Cálculo con elementos finitos de la impedancia armónica en cables tripolares submarinos

Proyecto Fin de Carrera Ingeniería de Telecomunicación Formato de Publicación de la Escuela Técnica Superior de Ingeniería Autor: F. Javier Payán Somet Tutor: Juan José Murillo Fuentes Dep. Teoría de la Señal y Comunicaciones Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, 2013 Trabajo fin de Máster / Master Thesis Máster en Sistemas de Energía Eléctrica Cálculo con elementos finitos de la impedancia armónica en cables tripolares submarinos Calculation of the Harmonic Impedance on Submarine Three-Phase Cables using Finite Element Method Autor: Ángel García Rivera Tutores: Pedro L. Cruz Romero y Juan Carlos del Pino López Dpto. Ingeniería Eléctrica Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, Junio, 2022 Trabajo fin de Máster / Master Thesis Máster en Sistemas de Energía Eléctrica Cálculo con elementos finitos de la impedancia armónica en cables tripolares submarinos Calculation of the Harmonic Impedance on Submarine Three-Phase Cables using Finite Element Method Autor: Ángel García Rivera Tutores: Pedro L. Cruz Romero y Juan Carlos del Pino López Profesores Titulares, Doctores Dpto. Ingeniería Eléctrica Escuela Técnica Superior de Ingeniería Universidad de Sevilla Sevilla, Junio, 2022 Trabajofin de Máster / MasterThesis: Calculation of the harmonic Impedance on Submarine Three-Phase Cables using Finite Element Method Autor: Ángel García Rivera Tutores: Pedro L. Cruz Romero y Juan Carlos del Pino López El tribunal nombrado para juzgar el trabajo arriba indicado, compuesto por los siguientes profesores: Presidente: Vocal/es: Secretario: acuerdan otorgarle la calificación de: El Secretario del Tribunal Fecha: Agradecimientos En ciertas situaciones no sabemos de lo que podemos llegar a ser capaces. Mientras estudiaba el grado en ingeniería eléctrica, nunca me imaginé haciendo un máster de ningún tipo, ya que en esa época lo único que me importaba era terminar la carrera lo antes posible y encontrar un trabajo para poder independizarme y dejar de ser una carga para la familia (por lo menos económica). Sin embargo, tras haber vivido y trabajado tres años en Hamburgo, me veo escribiendo este trabajo de fin de máster y, en cierto modo, me he sentido retrotraído a la época del grado durante este año. Durante este tiempo me ha ayudado mucho saber que tenía grandes apoyos, tanto familiares, como de amigos. Por eso mismo, y aunque ya se lo agradeciera en mi TFG, quiero agradecerle a mi madre su apoyo durante todo este tiempo. También a mis tíos, por su orientación para llegar hasta aquí, desde que empezara ese ciclo superior en el I.E.S. Los Viveros. A mis dos tutores por el tiempo que han dedicado a guiarme con este trabajo fin de máster. A mi hermano, por el apoyo económico después de terminar el máster y hasta que encontré otro trabajo. Por último, a la futura Dr. Lena Rölle por estar siempre ahí y ser la mejor persona del mundo, y a su familia, por el apoyo tanto moral como económico. Wi r wissen manchmal nicht, was wir wirklich tun können. Als ich noch meinen Bachelor in Elektrotechnik und Bauingenieurwesen studierte, hätte ich nie gedacht, dass ich dazu noch einen Master machen würde, da ich damals nur an das Abschließen des Studiums denken konnte und daran, einen Job zu finden. Dennoch, nach vier Jahren, in denen ich in Hamburg gearbeitet und gewohnt habe, finde ich mich hier nochmal beim Schreiben der Masterarbeit. Manchmal habe ich mich an die Bachelorzeit erinnert gefühlt, da ich wieder Student und ohne Job war, aber ich habe mich wohlgefühlt, weil ich die ganze Zeit viel Unterstützung gehabt habe, sowohl von Familie, als auch von Freunden. Deswegen möchte ich meiner Mutter dafür danken, dass sie mich immer unterstützt. Auch möchte ich mich bei meinem Onkel und meiner Tante für die Orientierung meiner Ausbildung bedanken. Außerdem möchte ich meinen beiden Tutoren dafür danken, dass sie sich so viel Zeit genommen haben, um mich bei dem Prozess der Masterarbeit zu begleiten. Ebenfalls möchte ich meinem Bruder danken für die finanzielle Unterstützung nach dem Master bis ich eine neue Stelle fand. Zuletzt möchte ich mich bei der zukünftigen Dr. Lena Rölle bedanken, weil sie immer für mich da gewesen ist und das Beste, was mir je passiert ist. Genauso bedanke ich mich bei ihrer Familie für die finanzielle und die moralische Unterstützung. Ángel García Rivera Sevilla, 2022 I Resumen El objetivo principal de este trabajo de fin de máster es abordar un tema actual y emergente, como es el cálculo de impedancia armónica o rango armónico propio de cables submarinos de alta tensión en corriente alterna, debido al creciente interés que este tema ha despertado en los últimos años. Actualmente, se emplean mucho para conectar parques eólicos marinos (off-shore) a la red de distribución más cercana en tierra (on-shore), o para interconexiones eléctricas a través de líneas submarinas, especialmente para conectar islas con continentes que se encuentren relativamente cerca geográficamente, las cuales implementan este tipo de soluciones para mejorar la fiabilidad y calidad del suministro eléctrico dentro de las mismas. El mismo principio puede ser aplicado para interconexiones entre países, separados por una masa de agua, bien sean mares, u océanos. Puesto que una solución aérea de transmisión de potencia en este tipo de escenarios no es viable debido al coste de ejecución y mantenimiento de la misma, cada vez se recurre más a una transmisión subacuática de potencia, pudiendo ser esta potencia transmitida en corriente continua (HVDC) o en corriente alterna (HVAC). El trabajo se orientará en una dirección en la que, mediante el uso de COMSOL, un software que trabaja con herramientas de elementos finitos, se tratará igualar o mejorar los resultados publicados hasta la fecha, con la mínima carga computacional. De esta forma, este trabajo estudiará y comparará desde los métodos clásicos de cálculo de impedancia mediante el tedioso empleo de las funciones de Bessel, como ya hizo Pollaczek [ 22 ], hasta métodos más modernos con la utilización del mencionado software COMSOL, el cual ofrece a día de hoy mucha rapidez de cálculo, a la vez que exactitud, pero no se suele usar mucho para estudios de frecuencia debido a los tiempos computacionales. La motivación para esto es simple y directa: de manera clásica, cuando se trata de calcular impedancia en cables, se solía tratar el problema como un escenario en dos dimensiones, es decir, una serie de circunferencias concéntricas que van sufriendo efectos no deseados como el pelicular y proximidad, y los cuales se ven afectados directamente por las corrientes parásitas desatadas por el campo magnético alrededor de un conductor cargado con una corriente alterna. Este escenario es una aproximación válida, pero no es excesivamente precisa, ya que no contempla varios efectos que tienen lugar en el cable, situándonos en un escenario más realista. Este primer escenario llega a mejorar cuando se comienza a hablar de escenarios 2.5D, es decir, el mismo problema en 2D, pero con el añadido de considerar que en la armadura externa del cable no se induce corriente neta debido al trenzado simétrico del cable y al both-ends bonding (puesta a tierra de armadura y pantallas al principio y final del trayecto del cable). Esto mejora en algunas situaciones el cálculo e incluso lo aproxima para un rango de frecuencias determinado a la respuesta en 3D, pero para otros rangos no es muy acertado. Aun así, una mejora notable, la cual requiere más tiempo computacional que en 2D, pero mucho menos que en 3D. Finalmente, se plantean los escenarios en 3D, en los que se tiene en cuenta todo lo anterior, además del trenzado de armadura y fases y longitud de paso de las mismas y, con ello, se puede III Notation Chapter 2.1 - Thermal behaviour of the cable ρTthermal resistivity of the insulation (K.m/W) λ1 Ratio of losses in the metal screen to total losses in all conductors in the cable λ2 Ratio of losses in the armouring to total losses in all conductors in the cable di j Distance between two cable axes (m) d0 i j Distance between two projected cable axes from the method of images (m) FB Constant calculated using the method of images for one phase of a threephase single-core cable GGeometric factor whose behaviour is given by [1] hDistance from the surface of the ground to the cable axis ICurrent flowing in the conductor (A) nNumber of load-carrying conductors in the cable R AC resistance per unit of length at maximum operating temperature (Ω.m−1) R1Core conductor radius (Figure 2.1) R2Inner screen radius (Figure 2.1) R3Outer screen radius (Figure 2.1) R4Inner armour radius for a single-core cable (Figure 2.1) R5Outer armour radius for a single-core cable (Figure 2.1) R6Outer covering radius for a single-core cable (Figure 2.1) SSurface area of a conductor (m2) T1−4Thermal resistances from conductor to outside of the cable (Figure 2.1) WdDielectric losses per unit of length for the main insulation (W.m−1) WcEnergy dissipated from the surface area by convection (W.m−2) WrEnergy dissipated from the surface area by radiation (W.m−2) Chapter 2.2 - Impedance and admittance matrices αTTemperature coefficient (°C−1) εPermittivity (Ω.m) εIns Insulation Permittivity (Ω.m) εPPermittivity of the pipe (Ω.m) XI XII Notation µ0Permeability in a vacuum µPPermeability of the pipe µrRelative permeability ωAngular frequency ρResistivity (Ω.m) ρAResistivity of the armour(Ω.m) ρCResistivity of the conductor(Ω.m) ρ0 CModified resistivity of the conductor(Ω.m) ρeResistivity of the earth(Ω.m) ρPResistivity of the pipe(Ω.m) ρSResistivity of the screen(Ω.m) ρtheor Resistivity given at 20 °C(Ω.m) a Inner radius of the insulation b Outer radius of the insulation C Capacitance d Distance between the conductors for single-core cables DeDepth penetration of the earth fFrequency of the system (Hz) GMR Geometric mean radius of the conductor hDepth of the cable buried underground Jn(x)Bessel function of x, first kind and order n Kn(x)Bessel function of x, second kind and order n KpProximity effect coefficient KsSkin effect coefficient L Inductance (H) mAReciprocal of the complex penetration depth for the armour mCReciprocal of the complex penetration depth for the conductor meReciprocal of the complex penetration depth for the earth mPReciprocal of the complex penetration depth for the pipe mSReciprocal of the complex penetration depth for the screen Pc,i,p Potential coefficients between the pipe and ground, of the core and screen and between cables and the pipe RAC Single-core AC resistance (Ω.m−1) R3AC Three-core AC resistance (Ω.m−1) RDC(T)DC resistance for a given temperature (Ω.m−1) R1Radius over the conductor core R2Radius over the insulation R3Radius over the screen R4Radius over the screen insulation R5Radius over the armour for single-core cable R6Radius over the armour insulation for single-core cable RPinner Radius over the inner armour for three-core pipe cable RPout Radius over the outer armour for three-core pipe cable RPins Radius over the armour insulation for three-core pipe cable sDistance between conductor axes SCross-section of the conductor TTemperature of the conductor ysSkin effect factor ypProximity effect factor Notation XIII Chapter 2.3 - Sequence impedances λAArmour loss factor for three-core cables as per IEC 60287-1-1 a Phase change variable F Fortescue matrix [I012]Sequence current matrix [Iabc]Phases current matrix [U012]Sequence voltage matrix [Uabc]Phases voltage matrix UR,S,TThree voltage phasors U0,+,−Sequence voltage [Z012]Sequence impedance matrix [Zabc]Phases impedance matrix ZSel f Self impedance of the conductor ZSel f ,SSelf impedance of the screen ZMMutual impedance between cables ZM,SMutual impedance between conductor and screen WdFrequency-dependent dielectric losses Chapter 2.4 - Simple switching transients theory αAttenuation θSwitching angle ζDamping factor φPhase angle between current and voltage ωAngular frequency ω0Angular resonance frequency ωdDamped resonance frequency C Capacitance (F) cos(φ)Power factor e Euler’s number I Current in frequency domain I(t) Current in time domain ˙ IFirst derivative of the current with respect to time for the initial values If(t)Forced response in time domain for the solution from Laplace domain Ih(t) Homogeneous response in time domain for the solution from Laplace domain L Inductance p Derivative of a variable with respect to time R Resistance s Laplace’s complex frequency parameter t Time V(t) Voltage in time domain VpPeak voltage Chapter 2.5 - Traveling waves εPermittivity (F/m) ε0Vacuum permittivity (F/m) µPermeability (H/m) XIV Notation µ0Permeability in a vacuum (H/m) a,b Radius of the conductor and insulations respectively C Capacitance of the capacitor I(x,t) Distance and time dependent current L Inductance R Resistance t Time V(x,t) Distance and time dependent voltage v Propagation speed for a cable x Length of a cable Chapter 2.6 - Harmonic impedance C Capacitance given a distributed parameters power transmission line εPermittivity (F/m) l Length of the transmission line IS,RCurrent on the sending/receiving end of the transmission line VS,RVoltage on the sending/receiving end of the transmission line R1Radius over the conductor core R2Radius over the insulation ωAngular frequency µPermeability (H/m) Y Admittance (Ω−1.m−1) Z0Characteristic impedance (Ω−1.m−1) γPropagation constant Chapter 3 - Submarine cables γbBessel’s constant (≈1.7811) linked to the Euler’s constant γe ρEResistivity of earth(Ω.m) DECIGRE’s equivalent depth of earth return path for underground cables d Distance between conductors hsea Burial subsea depth for a "infinite sea model" Kn(x)Bessel function of x, second kind and order n msea Reciprocal of the complex penetration depth for the sea Rext External radius Zii Self-impedance of a conductor Zi j Mutual-impedance between two different conductors Zgmi j Mutual impedance between phases and ground Zsel f −sea Self undersea impedance Zsel f −mutual Sea-mutual impedance Chapter 4 - COMSOL applied to 2D and 3D impedance calculation δSkin depth of conductive elements ρResistivity of a given conductive material(Ω.m) ωAngular frequency µPermeability (H/m) 1 Introduction & Motivation Th e invention of cable technology can be dated as far back as 1830, and the first underground cable installation 50 years later, in 1880, in Berlin. This delay was caused by the need to find a dielectric material capable of withstanding the conductor’s heat and the strong electric field. In 1880, Ferranti achieved a multi-layer dielectric using lapped paper tapes. Later in 1917 improved by impregnating the paper dielectric with low-viscosity insulation oil under permanent pressure. The next improvement occurred in the 1960s with the introduction of cross-linked polyethylene (XLPE) as dielectric, allowing the operational temperatures up to around 90 ° C. Another standard technology was the high-pressure gas-filled (HPGF) pipe-type cable, being SF 6 the pressurized gas used for this (not as standard as XLPE cable) [ 10 ]. All in all, the basic design for a layout and the components of an HV cable have not changed much in the last century (figure 1.1). Figure 1.1 Typical XLPE single-core and three-core cable [10]. Going by the components, we find the conductor, which is typically made of copper (higher tensile strength and lower resistivity) or aluminium (lower density, thus lighter than copper) and carries the electrical current; the insulation (typically XLPE), which ensures the lack of electrical connection between the conductor and the cable’s screen, being able to withstand the cable’s electrical field for steady-state, as well as in transient conditions; the screen (frequently, but it does not have to be always present) is the metallic mesh that preserves the electric field outside of the cable, gives a return path for the charging current and leads the fault currents to earth; and last, the semiconductive layers, which sit between the conductor and the insulation, but also between the insulation and the metallic screen, ensuring a cylindrical electric field and avoiding the formation of gaps/ voids between these mentioned elements. Besides, Jackets or armours are often added for mechanical 1 2Capítulo 1. Introduction & Motivation reinforcement. Whether an underground or a submarine line, the three-core or Pipe-type cable is commonly used, although mostly in offshore scenarios. As we can deduce from its name, the three-core