Impedance and Admittance Calculations of a Three- Core Power Cable by the Finite Element Method

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1 Impedane and Admittane Calulations of a Three- Core Power Cable by the Finite Element Method Angelo. A. Hafner, Mauriio V. Ferreira da Lu, Walter P. Carpes Jr. Abstrat--The analytial modeling of a three-ore able system is hallenging beause of the non-onentri onfiguration of the omponents involved. Given these limitations, a 2D finite element modeling of the able is developed in order to obtain the values of the self, mutual and sequene impedanes and admittanes. To alulate the series impedane, a magneti vetor potential magnetodynami formulation is used and for the alulation of the parallel admittane, an eletri salar potential eletrostati formulation is applied. By alulating the series impedane of the inner ables, the influene of the mutual impedanes in all metalli elements involved is shown. The methodology is applied to a typial able of 300 mm² - 18/30 kv. The numerial results are ompared with analytial ones and with values supplied by the manufaturer for eah phase, validating the numerial modeling. Keywords: Submarine power able, 2D finite element method, impedane, admittane. 1 I. NOMENCLATURES I 0 ( ) Bessel funtions of first kind and order 0. I 1 ( ) Bessel funtions of first kind and order 1. Eletri ondutivity, in S/m. Eletri resistivity, in m. Eletri permittivity, in F/m. Magneti permeability, in H/m. Magneti relutivity, in m/h. SC Semiondutor M Matrix omposed by matrixes. m Matrix omposed only by salars. V v Vetor. (.,. ) Volume integral in of produts of salar or vetor fields. <.,. > Surfae integral on of produts of salar or vetor fields. Im(. ) Funtion that returns only the imaginary part of a omplex number. Eletrial angular frequeny. FEM Finite Element Method. Angelo A. Hafner is PhD student at the Department of Eletrial Engineering, Federal University of Santa Catarina, Florianópolis, Brail ( angelo.hafner@posgrad.ufs.br) Mauriio V. Ferreira da Lu is Professor at the Department of Eletrial Engineering, Federal University of Santa Catarina, Florianópolis, Brail ( mauriio.lu@ufs.br) Walter P. Carpes Jr is Professor at the Department of Eletrial Engineering, Federal University of Santa Catarina, Florianópolis, Brail ( walter.arpes@ufs.br) Paper submitted to the International Conferene on Power Systems Transients (IPST2015) in Cavtat, Croatia June 15-18, 2015 II. INTRODUCTION HE expansion of submarine transmission systems Trepresents a major trend due to the growth of the oil and offshore wind energy industry. The deployment of these systems at large distanes from the shore and in deep water requires kilometri strethes of submarine power ables. Fed equipment or systems, as well as ables, need to be adequately proteted in ase of short iruits, overloads and transients. An aurate able model is needed to aurately represent the waveforms of voltage and urrent on the load and the transmission line providing tehnial support for the hoie of the most suitable protetion to be adopted for eah situation. When one onsiders the able as single-ore, the phases distributed impedanes and admittanes of the able for a ertain range of frequenies an be alulated analytially applying lassial analytial formulae ((4), (6), and (11)). However, in three-ore ables, even at 50/60 H, the following aspets should be taken into aount when modeling: (i) proximity effet generated by the urrents of the entral ondutor; and (ii) urrent indued in the sheath and its effets on the entral ondutor impedane [1]. The non-onentri onfiguration of the trefoil formation (Fig 1) hampers the analytial modeling. Thus, a 2D finite element model is developed to obtain the values of the able series impedane and parallel admittane. Some studies about able modeling are presented in [1-6] and [16-22]. III. BASIC CHARACTERISTICS OF A SUBMARINE POWER CABLE The physial onstitution of submarine power ables is very similar to underground power ables. The main differene is that in the first, there are additional protetions for water entry (Fig. 1 and Table 1). In this setion, we briefly desribe the onstituent parts of the submarine power able as well as the aspets related to the alulation of its series impedane. Eah part of the inner able is desribed in Table 1. Small variations may our from one manufaturer to another. The able of this study is omposed of a set of three power inner ables in trefoil formation, as shown in Fig. 1. Parts 10-11, and onsist of insulating material, ondutor, and insulating, respetively. The ondutive layer, alled armor, has the main funtion of mehanially proteting the set.

