point out exactly where the break down has occurred due to the complexity of the power cable. In a perfect case scenario the power cable would have no influence on the appearance of the propagating pulse, and the cable break down would be one of two scenarios, either a short circuit (indicating low impedance) or the opposite, an open end (indicating high impedance). The generated initial pulse would then propagate through the cable, be reflected at the location of the cable failure (with opposite polarity if the failure was of a low impedance character), and propagate back through the cable undeformed. Also, if the impedance at the cable break down is not equal to zero, some part of the pulse would propagate through the location of the cable failure and be reflected at the other end of the cable. But, in reality things are not that perfect. As was mentioned above, the power cable will influence the propagating pulse due to attenuation and
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Figure 3: To the left we can see one end of an 80 km long HVDC power cable. The armour is bent back, exposing the lead sheath and the conductor. The conductor is here attached to a conductive band which is connected to earth. The figure to the right shows us a typical example of the result of a TDR measurement with the initial rectangular pulse and its reflection.
dispersion. Also, when sending a pulse through the power cable all the impedance discontinuities in the power cable will affect the pulse in such a way that some part of the pulse will be reflected before even reaching a potential fault location. Impedance discontinuities of this kind could for instance arise from cable joints, which could be one or many depending on if it is a sea cable or a land cable. To complicate things even more, each cable failure is unique so the cable break down is not always one of the two cases mentioned above, it could be similar to one of them or it could be something in between (neither low or high impedance).
4 The electromagnetic model
If a correct electromagnetic model of a specific power cable can be derived, it is anticipated that such a model can be very useful for fault localization, Partial Dis- charge (PD) surveillance and cable length measurements, to name a few areas of interest.
The power cable is composed of a series of different layers, with different mate- rials, designs, purposes and electric properties. Cable modeling with several layers has been considered in e.g. [2, 3, 10, 13]. However, there does not seem to be any systematic treatment on the general behavior and computational techniques with regard to the dispersion relation for a multilayered coaxial cable. The layers will in this thesis be distinguished by their thickness ρ (related to the distance from the center of the cable), real relative permittivity r, and conductivity σ. Each layer is indexed, starting with 0 at the innermost layer (the conductor) and ending with 8 at the outermost layer (air or sea water), see Figure 4.
The focus of this thesis is concerned with the first transversal magnetic mode of
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r0
r1
r2
r3
r4
r5
r6
r7 r8
σ0
σ1
σ2 σ3
σ4 σ5
σ6
σ7 σ8
ρ0 ρ1 ρ2
ρ3
ρ4
ρ5
ρ6 ρ7
Figure 4: Geometrical and material definitions of the multi-layered HVDC power cable, where ρ is the distance from the center of the cable, r is the real relative permittivity and σ is the conductivity.
order zero [6, 12], also called the quasi-TEM mode, here denoted TM01. There are other modes in the power cable, both TE and TM modes, but in the low-frequency domain these modes will essentially be cut-off.
The propagating electromagnetic wave must submit to certain boundary condi- tions related to the boundary between the different layers in the power cable. The boundary conditions can be expressed as A(γ)x = 0, where A(γ) is a square matrix containing the field components, γ is the propagation constant representing a partic- ular mode and x is a vector with expansion coefficients. Solving this equation using the determinant, i.e. detA(γ) = 0, which is referred to as the dispersion relation, will give us the propagating modes in the cable.
Finding solutions to the dispersion relation is in this thesis done numerically using the MATLAB software. The zeros to the function detA(γ) is found by plotting the function in the complex plane, see Figure 5. The zeros can be found visually and given an approximate complex value and the exact value of the propagation constant can then be found by normalized residue [9]. When the frequency is changed, the zero, representing the TM01-mode in Figure 5, will move to another location in the plot, so each frequency has an associated complex propagation constant. We are in this thesis interested in the propagation constant in a frequency range of 0−100 kHz.
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Im[γ]
Re[γ]
Figure 5: A plot of the dispersion relation |detA(γ)| in the complex plane. The imaginary part of the propagation constant on the vertical axis and the real part on the horizontal axis. Two zeros (modes) can be found in the plot, where the TM01- mode is in the lower left corner. Blue color is indicating a low value of |detA(γ)|, while red color is representing a high value.
