高压直流电缆

1 Abstract—The paper deals with harmonic analysis of HVDC submarine cables. The longitudinal series impedance matrix is computed with reference to armoured cables. The results are also compared with Finite Element models showing a very good agreement. This investigation can be considered the first step of more detailed further analysis about the propagation of harmonics on the d.c. and a.c. sides. Heretofore the topic seems to be almost neglected in technical literature so this paper investigates the harmonic behaviour of HVDC cable lines. Index Terms— HVDC cables, Harmonic analysis, Armoured cables, Impedance calculation. I. INTRODUCTION Projects and installations of off-shore wind farms are becoming more and more widespread, drawing the Power Electric Society attention to the HVDC option. The possible resonances and propagation of harmonic into the system need a very accurate modeling of submarine (also land) HVDC cables. The HVDC submarine cables are rather complex systems (multilayered structures) with, beyond the core conductor, a metallic shield and armour which is in contact with the sea water. Since the sheath and armour are multi- point bonded and earthed, this implies that induced currents circulate into them. For mass-impregnated paper insulated cables, also the presence of a metallic reinforcement must be taken into account. The following configurations have been tackled: ¾ Monopolar configuration with sea return; ¾ Monopolar one with metallic return (placed at great distance or in bundle configuration); ¾ Bipolar one. These models can be usefully applied also to single-core a.c. submarine cables (50 or 60 Hz). For the three-core a.c. cables the procedures must be suitably adapted and will be subject of further investigations. In section II the typologies of installation and the HVDC cable characteristics are defined; in section III the theoretical backgrounds of the matrix procedure are presented including the matrix passage from loop quantities to the usual series impedences and the elimination of grounded conductors; in section IV the computation results are shown and compared with a FE model. R. Benato and M. Forzan are with Department of Electrical Engineering, University of Padova, Italy Via Gradenigo, 6/A 35131. (e-mail: roberto.benato@unipd.it) M. Marelli, A. Orini and E. Zaccone are with Prysmian S.r.l., Milan, Italy, Via Chiese, 72 20126. II. HVDC CABLE INSTALLATION FEATURES A. HVDC Submarine installations The installation features of submarine HVDC cables are rather diverse. In general, the cables are buried at about 1,5 m under the seabed surface. The metallic return (if any) can be placed either in contact with the pole cable (bundle configuration) or very distant (up to 300 m) in order to avoid that trawling or anchoring activities can damage both. The sea water electrical resistivity can range between 0,2 Ω⋅m and 0,5 Ω⋅m depending upon the salt content. A possible range for the seabed electrical resistivity is 1÷10 Ω⋅m. Figure 1 Submarine HVDC cables (not to scale) in monopolar configuration with metallic return B. HVDC cables characteristics There are three great categories of HVDC cables: ¾ Mass-Impregnated (MI): Insulated with special paper, impregnated with high viscosity compound; ¾ Self-Contained Fluid Filled (SCFF): Insulated with special paper, impregnated with low viscosity oil; ¾ Extruded: Insulated with extruded polyethylene- based compound. MI Cables are the most used; they have been in service for more than 40 years and have been proven to be highly reliable. At present they are used for voltage levels up to 500 kV d.c. and conductor sizes are typically up to 2500 mm2. Self Contained Fluid-Filled Cables