2 where s is the original insulation permittivity, and a and b are the inner and outer radii of the insulation, respetively. Fig. 1. Set of inner ables in trefoil formation of a three-ore submarine power able. TABLE 1 PARTS OF A POWER INNER CABLE Item Component Material 1 Core Copper Stranded Wires 2 Water-bloking tape Humidity absorber SC tape 3 Condutor Shield SC tape 4 Insulation XLPE 5 Insulation Shield Humidity absorber SC tape 6 Water-bloking tape Humidity absorber SC tape 7 Sheath Copper wires 8 Water-bloking tape Humidity absorber SC tape 9 Jaket Polyethylene IV. HOMOGENIZATION OF THE CORE Beause the materials are omposed of several parts, homogeniation tehniques are applied to model them as solids used for both numerial and analytial approahes. Homogeniation in the ore due to the natural gaps of the stranded ondutor is made by the orretion (inrease) in resistivity sine the ore is now onsidered massive. This is done by applying: k (1) where ρ is the orreted resistivity of the entral ondutor, k is the area orretion fator (k = r 2 /A n ), r is the ore radius, and A n is the nominal area provided by the manufaturer s atalog. Homogeniation in sheath also depends on the omposition of the material used. Therefore, its orreted resistivity is given by: k (2) s s s V. HOMOGENIZATION OF THE INSULATION Between the ore and insulation and sheath and insulation, there are semiondutor tapes whih have the funtion of uniformly distributing the eletri potential. These three materials are homogenied as one. For this, a orretion must be applied to the insulation eletri permittivity given by: ln rs r s s (3) ln ba VI. SERIES IMPEDANCE The analytial modeling for non-onentri ondutors, like the ase of the three-ore ables, is a very omplex task. A full analytial omputation of a submarine able an be found using the pipe-type able formulas from [3]. As explained in [4] there are some approximation in this, beause of the representation of the armor. Subsetion A presents a way to alulate the phase impedane for a single-ore able, where only the impedane in a single inner able an be onsidered without mutual ouplings with any other metalli part of the able. However, in the numerial approah, all ouplings involved are regarded. A. Analytial Approah Aording to [3], [2], and [4], the impedane of eah ore per unit of length is given by: I0 r 2r I1 r, (4) where is the intrinsi medium impedane, given by: j. (5) The ore impedane have one real part that represents the ore resistane; and other part imaginary, that represents the internal ore indutane times the eletri angular pulsation. The phase indutane is given by the sum of internal and external indutane. For a single ore ondutor, the external indutane is given by: s r lext _1C ln. (6) 2 rs However (6) onsiders the urrent of entral ondutor returning by sheath. For three-ore ables, the urrent returns by other phases, reating a bigger area for the magneti flux, that now is the area between two phases. So (6) beomes: r lext _3C ln 2 D, (7) where D is the distane between ores in the trefoil formation. For other formations, the geometri mean must be applied. Equation (7) is aurate and frequently used for impedane alulation at 50/60 H. For bigger frequenies, it is neessary to onsider the sheath effets as overed by [5][6]. B. Numerial Approah In order to find the self and mutual impedanes of all metalli parts of the able, a 1 A urrent is applied at one metalli element and measured the voltage in this and others. The self impedane is found by dividing the indued voltage by the urrent at the element that the urrent is applied. The mutual impedanes are found by dividing the indued voltage at the elements that have no urrent by the urrent that

