12407 FACTA UNIVERSITATIS Series: Electronics and Energetics Vol. 37, No 2, June 2024, pp. 391 – 408 https://doi.org/10.2298/FUEE2402391S © 2024 by University of Niš, Serbia | Creative Commons License: CC BY-NC-ND Original scientific paper CORRECTION OF THE IEC FORMULA FOR THE EDDY- CURRENT LOSS FACTOR: THE CASE OF SINGLE-CORE CABLES IN TREFOIL FORMATION WITH METALLIC SCREENS BONDED AND EARTHED AT ONE END Marko Šućurović1, Dardan Klimenta2, Dragan Tasić3 1University of Kragujevac, Faculty of Technical Sciences, Department of Power Engineering, Republic of Serbia 2University of Priština in Kosovska Mitrovica, Faculty of Technical Sciences, Department of Power Engineering, Republic of Serbia 3University of Niš, Faculty of Electronic Engineering, Department of Power Engineering, Republic of Serbia ORCID iDs: Marko Šućurović https://orcid.org/0000-0001-9574-6101 Dardan Klimenta https://orcid.org/0000-0003-0019-8371 Dragan Tasić https://orcid.org/0000-0001-5957-9617 Abstract. The purpose of this paper is to propose and apply the correct formula for the eddy-current loss factor for the case of three single-core cables in trefoil formation with metallic screens and armourings bonded and earthed at one end. This metallic screen bonding design is contrasted to the design where metallic screens and armourings are bonded and earthed at both ends, that is, the eddy-current loss factor is contrasted to the circulating-current loss factor. Ampacity calculations are carried out for 12 different underground lines with power cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622), assuming that the 33 kV cables are installed directly in the soil without drying out. The ampacity is calculated analytically in accordance with IEC 60287-1-1 and IEC 60287-2-1, and numerically in accordance with IEC TR 62095. The numerical calculations are carried out to verify the accuracy of the proposed formula using the finite element method (FEM) in COMSOL 4.3. A validation of the proposed formula is conducted based on the manufacturer's technical data for the considered cables. The calculated ampacity values determined the incompleteness of the current IEC formula for the eddy-current loss factor, and verified the accuracy of the proposed one. Key words: ampacity, circulating-current loss factor, eddy-current loss factor, finite element method (FEM), metallic screen bonding design, power cable Received January 09, 2024; revised February 21, 2024; accepted March 21, 2024 Corresponding author: Dardan Klimenta University of Priština in Kosovska Mitrovica, Faculty of Technical Sciences, Department of Power Engineering, Republic of Serbia E-mail: dardan.klimenta@pr.ac.rs https://orcid.org/0000-0001-9574-6101 https://orcid.org/0000-0003-0019-8371 https://orcid.org/0000-0001-5957-9617 392 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ 1. INTRODUCTION The latest version of the IEC 60287-1-1 standard was published in May 2023 [1]. This standard retained the incomplete formula for the eddy-current loss factor for the case of three single-core cables in trefoil formation with screens and armourings from non- magnetic metals bonded and earthed at one end. Thus, this formula existed in the same form in earlier versions of the IEC 60287-1-1 standard. This formula has been causing difficulties for design-engineers when calculating the ampacity of underground cable lines for several decades and should be corrected adequately. The current form of the formula for the eddy-current loss factor recommended by IEC 60287-1-1 [1] for three single-core cables in trefoil formation with metallic screens bonded and earthed at one end or cross-bonded is based on the researches conducted in earlier years [2-4]. The accuracy of the formulas for the calculation of metallic screen and armour losses in the 1994 version of the IEC 60287-1-1 standard was discussed by Barrett and Anders in 1997 [5]. Factors affecting metallic screen losses in underground cable lines with screens bonded and earthed at both ends were considered in [6]. In this regard, Anders also pointed out in 1997 [7] that eddy current losses occur regardless of the metallic screen bonding design, although they are often ignored in metallic screens bonded and earthed at both ends [1], where they are assumed to be small in magnitude compared to circulating current losses. For various metallic screen bonding designs and different short