are used for very high voltages (qualified for 600 kV d.c.), for short connections and for voltage levels up to 500 kV d.c. Conductor sizes are up to 3000 mm2. Extruded Cables for HVDC applications are still under development; at present they are used for relatively low voltage levels (up to 200 kV d.c.), mainly associated with Voltage Source Converters (VSC), that could reverse the power flow without reversing the polarity of the cable. Harmonic Behaviour of HVDC cables Roberto Benato Member IEEE, Michele Forzan, Marco Marelli, Ambrogio Orini, Ernesto Zaccone Member IEEE ≈1,5 m ≈0,15÷300 m ρsea=0,2÷0,5 Ω⋅m ρseabed=1÷10 Ω⋅m 978-1-4244-6547-7/10/$26.00 © 2010 IEEE 2 Fig. 2 shows an HVDC armoured MI-cable: it is composed of a core conductor, a lead-alloy sheath, a steel reinforcement and a steel-wire armour. Consequently, it is a four metallic layered cable. In the monopolar configuration with metallic return, the return cable is usually XLPE-insulated: it does not need any metallic reinforcement so that it is a three metallic layered cable. Figure 2 MI-paper insulated HVDC cable III. THEORETICAL BACKGROUNDS FOR THE COMPUTATION OF LONGITUDINAL SERIES IMPEDANCE MATRIX The theory of coaxial cylindrical conductors (see fig. 3) is well known in the technical literature thanks to the fundamental contribution of Schelkunhoff [1]. Its application allows computing the longitudinal series impedance matrix of the loop circuits shown in fig. 4. These loop circuits are formed of: ¾ Loop 1 (L1): core conductor c and sheath s as return; ¾ Loop 2 (L2): sheath s and armour a as return; ¾ Loop 3 (L3): armour a and sea water as return. The voltages (uL1, uL2 and uL3) and currents (iL1, iL2 and iL3) of these loops are clearly shown in fig. 4: it considers the loop currents as flowing in the xth conductor and returning into the (x+1)th conductor (if x=armour, x+1 is the sea water). Figure 4 also shows the electrical quantities (currents and voltages) pertaining to the classical, usual phase-to-ground voltages (uc, us and ua) and currents injected into the conductor (ic, is and ia). Figure 3 Composition of the three-layered single-core cable: core, sheath, armour and insulating materials Figure 4 Specification of the loop (grey) and "usual" (black) currents and voltages for the three-conductor cable In the example of fig. 4, the single-core cable has three conductors (four if considering the sea water), three insulations and three loops. For generality purpose, the exposition can be referred to n loops (and n+1 conductors included the sea return path). The longitudinal series impedance matrix of the loops ZL (order n×n) has a tridiagonal structure with the self impedances of the loops laying in its main diagonal and the mutual impedances between the loops located in the off- diagonal elements as shown in fig. 5. Figure 5 The tridiagonal loop matrix ZL In general, the xth loop has the following self-impedance: 1x,inc1x/x,insx,outcL,L zzzz xx +−+− ++= (1) For the last loop (i.e. x ≡ n), it is: 1n,inc1n/n,insn,outcL,L zzzz nn +−+− ++= (2) where n+1 conductor is the sea water. The definitions of the abovementioned impedances are: Copper core d Semiconducting MI-paper insulation Semiconducting Lead-alloy sheath PE sheath Steel reinforcement Bedding Steel wire armour Serving PP yarn iL1 iL2 Insulation c/s Insulation s/a CORE c SHEATH s uL1 uL2 ic is ia uc us LOOP 1 LOOP 2 axis of the cable iL3 LOOP 3 ua SEA WATER Insulation a/sea ARMOUR a uL3 zLx,Lx zLx,Lx+1 0 0 0 zLx+1,Lx ... … 0 0 0 … … 0 0 0 … … zLn-1,Ln 0 0 zLn,Ln-1 zLn,Ln Lx Lx+1 . . . Ln-1 Ln Lx Lx+1 . . . Ln-1 Ln 0 ZL= x WATER core insulation core/sheath sheath insulation sheath/armour armour insulation armour/sea (if