3 produed this indution. C. Cable Impedanes Fig. 2 presents a three-ore able impedane diagram where: (i) the letters a, b, and represent eah ore; (ii) the numbers 1, 2, and 3 represent eah sheath; (iii) and g represents the armor. The voltage drop from g to g' on ondutors are: Vag Va g ' aa ab a a 1 a2 a3 ag Ia Vbg V b g ' ab bb b b 1 b2 b3 bg I b Vg V g ' a b g I V1 g V1 g' a1 b g I1 V2 g V 2 g' a2 b g I 2 V3g V3 g' a3 b g I3 Vg V g' ag bg g 1g 2g 3g gg I g (8) 2 s j s y12 (11) ln rs r where s is the insulation ondutivity and s is the orreted permittivity of the insulation. Beause of the high resistivity of the insulating materials, only the apaitanes on them are onsidered (see Setion VIII-B). The three-ore able apaitane diagram is shown in Fig. 3. In addition, a numerial approah is performed and ompared with the analytial results for this apaitane. However, for sheath-sheath and sheath-armor apaitanes, only numerial results are onsidered due to the nononentriity between these parts. The leakage urrents on insulations from g to g in phases are: I a I y y a s Vag I s 0 b I y y V b bg I 0 0 y 0 0 y I s V g (12) I1 I1 ys 0 0 yss yss yssv1 g I2 I 2 0 ys 0 yss yss y V ss 2g I3 I3 0 0 y s yss yss y ss V3 g where y, y s, y ss and y ss are, respetively, the ore selfadmittane, ore-sheath mutual admittane, sheath selfadmittane and sheath-sheath mutual admittane. Fig. 2. Representation of three-ore able impedanes. In [5] it is proved that the sequene impedanes for interonneted sheaths are: 2 a1 a2 aa ab (9) a 1 2a2 0 aa 2ab 2 If the sheaths are not interonneted at both terminals, or are interonneted only at one point (grounded or not), there is not irulation urrent and the sequene impedanes beome [1]: aa ab (10) 11 0 aa 2 ab VII. PARALLEL ADMITTANCE There are three kinds of admittanes on three-ore ables: (i) ore-sheath, (ii) sheath-sheath, and (iii) sheath-armor. Only the ore-sheath armor is analytially feasible, given by: a2 Fig. 3. The three-ore able apaitane diagram. By using the tehnique presented in [7] it is possible to get: y y y, (13) s a1 yss y11 ya 1 y1g y12 y1 3 (14) y y y (15) ss VIII. NUMERICAL MODELING USING FEM To perform the numerial modeling, the software Gmsh [8] and GetDP [9] are used. Gmsh is the pre and post-proessor and the GetDP is the solver. The problem is implemented in the software by two odes: one that defines the geometries and the mesh of the struture (.geo file) and other that defines the physial proprieties of the materials, the onstraints and the formulation to be used (.pro file). The eletrostati formulation used to alulate of the

4 parallel admittane is given by: grad V,grad V nd, V, V V D VF v (16) where V is the eletri salar potential, V' is the test funtion for salar potential, V is the volume harge density, n is the unit normal vetor exterior to, and n D is a onstraint on the eletri flux density assoiated with nonfixed potential boundaries D of the domain, e.g. on floating potential boundaries f [14]. F v () denotes the funtion spae defined on, whih ontains the basis and test funtions for both salar potentials V and V' [14]. At the disrete level, F v () is approximated with nodal finite elements. The harmoni magnetodynami formulation used to alulate of the series impedane is given by: vurl A,url A nhs, A j A, A H (17) grad V, A J, A 0, AF s a where is the eletri ondutivity defined on onduting parts of, A is the magneti vetor potential, A is the test funtion for vetor potential, J s is the soure eletri urrent density defined in s, and n H s is a onstraint on the magneti field assoiated with boundary H of the domain [15]. F a () denotes the funtion spae defined on whih ontains the basis and test funtions for both vetor potentials A and A. C. The Finite Element Approah Implementation Initially it is neessary to implement the surfaes (.geo file) from the model as shown in Table 2 and Fig. 4. In the same file, the mesh density fators must be inserted set in eah point of the geometri figure. Based on the geometry file, a mesh is