cable is three conductors embedded in a standard armour or pipe, and each core is similar to a single-core cable. It is relevant to mention that, although we are talking about several kinds of material, the principal value in which we will be interested is the electrical impedance and its evolution over the frequency domain. In the past years, the number and size of installed offshore wind farms have increased rapidly, and larger farms are being planned worldwide. Submarine cables are and will continue to be an essential part of this development. They are used as array cables between the generators, as export cables to connect the offshore generation farms with the onshore transmission grid and even as part of interconnections between different synchronous systems, countries or price areas. Alternating current, or AC transmission, is far more common than DC in terrestrial power networks. However, three-phase AC requires three separate conductors and produces three separate electromagnetic fields, which interact to produce inefficiencies in power transmission. In submarine cables, this leads to thicker conductors with more insulation. Because of the limitations of physics and economics, the transmission distance of an HVAC submarine cable does not exceed typically around 100 km. Of course, thinking about this distance limitation, one could think of a better solution with an HVDC cable because of the lack of reactive power compensation. However, this kind of solution is also expensive due to the two extra substations (a rectifier and an inverter), making this a not-so-optimal solution for offshore wind farms (OWF) or short-distance interconnections. Therefore, the selection between HVDC and HVAC configurations for long distances must be considered. Nowadays, more and more studies are focused on analyzing reactive power compensation using reactors for longdistance HVAC-cable connections from OWFs. It is also to be remarked that the longest HVAC until today is a 163 km cable in the North Sea connecting the Martin Linge platform (oil and gas) and Kollsnes, Norway. In this particular solution, a single three-core 145 kV HVAC cable can give a total capacity of 55 MW, with low electrical losses and a proper tensile strength to withstand the conditions of the North Sea. Large OWFs are based on extensive MV submarine cable collection networks connecting tens to hundreds of usually identical wind turbines placed in regular patterns with certain interspacing and connected to a substation which in large wind farms is usually also placed offshore where the voltage is raised from MV to HV for transmission through a submarine cable to a connection point in the public power system onshore [ 17 ]. Such arrangements are unusual compared to a typical power system expansion; it is also unusual compared to other power systemsóperating conditions and environmental constraints. The wind farm transformer substation is often connected through a single export cable to the public export grid onshore up to more than a hundred kilometres from the OWF. This significant distance between the generator and the public grid is also highly unusual for a power generation plant. Engineers need to know the impedance of these cables, execute the proper calculations and translate the math into reality. The harmonic impedance calculation is highly relevant in these offshore scenarios due to its high capacitance contribution to the appearance of unwanted overvoltages or resonances at specific frequencies. To adequately assess these problems, detailed and reliable cable models are required to develop time-domain/frequency-domain analyses on this type of cable. Taking a deeper look into these solutions, we cannot avoid considering the electromagnetic transients (EMT) in this scenario, such as lightning and breakers operation, which are a significant source of power failures in today’s power systems [ 2 ]. To study and mitigate these transients, there are several EMT simulation tools. EMT tools need broadband models for all network components, including cables. These broadband cable models can be obtained only if the per-unit-length (p.u.l.) impedance and admittance of the cable are accurately known [ 3 ] [ 21 ]. In most EMT tools, ground effects in underground cables are included through an approximation of Pollaczek’s formulas [ 22 ]. These formulae can only model a surrounding medium of two half-spaces, with the top layer being 3 air and the bottom layer being either soil or water [21]. In this thesis, a 2D method comparison will be carried out in chapter 2 with Bessel’s functions, along with the convenient theoretical background to support the results. After seeing that, we will focus on submarine cables in particular and the theoretical 2D calculation of those, throwing a comparison between underground and submarine cables. In chapter four, the software COMSOL will be presented concerning the purpose of this thesis and to approach a 3D form of study. This 3D study is addressed with a finite element method, taking into account some calculation details that the 2D method cannot cover, such as the influence of the armour twisting on the power cable impedance. Chapter five will present the chosen study case for chapters two and three and see the differences and why. Chapter six will conclude this thesis as a conclusion and recapitulation of the significant points. 2 Theoretical background Some classic theories about underground cables will be approached to place this thesis on the right path. Content such as electromagnetic transients, frequency domain or thermal behaviour of the cables needs to be explained to frame the right background, leading step by step to the aim of this thesis, which is the behaviour of submarine-cable impedances within a possible frequency variation range, given a wind generation scenario with AC exporting cables. A first and proper approach would be talking about Schelkunoff. In 1934 [ 24 ] he worked out comprehensive work on coaxial structures: he derived from Maxwell’s equations formulae for the impedances and admittances which link voltages and currents in such systems [ 5 ]. Back there, the initial assumptions were: •Take a homogeneous base system. •Ideal materials with constant conductivity, permittivity and permeability. • The longitudinal currents in the insulating materials are negligible compared to longitudinal currents in the conductors. After that, Pollaczek [ 22 ] studied the electromagnetic field propagation in the ground, filling the gap for modelling a buried single-core cable (Schelkunoff’s approach is limited to a case where the cable surrounding can be considered as an infinite medium). Now, modelling of three-phase transmission systems was achieved, assuming that the cables are parallel and that the electrical field at a cable surface is achievable using the superposition principle. Unfortunately, the formulae were (still is) not easy to handle since they involve (a lot of) Bessel’s functions to describe the behaviour of each layer in a cable, along with the undesired effects that occur when AC flows through the conductors. Simpler formulae based on Bessel’s functionsápproximations may be found in a paper from Wedepohl and Wilcox (1973) [ 26 ], valid up to about 100 kHz. As regards three-core cables, impedances and admittances may be derived the same way from, as we will see in this chapter, Maxwell’s equations, with additional assumptions. This concluded when Ametani proposed in 1979-80 a general formulation covering any type of cable based on these precedents [4]. It will be addressed in this chapter a short section on thermal behaviour, a section about introductory impedances/admittances, a third section about sequence impedances, a fourth section about basic switching transients, a fifth section about travelling waves within the time and frequency domain, a sixth section about the grounds of harmonic impedance calculation and the last one will address an analytical calculation of series/sequence impedances. 5 12 Capítulo 2. Theoretical background insulation and the screen: C=2πε ln R2 R1F.m−1(2.19) R2 is the radius over the insulation, including the semiconductor layers. Therefore the permittivity should suffice this equation: ε=εIns ln R2 R1 ln b aF.m−1(2.20) where εIns is the insulation permittivity, with a typical value of 2.3 and 2.5; "b" is the outer radius of the insulation and "a" is the inner radius of the insulation. The last basic important effect in transmission power lines is the inductance. This can be calculated with the expression (2.21): L=µ 2πln De GMRH.m−1(2.21) µ is the permeability of the conductor (in a vacuum, the relative permeability is normally one, except, in our case, for lead), De the depth penetration of the earth (calculated with the earth resistivity) and GMR the geometric mean radius of the conductor (normally equal to R1e−1/4). There is a specific formula for the return path earth resistance, and, although it depends on the type of soil (humidity, the density of soil, etc.), a general formula can be used (2.22) with an estimated depth of the earth return path (2.23): R0 E=ωµ0 8Ω.m−1(2.22) De≈659 ·rρe f[m](2.23) With all seen, the self impedance of a phase conductor with earth return according to the CIGRE formulae is the following: ZSel f =R0 E+RAC +jωXcΩ.m−1(2.24) where Xc=jωL , being L given by the expression (2.21). Now, the self impedance of metal screen with earth return will be: ZSel f ,S=R0 E+Rs+jωµ 2πln De rsΩ.m−1(2.25) where rs is the mean radius of the metal screen and Rs is the resistance of the screen. Then, the mutual impedance between phase conductor and metal screen of a cable with earth return is given by (2.26): ZM,S=R0 E+jωµ 2πln De rsΩ.m−1(2.26) Last, the mutual impedance between phases with earth return is given by (2.27): ZM=R0 E+jωµ 2πln De di j Ω.m−1(2.27) Now we will see how it all comes together when the total impedance of a three-phase cable needs to be accurately calculated. Thinking about the matrix composition, we realize instantly that the final matrix will have as many rows and columns as conductive elements in the cable. However, a 2.2 Impedance and admittance matrices 13 three-phase cable has three conductors per phase, which requires some changes in the procedures. The first significant change is in the calculation of the impedance and admittance of the cable, which will be next shown depending on the bonding type. 2.2.1 both-ends bonded cable A case with both-ends bonded cable will be detailed to learn about the impedance calculation in a cable. HVAC cables are commonly installed using three single-core cables. In this scenario, the cable series impedance becomes a 6 x 6 matrix or a 9 x 9 matrix, depending on whether the cable has armour. The matrix contains both the self-impedance and the mutual impedances. The first column and the first row of the matrix represent the core/conductor of Phase A, the second column and row represent the screen of Phase A, the third column and row represent the armour of Phase A, whereas the other columns and rows represent Phases B and C in the same order. Thus, the main diagonal is the self-impedance of the cores, screens and armours, whereas the upper and lower triangles are the mutual impedances, resulting in a symmetrical matrix, as we can see in (2.28) [ 10 ]. Z=                 Core 1 z}|{ ZC1C1 Screen 1 z}|{ ZC1S1 Arm.1 z}|{ ZC1A1 Core 2 z}|{ ZC1C2 Screen 2 z}|{ ZC1S2 Arm.2 z}|{ ZC1A2 Core 3 z}|{ ZC1C3 Screen 3 z}|{ ZC1S3 Arm.3 z}|{ ZC1A3 ZS1C1ZS1S1ZS1A1ZS1C2ZS1S2ZS1A2ZS1C3ZS1S3ZS1A3 ZA1C1ZA1S1ZA1A1ZA1C2ZA1S2ZA1A2ZA1C3ZA1S3ZA1A3 ZC2C1ZC2S1ZC2A1ZC2C2ZC2S2ZC2A2ZC2C3ZC2S3ZC2A3 ZS2C1ZS2S1ZS2A1ZS2C2ZS2S2ZS2A2ZS2C3ZS2S3ZS2A3 ZA2C1ZA2S1ZA2A1ZA2C2ZA2S2ZA2A2ZA2C3ZA2S3ZA2A3 ZC3C1ZC3S1ZC3A1ZC3C2ZC3S2ZC3A2ZC3C3ZC3S3ZC3A3 ZS3C1ZS3S1ZS3A1ZS3C2ZS3S2ZS3A2ZS3C3ZS3S3ZS3A3 ZA3C1ZA3S1ZA3A1ZA3C2ZA3S2ZA3A2ZA3C3ZA3S3ZA3A3                 (2.28) By making some assumptions, we can simplify the impedance matrix: • Since the three cables are equal, we can say: ZC1C1=ZC2C2=ZC3C3 ; ZS1S1=ZS2S2=ZS3S3 ; ZA1A1=ZA2A2=ZA3A3;ZC1S1=ZC2S2=ZC3S3and last ZC1A1=ZC2A2=ZC3A3 • The mutual inductance between phases depends on the distance. The distance between the core of Phase A and the core of Phase B is practically equal to the distance between the core of Phase A and the screen of Phase B; the same can be said for the other phases. Thus: ZC1C2≈ ZC1S2≈ZS1C2≈ZS1S2 ; ZC1C3≈ZC1S3≈ZS1C3≈ZS1S3 ; ZC2C3≈ZC2S3≈ZS2C3≈ZS2S3 ; the same reasoning can be applied for the armour of the conductors. • We can name each element of the series impedance matrix with a more comfortable nomenclature, this is: Z11,Z22,Z33 as the self-impedance of the core/screen/armour; Z12,Z13,Z23 as the mutual impedance between core-screen/core-armour/screen-armour; Zgm12,Zgm13,Zgm23 as the ground mutual impedance between phases. All of them will be further explained in this section when the loop equations between the cable layers are presented. 14 Capítulo 2. Theoretical background This would simplify our series impedance matrix (2.28), into (2.29): Z=                  Conductor1 z }| { Conductor2 z }| { Conductor3 z }| { Z11 Z12 Z13 Zgm12 Zgm12 Zgm12 Zgm13 Zgm13 Zgm13 Z12 Z22 Z23 Zgm12 Zgm12 Zgm12 Zgm13 Zgm13 Zgm13 Z13 Z23 Z33 Zgm12 Zgm12 Zgm12 Zgm13 Zgm13 Zgm13 Zgm12 Zgm12 Zgm12 Z11 Z12 Z13 Zgm23 Zgm23 Zgm23 Zgm12 Zgm12 Zgm12 Z12 Z22 Z23 Zgm23 Zgm23 Zgm23 Zgm12 Zgm12 Zgm12 Z13 Z23 Z33 Zgm23 Zgm23 Zgm23 Zgm13 Zgm13 Zgm13 Zgm23 Zgm23 Zgm23 Z11 Z12 Z13 Zgm13 Zgm13 Zgm13 Zgm23 Zgm23 Zgm23 Z12 Z22 Z23 Zgm13 Zgm13 Zgm13 Zgm23 Zgm23 Zgm23 Z13 Z23 Z33                  (2.29) The cable we are describing has for each phase three conductors (core, screen and armour) and three current loops. In the first loop, the current flow in the core and returns to the screen; in the second loop, the current flows in the screen and returns to the armour, whereas in the third loop, the current flows in the armour and returns to the ground [10]. Conductor series impedance This parameter is the internal impedance of the core, as shown in (2.30). It depends on the conductor’s resistivity, the conductor’s radius and penetration depth, as previously stated at the beginning of this second chapter. ZCouter(ω) = ρ0 CmC 2πR1·J0(mCR1) J1(mCR1)(2.30) where, ρCis the resistivity of the conductor, mCis the reciprocal of the complex penetration depth for the conductor given by (2.31), R1is the radius of the conductor, shown in the figure 2.4, Jn(x)is the Bessel function of x, first kind and order n. The equation assumes a solid conductor, but a cable often has a stranded conductor. Therefore, the resistivity must be corrected using equation (2.32), where ACis the cross-sectional area. mC=sjωµ ρ0 C (2.31) ρ0 C=ρC πR2 1 AC (2.32) 2.2 Impedance and admittance matrices 15 Figure 2.4 Cross-section of a single-core cable. Using the Bessel functions provides the most accurate analytical results but at the cost of more significant complexity. An approximation of (2.30) is given by (2.33), where k is an arbitrary constant used to optimize the formula for low frequencies, typically equal to 0.777 [10]. ZCouter(ω) = ρ0 CmC 2πR1·coth(mCR1k)+ ρ0 C πR2 11−1 2k(2.33) Insulation series impedance This is a result of the time-varying magnetic field in the inner insulation of the cable, and it is calculated through (2.34). This term is a function of the currents flowing in the core and screen and the magnetic field between them [10]. ZCS−ins(ω) = jωµins 2πlnR2 R1(2.34) where, µins is the permeability of the insulation, R2is the radius of the insulation. Screen and Armour Inner Series Impedance The screen inner series impedance is associated with the voltage drop on the inner surface of the screen when a current returns via the core. It is calculated by using (2.35) [10]. ZSinner(ω) = ρSmS 2πR2·J0(mSR2)K1(mSR3)+K0(mSR2)J1(mSR3) J1(mSR3)K1(mSR2)−J1(mSR2)K1(mSR3)(2.35) where, ρSis the resistivity of the screen, mSis the reciprocal of the complex penetration depth for the screen, 16 Capítulo 2. Theoretical background R3is the radius over the screen, Kn(x)is the Bessel function of x, of second kind and order n. This equation can also be simplified to (2.36), as it happened with equations (2.30) to (2.33): ZSinner(ω) = ρSmS 2πR2·coth(mS(R3−R2))−ρS 2πR2(R2+R3)(2.36) Now, following the same logic, we can define the same equation for the armour: ZAinner(ω) = ρAmA 2πR4·coth(mA(R5−R4))−ρA 2πR4(R4+R5)(2.37) Screen and Armour outer Series Impedance The screen outer series impedance is analogous to the screen inner series impedance, but for the outer surface of the screen instead of the inner surface. In other words, it is associated with the voltage drop on the inner surface of the screen when the current returns through the armour [ 6 ]. The same deduction and approximation will be followed for the next two equations. From this point on, every impedance formula will be given by their approximation due to the few difference from the exact formula, but it will be simulated with the original Bessel functions to be more accurate with the results. ZSouter(ω) = ρSmS 2πR3·coth(mS(R3−R2))−ρS 2πR3(R2+R3)(2.38) ZAouter(ω) = ρAmA 2πR5·coth(mA(R5−R4))−ρA 2πR5(R4+R5)(2.39) where, ρAis the resistivity of the armour, mAis the reciprocal of the complex penetration depth for the armour, R4is the radius over the screen insulation, R5is the radius over the armour. Screen and Armour Insulation Series Impedance Similar to the insulation series impedance, the screen insulation series impedance results from the time-varying magnetic field in the inner insulation of the cable [10]. ZSA−ins(ω) = jωµout−ins 2πlnR4 R3(2.40) ZAG−ins(ω) = jωµout2−ins 2πlnR6 R5(2.41) where, µout−ins is the permeability of the screen insulation, µout2−ins is the permeability of the armour insulation, R6is the radius over the armour insulation. 