circuits, induced voltages and currents in metallic screens of high voltage underground cables in steady-state due to the adjacent cables and metallic screens were estimated in [8]. Formulas for the correction factor of three single- core cables with metallic screens cross-bonded and with unknown minor section lengths in order to correct the corresponding IEC 60287-1-1 formulas were provided in [9]. In addition, metallic screen voltages, circulating current losses, eddy current losses and associated loss factors for three single-core cables in trefoil and flat formations and various metallic screen bonding designs were simulated in [10]. Thermal and electrical effects of losses in metallic screens of single-core cables using the finite element method (FEM) were analyzed in [11]. Minimization of metallic screen and armour losses in an underground distribution network was performed in [12]. The paper [13] reviewed the formulas for induced losses in metallic screens used by IEC 60287-1-1 for cables in trefoil formation, making focus on three-core submarine cables with non- magnetic armourings. An improvement of loss allocation in three-core armoured cables using the FEM was proposed in [14]. The analytical techniques used to analyze currents and losses induced in armourings of medium voltage single-core cables were reviewed in [15]. Increasing the ampacity of underground cable lines by optimizing their crossings with respect to the metallic screen bonding designs was discussed in [16]. An improved analytical method for the cable ampacity calculation taking into account losses in metallic screens and armourings was proposed in [17]. Obviously, there have been attempts to correct or improve certain parts of the IEC 60287-1-1 standard that refer to losses due to eddy currents. However, the IEC formula for the eddy-current loss factor in question remained a research gap to be addressed in this paper. According to the IEC 60287-1-1 standard [1], the eddy-current loss factor is the ratio of the losses due to eddy currents in one metallic screen per unit length to the losses in one conductor per unit length. In this paper, such an eddy-current loss factor is multiplied by the ratio of the effective cross-section of the conductor on one side and the sum of the effective cross-sections of the metallic screen and armouring on the other. The proposed Correction of the IEC Formula for Eddy-Current Loss Factor 393 definition of the eddy-current loss factor can be regarded as the main contribution of this study. The idea for such a definition was found in [18], where losses due to eddy currents were included per unit volume. Ampacity calculations are performed for 12 different underground lines with power cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) from [19, 20], assuming that their metallic screens and armourings are bonded and earthed at both ends and at one end. These ampacity calculations are made in accordance with the IEC 60287-1-1 standard [1], the IEC 60287-2-1 standard [21] and the IEC TR 62095 technical report [22] using the FEM in COMSOL 4.3 [23]. 2. PROBLEM FORMULATION AND INPUT DATA When the IEC 60287-1-1 standard is used to calculate the ampacities of power cables with metallic screens and armourings from non-magnetic materials, a problem is encountered with the application of the formula for estimating the eddy-current loss factor 1. According to the IEC 60287-1-1 standard [1] from 2023, each non-magnetic metallic screen and its associated non-magnetic armouring are usually connected in parallel to form an enlarged metallic screen with a larger cross-section and lower electrical resistance per unit length. In this particular case, the corresponding IEC formula for the eddy-current loss factor 1 does not provide a good estimate and should be adequately corrected. The losses due to eddy currents in metallic screens of an underground cable line occur when the metallic screens are bonded and earthed at both ends [5,7,9,13] – to a low extent, as well as when the metallic screens are bonded and earthed at one end or cross- bonded [1] – to a significant extent. In the case of metallic screens bonded and earthed at both ends, the circulating currents counteract the magnetic field caused by the conductor current, and hence, affect the losses due to eddy currents. However, according to the 2023 version of the IEC 60287-1-1 standard [1], losses due to eddy