5 Included papers
All four papers have the electromagnetic model for the HVDC power cable in com- mon but with different areas of application.
Paper I: Electromagnetic dispersion modeling and measurements for HVDC power cables
In this paper the electromagnetic model is derived and validated with experimental TDR data from an 80 km long HVDC power cable. The frequency range is in the low-frequency regime of about 0−100 kHz. Due to limitations in the single- mode transmission line model, between the power cable and the TDR measurement equipment, a mismatch calibration procedure is formulated. An example of length measurement is performed and analyzed using statistical methods based on the Cram´er-Rao lower bound.
Paper II: Low-frequency dispersion characteristics of the multi-layered coaxial cable
Paper II provides an exact asymptotic analysis regarding the low-frequency dis- persion characteristics of the multi-layered coaxial cable. The dispersion relation is analyzed using a layer-recursive description. It is proven that, if there is one isolating layer and one perfectly conducting outer layer, then an exact analytical expression for the dominating term of the propagation constant can be derived, based on an asymptotic expansion. Also, an example with an outer layer with finite conductivity and an infinite exterior region with finite non-zero conductivity is addressed.
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Paper III: Electromagnetic dispersion modeling and sensitivity analysis for HVDC power cables
As was done in paper II, this paper treats the electromagnetic model in a layer- recursive way. The paper also focus on a sensitivity analysis of the propagation constant with respect to the electrical parameters r (the real relative permittivity) and σ (the conductivity). The results showed that r was most sensitive to a change in the insulation layer, whereas σ in the lead layer was most sensitive compared to the other layers and their conductivities.
Paper IV: Electromagnetic dispersion modeling and measurements for HVDC power cables
This paper is a synthesis of paper I and III. The paper provides an electromagnetic model and a sensitivity analysis for long (10 km or more) power cables. Experimental time-domain measurement data is used to validate the electromagnetic model and the sensitivity analysis focuses on finding which electrical parameters that will have the greatest impact on the electromagnetic model. However, one new addition in this paper compared to paper I and III is the study of seawater as an infinite exterior region. It is showed that the conductivity of seawater has a slightly increased sensitivity at frequencies below 1000 Hz.
6 Conclusions
In this thesis an electromagnetic model of a HVDC power cable has been derived, where the associated dispersion relation relates the frequency to the propagation con- stant, and the relevant frequency range is between 0 and 100 kHz. The model was validated with experimental time-domain measurement data from an 80 km long HVDC power cable using a Time Domain Reflection (TDR) measurement equip- ment. The results showed that there is a mismatch between the power cable and the measurement equipment and that a single-mode transmission line model is not accurate enough. Therefore, a mismatch calibration procedure was formulated.
An asymptotic analysis regarding the low-frequency dispersion characteristics has been derived, and this was done using a layer-recursive approach. It was proven that an exact analytical expression for the dominating term of the propagation constant could be derived. The results from the asymptotic analysis were compared with the numerical solutions of the dispersion relation. It could be shown that the asymptotic approximation was valid only with frequencies below 1 Hz.
Finding the exact correct values of the electrical parameters in the power cable is important when a reliable electromagnetic model is derived, but finding these values are not always easy. Some electrical parameters affect the model more than others, and in order to find out which parameters affect the model the most a sensitivity study was performed. The study showed that, in a frequency range of 0-100 kHz, the variables influencing the model were the real relative permittivity in
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the insulation layer and the conductivities in the metallic layers (the lead shield, the inner conductor and the outer steel armour). The rest of the electrical parameters showed to have no, or very limited, affect on the model.
It is assumed that a “perfect” electromagnetic model will have many areas of ap- plication, e.g. in the fields of Partial Discharge (PD) surveillance, Power Line Com- munication (PLC) techniques, fault localization problems and power cable length measurements. To further improve the model, finding more accurate electrical pa- rameter values is of importance, and to investigate the connection between the power cable and the measurement equipment with its related excitation of higher order modes. Also, the incorporation of the cable armour consisting of steel wires could be of interest.
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