any) 3 zc-out,x = per unit length internal impedance of the xth coaxial conductor (subscript c-) with the current returning in the (x+1)th conductor (outer conductor whence the subscript out); zins,x/x+1 = per unit length impedance of the insulation (subscript ins) between the conductor xth and (x+1)th; zc-in,x+1 = per unit length internal impedance of the (x+1)th coaxial conductor (subscript c-) with the current returning in the xth conductor (inner conductor whence the subscript in); zc-in,n+1 = zself_sea = per unit length self sea-return impedance of the nth conductor. As is well known, the skin effect is considered by means of the expression of zc-out,x whose real part is the a.c. conductor resistance. For the off-diagonal mutual impedances between the loop Lx and the Lx-1 one of the matrix ZL the following relations are valid: x,mcL,LL,L zzz x1x1xx −−== −− (3) where the minus sign takes into account the opposing directions of the loop currents (e.g. iL2 is negative in the loop L1). Between Lx and Lx+2 loops there are no common branches so that the coupling mutual impedance is zero. In order to particularize this general exposition, the single- core cable of fig. 3 can be considered so that x=c, s, a where c=core, s=sheath and a=armour. It has: 321s,incs/c,insc,outcL,L zzzzzzz 11 ++=++= −− (4) 765a,inca/sheath,inss,outcL,L zzzzzzz 22 ++=++= −− (5) 11109sea_selfsea/a,insa,outcL,L zzzzzzz 33 ++=++= − (6) where the abovementioned impedances have been re-named z1,..,z11. The impedance zc-out (i.e. z1, z5 and z9 in (4)÷(6)) can be computed as: [ ])r(I)r(K)r(K)r(I Hr2 z in1ext0in1ext0 ext outc ⋅⋅⋅+⋅⋅⋅⋅⋅ ⋅=− σσσσπ σρ ; (7) whereas the impedance (z3 and z7) are given by: [ ])r(I)r(K)r(K)r(I Hr2 z ext1in0ext1in0 in inc ⋅⋅⋅+⋅⋅⋅⋅⋅ ⋅=− σσσσπ σρ ; (8) where: ω = 2πf = angular frequency [rad/s]; ρωμσ j= = reciprocal of the complex depth of penetration [1/m] of the conductor with absolute magnetic permeability μ [H/m] and electric resistivity ρ [Ω⋅m]; rin = inner radius of the considered conductor [m]; rext = outer radius of the considered conductor [m]; I0(x), I1(x) = first kind modified Bessel functions of order 0 and 1 respectively; K0(x), K1(x) = second kind modified Bessel functions of order 0 and 1 respectively; H = I1(σ⋅rext)⋅K1(σ⋅rin) - I1(σ⋅rin)⋅K1(σ⋅rext). If the core conductor is not hollow ( 0rin = ), eq. (7) becomes: )r(Ir2 )r(Iz ext1ext ext0 1 ⋅⋅⋅ ⋅⋅⋅= σπ σσρ . The impedances of insulating layers zins (z2, z6 and z10) are computed by: ⎟⎟⎠ ⎞ ⎜⎜⎝ ⎛⋅= in_ins ext_insins ins r r ln 2 jz π μω (9) where: μins = absolute magnetic permeability of insulating material (usually ≅ 4⋅π 10-7[H/m]); rins_ext = outer radius of the insulating material [m]; rins_in = inner radius of the insulating material [m]. It is worth reminding that the insulating layer between armour and sea is usually missing, and the armour is in contact with the sea water so that z10=0. With regard to the off-diagonal elements of the matrix ZL, the per unit length mutual impedance between L1 and L2 is given by; 4m_sL,LL,L zzzz 1221 =−== whereas for L2 and L3 the mutual impedance is: 8m_aL,LL,L zzzz 2332 =−== where: Hrr2 z extin x,mc ⋅=− π ρ . (10) As aforementioned zL1,L3=zL3,L1=0. The computation of the self-impedance taking into account the sea water as return (z11) would be rather complex also because the cables are laid in the sea-bed having a resistivity different from that of the sea itself (see fig. 1). It is worth remembering that exact formulae of the self earth-return impedance have been first derived from Pollaczek [2] in the hypothesis of cable buried in a semi-infinite earth. In the following, as in [3], the hypothesis of "Infinite Sea Model" (or Infinite Earth Model) is