currents should be ignored for the case of metallic screens bonded and earthed at both ends. Since the aforementioned incorrectness is analyzed, the neglect of eddy current losses in metallic screens bonded and earthed at both ends and other standardized assumptions must be used for the considered metallic screen bonding designs. The same assumptions are selected to compare the calculated results with those provided by the manufacturer in [19, 20]. Accordingly, the question is at which cross-section of the conductor the metallic screen bonding design [24] should be switched from the case of metallic screens bonded and earthed at both ends in accordance with Fig. 1(a) to the case of metallic screens bonded and earthed at one end in accordance with Fig. 1(b). In order to obtain the correct formula for 1, the following calculations for twelve different 33 kV underground lines with cables laid in trefoil formation will be performed: (i) ampacity calculations for cables with metallic screens and armourings bonded and earthed at both ends using the formula for the circulating-current loss factor 1 from the IEC 60287-1-1 standard [1]; (ii) ampacity calculations for cables with metallic screens and armourings bonded and earthed at one end using the formula for the eddy-current loss factor 1 from the IEC 60287-1-1 standard [1]; (iii) ampacity calculations for cables with metallic screens and armourings bonded and earthed at one end using a correct formula for the eddy-current loss factor 1; and (iv) ampacity calculations for cables with metallic screens and armourings bonded and earthed at both ends and at one end using the FEM in the Heat Transfer Module of COMSOL 4.3 [23]. The results of the ampacity 394 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ calculations (i) will be used to identify the cross-sections of conductors for which incorrect or significantly high values of losses due to circulating currents in enlarged metallic screens occur. In addition, these results together with the results of the ampacity calculations (iii) will be used to respond to the question at which conductor cross-section the metallic screen bonding design should be switched from one case to another. The results of the ampacity calculations (ii) will be used to demonstrate the incorrectness of the formula for the eddy- current loss factor 1 given in the IEC 60287-1-1 standard [1]. Numerical verification of the correctness of the current IEC formula for the circulating-current loss factor 1 and the proposed formula for the eddy-current loss factor 1 will be conducted based on the results of the ampacity calculations (iv). Fig. 1 A 33 kV underground cable line with (a) Metallic screens bonded and earthed at both ends; (b) Metallic screens bonded and earthed at one end For the purpose of quantifying the effect of the conductor cross-section on the losses in metallic screens, the ampacity calculations will be carried out with 12 power cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) whose nominal cross- section SC,n ranges from 70 to 1000 mm2. Service conditions as well as other constructional, electrical and rating data for these cables are taken directly from the manufacturer's technical documentation [19,20], or are estimated based on those data. Correction of the IEC Formula for Eddy-Current Loss Factor 395 According to [19,20], the following service conditions are considered: three single- core cables installed directly in the soil, in trefoil formation, with metallic screens and armourings bonded and earthed at both ends or at one end; installation depth to center of trefoil formation L = 800 mm; referent soil temperature rs = 15 oC; and thermal resistivity of the native soil t,s = 1.2 Km/W. Based on these conditions, it is evident that there is no drying out of the surrounding soil. The service conditions together with the details necessary for the FEM-based modeling (such as dimensions of the computational domain, boundary conditions, cable construction elements, etc.) are illustrated in Fig. 2. Fig. 2 Representation of the problem solved; (a) Computational domain referred to the design of a 33 kV underground cable line; (b) Dimensions of the construction elements of the Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) cable, i.e., diameters of 1 – conductor, 2 – conductor screen, 3 – insulation, 4 – insulation screen, 5 – metallic screen, 6 – bedding, 