2.2 Impedance and admittance matrices 17 Mutual Series Impedance of two loops The mutual series impedance is associated with two different current loops, equal for both. A first loop, where there is a voltage drop in the outer surface of the screen because of a current returning in the core, and a second loop where there is a voltage drop in the inner surface of the screen because of a current returning in the armour. There is also a mutual series impedance associated with the armour. In the first loop, there is a voltage drop in the outer surface of the armour because of a current returning in the screen, whereas in the second loop, there is a voltage drop in the inner surface of the armour because of a current returning in the ground [10]. ZS−mutual(ω) = ρSmS π(R2+R3)·cosech(mS(R3−R2)) (2.42) ZA−mutual(ω) = ρAmA π(R4+R5)·cosech(mA(R5−R4)) (2.43) Earth Series Self Impedance The earth series self-impedance is the parameter whose calculation arises more difficulties. Therefore it is also the one presenting the most prominent errors. The classic formula based on original developments made by Pollaczek is too complex, so instead, one of the most accurate ones is the one proposed by Saad, Gaba and Giroux in 1996. Zearth(ω) = ρem2 e 2πK0(meR4)+ 2 4+m2 eR2 4 e−2hme(2.44) Where, ρeis the resistivity of the earth, meis the reciprocal of the complex penetration depth for the earth, h is the depth of the cable. Mutual earth impedance The mutual earth impedance represents the mutual inductance between the cables. It results from a voltage drop in the earth adjacent to the cables inducing an e.m.f. in the cables when the current returns through the soil [10][5]. The exact approximation can be made here, but instead of R4, the distance between the conductors "d" will be introduced. Zearth−mutual(ω) = ρem2 e 2πK0(med)+ 2 4+m2 ed2e−2hme(2.45) Where, d is the distance between the conductors, h is the depth of the cable. Configuration of the loops - Single-phase cable Now we start building the relationship between all these impedances to represent a more accurate reality, closer to its actual behaviour. 18 Capítulo 2. Theoretical background Figure 2.5 Loop equations for a single-core cable [5]. The impedances of this loop are equal to the core self-impedance, plus the insulation series impedance and the screen inner impedance (2.46), which are the series of all the impedance in the path travelled by the current. The impedance of the screen-armour loop is similar, and it is equal to the screen outer series impedance, plus the screen outer-insulation series impedance and the armour inner impedance (2.47). Last, the impedance of the armour-ground is equal to the armour outer series impedance, plus the armour outer-insulation series impedance and the earth series impedance (2.48). Figure 2.5 shows the equivalent circuit for a single-phase cable’s three loops within the cable. To calculate each impedance, every loop has to be studied and, using Ohm’s law; we can obtain the respective impedances. ZL−CS =ZCouter +ZCS−ins +ZSinner (2.46) ZL−SA =ZSouter +ZSA−ins +ZAinner (2.47) ZL−AG =ZAouter +ZAG−ins +Zearth (2.48) There is also mutual impedance associated with the core-screen loop (2.49) and the screen-armour loop (2.50), both calculated directly using the equations previously explained: ZLm−CS =−ZS−mutual (2.49) ZLm−SA =−ZA−mutual (2.50) The last of the impedances is the mutual ground-return impedance between the armours of two phases, this is, Zgm12 =Zgm13 =Zgm23 =Zearth−mutual. 2.2 Impedance and admittance matrices 19 Configuration of the loops - Three-phase cable Now, with all this information about the impedances within the single-phase cable, we can define the matrix equations describing the loop system for a three-phase system:            VCS1 VSA1 VAG1 VCS2 VSA2 VAG2 VCS3 VSA3 VAG3            =            ZL−CS ZLm−CS 0 0 0 0 0 0 0 ZLm−CS ZL−SA ZLm−SA 0 0 0 0 0 0 0ZLm−SA ZL−AG 0 0 Zgm12 0 0 Zgm13 0 0 0 ZL−CS ZLm−CS 0 0 0 0 0 0 0 ZLm−CS ZL−SA ZLm−SA 0 0 0 0 0 Zgm12 0ZLm−SA ZL−AG 0 0 Zgm23 0 0 0 0 0 0 ZL−CS ZLm−CS 0 0 0 0 0 0 0 ZLm−CS ZL−SA ZLm−SA 0 0 Zgm13 0 0 Zgm23 0ZLm−SA ZL−AG                       ICS1 ISA1 IAG1 ICS2 ISA2 IAG2 ICS3 ISA3 IAG3            (2.51) If we now take this loop impedance matrix (ZLoop) and we operate it conveniently, taking into account the expression (2.52), which gives us the relation between the series and the loop impedance matrices, we can now calculate the series impedance matrix, where A is a transformation matrix for this described cable as follows: the only current in the core comes from the core-screen loop. Thus, the first entry should be one, and the remaining should be 0; the second row corresponds to the screen, then the first entry should be -1 (coming from the core-screen loop), and the second entry should be 1 (coming from the screen-armour loop), whereas the other remaining entries would be 0. Following this reasoning, the same logic can be applied to the rest of the matrix, giving us the expression (2.53). (Z) = (A)−T(ZLoop)(A)−1(2.52) where ZLoop is given by (2.51) and A is given by (2.53), being A=               1 0 0 0 0 0 0 0 0 −1100 000 00 0−1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 −1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 −110 0 0 0 0 0 0 0 −1 1               (2.53) The result of Z was given before by the expression in (2.29), where now we can define each element, taking into account the figure 2.4: •Z11 = ZL−CS +ZL−SA +ZL−AG +2ZLm−CS +2ZLm−SA = ZCouter +ZCS−ins +ZSinner +ZSouter + ZSA−ins +ZAinner +ZAouter +ZAG−ins +Zearth −2ZS−mutual −2ZA−mutual •Z22 = ZL−SA +ZL−AG +2ZLm−SA = ZSouter +ZSA−ins +ZAinner +ZAouter +ZAG−ins +Zearth − 2ZA−mutual •Z33 =ZL−AG =ZAouter +ZAG−ins +Zearth •Z12 = ZL−SA +ZL−AG +ZLm−CS +2ZLm−SA = ZSouter +ZSA−ins +ZAinner +ZAouter +ZAG−ins + Zearth −ZS−mutual −2ZA−mutual •Z13 =ZAouter +ZAG−ins +Zearth −ZA−mutual •Z23 =ZAouter +ZAG−ins +Zearth 20 Capítulo 2. Theoretical background Finally, the ground mutual impedance between phases ( Zgm12,Zgm13 and Zgm23 in the series impedance matrix) can be easily recognized as the mutual earth impedance given by (2.45). Shunt admittance matrix The shunt admittance matrix is obtained similarly. The screens of an HVAC underground cable are typically grounded at both ends and in between (if cross-bond), whereas the subsea cable is usually grounded at both ends. Thus, it is safe to assume that the screen potential is virtually zero all along the cable (although there is always a small voltage in the screen through the cable). As a result, the electric field is confined to each phase, and there is no shunt admittance between different phases. Figure 2.6 describes the equivalent capacitance circuit for a three-phase cable. Therefore, the admittance matrix for a both-ends grounded cable can be described as follows: Figure 2.6 Representation of the capacitance in the three-phase cable. Therefore, the admittance matrix for a both-ends grounded cable can be described as follows: Y=               YC1C1YC1S10 0 0 0 0 0 0 YS1C1YS1S1YS1A10 0 0 0 0 0 0YA1S1YA1A10 0 0 0 0 0 0 0 0 YC2C2YC2S20 0 0 0 0 0 0 YS2C2YS2S2YS2A20 0 0 0 0 0 0 YA2S2YA2A20 0 0 0 0 0 0 0 0 YC3C3YC3S30 0 0 0 0 0 0 YS3C3YS3S3YS3A3 0 0 0 0 0 0 0 YA3S3YA3A3               (2.54) If we now substitute the value of each element of the admittance matrix, taking into account that the conductance G is always considered zero, the substitutions shown in (2.55) are quite simple; thus, the admittance matrix in (2.54) can be simplified into (2.56). YCiCi =jωCCS YSiSi =jωCCS +jωCSA YAiAi =jωCSA +jωCAG YCiS j =jωCCS YSiA j =jωCSA (2.55) 2.2 Impedance and admittance matrices 21 Y=      jωCCS −jωCCS 0 0 0 0 0 0 0 −jωCCS jω(CCS +CSA)−jωCSA 0 0 0 0 0 0 0−jωCSA jω(CSA +CAG)0 0 0 0 0 0 0 0 0 jωCCS −jωCCS 0 0 0 0 0 0 0 −jωCCS jω(CCS +CSA)−jωCSA 0 0 0 0 0 0 0 −jωCSA jω(CSA +CAG)0 0 0 0 0 0 0 0 0 jωCCS −jωCCS 0 0 0 0 0 0 0 −jωCCS jω(CCS +CSA)−jωCSA 0 0 0 0 0 0 0 −jωCSA jω(CSA +CAG)       (2.56) 2.2.2 Cross-bonded cable As shown in figure 2.2 at the beginning of this chapter two within section 2.2 , the screen of each conductor is cross-connected in this type of cable. This modifies the (until now) calculated and demonstrated series-impedance matrix due to changes in the inductive coupling between the phases, being necessary to write three matrices (one for each minor section). The matrix describing the impedance of the first minor section is equal to the one describing a cable bonded at both ends, given by (2.28), whereas the matrices describing the other two minor sections are as shown in (2.57) and (2.58). The changes in the two matrices are related to the changes in the position of the screens, whereas the coupling between the cores and between armours remains unchanged. For example, notice that in the matrix describing the first minor section (2.22), the second column is the screen of Phase A, whereas, in the second minor section, it is correspondent to the screen of Phase B and Phase C in the third minor section. However, the first column describes the core of Phase A for all three matrices. ZS2=                 Core 1 z}|{ ZC1C1 Screen 2 z}|{ ZC1S2 Arm.1 z}|{ ZC1A1 Core 2 z}|{ ZC1C2 Screen 3 z}|{ ZC1S3 Arm.2 z}|{ ZC1A2 Core 3 z}|{ ZC1C3 Screen 1 z}|{ ZC1S1 Arm.3 z}|{ ZC1A3 ZS2C1ZS2S2ZS2A1ZS2C2ZS2S3ZS2A2ZS2C3ZS2S1ZS2A3 ZA1C1ZA1S2ZA1A1ZA1C2ZA1S3ZA1A2ZA1C3ZA1S1ZA1A3 ZC2C1ZC2S2ZC2A1ZC2C2ZC2S3ZC2A2ZC2C3ZC2S1ZC2A3 ZS3C1ZS3S2ZS3A1ZS3C2ZS3S3ZS3A2ZS3C3ZS2S1ZS3A3 ZA2C1ZA2S2ZA2A1ZA2C2ZA2S3ZA2A2ZA2C3ZA2S1ZA2A3 ZC3C1ZC3S2ZC3A1ZC3C2ZC3S3ZC3A2ZC3C3ZC3S1ZC3A3 ZS1C1ZS1S2ZS1A1ZS1C2ZS1S3ZS1A2ZS1C3ZS1S1ZS1A3 ZA3C1ZA3S2ZA3A1ZA3C2ZA3S3ZA3A2ZA3C3ZA3S1ZA3A3                 (2.57) ZS3=                 Core 1 z}|{ ZC1C1 Screen 3 z}|{ ZC1S3 Arm.1 z}|{ ZC1A1 Core 2 z}|{ ZC1C2 Screen 1 z}|{ ZC1S1 Arm.2 z}|{ ZC1A2 Core 3 z}|{ ZC1C3 Screen 2 z}|{ ZC1S2 Arm.3 z}|{ ZC1A3 ZS3C1ZS3S3ZS3A1ZS3C2ZS3S1ZS3A2ZS3C3ZS3S2ZS3A3 ZA1C1ZA1S3ZA1A1ZA1C2ZA1S1ZA1A2ZA1C3ZA1S2ZA1A3 ZC2C1ZC2S3ZC2A1ZC2C2ZC2S1ZC2A2ZC2C3ZC2S2ZC2A3 ZS1C1ZS1S3ZS1A1ZS1C2ZS1S1ZS1A2ZS1C3ZS1S2ZS1A3 ZA2C1ZA2S3ZA2A1ZA2C2ZA2S1ZA2A2ZA2C3ZA2S2ZA2A3 ZC3C1ZC3S3ZC3A1ZC3C2ZC3S1ZC3A2ZC3C3ZC3S2ZC3A3 ZS2C1ZS2S3ZS2A1ZS2C2ZS2S1ZS2A2ZS2C3ZS2S2ZS2A3 ZA3C1ZA3S3ZA3A1ZA3C2ZA3S1ZA3A2ZA3C3ZA3S2ZA3A3                 (2.58) Now that the three minor sections are detailed, the total impedance matrix will be given by summing and dividing them by three. This average calculation is only valid when the cross-bonding is ideal, i.e., a perfect transposition of the screens. The same logic can now be followed for the admittance matrix as stated with the both-ends bonded configuration. 28 Capítulo 2. Theoretical background So as we did with the voltage matrix, it is possible to do the same with the phase-current and sequence-current matrices: [I012] = [F]·[Iabc]⇒[Iabc] = [F]−1·[I012](2.86) The sequence-impedance matrix can be solved from the phase-impedance matrix as follows: [Uabc] = [Zabc][Iabc](2.87) which we can arrange using the same logic from (2.84) and (2.86) to obtain (2.88): [F]−1[U012] = [Zabc][F]−1[I012]⇒[U012] = [F][Zabc][F]−1[I012](2.88) Therefore we can name Z012 as the sequence-impedance matrix described below as: [Z012] = [F][Zabc][F]−1= 1 3  1 1 1 1a a2 1a2a   ZSel f ZMZM ZMZSel f ZM ZMZMZSel f    1 1 1 1a2a 1a a2 =  ZSel f +2ZM0 0 0ZSel f −ZM0 0 0 ZSel f −ZM  (2.89) where, •Z0=ZSel f +2ZM •Z+=ZSel f −ZM •Z−=ZSel f −ZM This sequence-impedance defines the behaviour of the impedance, and, since mutual impedances between sequences do not exist, the three-sequence impedance is considered decoupled; therefore, it can be correctly studied using three single-phase circuits. The bonding type is essential to keep in mind because the positive-sequence impedance depends on the type of bonding used in the cable (zero-sequence impedance does not depend on the bonding, except for single-point bonding cables, but, as it happens, this type of bonding is not much used for HV cables). For a single-core cable, the positive-sequence impedance of a cross-bonding can be calculated by (2.90), the both-ends bonding cable by (2.91) and the zero-sequence impedance by (2.92) as follows: Z+ Cross = (ZSel f −ZM)(2.90) Z+ Both−ends = (ZSel f −ZM)−(ZM,S−ZM)2 ZSel f −ZM (2.91) Z0= (ZSel f +2ZM)−(ZM,S+2ZM)2 ZSel f +2ZM (2.92) If there were three-core cables, the formulae would be slightly altered due to the external armour loss factor around the conductors for a submarine cable (typically, both-ends bonding). The equation would be the one given by (2.91) but adding the term λA·Rc. The value of the impedances comes from the CIGRE formulae, as it can be deeply explained in [ 5 ] and it was already explained at the beginning of this section from (2.22) to (2.27), where ZSel f is the self-impedance of the conductor, ZSel f ,S is the self-impedance of the screen, ZM is the mutual-impedance between phases and ground, and ZM,S is the mutual-impedance between the conductor, screen and ground. 2.4 Simple switching transients theory 29 Regarding the sequence impedance, we also need to consider the dielectric losses, as we have already done with the circulating current losses within the bonding techniques. Due to the high voltage levels of transmission, the losses can be significant because these are proportional to the square of the voltage: Wdielectric =ω·C·U2 0·tan(δ)(2.93) So we can conclude this point by saying that the dielectric losses are also dependent on the capacitance of the dielectric, the frequency and the loss factor tan(δ). 2.4 Simple switching transients theory In order to keep introducing the theoretical frame of the thesis, the transient phenomenon that occurs when closing or opening a circuit will be addressed. An RL/RC circuit is a low-order circuit, and due to its simplicity, it is a good start for an introduction to the world of transients, although this thesis will not be facing this study much further since the main aim is the harmonic impedance within the frequency domain. Due to the theoretical frame of this thesis, both DC and AC circuits will be shown, but since we are interested in the evolution regarding the frequency, we will be focusing on the AC-source scenarios. 2.4.1 DC-Source: RL circuit Figure 2.8 DC-source RL circuit. The first RL circuit will be developed and calculated for a DC source to initiate this section. Due to the lack of interest, we will jump right to the equation after the differential equation of the circuit has been solved. The expression (2.94) shows it: I(t) = V R1−e−R Lt(2.94) We can see that the current will start at zero and increase up to V/R with a time constant τ=L R . Typically, it is considered that after τ seconds, the current will reach 63% of its steady-state value, and after 5τ seconds, the current will reach its steady-state, as we can see in figure 2.9. Another critical approach is that the current cannot change instantaneously because of the conservation of the moment in the magnetic flux of the inductor; that is, the magnetic flux has to be continuous. 