# Comparison of Power Título Transmission Line Models with Finite Elements

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1 CEPEL Comparison of Power Título Transmission do trabalho Line Models with Finite Elements Carlos K. C. Arruda, Anny A. Silveira, Fernando C. Dart Autor/apresentador Área Departamento de Linhas e Estações

2 Introduction Extra-high voltage transmission lines (TLs) Ultra-high voltage TLs (> 765 kv AC or > 600 kv DC) Unknown issues Experimental lines - costly Unconventional TLs Project optimization for existing voltage levels Recapacitation/ uprating High Surge Impedance Line - HSIL (a.k.a. Linha de potência natural elevada LPNE)

3 Introduction Some project criteria Electric field at conductor surface Corona effect Current density Skin effect Electromagnetic field at ground level Environmental restrictions Impedance and admittance (balanced and unbalanced) circuit model: load flow, fault calculations etc. Analytical models

4 Introduction HSIL test line - FURNAS

5 Objectives Compare some aspects regarding TL design using analytical models and Finite Elements (COMSOL) Advantagens and limitations shall be discussed for both methods

6 Practical aspects in high voltage transmission lines Governing equations Basic circuit relations Δvv 1 zz 11 zz 12 zz 13 Δvv 2 = zz 21 zz 22 zz 23 Δvv 3 zz 31 zz 32 zz 33 Skin effect Electric field ii 1 ii 2 ii3 ddvv dddd = ZZ ii = (RR + jjωωll)ii dddd dddd = jjjjcc vv q Z i = jωμ K 1 (ρ 0 )I 0 (ρ 1 ) + K 0 (ρ 1 )I 1 (ρ 0 ) 2πr 0 I 1 (ρ 1 )K 1 (ρ 0 ) I 1 (ρ 0 )K 1 (ρ 1 ) vv 1 pp 11 pp 12 pp 13 qq 1 vv 2 = pp 21 pp 22 pp 23 qq 2 vv 3 pp 31 pp 32 pp 33 qq 3 = P 1 v = CC vv E kx = q rx + jq ix 2πε x m x m2 + (h k h m ) 2 x m2 x m + (h k + h m ) 2 E ky = q rx + jq ix 2πε h m h k x m2 + (h k h m ) 2 h m + h k + (h k + h m ) 2 x m2 E x = E kx k E y = E ky k = E rx + je ix = E ry + je iy

7 Proposed test case Nominal voltage level Current per phase Conductor 500 kv, phase-phase RMS 1000 A Phase: ACSR Tern Ground wire: EHS 3/8" Number of conductors 6 per phase, 2 ground wires Bundle radius (ellipsis major/ minor axis) Bundle asymmetry (relation between axis) Height at tower Outer phases: 1.4/ 1.12 m Center phase: 0.6/ 0.48 m 0.8 Outer phases: 29 m Center phase: 26.5 m Ground wires: 39 m Sag at nominal conditions Distance between phases (bundle centers) Phases: m Ground wire: 7.64 m 6.5 m Test case profile, conductor positions at tower (color) and mid-span (gray)

8 Modeling Some peculiarities: 2D approximation Valid for studies at midspan higher electric fields at conductors and ground level Three phase system Simultaneous interaction Three leg coil in magnetic field model? Unbalanced system return to ground ACSR cables (aluminum wires with steel core) Magnetic interaction, relevant for Joule losses Domain dimensions From 1 cm (cable diameter) up to 100 m (cable height or right-ofway width) or 100 km (line length) Infinite domain consideration Ground resistivity Infinite domain with current return?