7 – armouring, and 8 – oversheath Data on cables with conductors of cross-sections from 50 to 630 mm2 are given in [19], while data on cables with conductors of cross-sections from 70 to 1000 mm2 are given in [20]. For conductors having cross-sections from 70 to 630 mm2, the ampacity values from [19] are higher than those provided by the same manufacturer in [20]. In [19], for cables with 800 mm2 and 1000 mm2 conductor cross-sections, the following note is provided: “Please refer to our technical department for further information”. In addition to this, the service conditions are the same in [20] as they are in [19]. In this regard, the BS 6622:2007 standard [25] states the following: “In special circumstances it may be necessary to employ cross bonding or single-point bonding and in these cases recommendations should be sought from the manufacturer.” Therefore, in [20] it was certainly taken into account that metal screens and armourings of cables with 800 mm2 and 1000 mm2 conductor cross- 396 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ sections are bonded and earthed at one end. Based on this, the same assumption is accepted in this paper for cables with 800 mm2 and 1000 mm2 conductor cross-sections. According to Fig. 2(a), s is the interaxial spacing between cables in trefoil formation in mm, and d8 is the outer diameter of a cable in mm. Further, based on Fig. 2(b), d1 is the diameter of a circular stranded copper conductor complying with IEC 60228 Class 2 [26] in mm; d2 is the outer diameter of a conductor screen made of semi-conducting cross- linked polyethylene (XLPE) in mm; d3 is the outer diameter of XLPE insulation in mm; d4 is the outer diameter of a semi-conducting XLPE insulation screen in mm; d5 is the outer diameter of a copper tape screen (CTS) in mm; d6 is the outer diameter of polyvinyl chloride (PVC) bedding in mm; d7 is the outer diameter of aluminium wire armouring (AWA) in mm, and d8 is the outer diameter of a PVC oversheath (i.e., a cable) in mm. Table 1 outlines these diameters (di, i = 1,2,…8 in mm), maximum AC resistance of conductors at their continuously permissible temperature C,cp = 90 oC per phase and unit length of a cable (RC,l in /m), and tabulated ampacity values (IC,T in A) for the 12 cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622). The parameter Ss,n appearing in Table 1 represents the nominal cross-section of a metallic screen in mm2. Table 1 Constructional, electrical and rating data on power cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) SC,n/Ss,n d1 d2 d3 d4 d5 d6 d7 d8 RC,l IC,T (mm2)/(mm2) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (mm) (/m) (A) 70/16 9.8 11 27 30.366 30.7 33.1 37.1 41.5 342 265 95/16 11.5 12.7 28.7 32.184 32.5 34.9 38.9 43.5 247 315 120/16 12.8 14 30 33.698 34 36.4 40.4 45 196 355 150/25 14.3 15.5 31.5 34.642 35.1 37.7 42.7 47.5 159 395 185/25 15.9 17.1 33.1 36.466 36.9 39.5 44.5 49.5 127 445 240/25 18.4 19.6 35.6 38.488 38.9 41.5 46.5 51.5 97.4 505 300/25 20.5 22.1 38.1 41.114 41.5 44.3 49.3 54.5 78.3 560 400/35 23.2 25.2 41.2 43.794 44.3 47.1 52.1 57.5 62.2 625 500/35 26.2 28.2 44.2 47.936 48.4 51.4 56.4 62 49.6 685 630/35 30.3 32.3 48.3 51.77 52.2 55.2 60.2 66 39.9 750 800/50 34.7 36.7 52.7 56.034 56.6 59.8 64.8 71 33.1 810 1000/50 38 41.3 57.3 60.172 60.7 64.1 69.1 75.5 28.4 855 Thermal, electrical and dielectric properties of materials used for IEC- and FEM- based modeling are shown in Table 2. The values for these material properties are taken from the standards IEC 60287-1-1 [1], IEC 60287-2-1 [21] and IEC 60287-3-1 [27], as well as from the available literature. The meanings of the parameters that appear in Table 2 are as follows: kt is the thermal conductivity of the materials used in W/(Km); e is the electrical resistivity of metallic screen or armouring material at the corresponding operating temperature in m; 20 is the temperature coefficient of electrical resistivity at 20 °C in 1/K; r is the relative permittivity of the XLPE insulation; and tan  is the loss factor for the XLPE insulation at a system frequency f = 50 Hz. Correction of the IEC Formula for Eddy-Current Loss Factor 397 Table 2 Thermal, electrical and dielectric properties of materials used for modeling Material kt e 20 r tan [W/(Km)] (m) (1/K) (–) (–) Aluminium at 20 °C 239 2.8410-8 0.00403 – – Copper at 20 °C 385 1.724110-8 0.00393 – – XLPE 0.286 – – 2.5 0.004 PVC 0.167 – – – – Native soil 0.83 – – – – 3. IEC- AND FEM-BASED METHODS FOR CALCULATING THE CABLE AMPACITY 3.1. The current IEC 60287-based method For three single-core cables installed directly in the soil without drying out, in trefoil formation, whose screens and armourings are made from non-magnetic metals, in parallel, bonded and earthed at both ends or at one end, the formula for the calculation of the cable ampacity IC (in A) based on IEC 60287-1-1 and IEC 60287-2-1 reduces to [1,21,28]: 5.0 432111, 4321,, )])("'1([ )5.0(         +++++ +++−− = TTTTR TTTTW I lC ldrscpC C   (1) where Wd,l represents the dielectric losses per phase and unit length of a cable in W/m, T1 is the thermal resistance of the layers between one conductor and the corresponding metallic screen in Km/W, T2 is the thermal resistance of the PVC bedding in Km/W, T3 is the thermal resistance of the PVC oversheath in Km/W, and T4 is the thermal resistance of the surrounding soil in Km/W. The remaining parameters appearing in Equation (1) are as follows: RC,l maximum AC resistance of one conductor at C,cp = 90 oC in /m (from Table 1), 1 is the ratio of the losses due to circulating currents in one enlarged metallic screen to the losses in one conductor, and 1 is the ratio of the losses due to eddy currents in one enlarged metallic screen to the losses in one conductor. According to the IEC 60287-1-1 standard [1], 1 is equal to zero for metallic screens and armourings bonded and earthed at both ends, and 1 is equal to zero for metallic screens and armourings bonded and earthed at one end (or cross-bonded). For three single-core cables in trefoil formation with metallic screens and armourings bonded and earthed at both ends, the circulating-current loss factor 1 is given by [1,28]:         + == 22 , 2 , , , , 1 )2( )2( ' fMR fM R R W W lselC lse lC lse    (2) where Wse,l represents the losses due to circulating currents in an enlarged metallic screen per phase and unit length of a cable in W/m, WC,l represents the losses in a conductor per phase and unit length of a cable in W/m, Rse,l is the equivalent resistance of metallic screen and armouring in parallel in /m, f = 50 Hz is the system frequency, 398 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ         = − sed s M 2 ln102 7 (3) is the mutual inductance between a conductor and an enlarged metallic screen in H/m; and 8 )()( 2 76 2 54 dddd dse +++ = (4) is the mean diameter of an enlarged metallic screen in mm. The AC resistance of any metallic screen at its maximum operating temperature s,max (in °C) per phase and unit length of a cable (Rs,l in /m) is calculated using an effective cross- section of the metallic screen 4/10)( 62 4 2 5, −−= ddS effs  (in m2). The maximum operating temperature s,max in °C is estimated by means of the following formula:         −= 1 4 , 2 ,, ,max, ln 2 d d k IR XLPEt TClC cpCs   , (5) assuming 0, =ldW W/m for 33 kV cables. The AC resistance of an armouring at its maximum operating temperature A,max (in °C) per phase and unit length of a cable (RA,l in /m) is calculated using an effective cross-section of the armouring 4/10)( 62 6 2 7, −−= ddS effA  (in m2). The maximum operating temperature A,max in °C is estimated by means of the following formula:                 +        −= 5 6 ,1 4 , 2 ,, ,max, ln 1 ln 1 2 d d kd d k IR PVCtXLPEt TClC cpCA   , (6) assuming 0, =ldW W/m and 0"' 11 =+ for 33 kV cables. For three single-core cables in trefoil formation with metallic screens and armourings bonded and earthed at one end (or cross-bonded), the eddy-current loss factor 1 is given by [1]:         +++== 12 44 1 210 , , , , 1 1012 )1(" se gs lC lse lC lse C R R W W   (7) where Wse,l represents the losses due to eddy currents in an enlarged metallic screen per phase and unit length of a cable in W/m, )6.110(1 3 71 74.1 7 −         += −d d C se gs  , (8) se f    7 2 1 10 8 = , (9) effAlAeffsls effAlAeffsls se SRSR SRSR ,,,, ,,,, + = is the electrical resistivity of an enlarged metallic screen at its operating temperature in m, 2/)( 47 ddse −= is the thickness of an enlarged metallic Correction of the IEC Formula for Eddy-Current Loss Factor 399 screen in mm, and 0, Cgs, 1, 1 and 2 are the appropriate coefficients. 