2.4.2 AC-Source: RL circuit As seen before, the first step is to apply the Laplace transform (2.96) to the circuit equation (2.95): Vpsin(ωt) = RI +LdI dt (2.95) 30 Capítulo 2. Theoretical background Figure 2.9 DC-source current-time evolution. Figure 2.10 AC-source RL circuit. Vp ω s2+ω2=I(R+sL))−I(0)(2.96) For simplification, it is considered that I(0) = 0A . So if we now clear the current (I), we will obtain the following expression: I=Vp ω L·1 s2+ω2·1 s+R L (2.97) Operating conveniently, solving the mathematical problem and calling a=R L , we can obtain this expression: I=Vp ω L·1 a2+ω2·−s s2+ω2+a s2+ω2+1 s+a(2.98) Now we can apply the inverse Laplace transform leading to: I(t) = Vp ω L·1 a2+ω2·−cos(ωt)+ a ωsin(ωt)+ e−at(2.99) On the other hand, it is also known that the power factor cosφ=R √R2+(ωL)2 , so if we arrange a bit this expression we will obtain: cosφ=R LqR L2+ω2 =a √a2+ω2(2.100) 2.4 Simple switching transients theory 31 Therefore, in this context, we can also define the sinφ and the tanφ as (2.101) and (2.102) respectively: sinφ=ω √a2+ω2(2.101) tanφ=sinφ cosφ=ω a(2.102) Thus, we can describe again the expression (2.99) by replacing (2.100) and (2.101) into the first one, obtaining: I(t) = VP L·1 √a2+ω2(−sin(φ)cos(ωt)+cos(φ)sin(ωt)+sin(φ)e−at )(2.103) we can simplify (2.103) even more if we apply the following trigonometric relation: sin(ωt−φ) = cos(φ)sin(ωt)−sin(φ)cos(ωt), therefore: I(t) = VP L·1 √a2+ω2(sin(ωt−φ)+sin(φ)e−at)(2.104) We now undo the expression a=R L, introduce this in (2.101) and we do φ=arctanω a: I(t) = VP pR2+(ωL)2sinωt−arctanωL R−sin−arctanωL Re−R Lt(2.105) Supposing that the switching can be done at any time (it rarely will happen at zero-voltage), it will be assumed that the variable θ is the switching angle, being, then, the next expression a more accurate one: I(t) = VP pR2+(ωL)2sinωt+θ−arctanωL R−sinθ−arctanωL Re−R Lt (2.106) Assuming that a shunt reactor is basically an RL circuit with a much larger inductance value than resistance value, we can now simplify/approximate the last equation for its study, having now in (2.107) a steady-state component (sin(ωt+θ−π 2)), and a transient component (sin(θ−π 2)e−R Lt): I(t)≃VP pR2+(ωL)2sinωt+θ−π 2−sinθ−π 2)e−R Lt(2.107) The forced regime is a sinusoidal wave oscillating at power frequency with a phase difference of approximately 90 ° to the voltage and does not depend on the switching instant or conditions. The homogeneous regime is a decaying DC-current whose amplitude depends on the initial conditions. Moreover, the energy stored is typically zero, but the switching instant can be any. The typical response of this kind of circuit is the following: Due to the blue line in the figure 2.11 (DC-component), the resultant AC-wave is the red one, which in comparison with the black one ( Is−s(t) steady-state component) is noticeable different, and one can see the effect of the transients on this RL-circuits. Another approach to consider is the value of the DC component because, in this case, a positive value was taken, but in other cases, a negative value could have taken place. This will depend on the derivative of the voltage wave function. On the other hand, if the energization would take place at the peak voltage ( θ=±90° ), the transient component would be zero, and the current is only the steady-state component. 32 Capítulo 2. Theoretical background Figure 2.11 I(t) when using an AC voltage and energising at zero voltage. 2.4.3 DC-Source: RC circuit Now that we have seen how the current is deduced on the RL circuit, we use the same logic with the RC circuit, which is also a first-order circuit. These kinds of circuits are important with an AC source because of the reactive compensation of capacitor banks. Figure 2.12 DC-source RC circuit. The steady-state behaviour of an RC-load is an open circuit for the current with a DC-source. Due to the lack of interest in this thesis, this scenario will not be further studied. Instead, we will continue with the AC-source circuit for this matter. 2.4.4 AC-Source: RC circuit In this case, we will follow the same mathematical procedure, using the Laplace transform but considering that the switch can be closed at any given instant. 2.4 Simple switching transients theory 33 Figure 2.13 AC-source RC circuit. Vpsin(ωt+θ) = RI +1 CZIdt (2.108) Maturing this equation and applying the Laplace transform, we will obtain the following expression: I=Vp s·aC s+aωcos(θ) s2+ω2+s·sin(θ) s2+ω2,where a =1 RC (2.109) Then, we divide the second term of (2.109) in two parts as in (2.111) shown: I=Vp·aCsωcos(θ) (s2+ω2)(s+a)+s2sin(θ) (s2+ω2)(s+a)(2.110) (1):sωcos(θ) (s2+ω2)(s+a) (2):s2sin(θ) (s2+ω2)(s+a) (2.111) Now we apply the partial fraction method to this, obtaining (2.112), thus, replacing (2.112) into (2.110) we have the final expression in Laplace domain (2.113): (1):ωcos(θ) ω2+a2sa +ω2 s2+ω2−a s+a (2):sin(θ) ω2+a2sω2−aω2 s2+ω2+a2 s+a(2.112) I=Vp aC a2+ω2ωcos(θ)sa +ω2 s2+ω2−a s+a+sin(θ)sω2−aω2 s2+ω2+a2 s+a (2.113) We now use the inverse Laplace transform to obtain (2.114) in the time domain: I(t) = Vp aC a2+ω2[ωcos(θ)(a cos(ωt)+ωsin(ωt)−ae−at)] +Vp aC a2+ω2sin(θ)(ω2cos(ωt)−aωsin(ωt)+a2e−at)(2.114) 34 Capítulo 2. Theoretical background Then, as shown in (2.100), (2.101) and (2.102), the same logic will be applied for the power factor of the RC-circuit, being in this case cos(φ) = R √R2+( 1 ωC)2. cos(φ) = 1 q1+( 1 ωRC )2⇒cos(φ) = 1 1 ωqω2+( 1 RC )2⇒cos(φ) = ω √ω2+a2(2.115) So now we can conclude with the RC circuit as well: sin2(φ)+ cos2(φ) = 1⇒sin(φ) = s1−ω2 a2+ω2⇒sin(ω) = a a2+ω2(2.116) tan(φ) = sin(φ) cos(φ)=a ω(2.117) We now substitute (2.115) - (2.117) in (2.114), leading to (2.118): I(t) = Vp aC √a2+ω2ωcos(θ)(sin(φ)cos(ωt)+cos(φ)sin(ωt)−sin(φ)e−at) +Vp aC √a2+ω2ωsin(θ)cos(φ)cos(ωt)−sin(φ)sin(ωt)+ a ωsin(φ)e−at (2.118) We use the dual trigonometric relation used in (2.105) and shown in (2.119) for the RL-circuit, and use the expression (2.118), leading to (2.120): sin(φ)cos(ωt)+cos(φ)sin(ωt) = sin(ωt+φ) cos(φ)cos(ωt)+sin(φ)sin(ωt) = cos(ωt+φ)(2.119) I(t) = Vp aC √a2+ω2ωcos(θ)(sin(ωt+φ)−sin(φ)e−at ) +Vp aC √a2+ω2ωsin(θ)cos(ωt+φ)+ a ωsin(φ)e−at (2.120) We introduce the term φ=arctan(1 ωRC ) in (2.118), and approximate arctan(1 ωRC ) = π 2 because, either a cable or a capacitor bank, C will be in the order of micro-farads, so we obtain (2.119): I(t) = Vp qR2+( 1 ωC)2cos(θ)sinωt+π 2−e−1 RC t+sin(θ)cosωt+π 2+1 ωRCe−1 RC t (2.121) Also, due to the low value of C, τ will be a low value since τ=RC . This tells us that the energization transient will be damped in some micro-seconds. For this matter, the energization at zero voltage would mean that θ=0°, so the current would be: I(t) = Vp qR2+( 1 ωC)2hsinωt+π 2−e−1 RC t (2.122) This also means that the current starts at zero and rises while the transient component of the current is being damped; therefore, the current rises to a value approximately equal to the peak 2.4 Simple switching transients theory 35 value in micro-seconds. On the other hand, if the energization takes place at a peak voltage would mean that θ=90° ; thus, the following expression will be the result: I(t) = Vp qR2+( 1 ωC)2cosωt+π 2+1 ωRCe−1 RC t (2.123) In this case, it would be appropriate to distinguish between electrical and magnetic fields since the capacitor stores energy in the electric field (the inductor, on the contrary, stores energy in the magnetic field). A change in the electric field requires a change in the voltage/ charge, which would require infinite current, which is impossible. That tells us that there must be conservation of charge at the capacitor. As we can see in figure 2.14, the capacitor is initially uncharged and starts to charge when the switch closes. At this initial moment, I(t=0+)≃Vp R , but while the capacitor charges, the voltage at the resistor drops; therefore, the current drops as well until the steady-state is reached. Figure 2.14 AC-source RC circuit - Module representation of I/V-t. The dual scenario takes place at the inductor, where a change in the current is opposed by an electromotive force (emf), i.e., a voltage. Therefore, there cannot be an instantaneous change of current since it would imply infinite voltage. It is also to emphasize that during the energization of a capacitor bank (RC circuit), a large current, both high in magnitude and frequency, may appear if this is not prevented. 2.4.5 DC-source: RLC circuit RLC circuits are a more accurate example of much electrical equipment in real life. For example, although a capacitor bank can be described as a resistor and a capacitor, there is always some inductance between the two elements, becoming an RLC circuit instead of an RC. It is, therefore, a second-order circuit with a more complex transient behaviour due to the presence of both electric and magnetic fields. 36 Capítulo 2. Theoretical background Figure 2.15 AC-source RLC circuit. Again, since we are not aiming for the DC point of view, it will be simplified and commented taking into account the figure 2.16 shown below: Figure 2.16 DC-source RLC - transient damping [15]. To sum it up, depending on the values of the parameters RLC, we can expect any transient response from figure 2.16. What we can conclude is: • The more significant the damping factor is ( ζ ), the more damped response we will have, but it will take more time to reach the steady-state value. • The smaller the damping factor is, the more the signal will oscillate, reaching the steady-state value faster but oscillating until it stabilizes. • The red line in figure 2.16 represents the critically damped response, which is desirable to find a good ratio of fast-response/few damping for some applications. This damping factor will be directly proportional to the resistance value, but it also has to be taken into consideration regarding the losses of the circuit, i.e., the higher the resistance value is, the more losses the system will have. 2.4 Simple switching transients theory 37 Figure 2.17 AC-source RLC circuit. 2.4.6 AC-source: RLC circuit First, we write the equation that describes the circuit on the time domain, taking into account the same procedure that was taken in (2.95) and (2.108): Vpsin(ωt+θ) = RI +LdI dt +1 CZIdt, Vpωcos(ωt+θ) = Ld2I dt2+RdI dt +I C (2.124) Now, as we have done before, we will apply the Laplace Transform to have the required information within the frequency domain: Vpωscos(θ) s2+ω2−ωsin(θ) s2+ω2=I s2L+sR +1 C!−sLI(0)−L˙ I(0)−RI(0), Vpωscos(θ) s2+ω2−ωsin(θ) s2+ω2=I s2L+sR +1 C!−LVp(0) L (2.125) We could solve this again in the Laplace domain, but it is unnecessary. The expression for the current will attend to a forced and a homogeneous response. I(t) = If(t)+Ih(t)(2.126) Therefore, it is easy to realize that the forced component is the steady-state component, which can be expressed as: If(t) = Vp qR2+(ωL−1 ωC)2 sin ωt+θ−arctan ωL−1 ωC R!! (2.127) For the homogeneous component, we need to deduce the response. It is similar to the DC-source response, although now Vp(0) is not zero since it depends on the switching instant. The following expression (2.128) continues to be valid, but Ih(0)and ˙ Ih(0)need to be calculated. Ih(0) = A1+A2 ˙ Ih(0) = p1A1+p2A2)(2.128) 44 Capítulo 2. Theoretical background Figure 2.22 Comparison between bonding techniques - 132 kV Three-core cable without armour. As seen in figure 2.22, the cross bonding offers a bigger response with lower frequencies than the both-ends bonding, being the first harmonic peak even more than six times bigger. It is also to underscore that, even for higher frequencies, the value of the cross-bonded impedance is still relatively high. This is one of the main reasons why this bonding technique is not used for AC submarine power cables. Also, the cross bonding cable has more peaks than the both-ends one, meaning a more dangerous frequency spectrum to consider when operating. 2.7 Analytical calculation of cable impedances It has been learnt so far that we can achieve this analytical calculation of cables in different ways, so for this thesis, two different approaches will be taken into consideration: the one explained at the beginning of Section 2.2 (CIGRE formulae), and a most elaborated one approached from Section 2.2.1 to Section 2.2.3 which takes into consideration the formulae from different authors of whom we spoke about during these mentioned sections. Since the formulae from CIGRE are not optimal for frequency spectrum studies, they will only be addressed for the first case and then discarded for this thesisápproach. Now, to see a deep development of an impedance calculation, there will be three cases: 1. Underground three-core cable without steel armour (most common underground cable) 2. Underground three-core cable with solid steel armour 3. Underground three-core cable with stranded steel armour The calculations were self-made using Matlab as a mathematical tool for this 2D underground cable study. 2.7.1 Case 1: Underground three-core cable without steel armour As an introduction to the analytical calculation, this first scenario will be described: the core of the cable is a solid conductor made of copper, and the screen is a solid conductor. The needed values 2.7 Analytical calculation of cable impedances 45 for the calculation are given in Table 2.2 and shown in figure 2.23. Since the first cable will not have an external steel armour, the given values of the table for the armour will not be considered for this case. Table 2.2 132 kV Cable information [9]. Thickness (mm) Diameter (mm) ρtheor(Ω.m)µr Copper conductor - 34.5 1.72 ·10−81 XLPE (C-S) 20.5 75.5 - 1 Lead screen 2.5 80.5 2.14 ·10−71 XLPE (S-P) - 198.4* - 1 Steel armour 5.6 204* 1.38 ·10−7300 XLPE (P-G) 2.8 215.2* - 1 *Diameter measured from the geometrical center of the armour. Figure 2.23 Section of a three-core cable. CIGRE Formulae Since we have the required information, the calculation will proceed as follows: RDC =ρCond πR2 1 =1.72 ·10−8 π(17.25 ·10−3)2=1.84 ·10−5[Ω.m−1](2.6.1) Since the working temperature of the cable is usually 90 ° C, we will be considering the temperature coefficient to correct the DC resistance: RDC(90◦C) = 1.84 ·10−5(1+3.93 ·10−3(90 −20)) = 2.3461 ·10−5[Ω.m−1](2.6.2) Now to add the skin and proximity effects we need to calculate the equations shown from (2.12) 46 Capítulo 2. Theoretical background to (2.17), taking into consideration that the skin and proximity effect factors for each material and layout are ks=kp=1 . In this case, the impedance value will be evaluated at 50Hz, but we will see the frequency domain of the impedance in order to compare it with other cases. The distance between the phases di j =0.087m. x2 s=8πf RDC(90◦C)10−7ks=8π50 2.3461 ·10−510−7=5.3565 =x2 p(2.6.3) ys=x4 s 192 +0.8x4 s =0.13347 (2.6.4) yp=x4 p 192 +0.8x4 p2R1 di j 2 ·  0.312 2R1 di j 2 +1.18 x4 p 192+0.8x4 p+0.27   =0.06241 (2.6.5) Therefore, the AC resistance of the three-phase conductor at 50Hz will be: RCond =2.3461 ·10−5(1+1.5(0.13347 +0.06241)) = 3.0354 ·10−5[Ω.m−1](2.6.6) Now for the calculation of the reactance we need to know the depth penetration of the earth and the geometrical mean radius of the conductor, supposing ρearth =100[Ω.m]: De=659rρearth f=931.966[m](2.6.7) XCond =2πfµ 2πlnDe GMR=2π50 4π·10−7 2πln931.966 0.01725e−1 4=7.004 ·10−4[Ω.m−1](2.6.8) The same procedure will be applied for the screen ((2.6.9) to (2.6.11)). The external armour will not be considered due to the lack of interest regarding the approximated method because its value will be considered near zero since it would be grounded. Since the screen is made of lead, the calculation will be as follows: RScreen,DC =ρScreen π(R2 3−R2 2)=2.14 ·10−7 π((40.25 ·10−3)2−(37.75 ·10−3)2)=4.6196·10−5[Ω.m−1](2.6.9) RScreen,DC(90◦C) = 3.493 ·10−4(1+4.03 ·10−3(90 −20)) = 5.9228 ·10−5[Ω.m−1](2.6.10) XScreen =2πfµ 2πlnDe GMRS=2π50 4π·10−7 2πln 931.966 R2+R3 2!