9 Modeling considerations Three-phase balanced line

10 Modeling considerations Three-phase unbalanced line/ single phase

11 Modeling considerations Domain-circuit interaction A TL, by definition, connects two systems The ground node is not in the same potential, like case (b) numerical methods demands exactly one reference node Case (a) is acceptable for short lines or balanced system (single phase reduction) Proposed case (c) should consider correctly the earth return (good for fault simulations) not supported in Comsol

12 Modeling COMSOL Configuration Infinite elements (open domain) Single coil representing each current injection, grouped by bundles, A return coil for ground, for unbalanced loads, Separation of real ground bounded by a ideal ground underneath a infinite element layer, Representation of steel core of ACSR cables for proper skin effect observation, Meshing of cables and ground surface, layer refinement in one conductor. Using the same geometry, it's possible to build a number of meshes, specific for each study Study outputs Admittance matrix ( electrostatics, frequency domain) Electric field at conductor ( electrostatics, frequency domain) Impedance (magnetic fields, frequency domain) EM fields at ground level (magnetic fields and electrostatics, time dependent and frequency domain)

13 Meshing 2D plot of skin effect in ACSR cable (current density, A/mm²), 100 khz

14 Results Skin effect Current density profile at phase conductor for selected frequencies [A/mm²] Equivalent resistance for an ACSR cable frequency sweep [Ω/m]

15 Results Capacitance Matrix Capacitance Matrix, analytic [nf/km] Capacitance Matrix, FEM [nf/km]

16 Results Electric field at Ground Level Parametric plot of electric field near ground level [kv/m], at points below each phase Electric field profile near ground [kv/m]

17 Results EM Field near conductors Electric field distribution for a commercial 18 bundle (45.7 cm) Electric field intensity at outer phase [kv/cm]

18 Results EM Field near conductors Electric field around outer phase conductors (filled line) and center phase (dashed line)

19 Results Magnetic Field Magnetic lines for 60 Hz current with 10 Ω/m soil, maximum at center phase Magnetic lines for 100 khz current, same conditions

20 Conclusions Meshing is a relevant aspect, which must be made with caution. The boundary layers option is of great value for observe the skin effect. The work arises some questions regarding established premises in both sides, FEM and analytical, such the homogeneous ground in unbalanced systems and the inductance calculation. The authors suggests further development in the circuit models inside COMSOL: Two uncoupled circuits (no common ground/ zero port) connected through a 3d model with ports 2d model, but with distinguishable in/ out port

21 Future research Precise modeling of ACSR conductor, including each wire, in 2D and 3D, using helicoidal objects, to investigate the transformer effect, Interaction with two independent circuit models and a FEM domain, including considerations about remote ground reference, Extrusion of 2D solution for a 3D model, as an analogy of 3D revolve, 2D magnetic model with assumptions of in and out terminals for a same domain.

22 References 1. Guimarães, R. P., Miller, M. D. Brazilian Transmission System: A Race for the Future. Electrical Transmission and Substation Structures, (2012). 2. Régis Jr, O., et al. Expanded Bundle Technique: The Application of HSIL TL Concept to Increase the Capacity of Overhead Lines, CIGRÉ Paper (1998). 3. Araújo, M. C., et al. Eletronorte and the Challenge of Long-distance Transmission in Brazil, CIGRÉ Paper B2-111 (2008). 4. Stevenson, W. D. Elements of Power System Analysis, 4 th Ed. (1982). 5. EPRI AC Transmission Line Reference Book 200 kv and Above, 3 rd Ed., EPRI, Palo Alto, CA (2005). 6. Carson, J. R., et al. Wave Propagation in Overhead Wires with Ground Return, Bell System Technical Journal 5, (1926). 7. Deri, A., Tevan, G., Semlyen, A. and Castanheira, A. The Complex Ground Return Plane a Simplified Model for Homogeneous and Multi-layer Earth Return, IEEE Transactions on Power Apparatus and Systems, (1981). 8. Lorenzen, H., Timmerberg, J., Mylvaganam, S. Calculation of Cable Parameters for Different Cable Shapes, COMSOL Conference, Hannover (2008). 9. Pelster, W. Ampacity simulation of a high voltage cable to connecting off shore wind farms, COMSOL Conference, Milan (2012). 10. EL-Fouly, T., et al. Power Transmission Lines Generated Electric and Magnetic Fields Calculations, COMSOL Multiphysics User's Conference, Boston (2005).

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