built by Gmsh. Fig. 5(a) shows the mesh of the whole alulation domain while Fig. 5(b) shows the mesh of the lower left inner able of the three-ore able, respetively, for the alulation of series impedane. Both figures are shown with the aim of highlighting the mesh density utilied. Region A is a neessary region in order to avoid domain trunation errors, where the magneti vetor potential on its outer irle is ero. The physial onstants values in this region are the same values as region B. When alulating the parallel apaitane the eletri salar potential at the armor is set to ero. TABLE 2 PARTS OF A POWER INNER CABLE Item Radius [mm] IX. METHODOLOGY The physial and geometri data able is obtained from manufaturer s atalog [10], for a three-ore able in trefoil formation. Due to the omplexity of the able geometry, some simplifiations like homogeniation are required. In addition, it is imperative that some orretion fators be applied before starting the simulation as explained in Setion II. A. Physial and geometry onstants used in the model At the entral ondutor, the opper resistivity is onsidered (17.24 nω m). It is then orreted for a temperature of 90 C followed by an equivalent area (homogeniation) resulting in a resistivity ρ of nω m. The transversal magneti permeability used for all materials is onsidered μ 0, even the armor, beause it is omposed of wires that are not in diret ontat [11]. The able is onsidered as totally surrounded by seawater with a ondutivity of 5 S/m [12]. B. Simplified diagram of the able The parts onsidered for the able model (analyti and numeri) are all solids and represented by Table 2 and Fig. 4. Fig. 4. Set of inner ables in trefoil formation of a three-ore submarine power able. (a) (b) Fig. 5. Diagram and mesh for alulation of the series impedane. In (a) is the domain of alulation and in (b) the mesh detail of one power inner able.

5 D. Obtaining the mutual and self-impedanes and admittanes In order to find the self and mutual impedanes of all metalli parts of the able, the iruit presented in Fig. 6 is implemented and the tehnique explained in Setion IV-B is applied. The sequene impedanes are also obtained where three short-iruited ores are fed by a 1 V / 50 H three-phase sinusoidal soure (Fig. 7). For this implementation, two onsiderations are made: (i) with the sheath and armor opened; and (ii) with all sheaths interonneted at both ends and these onneted to the respetive armor end. The armor potential is onsidered floating (Fig. 7). The representation of the diagrams shown in Fig. 6 and 7 illustrates as the eletrial iruits are onsidered in GetDP. However, the modeling is arried out in two dimensions. In order to find the parallel apaitane we apply the Maxwell Capaitane Matrix onept [7]. Firstly, a 1 V potential is applied on the ore and ero on all other parts. The result is the ore self-apaitane. After that it is applied a 1 V on the sheath and ero on all other parts (Fig. 8). From this measurement we find the sheath s self-apaitane whih is the sum of sheath-ore, sheath-sheath (2 times), and sheatharmor apaitanes. Fig. 6. Ciruit diagram implemented in GetDP to determine the ore self and mutual impedanes. Fig. 7. Ciruit diagram implemented in GetDP to determine the phase sequene impedanes. Fig. 8. Ciruit diagram implemented in GetDP to determine the sheath selfapaitane. To find the sheath 1-sheath 2 apaitane, is imperative, for instane, to apply 1 V to ore a, sheath 1, ore, sheath 3, and armor, and ero on all other parts. The apaitane sheath 1- sheath 2 is obtained with basis on the eletri flux that goes out from surfae sheath 1. A similar proedure is applied to find ore-sheath and sheath-armor apaitanes. Finally, the numerial results are ompared with the analytial ones and also with the values supplied by the manufaturer for eah phase, validating the numerial modeling. X. RESULTS