=6.3344·10−4[Ω.m−1](2.6.11) The self-impedance is then given by the summation of the conductor’s/screen’s resistance plus the conductor’s reactance and the ground impedance, given by the classic Carson’s formula (2.6.12): Re=2πfµ0 8=4.9348 ·10−5(2.6.12) 2.7 Analytical calculation of cable impedances 47 Therefore we can describe the following self impedances for each metallic layer of the cable: RSel f ,C=RCond +Re+jXCond =7.9702 ·10−5+j7.004 ·10−4[Ω.m−1](2.6.13) RSel f ,S=RScreen(90◦C)+Re+jXScreen =1.0858 ·10−4+j6.3344 ·10−4[Ω.m−1](2.6.14) The mutual impedance between phases ZM,Cond and the mutual impedance between conductor and screen ZM,Screen can be calculated as follows: ZM,Cond =Re+jωµ 2πlnDe s=4.9348 ·10−5+j5.8303 ·10−4[Ω.m−1](2.6.15) ZM,Screen =Re+jXScreen =4.9348 ·10−5+j6.3344 ·10−4[Ω.m−1](2.6.16) Thus the positive and zero sequence impedance of the cable will be the following, considering a both-ends bonding configuration, which will be the same for every cable in this section. Z+= (ZSel f ,C−ZM,Cond)−(ZM,Screen −ZM,Cond)2 RSel f ,S−ZM,Cond =5.5237 ·10−5+j9.6196 ·10−5[Ω.m−1] (2.6.17) Z0= (ZSel f ,C+2ZM,Cond)−(ZM,Screen +ZM,Cond)2 RSel f ,S+ZM,Cond =1.3856 ·10−4+j6.5282 ·10−4[Ω.m−1] (2.6.18) Bessel function’s method This will be the method where every impedance will be calculated, as it was stated in Section 2.2 . Since we are studying a case with a solid screen, there will not be resistivity correction, therefore, the method will follow the equations stated from (2.24) to (2.39), only the equations regarding the armour will be avoided in this case: mC=sjωµ ρC =rj2π50 ·4π·10−7 1.72 ·10−8=107.127 +j107.127[m−1](2.6.19) mS=sjωµ ρS =rj2π50 ·4π·10−7 2.83 ·10−8=83.5164 +j83.5164[m−1](2.6.20) So now we can calculate every impedance of the cable described before: ZCouter =2.2154 ·10−5+j1.4130 ·10−5[Ω.m−1] ZCS−ins =j4.9623 ·10−5[Ω.m−1] ZSinner =2.6448 ·10−4+j1.79 ·10−6[Ω.m−1] ZSouter =2.6448 ·10−4+j1.649 ·10−6[Ω.m−1] ZSP−ins =j3.6521 ·10−5[Ω.m−1] ZPG−ins =j1.7489 ·10−6[Ω.m−1] ZS−mutual =2.6447 ·10−4−j8.5861 ·10−7[Ω.m−1] Zearth ≈4.9524 ·10−5+j5.7459 ·10−4[Ω.m−1] 48 Capítulo 2. Theoretical background Zearth−mutual ≈4.9524 ·10−5+j5.8284 ·10−4[Ω.m−1] With that, we can set the series-impedance matrix, also explained in Section 2.2: [Z] =         0.0717 +0.682 j0.0496 +0.6154 j0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j 0.0496 +0.6154 j0.314 +0.6145 j0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j 0.0495 +0.583 j0.0495 +0.583 j0.0717 +0.682 j0.0496 +0.6154 j0.0495 +0.583 j0.0495 +0.583 j 0.0495 +0.583 j0.0495 +0.583 j0.0496 +0.6154 j0.314 +0.6145 j0.0495 +0.583 j0.0495 +0.583 j 0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j0.0717 +0.682 j0.0496 +0.6154 j 0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j0.0495 +0.583 j0.0496 +0.6154 j0.314 +0.6145 j         mΩ/m (2.6.21) From this matrix, we can state that: • The matrix is symmetric 6x6 (if armour were present, it would be 7x7 due to the three embedded conductors within the armour, as we will see in the two other cases). •We can distinguish four elements: –Self-impedance of the core: 0.0717 +0.682 j[mΩ/m], –Screen self-impedance: 0.314 +0.6145 j[mΩ/m], –Core-screen mutual impedance between phases and earth: 0.0496 +0.6154 j[mΩ/m], –Mutual impedance between phases and earth: 0.0495 +0.583 j[mΩ/m]. • We can see that the self-impedances are larger than the mutual impedances, as it was to expect. • Also, the mutual impedance in the same phase is larger than the mutual impedance between phases, being the difference mainly in the imaginary part of the element due to the inductive coupling. Now, for the sequence impedance we will apply the theory studied in Section 2.3 , starting from the expression (2.81), which can also be described as (2.6.22): [Uabc] = [Zabc][Iabc](2.81) VC VS=ZCZCS Zt CS ZSIC IS(2.6.22) where, •VCand ICare the voltages and currents of the conductors, •VSand ISare the voltages and currents of the screens, •ZCand ZSare the impedances of the conductors and screens, respectively. •ZCS is the mutual impedance matrix between conductors and screen in each phase. Since screen (and armour, when it is present) are all interconnected at both ends in submarine power cables, VS can be assumed to be zero to simplify the equation and deduce further the sequence impedance [18]: VC= (ZC−ZCS ·Z−1 S·Zt CS)IC=Zabc ·IC(2.6.23) For our case, taking the right values from the impedance matrix, arranging and operating it, Z012 will be obtained (2.6.24) 2.7 Analytical calculation of cable impedances 49 •ZC=  0.0717 +0.682 j0.0495 +0.583 j0.0495 +0.583 j 0.0495 +0.583 j0.0717 +0.682 j0.0495 +0.583 j 0.0495 +0.583 j0.0495 +0.583 j0.0717 +0.682 j  •ZCS =  0.0496 +0.6154 j0.0495 +0.583 j0.0495 +0.583 j 0.0495 +0.583 j0.0496 +0.6154 j0.0495 +0.583 j 0.0495 +0.583 j0.0495 +0.583 j0.0496 +0.6154 j  •ZS=  0.314 +0.6145 j0.0495 +0.583 j0.0495 +0.583 j 0.0495 +0.583 j0.314 +0.6145 j0.0495 +0.583 j 0.0495 +0.583 j0.0495 +0.583 j0.314 +0.6145 j  [Z012] =   0.2782 +0.1029 j0 0 0 0.0261 +0.0985 j0 0 0 0.0261 +0.0985 j mΩ/m(2.6.24) Therefore the positive sequence impedance at 50Hz will be Z+=2.61·10−5+9.85·10−5[Ω.m−1] and the zero sequence impedance Z0=2.782 ·10−4+1.029 ·10−4[Ω.m−1] , whereas the simplified analytical calculation was Z+=5.5237 ·10−5+j9.6196 ·10−5[Ω.m−1] and Z0=13.856 ·10−5+ 65.282 ·10−5[Ω.m−1] . Although we only can see small differences when it comes to the positive sequence impedance, the zero sequence impedance, on the other hand, shows a greater difference. Then, when we study the frequency range for the two different methods, we can also see important discrepancies along the spectrum, as shown in figure 2.24 and figure 2.25: Figure 2.24 Method comparison for an underground cable without external armour - Z+and L+. Therefore, as was already mentioned, due to the reduced distances between the conductors within a three-core cable, the CIGRE formulae are not optimal for this type of study. Nonetheless, it is to recognize that this method, although a bit more basic, is approximate for a given frequency. Thus, this method will not be further approached. 2.7.2 Case 2: Underground three-core cable with solid steel armour Since we will be using the same cable and the main difference is given by adding an external armour around the XLPE filling, the consequent calculation will follow, taking into account this 50 Capítulo 2. Theoretical background Figure 2.25 Method comparison for an underground cable without external armour - Z0and L0. modification and following the equations given in section 2.2 . The new impedances (considering the same conductor and screen impedances, which were not modified) due to the external steel armour are the following: ZSP−ins =j3.6521 ·10−5[Ω.m−1] ZPinner =3.8858 ·10−5+j3.8003 ·10−5[Ω.m−1] ZP−mutual =3.8445 ·10−5−j5.7479 ·10−7[Ω.m−1] ZPouter =3.8458 ·10−5+j1.1191 ·10−6[Ω.m−1] ZPG−ins =j1.6567 ·10−6[Ω.m−1] ZPipe−mutual =3.8458 ·10−5+j1.8142 ·10−5[Ω.m−1] Therefore, the series-impedance matrix will be defined as follows: [Z] =          0.0721 +0.722 j0.05 +0.655 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.578 j 0.05 +0.655 j0.314 +0.654 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.578 j 0.0495 +0.597 j0.0495 +0.597 j0.0721 +0.7219 j0.05 +0.655 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.578 j 0.0495 +0.597 j0.0495 +0.597 j0.05 +0.655 j0.314 +0.654 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.578 j 0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0721 +0.7219 j0.05 +0.655 j0.0495 +0.578 j 0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.0495 +0.597 j0.05 +0.655 j0.314 +0.654 j0.0495 +0.578 j 0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.088 +0.577 j          mΩ/m(2.6.25) Thus we can write the positive-sequence impedance matrix being: [Z012] =   Z00 0 0Z+0 0 0 Z− =  0.1194 +0.1197 j0 0 0 0.035 +0.1224 j0 0 0 0.035 +0.1224 j mΩ/m (2.6.26) The sequence impedance values are affected due to the armour around the conductors, increasing the mean value of the positive sequence impedance. This is a real scenario for a submarine cable since underground cables are generally armour-less at lower voltages. Still, this needs to be approached for a fair comparison between underground and submarine environments. Now, as happens with submarine cables, the exterior armour is not a solid one. Instead, a stranded armour will be addressed, and then we will compare the studied cases. 2.7 Analytical calculation of cable impedances 51 2.7.3 Case 3: Underground three-core cable with stranded steel armour Now, having the resistivity of the armour corrected and following the same procedure as before, the series-impedance matrix is given by (2.6.28) in mΩ/m and the sequence-impedance matrix by (2.6.29). ρStranded−Arm =ρSteel 2.82 104.82−99.22Nwire =75.49 %ρSteel (2.6.27) Since the armour of this cable is made of 110 steel wires, the correction is carried out as in (2.6.27), resulting the consequent series impedance matrix: [Z] =          0.0722 +0.7219 j0.05 +0.655 j0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0495 +0.578 j 0.05 +0.655 j0.314 +0.654 j0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0495 +0.578 j 0.0496 +0.5967 j0.0496 +0.5967 j0.0722 +0.7219 j0.05 +0.655 j0.0496 +0.5967 j0.0496 +0.5967 j0.0495 +0.578 j 0.0496 +0.5967 j0.0496 +0.5967 j0.05 +0.655 j0.314 +0.654 j0.0496 +0.5967 j0.0496 +0.5967 j0.0495 +0.578 j 0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0722 +0.7219 j0.05 +0.655 j0.0495 +0.578 j 0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.0496 +0.5967 j0.05 +0.655 j0.314 +0.654 j0.0495 +0.578 j 0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0495 +0.578 j0.0786 +0.5774 j          mΩ/m(2.6.28) Thus the new positive-sequence impedance matrix will be: [Z012] =   Z00 0 0Z+0 0 0 Z− =  0.1078 +0.1262 j0 0 0 0.0350 +0.1223 j0 0 0 0.0350 +0.1223 j mΩ/m (2.6.29) Here we can see a reduction in the element Z33 which impedance answers directly to the armour self impedance. This was nothing but logic since the resistivity of a solid cylinder is more significant than a stranded armour due to the reduction of area. Although this is only a simple view of the study, we can get an idea of how the presence of the armour affects the diverse values of the cable’s impedance. For a more thorough sight, a frequencyspectrum comparison will be now addressed among the three variations of the same cable: Figure 2.26 Comparison: R+ , R0 , L+ and L0 - Solid screen and solid/stranded armour - Comparison. There are few differences between solid and stranded armour regarding an analytical calculation within the frequency domain. This can be noticed in figure 2.26. We can see that the positive 52 Capítulo 2. Theoretical background sequence resistance is lower in the non-armoured cable due to the lack of armour, and between the two armoured cables, there is almost no difference. On the other hand, if we look at the zero sequence resistance, we notice that the non-armoured cable is higher than the two armoured since there are fewer layers between the conductors and the earth-return path of the current. Also, the stranded-armoured cable has a lower zero sequence resistance due to the lower resistivity of its armour layout. A similar approach can be set for the positive and the zero sequence inductances. For a better understanding, the following tables are shown for the three described scenarios: Table 2.3 Comparison between 132 kV cable layouts at 50 Hz. No arm. Solid arm. Stranded arm. R+0.0261 0.035 0.035 R00.2782 0.1194 0.1078 L+3.1341 ·10−43.895 ·10−43.893 ·10−4 L03.274 ·10−43.8087 ·10−44.016 ·10−4 *Values in mΩ/m&mH/m. Table 2.4 Comparison between 132 kV cable layouts at 100 Hz. No arm. Solid arm. Stranded arm. R+0.04395 0.07322 0.07321 R00.2903 0.1593 0.15413 L+3.0085 ·10−43.5835 ·10−43.5802 ·10−4 L02.3202 ·10−43.321 ·10−43.466 ·10−4 *Values in mΩ/m&mH/m. Table 2.5 Comparison between 132 kV cable layouts at 1000 Hz. No arm. Solid arm. Stranded arm. R+0.3194 0.3385 0.3382 R00.3463 0.3444 0.34467 L+1.9132 ·10−41.8513 ·10−41.8515 ·10−4 L01.7608 ·10−41.8062 ·10−41.8065 ·10−4 *Values in mΩ/m&mH/m. With this, the general approach for a three-core underground cable with and without external armour has been studied and concludes this chapter. After this theoretical chapter, submarine cables will be the focus of this thesis from now on, both for 2D studies (chapter three) and 3D (chapter five). 3 Submarine cables The majority of today’s offshore power generation facilities consist of wind turbines on fixed foundations, i.e., offshore wind parks (OWP). They stand typically in shallow waters like the north sea in northern Europe, which usually are less than 40 m deep. The technical lifetime of submarine cables is usually 30-40 years. However, a large variation between different cables has been experienced. While a few submarine systems have been abandoned after less than 20 years due to repeated outages, other systems are still operating after 50 years. The classification of the existent submarine cables is the following[6]. • Array cables: They are used to connect individual turbines. The array cables connect the turbines if there is an offshore transformer station. Array cables are also called "inter-array cables", "in-field cables", or "collector cables". Array cables are, in all known cases, threecore AC cables with rated voltages of up to 36 kV. Array cables have a steel wire armour for increased tensional force and mechanical protection. • HVAC Export cables: If there is no offshore platform with a step-up transformer, the MV array cables will also have the function of an export cable. The array cables would then transport the collected power from a group of the turbines to an onshore reception point. If there is an offshore transformer station, the export cable transports the collected power from the offshore transformer station to the onshore or offshore converter HVDC station. Export cables are usually three-core AC cables. If there is an offshore transformer, the export cables operate at around 145 kV AC, with some systems having up to 245 kV system voltage. Also, Single-core submarine cables are used in some interconnections and may be used in the future as export cables. • HVDC Export cables: they are also export cables since they transport the power to shore. HVDC export cables connect an offshore AC/DC converter station to a corresponding converter station onshore. They are typically single-core cables with extruded insulation and metallic radial water barrier. A pair of these cables make up a circuit and is commonly installed as a bundle. HVDC cables with mass-impregnated paper or PPL insulation are also suitable for HVDC export. Since the purpose of this thesis is to dive deeper into the HVAC submarine cables, and since we already explained the elements of a typical three-phase underground line, let us see the more specific situation of a three-core AC submarine cable. Modern three-core cables for the connection of offshore wind turbines often carry optical fibres for data transmission or temperature measurement, in addition to the electrical conductors, as shown in figure 3.1. Furthermore, although this is nowadays built that way, it is not the purpose of this thesis to consider the communication cable. These cables use optical fibre technology to carry 53 60 Capítulo 3. Submarine cables loaded. However, before that, as we already did in the previous chapter, the capacitance needs to be calculated so that we can build the PI-section for a 100 km length cable: C=2πε lnR2 R1 (3.10) With the calculated capacitance, the consequent frequency-dependent admittance is achieved with (3.11) Y(f) = j2πfC (3.11) With that, we can now calculate with MATLAB and show the harmonic response of the three cables within the frequency domain: Figure 3.7 Comparison of the harmonic impedance for the three described cables. There is quite a significant amount of things to see in figure 3.7: • The positive-sequence impedance (complex sum of both R and X) is directly proportional to the voltage bounce-response seen from the transmitting end for different frequencies through its spectrum • For the two armoured cables, the positive-sequence resistance is lower than the non-armoured cable. This happens because the induced currents on the non-armoured cable return through the screens to the ground, which is nearer to the cores than the armour from the cores, inducing less magnetic field (less active losses), resulting in a lower peak than the non-armoured cable. • Concerning the zero-sequence resistance, the armoured cables have higher values in the first peak because the impedance shows a behaviour directly related to the medium and the current return path, which is higher with an armour. • As an interesting detail, the peak of the resistances are the decreasing zeros of the reactances, and the minimums are the creasing zeros of the reactances. • It is also to underline that the peaks of both the resistance and the reactance are being damped with the increasing frequency, given the effect of the line resistance. 