0 is a dimensionless coefficient, Cgs is expressed in [mrad/(s)]0.5, 1 is expressed in [rad/(ms)]0.5, while the units of the coefficients 1 and 2 can be obtained based on the given units. Furthermore, for three single-core cables in trefoil formation is [1]: 7 , 10 2 −= lsR f m  (10) 2 2 2 0 21 3               + = s d m m se (11) where m is a parameter in mrad/(s), )66.192.0( 45.2 1 2 )33.014.1( +       += m se s d m and 02 = for 1.0m , (12) and 01 = and 02 = for 1.0m . (13) 3.2. A corrected IEC 60287-based method Specifically, the corrected IEC-based method for the calculation of the cable ampacity IC involves replacing Equation (7) with the following formula:         +++ + = 12 44 1 210 , , ,, , 1 1012 )1(" se gs lC lse effAeffs effC C R R SS S   (14) where 4/10 62 1, −= dS effC  is the effective cross-section of a conductor in m2. All other equations in this model are the same as in the IEC-based model from Section 3.1. The ratio )/( ,,, effAeffseffC SSS + is found according to the ratio of the volume power of heat sources in one enlarged metallic screen Wse,v (in W/m3) to the volume power of heat sources in one conductor WC,v (in W/m3) as follows: lC lse effAeffs effC effClC effAeffslse vC vse W W SS S SW SSW W W , , ,, , ,, ,,, , , / )/( + = + = (15) where the term lClse WW ,, / represents Equation (7) that comes from IEC 60287-1-1 [1]. 3.3. The current IEC TR 62095-based method Two-dimensional steady-state heat transfer through the computational domain in Fig. 2(a) is governed by the following second-order partial differential equation [29,30]: vttt W y k yx k x k =        −   +        −   =−   )( (16) where kt is the thermal conductivity in W/(mK);  is the unknown nodal temperature in K; x and y are the Cartesian spatial coordinates in m; and Wv is the volume power of heat sources in W/m3. 400 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ For the purpose of thermal analysis using the FEM in COMSOL 4.3, any cable of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) needs to be represented by an equivalent construction consisted of the copper conductor, XLPE insulation, equivalent copper screen and PVC oversheath with outer diameters d1, d4, d7 and d8, respectively. The equivalent cable construction is based on the following IEC 60287-1-1 standard instruction [1]: “Where screening layers are present, for thermal calculations metallic tapes are considered to be part of the conductor or sheath while semi-conducting layers (including metallized carbon paper tapes) are considered as part of the insulation. The appropriate component dimensions must be modified accordingly.” This means that the conductor and insulation screens are added to XLPE insulation, and materials of the copper screen, PVC bedding and aluminium armouring are modeled by the equivalent metallic screen having thermal conductivity of copper. Since the PVC bedding is included in the equivalent metallic screen, it means that the thermal resistance T2 should be added to the thermal resistance T3, that is, an equivalent thermal conductivity of the PVC oversheath should be determined. In that case, the equivalent thermal conductivity of the PVC oversheath PVCtk ,' in W/(Km) is         + = 7 8 32 , ln )(2 1 ' d d TT k PVCt  (17) The volume power of heat sources in a conductor WC,v in W/m3 is given by [29,30]: 62 1 2 , , , , 10 4 − == d IR S W W ClC effC lC vC  (18) where the cable ampacity IC can be replaced with the tabulated ampacity value IC,T, and the diameter d1 is in mm. The volume power of heat sources located between a conductor and an equivalent metallic screen Wd,v in W/m3 is given by [29, 30]: 62 1 2 4 , , 10)( 4 −− = dd W W ld vd  (19) where the diameters d1 and d4 are in mm. The equivalent volume power of heat sources located between a semi-conducting XLPE insulation screen and a PVC oversheath W se,v in W/m3 is given by [29, 30]: 62 4 2 7 2 ,1 62 4 2 7 ,1 62 4 2 7 , , 10)( 4 10)( 4 10)( 4 ' −−− − = − = − = dd IR dd W dd W W ClClClse vse      (20) where the cable ampacity IC can be replaced with the tabulated ampacity value IC,T, 1 = 1 for the case when enlarged metallic screens are bonded and earthed at both ends, 1 = 1 for the case when enlarged metallic screens are bonded and earthed at one end, and the diameters d4 and d7 are in mm. The loss factor 1 = 1 corresponds with Equation (2), while the loss factor 1 = 1 corresponds with Equation (7) or Equation (14). The top side of the computational domain in Fig. 2(a), i.e., the earth surface is modeled by rs = (21) – constant temperature boundary condition, or )()( rsct hkn  −=−  (22) Correction of the IEC