AND VALIDATION The presentation of results is divided into two parts: (i) analysis of impedanes, and (ii) analysis of admittanes. Validations are made by omparison with analytial methods, when possible, and with manufaturer s atalog [10] for the frequeny of 50 H. A. Series impedane At 50 H when we apply a urrent of 1150 A / 50 H to the ore a (Fig. 9), we obtain the indued voltages shown in Table 3. Dividing the indued voltage in eah metalli part of the able by the urrent (imposed on ore a) that originated them; we obtain the ore self-impedane and the mutual impedane between the respetive ondutive part and the ore (Table 4). Beause it is a able in trefoil formation (symmetri onfiguration), the same values are repeated when urrent is applied only in the ore b or. TABLE 3 INDUCED VOLTAGE IN ALL CABLE CONDUCTIVE PARTS WHEN CORE A IS FED BY 1150 A / 50 HZ Voltage at: Modulus [mv/km] Angle [] Core a Core b Core Sheath Sheath Sheath Armor g TABLE 4 CORE SELF-IMPEDANCE, MUTUAL BETWEEN CORES, MUTUAL CORE-SHEATH, AND MUTUAL CORE-ARMOR Impedane Resistane [m/km] Indutane [H/km] aa ab a a a a ag One noties a great similarity in the values of mutual impedanes between ores and between ore and sheaths of other ores. In other words ( ab = a ) ( a2 = a3 ), as desribed

6 in Setion 3.3 of [1]. The same proess is repeated but now the urrent is applied to sheath 1 and the indued voltages in all the metalli elements of the able are alulated. From this proess Table 5 is formulated for sheath s self and mutual impedanes. Finally, the alulation is repeated applying urrent at the armor and alulating the other indued voltages, resulting in Table 6. As expeted, independent of where the urrent is applied, the mutual impedanes are always the same as evidened in Tables 4, 5, and 6. Similarly we obtain the value of the positive sequene resistane: R m km. (19) The positive sequene impedane are also determined when three balaned voltages are applied, displaed 120 degrees from eah other, with the three ores short-iruited and the sheaths and armor opened (Fig. 10). Values equal to those found in (18) and (19) are obtained. (a) (b) Fig. 9. (a) Current density [A/mm 2 ] and (b) Magneti flux [Wb/m] used for the alulation of ore self and mutual sequene impedanes via FEM. TABLE 5 CORE-SHEATH MUTUAL IMPEDANCE, SHEATH SELF, AND MUTUAL SHEATH-ARMOR Impedane Resistane [m/km] Indutane [H/km] 1a b g TABLE 6 CORE-ARMOR MUTUAL, SHEATH-ARMOR MUTUAL, AND ARMOR SELF-IMPEDANCE Impedane Resistane [m/km] Indutane [H/km] ga gb g g g g gg If the sheaths are interonneted only at one of the ends (whether grounded or not), only the mutual impedanes between ores influene the phase positive sequene impedane, whih for the indutane an be obtained from (10): L mh km (18) The series indutane value of the able in the manufaturer s atalog [10] is 0.36 mh/km, whih validates the auray of the method used. (a) (b) Fig. 10. (a) Current density [A/mm 2 ] and (b) Magneti Flux [mwb/m] used for the alulation of positive sequene impedane diretly via FEM. Finally, an analytial approah is made by applying of (4) and (7). Table 7 presents a omparison of the results obtained for the positive sequene resistane and indutane between the adopted approahes. The error is found by taking the referene value provided by the manufaturer. The manufaturer did not provide the distributed able resistane value. If the sheaths are onneted at both ends, the distributed positive sequene series resistane and indutane would be: R + = 77.3 m/km and L + = H/km. TABLE 7 INDUCTANCE PROVIDED BY THE MANUFACTURER, CALCULATED ANALYTICALLY, AND CALCULATED VIA FEM R + [m/km] L + [mh/km] L + Error [%] Manufaturer Analytial Numeri The inrease in resistane ours beause, when the sheaths are interonneted, a irulation path is reated for the indued urrents. The introdution of an effet in the ore urrent distribution is therefore due to the sheath's urrent inreasing the proximity effet in the respetive ore ompared to the ase where the sheaths are not interonneted. As the frequeny inreases, this effet is inreased [13]. Knowing the resistanes and indutanes (selves and mutual) found for all ondutive parts of the able, the impedane matrix an be mounted: ore oresheath orearmor Z ore sheath sheath sheatharmor (20) orearmor sheatharmor armor