3.3 Analytical calculation for a submarine cable 61 Due to the limitations of a 2D study, the armour has no more influence than having extra resistance in the cable and offering a better return current path. This is not the case for a 3D study, where the twisting of the conductors and stranded armour influences the calculation result. Therefore, the theoretical part of this thesis has been approached with some study cases to bring some light and precedent around the primary purpose. It has been proved that the simulation of these studies with an analytical approach is not quite optimized, although it can be quite accurate for some frequencies; thus, the proposal of approaching this theme within a 2.5D and 3D scenario with another finite element software such as COMSOL, as will be from now on considered. 4 COMSOL applied to 2D and 3D impedance calculation Comsol is a cross-platform finite element analysis, solver and multiphysics simulation software that allows conventional physics-based user interfaces and coupled systems of partial differential equations [ 25 ]. Many researchers use COMSOL since it provides an integrated development environment and unified workflow for electrical, mechanical, fluid, acoustics, and chemical applications. However, first things first: we have already talked about finite element methods (FEM, from now on) programs such as COMSOL. A FEM is a popular method for numerically solving differential equations arising in engineering and mathematical modelling [ 14 ]. To solve a problem with FEM, it subdivides an extensive system into smaller, simpler and smaller parts called finite elements. A particular space discretization achieves this, which is implemented by constructing a mesh of the object: the numerical domain for the solution, which has a finite number of points. The larger the number of space dimensions in an area, the more calculating time the program will need, but better results accuracy will be achieved with this. Therefore, COMSOL tries to find a balance between these by having a proven well-balanced solving equation algorithm. In this chapter, we will briefly introduce the software regarding impedance calculation in submarine cables. 4.1 Interface Figure 4.1 View of the interface when starting COMSOL. 63 64 Capítulo 4. COMSOL applied to 2D and 3D impedance calculation Once the program has been installed, we can look at the Application Libraries that we have installed in the process. This library contains lots of tutorials for diverse technical areas. For our case, the AC/DC Module offers different categories and, more precisely, if we click on cables, a menu with some examples will be shown, as shown in figure 4.1. In order to simplify this chapter and cut to the chase of our aim, COMSOL will only be approached for the mentioned use, although, as said, the software can work in many different areas within engineering. At the upper area or the interface, we can see there is a pop-up menu with individual commando options for each part of the calculation (figure 4.2). These will be our most used tools for our case of study. As shown, let us Figure 4.2 View of the upper menu within COMSOL. open example number five, "submarine_cable_05_bonding_inductive". Once open, we should see something like in figure 4.3, where on the left side, we have a menu with some parameters (settings), and on the right side, we have the representation of the cable section. Here, we can see the structure of the cable: a three-core cable with solid sheaths around and a stranded armour around, showing how intense the magnetic flux density is on the example cable. For this, geometry and electrical parameters had to be configured prior to the representation. Figure 4.3 View of the interface with an open example. However, let us go rewind back a bit so that we can have a better understanding of the interface. Model Builder & Settings As we can see in figure 4.4, there are four sub-categories within the model builder, which, when we click on them, we will obtain different parameters on the settings menu. The main category is the file’s name, in this case, "submarine_cable_04_inductive_effects.mph". The four sub-categories are: • Global parameters: it will allow defining parameters such as the cable’s geometry, electromagnetic parameters, materials, etcetera. • Component: The complete model will be interpreted here as an entire system, along with the relevant equations that we want to be involved with. •Study: Here, we will set which kind of study we want to develop. 4.2 COMSOL - 2D & 2.5D 65 • Results: In this tab, we will obtain the wished results, whether a graphic or the relevant calculations we want to execute. Figure 4.4 General working schema - COMSOL. The first two sub-categories correspond to the modelling and pre-processing of the model; the third sub-category belongs to the processing phase, and the last one corresponds to the post-processing of the model, where the result can adopt different formats. The software will only solve one variable; that is, we can only ask for one parameter to be our result; COMSOL, on the other hand, offers the possibility of changing this parameter by operating it to get another wished parameter, not shown as default. Moreover, an interesting detail is that one can load his own ".dwg" file and COMSOL will interpret it and translate it into its language. COMSOL has a solid database, where lots of technical values are saved when we select some material, the shape of a domain. This information can also be modified if it is required. Figure 4.5 Physics for the problem - COMSOL. 4.2 COMSOL - 2D & 2.5D A 2D model will be first addressed to compare with our theoretical method, and then a 2.5D will be approached. These two methods have essential differences when using finite-element software such as COMSOL since the 2D does not consider the armour’s or lay length’s twisting. Therefore, 66 Capítulo 4. COMSOL applied to 2D and 3D impedance calculation COMSOL will interpret the armour as a conductor material carrying a magnetic field all along the cable length for a 2D study. This is not the case in a real scenario; therefore, it is an approximation less veritable than the 2.5D (later explained) or the 3D study. The example pre-designed within COMSOL is a 500 mm2 , 220 kV cable with an effective nominal current of 655 A. The rest of the information is already given in the parameters. Moving on, another critical thing to input in COMSOL is the physics of the problem; that is, it is required to define which equations will take place for the selected study. As shown in figure 4.5, for our problem, the following theoretical equations need to be considered in COMSOL. If we look at the left window in figure 4.5, we will see the different layers of the cable that will generate a current and, therefore, an induced magnetic field. These need to be configured as "coils" so COMSOL understands them as elements with a current source (magnetic field source, after all). Thus, for this example, the three conductors will simulate their nominal current through themselves; that is, their main design current with a 120 ° angle. Regarding the screens, on the other hand, if we do not change their configuration, they are usually configured as both-ends bonding, which is of utmost importance to consider if we would like to study other bonding techniques. The three single-phase AC resistances have already been selected for this built-in example. After clicking on "build all" and then on "compute" within the "Study 1", we can see the results, such as the resistance of the cable and phase inductance, as shown in figure 4.6. We have to click on them to see the values for an already given frequency (f0=50 Hz in this case). The results are: •R1 AC =43.307mΩ/km, •R2 AC =43.282mΩ/km, •R3 AC =43.246mΩ/km, Also, we can see that, as default within this pre-built example, the DC resistance at 20 ° C was calculated. Figure 4.6 Studying results - AC Resistance of the cable. As we can also see in figure 4.6 in the "expressions" area, a mathematical expression describes the positive-sequence resistance of each phase. If we want the program to show us the complete positive-sequence AC impedance of the cable, we need the arithmetical average of the three positivesequence resistances (also divided by 1-meter distance, so we obtain as units mΩ/m , which can also be modified as mΩ/km ). The difference in the values for the three phases is given because, due to the built mesh for a given solution, the mesh of one conductor may not be precisely equal to the other two conductors. Therefore, the software calculates considering the different meshes for each 4.2 COMSOL - 2D & 2.5D 67 phase, as it usually happens with other finite-element software. Figure 4.7 Mesh concept 1. Moreover, the discretization of the modelling space via a mesh serves two purposes: first, the mesh is used to approximate the CAD geometry; second, the approximate solution to the problem is solved at discrete points in space defined by this mesh [ 25 ]. Regarding the solution for our cable, the size of the mesh needs to be taken into consideration. As we can see in figure 4.7, figure 4.8 and figure 4.9, in order to define a proper mesh, we need to know which effects are going to take place, as we already studied in chapters two and three; since the density of current will be more located at the edges of the conductors, the mesh needs to be accurate and refined enough, so this will not go unnoticed by the software when calculating each finite-element for each frequency, given the studied effects on the conductors. As shown in Figure 4.7, we can begin to appreciate the level of refinement for the conductive elements (also shown in more detail in Figure 4.8 and Figure 4.9) and the not much refined discretized elements for the external triangles of the cable representing the medium, in this case, subsea water, where not much needs to be calculated, in comparison with the conductors and their surroundings. As shown in Figure 4.8, we do not need much accuracy when the software goes further from the conductive elements. Looking a bit closer to the conductor and the external armour, we can see that the mesh needs to be smaller to be more accurate with the calculations (which, therefore, also costs more computational calculation, thus time to simulate). If we zoom more into the external armour (Figure 4.9), we can appreciate the measurement of each discretized triangle around and within each strand of armour, even more knowing that the measurement unit at each axis is given in meters. We can also define the diameter of the medium by setting how big this circle will be considered. As it is to imagine, the larger the diameter of the medium is, the more computational cost it will take, and the more accurate the result will be. As we can see in figure 4.10, since this is a pre-designed cable within COMSOL, the proportions of the external medium diameter will be given as shown. Regarding our main aim for this thesis, We can select a range of frequencies instead of a single frequency to see an evolution of either the resistance or the inductance against the frequency spectrum. Therefore, if we want to see the evolution of the parameters of the cable within the frequency domain, we set an interval of frequencies (figure 4.11), click on compute, and then continue with our derived values for the AC resistance. After this, we arrange the average of the 68 Capítulo 4. COMSOL applied to 2D and 3D impedance calculation Figure 4.8 Mesh concept 2. Figure 4.9 Mesh concept 3. three resistances as earlier stated, we evaluate this, and finally, we can graphic the results (figure 4.12 and figure 4.13): Something important to highlight is the given frequency range: when we define it as we did in figure 4.11, the simulations take too long (up to hours, depending on the number of steps per decade when simulating, which, if there is not enough RAM/memory swapping, would crash the simulation). This problem can be easily solved by selecting different frequency intervals to evaluate the solution. The one I selected is: 10ˆ { range(log10(0.1),1/50,log10(100000)) } . This means, starting on 0.1Hz and having fifty steps per decade, evaluate the frequency up to 100kHz. This takes considerably less time to evaluate and gives us a good approximation without taking that much computational time. A problem arises when a larger number of samples per decade is required for more accuracy. In that case, either a high quantity of RAM/swapping memory or separate the simulations into smaller intervals. Still, it would cost an excessive amount of time. 4.2 COMSOL - 2D & 2.5D 69 Figure 4.10 Mesh concept 4. Figure 4.11 Preparing simulation - Frequency interval. 4.2.1 Sequence Impedance 2D Calculation As we already said, in this 2D study, the armour is understood by COMSOL as any other conductor, where currents injected in the three primary conductors of the cable and, therefore, induced currents will circulate through the external steel armour. After having simulated, we can now see both the resistance and inductance of the positive sequence over the frequency, as shown in figure 4.12: We are now recognizing the shape of both graphics since they show us a similar behaviour as it already did when calculating with the analytical method, both in chapters two and three. Also, if we want to study the zero sequence of the same cable, we need to change the value of the current given to the coils, so, instead of each current with a 120 ° phase displacement, we want to simulate the zero sequence by giving these currents all the same phase displacement. Once this has been changed, we repeat the same procedure, and we obtain the following figure 4.13: 76 Capítulo 4. COMSOL applied to 2D and 3D impedance calculation (a) Mesh for skin depth at 50Hz. (b) Mesh for skin depth at 5000Hz. Figure 4.24 Cable Analyzer mesh adaptation - Comparison. figure 4.25 that the 3D version that COMSOL made is adjusting the mesh, but the time elapsed is not entirely optimized for this kind of study. Figure 4.25 Cable Analyzer mesh - Armour zoom at 5000Hz. When trying to simulate with the mentioned model from COMSOL, it takes over six hours and it never finishes the simulation, showing the following error message: Figure 4.26 Error message after six hours simulation. That is why, as we did before with 2D studies, we will adapt the program and the mesh depending 4.3 COMSOL 3D 77 on the skin depth at each frequency to look for a better way to simulate 3D submarine cables. Therefore, applying similar thinking, as we did with the 2D model, we will see how our mesh will adapt itself to the 3D model, and we will also demonstrate that not only the program will run the entire simulation, but it also will be done in much less time than the prognostic of COMSOL’s Cable Analyzer. The method of calculation within COMSOL was already developed in [ 8 ] by PhD. Juan Carlos del Pino López, professor at "Universidad de Sevilla". He has been studying the development of 3D models for submarine power cables for years. It also needs to be said that the design of this mesh adaptation is not a goal of this thesis, but the comparison between the most current 3D studies and the traditional theoretical 2D and the improved 2.5D methods to show how accurate they are up to date, and for that purpose, the use of this developed model is reasonably practical. Of course, what is inside of COMSOL and this improvement had to be learnt to put the results to use. Still, the recognition for this adaptation shall go to PhD. Juan Carlos del Pino López. Table 4.2 Time comparison 2. Time elapsed per simulation Simulation completed 2D no mesh correction 196 s Yes 2.5D no mesh correction 293 s Yes 2.5D + mesh correction 3535 s Yes 3D Cable analyzer (COMSOL) >21600 s No 3D J.C. del Pino 13210 s Yes *All simulations were made taking 50 samples per decade Again, the simulations were made with a PC Desktop equipped with the already described characteristics in Table 4.1. Furthermore, as much as the modified 3D takes over three and a half hours, if the model of COMSOL had finished the simulation, it would have taken probably four or five times that time, taking into account the projection of time it took and how many frequencies it could analyze until it crushed. Thus, not a bad improvement for that matter. Figure 4.27 2D, 2.5D & 3D comparison. As we can see in the table 4.2, 3D studies take much more time than 2D and over three times more 78 Capítulo 4. COMSOL applied to 2D and 3D impedance calculation than 2.5D studies. Let us now compare each scenario’s results so we can better see each technique in figure 4.27. As we can see, there are essential differences among the studied cases, but especially the similitude between 2.5D with mesh correction and the 3D with mesh correction, which tells us how accurate 2.5D studies can be with this kind of scenario. Also, we can appreciate that between 2D and 2.5D without mesh correction, the zero-sequence response is practically equal, whereby the positive-sequence there are remarkable differences for middle/high-range frequencies (from 100 Hz ahead). Aside from these facts, without forgetting the main aim of this thesis, once we have the positiveand zero-sequence resistance and inductance, we can go back to MATLAB to calculate the harmonic impedance with a distributed parameter equation system, as we already did in chapters two and three. For this, since we will focus on a lower range of frequencies, we start simulating with COMSOL with more samples per decade to have a clearer sight of the figure, and then we calculate the harmonic impedance range with MATLAB. All the scenarios were simulated for a given distance of 100 km, as we already did in chapter three, but with the default 220 kV cable within COMSOL. Once this is done, we can plot the results: Figure 4.28 Harmonic impedance - 2D, 2.5D & 3D comparison. We can conclude by taking a look at the figure 4.28 that there are relevant differences among the studies carried out: • The shown peak of the R+ OC is more significant when simulating in 2D than when simulating in 2.5D with mesh correction or with the 3D method. This is due to the most approximated reality of the 3D method against the 2D, even when using COMSOL. • There are some differences in the frequencies where each peak is shown. The similitude is more significant, of course, which tells us that these four have a good approximation from this point of view. • As we already studied at the end of chapter 3, the peaks of the resistances are coincident with the reactances going over zero. • We can conclude that, for this precise cable, the best method was 2.5D with mesh adaptation since it shows a similar outcome as the 3D method does but takes much less time, as shown in table 4.2. 