assumed. As it will be demonstrated also with the comparison of FEM, it is a really strong assumption. Attention 4 must be paid, since the cable can be considered as surrounded by an infinite sea in all directions around it, when the penetration depth is much less than the burial depth (measured by the sea surface): ]m[depthburial f 5032d )Hz( )m(sea sea sea <<≅= Ω ρ σ This approximation could not be valid when the cable is installed in very swallow water. The impedance z11 can be inferred by (12) under the hypothesis that the sea is a tubular conductor with infinite radius (rext→∞) and that the inner radius is equal to the cable external one (see fig. 6). If the armour is in contact with the water, the cable outer radius coincides with the armour external radius (rout_cable=re_armor). Since [ ] 0)rσ(Klim ext1 rext =⋅ ∞→ the relation (8) becomes: [ ][ ] )r(K )r(K r2)r(K)r(I)r(K)r(I )r(I)r(K)r(K)r(I r2 z in1 in0 inext1in1in1ext1 ext1in0ext1in0 in inc ⋅ ⋅⋅⋅ ⋅=⋅⋅⋅−⋅⋅⋅ ⋅⋅⋅+⋅⋅⋅ ⋅ ⋅=− σ σ π σρ σσσσ σσσσ π σρ So that the self-sea impedance z11 is equal to: 11 cable_outsea1 cable_outsea0 cable_out seasea sea_self z)r(K )r(K r2 z = ⎥⎥⎦ ⎤ ⎢⎢⎣ ⎡ ⋅ ⋅ ⋅= σ σ π σρ (11) Figure 6 Infinite sea model Finally, the series impedance of the three loops of the single- core cable of fig. 3 is given by: ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ ++− −++− −++ = ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ = 111098 87654 4321 ,LL,LL ,LL,LL,LL ,LL,LL zzzz0 zzzzz 0zzzz zz0 zzz 0zz 3323 322212 2111 LZ .(12) The longitudinal series impedance matrix Z of the single-core cable of fig. 3 is given by: ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ = a,as,ac,a a,ss,sc,s a,cs,cc,c zzz zzz zzz Z . (13) In the scientific literature, there are cumbersome algebraic formulations in order to express the elements of series impedance matrix Z starting from those of the loop matrix ZL i.e. 1110987654321c,c zzzz2zzzz2zzzz +++−+++−++= ; 111098765s,s zzzz2zzzz +++−++= ; 1110987654c,ss,c zzzz2zzzzzz +++−+++−== ; 111098s,aa,sc,aa,c zzzzzzzz +++−==== ; 11109a,a zzzz ++= . In the next subsection, a novel and elegant matrix algorithm will be developed so avoiding the abovementioned relations. A. Matrix procedures to pass from ZL to Z The passage from the loop impedance matrix ZL to the usual longitudinal series impedance matrix Z can be derived by making extensive use of the matrix formalism rather than of the abovementioned cumbersome algebraic expressions. In fact, between the loop voltages (uL1, uL2 and uL3) and the voltages with respect to ground (uc, us and ua), the following obvious relations can be written: scL uuu 1 −= asL uuu 2 −= aL uu 3 = . Analogously, between loop and conductor currents, it can be written: 1Lc ii = 12 LLs iii −= 23 LLa iii −= Both the relation sets can be elegantly synthesized in the matrix forms: UTU L ′= and LITI = (14) where: , i i i 3 2 1 L L L ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎣ ⎡ =LI , u u u 3 2 1 L L L ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎣ ⎡ =LU , i i i a s c ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ =I . u u u a s c ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ =U and the real matrix T (T' is its transpose) is given by: ⎥⎥ ⎥ ⎦ ⎤ ⎢⎢ ⎢ ⎣ ⎡ − −= 110 011 001 T . The longitudinal series loop impedance ZL can be expressed by the obvious: LL L IZU ⋅=∂ ∂− x (15) By substituting eqs. (14) in (15), it yields: ITZUT 1L −⋅=∂ ⋅′∂− x = ( )[ ] ITZTU 1L1 ⋅⋅′=∂∂− −−x (16) rin rext →∞ sea water 5 where the matrix Z can be computed by means of loop matrix ZL. 1 L 1' TZ)(TZ −− ⋅⋅= [Ω/m] (17) The great advantage of the matrix formalism, as expressed by G. Kron in [4], is the possibility of generalization