7 where, ore, sheath, ore-sheath, armor, ore-armor, sheath-armor, are, in /m: 75.7 j407 j295 j295 ore j j407 j295 (21) j295 j j j347 j295 j295 sheath j j347 j295 (22) j295 j j j347 j295 j295 oresheath j j347 j295 j295 j j347 (23) armor 616 j285 orearmor j295 (24) j295 sheatharmor B. Parallel admitane Beause the insulating material has a high resistivity, the branh that represents the parallel ondutane an be negleted, whih an be seen already at 50 H by applying (11) to the able under onsideration (XLPE insulation), where a = 11.4 mm and b = 19.4 mm. y j82.184ns m (25) The ore-sheath apaitane is Im (y12) / = pf/m, very lose to the value provided by the manufaturer s atalog [10], whih is 0.26 F/km. The ore-sheath apaitane is also alulated through finite element tehnique, obtaining the value of pf/m, whih is exatly the value found by the analytial method (also very lose to the value provided by the manufaturer). Fig. 11 shows the eletri field in the region under analysis (as well as in sheath-sheath and sheath-armor regions). Aording to (12), the parallel apaitane matrix is: Y jc, (26) where C, in pf/m, is: C (27) Region TABLE 8 CAPACITANCE OF THE THREE-CORE CABLE IN STUDY Numeri [F/km] Analytial [F/km] Manuf. [F/km] Error [%] Core-Sheath Sheath-Sheath Sheath-Armor XI. CONCLUSIONS Spae in manufaturer s able atalogs is typially dediated only for distributed positive sequene indutane and apaitane values at industrial frequeny (50 or 60 H). In [13] it is presented for the same able of present study, the behavior of positive sequene impedanes for a frequeny range from 20 H to 20 kh. In the present work, it was onsidered 50 H, with the improvement to find the parallel admittane and the speifiity in relation to mutual oupling between phases, thereby allowing to get the sequene impedanes. Similar to what was done in referene [13], in future works the goal will be to evolve the work presented in this paper by examining the frequeny range from 20 H to 20 kh. We will intend: (i) to simulate underground ables with grounded ends; and (ii) to apply the same modeling of this paper without the appliation of homogeniation tehniques (suh as those applied in the entral ondutor and sheath in Setion III). It is expeted that through this study, an inrease in the auray of the model s response, espeially at high frequenies, may be ahieved. Moreover, field measurements have to be made for validation. XII. ACKNOWLEDGEMENT The authors would like to gratefully aknowledge the PETROBRAS for the finanial support to this researh effort. Fig. 11. Eletri field [V/m] lines when a 1 V potential is applied at the sheath 1 and 0 V to all other metalli parts of the able, to obtain the sheath self-apaitane. Finally, the apaitane sheath-sheath and sheath-armor by the finite element method are alulated. Table 8 shows the ables apaitanes values between ore and sheath, sheath and sheath and sheath and armor, as also the error of measurement, onsidering the value of the manufaturer [10] as referene. XIII. REFERENCES [1] F. F. Da Silva and C. L. Bak, Eletromagneti Transients in Power Cables. Springer London, Limited, [2] S. A. Shelkunoff, The Eletromaneti Theory of Coaxial Transmission Lines and Cylindrial Shields, Bell Syst. Teh. J., p. 47, [3] A. Ametani, A General Formulation of Impedane and Admittane of Cables, IEEE Trans. Power Appar. Syst., vol. PAS-99, no. 3, pp , May 1980.

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