4.3 COMSOL 3D 79 • As we already stated in chapter three, it is interesting to notice that the first impedance peak comes first when the receiving end is shorted (full load demand at receiving end) than for the open circuit scenario (no load demand at receiving end). To be more thorough, we will compare each positiveand zero-sequence harmonic resistance for each given scenario to better see the four suggested scenarios. This is given in figure 4.29. Figure 4.29 Harmonic resistance - all scenarios comparison. Here we can appreciate the few differences between 2D and 2.5D without mesh adaptation, and of course, between 2.5D with mesh adaptation and 3D. Of course, it is to be considered as well that there are massive differences between 2D and 3D, meaning a more conventional calculation for the 2D, which would mean a more secure study in reality since the worst-case scenario would be covered. Twisting of phases and external armour in 3D simulations Figure 4.30 shows a general visual representation of lay length phase and armour layout; that is, the phases twisting have a direct mathematical relationship with the armour twisting, also explained in [18]. Figure 4.30 External armour twisting lay [18]. 80 Capítulo 4. COMSOL applied to 2D and 3D impedance calculation It is also to be said that the twisting lay of the armour and the conductors are not easy to obtain since this data is most usually not available, often not even from the cable manufacturers. Luckily, COMSOL gives their values for the shown 220 kV cable model for our default case; that is, the armour twisting lay La=3.08m , and the phase twisting lay Lc=3.46m . Therefore, La/Lc=0.89 . It is important to clarify that the twisting direction of the external armour is always set counterclockwise twisting (contralay) with respect to the clockwise (unilay) phase twisting (figure 4.23) usually in the opposite direction to achieve torsion stability. Despite this, an unilay layout will also be now addressed, so we can understand the impact that this would imply. Figure 4.31 Twisting comparison - 220kV COMSOL. So, to be thorough, let us study the impact on the parameters mentioned, that is, twisting direction (unilay and contralay) and armourand phase lay length variation. The following figure shows a comparison among these scenarios for the same default cable studied in chapter four. For the lay length variation, La=2.5 and La=4.5 were respectively considered since COMSOL showed errors by using a lay length value lower than 2.5 or higher than 5. Lcwas not modified. Considering figure 4.31 there is not much to comment about, for example, that there are few differences among the cases for the positive-sequence impedance, but if we zoom up a bit (figure 4.32) we can appreciate relevant differences: it would be expected that the lowest resistance value would come from the scenario with the bigger La/Lc ratio since there is less armour along the cable, but interestingly, the unilay scenario obtains the lowest resistance and, therefore, the inductance of all. This scenario is not considered since an unilay configuration of the cable offers much less mechanical robustness to the whole cable, which is necessary for submarine installations deep under the sea/ocean. On the other hand, the lower twisting lay length of the armour results in higher resistance in both sequences since there is more armour around the conductors. The opposite effect occurs with the inductance since less current will be induced from less external armour around the conductors. No scenario varying the twisting lay length of the conductors was simulated since we can extrapolate it from the results, keeping in mind what happens when varying the twisting lay length of the external armour, that is, if we reduce the twisting lay length of the conductors, the ratio La/Lc will go higher, therefore a lower value of both resistance and inductance, and vice-versa. Of course, we must remember that the often used configuration of lay/contralay gives more torsion stability when placed in the water and moved by the maritime currents. Therefore, to conclude this fourth chapter, another topic to consider is the disposition of the 4.3 COMSOL 3D 81 Figure 4.32 Twisting comparison (zoom) - 220kV COMSOL. conductor’s material: In real-world applications, solid copper/aluminium conductors are generally not used but are stranded. This is not being considered for this thesis since it was already demonstrated that it makes almost no difference when simulating, although the theory tells us that both proximity and skin effects would be drastically reduced when using stranded conductors. Since I already tried both layouts in COMSOL and obtained practically the same outcome, solid conductors will only be addressed. Having explained that, chapter five will now follow up with the simulation of the already studied 132 kV cable, along with a 150 kV and this 220 kV cable, and we will compare the results thrown by COMSOL with chapter two’s studied theoretical approach. 5 Harmonic impedance: comparison of methods In this chapter, we will compare what we have already studied in chapters two and three with what we have already seen in chapter four; that is, we will compare the most interesting harmonic impedance calculation methods for 2D, 2.5D and 3D scenarios. The relevant information we need can be described as we already did in chapters two and three, or as it is often also described in COMSOL (figure 5.1), which can be even more convenient. Figure 5.1 Cable dimensions. As we already did before, the methodology to follow should be known, i.e., first, we will study the sequence impedance of the cables, and with that, we go to MATLAB and calculate their selfharmonic response within the frequency domain for a distributed parameters power line system. Therefore, once again, we will remember the relevant information for our cables shown in the following table: Table 5.1 Cables information. S (mm2)In(A) Dc*Ds*es*Rt* N Da*da*La(m) Lc(m) 132 kV 800 900 34.5 82.5 2.5 50.23 110 204 5.6 3.395 2.624 150 kV 630 650 30.5 80.3 2.4 49.02 105 195.47 5.6 3.128 2.927 220 kV 500 655 26.2 83.4 2.9 51.5 110 205.6 5.6 3.084 3.46 *Values given in mm. The 132 kV cable was presented by Chippendale in [ 9 ], the 150 kV cable by Filipe da Silva in [ 20 ], and the 220 kV cable was already addressed in chapter four as the default cable given by 83 84 Capítulo 5. Harmonic impedance: comparison of methods COMSOL. Having this purpose in mind, we will consider the following studied methods: •2D analytical method with MATLAB (chapter three - stranded external armour) •2D method with COMSOL •2.5D method with COMSOL (with frequency-dependent mesh adaptation) •3D method with COMSOL (Juan Carlos del Pino COMSOL’s cable-method) After comparing the analytical results, we would compare in addition the computational time required for each method, considering how accurate the results are with the time elapsed to obtain them, and also we would see the the relative errors when comparing these methods with the 3DCOMSOL method, since the paper [ 8 ] reveals that Juan Carlos del Pino COMSOL’s model is quite accurate in comparison with the experimental measured values, we would be thorough by considering that our 3D-COMSOL method offers the nearest result to the experimental values measured. 5.1 132 kV Cable Comparison Figure 5.2 132 kV Chippendale Cable dimensions. In order to make this document lighter to read, there will not be calculation procedures in this chapter since these were already explained in chapters two, three and four for each scenario. Still, going back to chapters two and three, we already studied this 132 kV cable with MATLAB and saw the effect of having solid/stranded conductive elements and the current return-path medium (soil or seawater) affecting the zero sequence impedance, among other aspects. Thus, here we can suppose already the results we are going to obtain: we could rush into saying that the 2D method with COMSOL will be more similar to the analytical method that we studied in chapter three using only MATLAB, but, as we can see in figure 5.3, the classical theoretical model given from the already known Bessel functions is more similar to the 3D model simulated with COMSOL than one could expect. Without further ado, let us show the results in figure 5.3 and discuss them for this case: Here we can see both positiveand zero-sequence resistance and inductance responses over the frequency. There are plenty of things to comment about this: 5.1 132 kV Cable Comparison 85 Figure 5.3 132 kV Methods Comparison. • We can see that, in some graphics, for example, the positive-sequence resistance that the first analytical method gives a better response over frequency than the 2D method given by COMSOL. This is, nonetheless, eye-catching since this first method takes much less time to obtain than any studied COMSOL scenario. • There are other situations, for example, the zero-sequence resistance, where the analytical method differs from the rest at some frequencies, but it has important similarities at the rest of the frequencies. • It has some sense that the 2D COMSOL model offers a lower positive-sequence resistance at high frequencies since the model within COMSOL considers the external armour as another conductive element where magnetic fields will be induced and, given the temperature effects within COMSOL, the resistance will drop due to the result of having those temperature equations in the model without any further consideration on the external armour. • The same (inverse) logic can be applied to the inductance since, in this model, there is less induction considered. What catches the eye is the analytical model with MATLAB, which shows more similar results to the 3D simulation. Still, since we are moving around logarithmic scales, let us compare the margin of error of each particular method shown in figure 5.4, comparing all against the 3D-COMSOL method with the pertinent validation learnt from [ 8 ], which is by far one of the most accurate and established methods in the scientific community up to date. Here we can appreciate how far the studied models are for some frequencies compared to the 3D-COMSOL model. On the other hand, we can see that the 2.5D model is often close to the results from the 3D-COMSOL model, and so does the Bessel function’s method, which is not to be ignored. The 2D-COMSOL model is the one that raises the most doubts, whether this method should even be considered for this purpose, given the results. Although the 2.5D COMSOL method is the most accurate compared to some scenarios, we also need to keep in mind the time elapsed for simulation vs method. This can be appreciated in table 5.2. To go a bit further, let us study this cable’s harmonic response analyzed with each technique. For this, shown in figure 5.5, we studied in MATLAB for a distributed parameters model the response 92 Capítulo 5. Harmonic impedance: comparison of methods Taking a quick look, a similar outcome is obtained for this cable, where we find accurate results from our analytical model in some frequencies, in this case, for both positiveand zero-sequence impedances. Moreover, again, the 2D COMSOL model is less accurate when compared with the other methods. The only difference here is the time elapsed on the 2D COMSOL method, which will be shown later. For better sight, relative errors will now be shown in 5.14. Figure 5.14 220 kV Cable - Error (%) Method’s comparison. Here we can appreciate how far/close each method is for this 220 kV submarine cable. As we could see before, 2D-COMSOL is far from accurate, whereas the 2.5D and the Bessel function’s methods are quite accurate depending on which frequencies are being studied. As we did with the 132 kV and 150 kV cable, We see the most significant differences in the inductance response, reaching even 60% of inaccuracy, where 2.5D has around 35% at the most. Figure 5.15 220 kV Zoc & Zsc Comparison. 5.3 220 kV Cable Comparison 93 Let us now look at the harmonic impedance (figure 5.15) for both short circuit and open circuit at the receiving end of a transmission line with a distributed parameters line model. Again, the 2D COMSOL method has the most distant response compared to the others. We now can confirm the pattern regarding the analytical method using Bessel’s functions and the 2D COMSOL method with respect to the other two: the analytical one is quite accurate compared to the 2.5D or 3D COMSOL methods, taking much less computing time to obtain this response. Still, let us compare only the positiveand zero-sequence resistance in these two already described scenarios: Figure 5.16 220 kV Positive- & zero-sequence - Roc & Rsc Comparison. As we can finally confirm by looking at figure 5.16, the 2D COMSOL method is not entirely accurate compared with the other three. On the other hand, the first method implemented with MATLAB (Bessel’s functions) gives us quite an accurate answer regarding obtaining the harmonic impedance over the frequency, taking much less time than 2.5D or 3D COMSOL methods. All in all, the 2.5D COMSOL method is the one that offers much more accuracy by not sacrificing much computing time on it. This can also be seen in table 5.6. Table 5.6 Time comparison 220 kV Cable. Time elapsed per simulation MATLAB Analytical method 271.2 s 2D COMSOL 177 s 2.5D with mesh correction 2562 s 3D COMSOL 14851 s *All simulations with COMSOL took 50 samples per decade. MATLAB took one million samples. Last, the relative errors against the 3D-COMSOL method will be shown in table 5.7, where the first resonance frequency is reached in each case shortand open circuit at the receiving end of a distributed parameters transmission power line. Thus, if we look for the best compromise for results/computing time efficiency, we should go for the 2.5D COMSOL scenario. This method also shows that the higher the voltage of the cable, the more accurate this method is since it kept being more and more accurate as we studied the two 132 and 150 kV cables prior to this 220 Kv cable. 94 Capítulo 5. Harmonic impedance: comparison of methods Table 5.7 Error (%) Methods vs. 3D-COMSOL for 220 kV Cable. R+ OC(f=1124Hz)R0 OC (f=1138Hz)R+ SC (f=544Hz)R0 SC (f=550Hz) Bessel function’s method 2.937% 8.793% 3.166% 7.435% 2D COMSOL 161.495% 117.46% 105.84% 63.909% 2.5D with mesh correction 6.748% 5.899% 4.361% 3.177% Table 5.8 Error (%) Methods vs. 3D-COMSOL at relevant frequencies for 132 kV Cable. 