from one single-core cable to n single-core ones. For example by considering two single-core cables as shown in fig. 7, the eq. (17) is still true but with new meanings of the involved matrices i.e. 110000 011000 001000 000110 000011 000001 − − − − =T and ZL: where the new mutual impedance between the loop L3 of P and the loop L3 of R have not been yet defined. Figure 7 Two single-core cables: pole (P) and metallic return (R) or second pole All the other loop mutual impedances between P and R are zeroed since the loop L1 and L2 in each cable do not generate magnetic field external to the cables themselves (e.g. the magnetic field generated by iL1 in the core is zeroed by the returning current iL1 in the sheath). The computation of these mutual impedances can be performed always with the Infinite Sea Model (for the demonstration see [5]), namely: zm-PR=zm-RP= )r(K)r(Krr2 )d(K R_extsea1P_extsea1R_extP_ext ijsea0sea ⋅⋅⋅⋅⋅ ⋅⋅ σσπ σρ (18) where: dij=spacing between cable axes [m]; rext_P=outer radius of the P cable [m]; rext_R=outer radius of the R cable [m]. B. Elimination of grounded conductors by a novel technique The technique of elimination of grounded conductors is well known in the scientific literature. It is only valid in the hypothesis that the voltages along the entire length of the conductors to be eliminated are zero. It is worth remembering that for HVDC cables, since the sheath and the armour are bonded and earthed at both ends (bonded also discretely at intermediate intervals every 5÷10 km or separated by a continuous semiconducting layer), and the armour is always in contact with the sea water, their voltages can be considered negligible along the entire cable length. The present procedure can be performed either on one single-core cable or n single- core cables. Once the series impedance matrix Z has been computed, its inverse matrix Y must be firstly obtained as shown in fig. 8. The advantage of the matrix Y is that it can be directly multiplied by the vector ⎥⎥ ⎥⎥ ⎥⎥ ⎥⎥ ⎦ ⎤ ⎢⎢ ⎢⎢ ⎢⎢ ⎢⎢ ⎣ ⎡ = 0 0 u 0 0 u R P c c U so that it is equivalent to the element extraction of fig. 8. Figure 8 Elimination of grounded conductors (s, a) for two single-core cables zL1,L1-P z L1,L2-P 0 z L2,L1-P z L2,L2-P z L2,L3-P z L3,L2-P z L3,L3-P zm-PR 0 0 0 0 0 0 0 0 0 zm-RP 0 0 0 0 0 0 0 0 zL1,L1-R zL1,L2-R 0 zL2,L1-R zL2,L2-R zL2,L3-R zL3,L2-R zL3,L3-R 0 P R P R ZL= yc,c-P y c,s-P y c,a-P y s,c-P y s,s-P y s,a-P y a,s-P y a,a-P y a,c-P ycP,cR ycP,sR ycP,aR yc,c-R yc,s-R yc,a-R ys,c-R ys,s-R ys,a-R ya,s-R ya,a-R ya,c-R P R P R Y= ysP,cR ysP,sR ysP,aR yaP,cR yaP,sR yaP,aR ycR,cP ycR,sP ycR,aP ysR,cP ysR,sP ysR,aP yaR,cP yaR,sP yaR,aP Y=Z-1 YRED yc,c-P ycP,cR ycR,cP yc,c-R c s a s a c c c s a s a iL3 - P iL3 - R P R dij 6 Therefore, the elements not to be eliminated (in grey in fig. 8) can be extracted from this matrix and form a new reduced matrix YRED. In order to obtain the final reduced impedance ZRED without the grounded conductors (but taking into account their electrical effects), it is sufficient to invert YRED namely: ZRED= 1REDY − . For the case of one single-core cable, this procedure gives obviously the same result of the cumbersome procedures available in literature. C. Steel wire armour modelling The armour is often composed of n steel wires with a given diameter Φ: it is also stranded with a laying pitch p. The armour can be modelled as a tubular conductor having the inner radius equal to that of armour and the outer radius computed so that the cross-section of tubular conductor is equal to that of all wires S= 4 n 2Φπ⋅ . The per unit length resistance is given by: ⎥⎦ ⎤⎢⎣ ⎡⋅⋅⋅ ⋅= kmcos 1 n 4r 2 armour dc,armour Ω δΦπ ρ where δ=laying angle (see f