Formula for Eddy-Current Loss Factor 401 – convection boundary condition [29,30]. In addition, the left-hand, bottom, and right- hand sides of the computational domain in Fig. 2(a) are modeled by 0)( =− tkn  (23) – adiabatic boundary condition [29,30]. In Equations (21-23),  is the unknown temperature of the earth surface in K; rs is the known temperature of the earth surface in K, or the known temperature of the air along the earth surface in K in accordance with IEC 60287- 1-1 [1] and IEC TR 62095 [22]; n  is the outwards-oriented normal vector of the constant temperature and convection boundaries; and hc = 250 W/(m2K) is the heat transfer coefficient due to forced convection in accordance with IEC TR 62095 [22]. In addition, due to the absence of the radiation boundary condition, Equation (16) is linear. 4. RESULTS AND DISCUSSION The values of the thermal resistances and dielectric losses appearing in Equation (1) are listed in Table 3. In addition to these values Table 3 contains the equivalent thermal conductivity of the PVC oversheath calculated using Equation (17) for the purpose of FEM-based steady-state thermal modeling in COMSOL 4.3. Table 3 Thermal resistances, equivalent thermal conductivity and dielectric losses for power cables of the type Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) SC,n/Ss,n T1 T2 T3 T4 k't,PVC Wd,l (mm2)/(mm2) (Km/W) (Km/W) (Km/W) (Km/W) [W/(Km)] (W/m) 70/16 0.630 0.072 0.107 2.225 0.100 0.070556 95/16 0.573 0.068 0.107 2.198 0.102 0.077709 120/16 0.539 0.065 0.103 2.178 0.102 0.083128 150/25 0.493 0.068 0.102 2.147 0.100 0.089340 185/25 0.462 0.065 0.102 2.124 0.102 0.095927 240/25 0.411 0.062 0.097 2.101 0.102 0.106156 300/25 0.388 0.062 0.096 2.069 0.101 0.116326 400/35 0.354 0.058 0.094 2.038 0.103 0.128877 500/35 0.336 0.057 0.090 1.995 0.102 0.140977 630/35 0.298 0.053 0.088 1.959 0.104 0.157458 800/50 0.267 0.052 0.087 1.917 0.104 0.175093 1000/50 0.256 0.052 0.084 1.882 0.103 0.193488 Table 4 shows the circulating-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the tabulated ampacity values and other service conditions taken from [20], and metallic screens and armourings bonded and earthed at both ends. Based on the maximum conductor temperatures from the last two columns of Table 4 (which are higher than the continuously permissible temperature of 90 °C), it is obvious that there is something illogical about the cable ampacities provided by the manufacturer, or about the metallic screen bonding design. In particular, this means that the volume powers of heat sources in the conductors and equivalent metallic screens are not adequate and the reason for this should be identified. In this regard, the volume powers of heat sources in the XLPE insulations do not contribute to this illogicality. 402 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ Table 5 shows the circulating-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the ampacities calculated in accordance with IEC 60287-1-1 [1], other service conditions taken from [19,20], and metallic screens and armourings bonded and earthed at both ends. Table 4 Circulating-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the tabulated ampacity values taken from [20], and metallic screens and armourings bonded and earthed at both ends SC,n/Ss,n IC,T '1 WC,v Wd,v W'se,v C,max * C,max ** (mm2)/(mm2) (A) (–) (W/m3) (W/m3) (W/m3) (°C) (°C) 70/16 265 0.060335 318402.0 147.7 4061.0 91.495 91.556 95/16 315 0.084997 235956.8 149.4 5555.9 92.390 92.453 120/16 355 0.107885 191956.5 150.3 6832.4 92.758 92.823 150/25 395 0.161023 154464.7 151.3 8161.0 94.222 94.290 185/25 445 0.203234 126660.0 152.1 10004.5 96.181 96.253 240/25 505 0.265159 93414.9 153.0 12315.7 96.472 96.547 300/25 560 0.333637 74394.4 153.8 14093.4 97.678 97.756 400/35 625 0.428533 57475.7 154.4 16644.1 99.838 99.921 500/35 685 0.540382 43168.8 154.9 18132.8 99.751 99.953 630/35 750 0.672362 31125.8 155.5 20355.4 100.645 100.734 800/50 810 0.820678 22964.1 155.8 21423.5 102.309 102.403 1000/50 855 0.953514 18306.0 156.1 21838.8 102.380 102.477 * Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the constant temperature boundary condition. ** Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the convection boundary condition. Table 5 Circulating-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the ampacities calculated in accordance with IEC 60287-1-1 [1], and metallic screens