132 KV Cable R+ OC(f=1082Hz)R+ OC(f=2205Hz)R+ OC(f=3333.5Hz) Bessel function’s method 6.978% 5.83% 5.976% 2D COMSOL 235.927% 418.94% 564.65% 2.5D with mesh correction 12.317% 19.409% 13.732% R0 OC(f=1094Hz)R0 OC(f=2225.8Hz)R0 OC(f=3355.8Hz) Bessel function’s method 3.445% 1.84% 4.409% 2D COMSOL 141.556% 308.33% 469.58% 2.5D with mesh correction 5.439% 21.697% 24.68% R+ SC(f=527Hz)R+ SC(f=1642.8Hz)R+ SC(f=2770Hz) Bessel function’s method 9.171% 6.405% 5.836% 2D COMSOL 152.738% 325.87% 500.63% 2.5D with mesh correction 7.434% 15.31% 18.763% R0 SC(f=524Hz)R0 SC(f=1659.6Hz)R0 SC(f=2789.1Hz) Bessel function’s method 9.952% 0.075% 3.555% 2D COMSOL 82.434% 220.487% 396.19% 2.5D with mesh correction 5.288% 15.183% 24.286% To summarise this section, a comparison among the relative errors will be shown for each cable and certain frequencies, where the error is most relevant depending on the chosen method. For this, tables 5.8, 5.9 and 5.10 will be shown. Table 5.9 Error (%) Methods vs. 3D-COMSOL at relevant frequencies for 150 kV Cable. 150 KV Cable R+ OC(f=1124Hz)R+ OC(f=2228.7Hz)R+ OC(f=3373.3Hz) Bessel function’s method 9.447% 11.099% 11.434% 2D COMSOL 106.924% 310.92% 439.48% 2.5D with mesh correction 5.462% 6.249% 6.038% R0 OC(f=1099.6Hz)R0 OC(f=2251.4Hz)R0 OC(f=3398.1Hz) Bessel function’s method 4.202% 4.436% 8.057% 2D COMSOL 101.205% 220.834% 354.004% 2.5D with mesh correction 4.589% 8.191% 8.839% R+ SC(f=544Hz)R+ SC(f=1656.7Hz)R+ SC(f=2800.5Hz) Bessel function’s method 7.105% 10.674% 11.264% 2D COMSOL 117.102% 243.101% 379.98% 2.5D with mesh correction 6.524% 6.067% 6.362% R0 SC(f=523Hz)R0 SC(f=1675Hz)R0 SC(f=2823.9Hz) Bessel function’s method 15.523% 1.5% 6.816% 2D COMSOL 58.509% 157.415% 290.05% 2.5D with mesh correction 2.161% 6.76% 8.639% Also, we realize that, as the error decreases when simulating a 2.5D scenario for higher voltage 5.4 Harmonic impedance comparison 132 kV vs 150 kV vs 220 kV 95 Table 5.10 Error (%) Methods vs. 3D-COMSOL at relevant frequencies for 220 kV Cable. 220 KV Cable R+ OC(f=1124Hz)R+ OC(f=2295Hz)R+ OC(f=3467.9Hz) Bessel function’s method 2.937% 9.904% 12.518% 2D COMSOL 161.495% 321.16% 408.484% 2.5D with mesh correction 6.748% 9.84% 10.92% R0 OC(f=1138.7Hz)R0 OC(f=2310.2Hz)R0 OC(f=3480.7Hz) Bessel function’s method 8.793% 14.62% 15.335% 2D COMSOL 117.46% 289.097% 492.587% 2.5D with mesh correction 5.899% 9.946% 10.774% R+ SC(f=544Hz)R+ SC(f=1710.7Hz)R+ SC(f=2882.1Hz) Bessel function’s method 3.166% 7.204% 11.827% 2D COMSOL 105.84% 231.933% 414.79% 2.5D with mesh correction 4.361% 8.812% 10.757% R0 SC(f=550Hz)R0 SC(f=1725Hz)R0 SC(f=2895.4Hz) Bessel function’s method 7.435% 12.075% 15.452% 2D COMSOL 63.909% 194.289% 391.78% 2.5D with mesh correction 3.177% 8.213% 10.813% cables, the contrary occurs when simulating with the Bessel function’s method, since we obtained relatively low errors for our 132 kV cable, but they went up for 150 kV and 220 kV cables. 5.4 Harmonic impedance comparison 132 kV vs 150 kV vs 220 kV Lastly, a comparison of each cable’s response will be addressed, although it is not entirely fair since, as we mentioned, each cable is designed for a specific task. Still, let us look at the harmonic impedance comparison for both described scenarios with openand short-circuit at receiving end. First, we will look at the first studied method and self-written in MATLAB using Bessel’s functions. Figure 5.17 Cable comparison Positive- & zero-sequence - Zoc & Zsc - Analytical. 96 Capítulo 5. Harmonic impedance: comparison of methods Let us now look at the 2D COMSOL method and compare it with figure 5.17. Figure 5.18 Cable comparison Positive- & zero-sequence - Zoc & Zsc - 2D COMSOL. Here we can see that the simulations do not respect the logical order of each cable; that is, the bigger the cable, the bigger the harmonic impedance outcome. Of course, the 220 kV cable has a more significant impedance, but sometimes the 132 kV cable has a higher impedance than the 150 kV cable, which does not answer logic. The only motive for this is the consideration of the 2D COMSOL model; that is, the overall resistance is "lower" for the 150 kV cable due to the lower number of wires (105 instead of 110) in the external armour and the thinner screens (2.4 mm instead of 2.5 mm on the 132 kV cable and 2.9 mm on the 220 kV cable). Therefore, to conclude this chapter, let us now also see the 2.5D and 3D method’s comparison: Figure 5.19 Cable comparison Positive- & zero-sequence - Zoc & Zsc - 2.5D COMSOL. 5.4 Harmonic impedance comparison 132 kV vs 150 kV vs 220 kV 97 Figure 5.20 Cable comparison Positive- & zero-sequence - Zoc & Zsc - 3D COMSOL. With this, we can conclude this chapter by saying that the best and most accurate results are obtained by the 3D COMSOL method, followed by the 2.5D COMSOL method and, depending on the frequencies, closely followed by our first analytical method, including Bessel’s functions. Of course, we cannot discard the 2D COMSOL method for harmonic impedances, but we can firmly conclude that this method is on the most conventional side and does not show the relevant frequencies where resonance can appear. Still, it comes to mind how accurate sometimes the Bessel function’s method can be, considering the time elapsed to obtain these results. 6 Conclusions This thesis aimed to address the issue of calculating harmonic impedances on undersea AC power cables. This was carried out by giving some theoretical background from classical authors and explaining how difficult it is to calculate regular underground power transmission cables. Also, a classical method using Bessel’s functions was approached and implemented in MATLAB so that we could start with some calculations. After this, the software COMSOL was presented and detailed in deep. Here we could approach 3D cable studies, which gives us a more realistic scenario taking into account both directions of the twisting of armour and conductors and also getting to know how the twisting lay of each element would respond if modified. With that, we were able to study how a more realistic "real" cable would behave when energized to simulate a steady state to give us the possibility to calculate the positiveand zero-sequence series impedances and then with that calculate the harmonic response by considering a distributed parameters power transmission line. After this, in chapter five, we could draw general conclusions when comparing each studied scenario with COMSOL. The ones I found the most relevant are: • Despite its complexity, Bessel’s functions are still one of the best options for cable impedance calculations, and one is not equipped with the best technology around. It will not be the most accurate, but it will give a decent approximation. • Although 3D COMSOL Juan Carlos del Pino’s method, detailed in [ 8 ], is a step forward given how thin his cable model is, it still requires either a good amount of hardware power or takes too much time for one person without any university facilities to work with. It is also essential to remember that COMSOL is not yet optimized for proper CPU usage, as explained in chapter four since the software uses under 60% of the power the PC system offers. • The best compromise would be going for the 2.5D COMSOL method since it gives the most accurate response when compared with the 3D COMSOL method but uses much less time and resources than this one. • Although a specific study aims to be on the safe side by estimating the worst response possible for a submarine cable, I would discard the 2D COMSOL method since not only does it give a response quite distant from the actual measurement, but also there is a specific frequency displacement over the frequency domain that misrepresents the relevant harmonic response of the cable, which is not to be ignored. This could lead to expecting resonance at frequencies where there are none, and vice-versa. • This thesis was not aiming to improve available 3D calculation methods with COMSOL. One of the main aims was to compare and demonstrate that, although antiquated, Bessel’s functions implemented (in my case) with MATLAB is an option not to be easily underestimated. 99 100 Capítulo 6. Conclusions • Mesh adaptation within COMSOL was achievable only by adding the frequency-dependent equations to each conductive material to simplify the mesh calculations when the frequency gets higher. Without this (COMSOL’s default cable analyzer), the simulations would have taken much more time (in my case, this needed too much RAM, therefore crashed without ending the simulation, but if I had had enough RAM in my PC system, it would have taken double the time elapsed than with Juan Carlos del Pino’s layout). • Evaluating the errors when compared with the 3D COMSOL model offers a fair idea of how close/far each method is from this one. The bigger the voltage of the cable is, the less reliable the Bessel function’s method is, and the fewer error results from the 2.5D COMSOL method. Also, for future lines of work towards this aim or similar, I would suggest some other points of view to take into consideration: • I would go further into COMSOL since it is a powerful software that can still be further exploited to achieve a faster simulation result. • Experiment with more considerable cable distances and public the results, so other colleagues from international universities with better facilities can prove if these are feasible. • With COMSOL, it would have also been interesting to work with stranded conductors (phases) instead of solid ones since, theoretically, stranded conductors do not suffer from skin effect due to the smaller diameter of each wire. • Although it is directly related, a thesis about the relation between harmonic impedance and active power losses would also be engaging, in my opinion. • A thesis more focused on the optic fibre cable within power transmission lines would also be interesting. • It would also be most interesting if other students took these results published in this thesis and continued with some transitory models with DigSilent or PSCAD; although it is known that this two software are not open-sourced, therefore it would be expensive for a student to work with these. This way, the appearances of unwanted overvoltages or resonances at specific frequencies could be better approached in a time-based simulation scenario when disturbing the system with a high-frequency signal. • Finally, it would be interesting to read some thesis or line of work where offshore power transmission cables and battery energy storage could be simulated in a unique scenario. For that, I am again afraid it would be necessary a more expensive software such as DigSilent or PSCAD. Apéndice A MATLAB codes The used MATLAB-code for calculation of series-impedance, sequence-impedance and harmonic impedance will be shown. A.1 Three-core submarine impedance with external stranded-armour for symmetric trefoil layout Código A.1 Three-core submarine cable with solid screens and pipe. %Three-phase Pipe cable clear all %Datos Rt=50.23;%mm Na=110;%Número de hilos de la armadura Da=204;%mm da=5.6;%mm le=100e3;%100 km longitud Rc=Rt/1000; rho_cobre=1.72e-8;%Ohm*m rho_plomo=2.14e-7;%Ohm*m rho_acero=1.38e-7;%Ohm*m rho_earth=100;%Ohm*m rho_agua=0.2;%Ohm*m mu_r_acero=300; mu_r_cobre=1; mu_r_plomo=1; mu_0=pi*4e-7; epsilon_ins=2.5; d=0.087; %Pipe R1=0.01725; R2=0.038; R3=0.04125; Ra=R1+0.0015; Rb=R2+0.0015; 101 List of Tables 2.1 Experimental values of ksand kp11 2.2 132 kV Cable information [9] 45 2.3 Comparison between 132 kV cable layouts at 50 Hz 52 2.4 Comparison between 132 kV cable layouts at 100 Hz 52 2.5 Comparison between 132 kV cable layouts at 1000 Hz 52 4.1 Time comparison 1 73 4.2 Time comparison 2 77 5.1 Cables information 83 5.2 Time elapsed - Series Sequence-impedance 132 kV Cable 87 5.3 Error (%) Methods vs. 3D-COMSOL for 132 kV Cable 88 5.4 Time comparison 150 kV Cable 90 5.5 Error (%) Methods vs. 3D-COMSOL for 150 kV Cable 91 5.6 Time comparison 220 kV Cable 93 5.7 Error (%) Methods vs. 3D-COMSOL for 220 kV Cable 94 5.8 Error (%) Methods vs. 3D-COMSOL at relevant frequencies for 132 kV Cable 94 5.9 Error (%) Methods vs. 3D-COMSOL at relevant frequencies for 150 kV Cable 94 5.10 Error (%) Methods vs. 3D-COMSOL at relevant frequencies for 220 kV Cable 95 109 Índice de Códigos A.1 Three-core submarine cable with solid screens and pipe 101 111 Bibliografía [1] IEC Standard 60287-2-1:2015, Electric cables—calculation of the current rating—part 2-1: Thermal resistance—calculation of thermal resistance—general, 2015, IEC, Geneva. [2] Y. Baba A. Ametani, N. Nagaoka and T. Ohno, Power system transients: Theory and applications, Boca Raton, FL, USA: CRC, 2013. [3] B. Gustavsen A. Morched and M. Tartibi, A universal model for accurate calculation of electromagnetic transients on overhead lines and underground cables, IEEE Trans. Power Del., vol. 14, no. 3, pp. 1032–1038, Jul. 1999. [4] A. Ametani, A general formulation of impedance and admittance of cables, IEEE transactions on power apparatus and systems, vol. pas-99, no. 3, May/June 1980. [5] Cigre Working Group B1.30, Cable systems electrical characteristics, April, 2013. [6] Cigre Working Group B1.40, Offshore generation cable connections, February 2015, Cigre. [7] Chang Hsin Chien and Richard WG Bucknall, An assessment of the influence of bonding on the harmonic performance of AC XLPE insulated subsea cable transmission system, Journal of Marine Engineering and Technology, No. A8, 2006. [8] J. C. del Pino-López and P. Cruz-Romero, Experimental validation of ultra-shortened 3d finite element models for frequency-domain analyses of three-core armored cables, IEEE Transactions on Power Delivery, Vol. X, NO. X, September 2021. [9] Dr James Pilgrim MIEEE Dr Richard Chippendale, The tony davies high voltage laboratory. deliverable number: 2.0, ref: Ore/15/57, tdhvl ref: 15242-re2-v6. comparison of IEC values with 2.5d FEA results and measurements, proceedings of the CIGRÉ session, 2015. [10] Claus L. Bak Filipe F. Da Silva, Electromagnetic transients in power cables, 2013, Springer. [11] C.L. Fortescue, Method of symmetrical co-ordinates applied to the solution of polyphase networks A.IE.E. trans vol. 37 part ii, 1918. [12] G. Luoni G. Bianchi, Induced currents and losses in single-core submarine cables, IEEE Trans. On Power Apparatus and Systems, Vol. PAS-95, n1, January/February 1976. [13] https://en.wikipedia.org/wiki/Ampacity, Ampacity, April 2021. [14] https://en.wikipedia.org/wiki/Finite_element_method, Finite element method, February 2022. [15] https://en.wikipedia.org/wiki/RLC_circuit, Rlc circuit, December 2021. 113 114 Bibliografía [16] https://en.wikipedia.org/wiki/Submarine_communications_cable, Submarine communications cable, August 2021. [17] J. Holbøll K. H. Jensen I. Arana Aristi, A. H. Nielsen and Sørensen (fratrådt), Switching overvoltages in offshore wind power grids, T. (2011), Kgs. Lyngby: Technical University of Denmark (DTU). [18] M. Hatlo J. C. del Pino-López and P. Cruz-Romero, A 3D parametric analysis of three-core armored power cables series impedance, 2018 International Conference on Smart Energy Systems and Technologies (SEST), 2018. [19] R. Stølan J.J. Bremnes, G. Evenset, Power loss and inductance of steel armoured multi-core cables: Comparison of IEC values with 2.5d FEA results and measurements, proceedings of the CIGRÉ session, 2010. [20] C.S. Hansen L. Martedal T. Ebdrup M. Baù, N. Viafora and F. da Silva, Steady state modelling of three-core wire armoured submarine cables: Power losses and ampacity estimation based on fem and IEC, September 2016. [21] J. R. Marti, Accurate modelling of frequency-dependent transmission lines in electromagnetic transient simulations, IEEE Trans. Power App. Syst., vol. PAS-101, no. 1, pp. 147–157, Jan. 1982. [22] F. Pollaczek, On the field produced by an infinitely long wire carrying alternating current, Elektrische Nachrichtentech., vol. 3, pp. 339–359, 1926. [23] Manuel Reta-Hernández, Electric power engineering handbook (second edition), chapter 13, Taylor & Francis Group, LLC, 2006. [24] S.A. Schelkunoff, The electromagnetic theory of coaxial transmission lines and cylindrical shields, bell system technical journal 1934, n°13. [25] COMSOL Multiphysics ® v.5.6. (2021), Comsol ab: Stockholm, sweden. [online]., Available: http://www.comsol.com. [26] L.M. Wedepohl and D.J. Wilcox, Transient analysis of underground power-transmission systems - system-model and wave propagation characteristics, proc. I.E.E, vol 120, February 1973.