and armourings bonded and earthed at both ends SC,n/Ss,n IC '1 WC,v Wd,v W'se,v C,max * C,max ** (mm2)/(mm2) (A) (–) (W/m3) (W/m3) (W/m3) (°C) (°C) 70/16 262.321 0.060334 311997.9 147.7 3979.3 89.962 90.021 95/16 310.168 0.084997 228772.7 149.4 5386.7 89.042 90.103 120/16 348.664 0.107885 185145.0 150.3 6589.9 90.009 90.072 150/25 384.343 0.161023 146242.3 151.3 7726.6 90.021 90.086 185/25 427.668 0.203234 116985.7 152.1 9240.4 90.005 90.071 240/25 484.349 0.265159 85931.2 153.0 11329.1 89.972 90.041 300/25 533.130 0.333637 67426.5 153.8 12773.4 89.969 90.040 400/35 587.154 0.428533 50725.7 154.4 14689.4 89.921 89.994 500/35 643.392 0.540382 38083.9 154.9 15996.9 89.924 89.999 630/35 700.680 0.672362 27166.7 155.5 17766.3 89.813 89.891 800/50 749.603 0.820678 19667.2 155.8 18347.8 89.849 89.930 1000/50 790.828 0.953514 15661.2 156.1 18683.6 89.838 89.921 * Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the constant temperature boundary condition. ** Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the convection boundary condition. Correction of the IEC Formula for Eddy-Current Loss Factor 403 From Table 5, it is evident that the maximum conductor temperatures are approximately equal to the continuously permissible temperature of 90 °C. Thus, the IEC-based ampacity calculation was carried out correctly. In addition, it is found that the cable ampacity values are lower than those given by the manufacturer in [20] and that the circulating- current loss factors remained unchanged. Consequently, it remains that the cable ampacity values from [20] were not obtained in accordance with IEC 60287-1-1 [1], or that, at some conductor cross-section, the metallic screen bonding design was switched from the case shown in Fig. 1(a) to that of Fig. 1(b). Table 6 outlines the eddy-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the ampacities calculated in accordance with IEC 60287-1-1 [1] using the current IEC formula for 1, other service conditions taken from [19,20], and metallic screens and armourings bonded and earthed at one end. Table 6 Eddy-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the ampacities calculated in accordance with IEC 60287- 1-1 [1] using the current IEC formula for 1, and metallic screens and armourings bonded and earthed at one end SC,n/Ss,n IC "1 WC,v Wd,v W'se,v C,max * C,max ** (mm2)/(mm2) (A) (–) (W/m3) (W/m3) (W/m3) (°C) (°C) 70/16 266.978 0.014608 323172.7 147.7 997.9 89.953 90.012 95/16 317.977 0.020635 240437.8 149.4 1374.4 90.036 90.096 120/16 359.743 0.026621 197120.0 150.3 1731.2 90.006 90.068 150/25 401.872 0.043799 159886.0 151.3 2297.8 90.013 90.077 185/25 452.068 0.055645 130715.8 152.1 2826.9 89.992 90.058 240/25 519.872 0.074165 98998.0 153.0 3650.6 89.958 90.026 300/25 580.642 0.096986 79980.1 153.8 4404.4 89.972 90.041 400/35 651.733 0.129025 62497.7 154.4 5449.2 89.930 90.001 500/35 727.254 0.171481 48658.8 154.9 6485.9 89.932 90.006 630/35 808.811 0.219644 36198.6 155.5 7733.3 89.836 89.912 800/50 879.755 0.286259 27089.6 155.8 8815.2 89.865 89.943 1000/50 939.099 0.348455 22084.3 156.1 9628.1 89.859 89.940 * Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the constant temperature boundary condition. ** Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the convection boundary condition. Based on Table 6, the maximum conductor temperatures are close to the continuously permissible temperature of 90 °C. It seems that the ampacity calculations were carried out as required by the IEC 60287-1-1 standard. However, for conductor cross-sections equal to or larger than 400 mm2, it is found that the ampacities are significantly higher than those provided by the manufacturer in [20]. In addition, all the values of the eddy-current loss factor are lower than 0.35. The eddy-current loss factors are significantly lower than the corresponding circulating-current loss factors. Thus, the metallic screens and armourings for all conductor cross-sections had to be bonded and earthed at one end instead of both. This does not agree with the service conditions given in [20] and indicates that something is wrong with the current IEC formula for the eddy-current loss factor. It also follows that the ampacities provided by the manufacturer in Table 4 for all conductor cross-sections are obtained as approximate mean values of the ampacities from 404 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ Tables 5 and 6 (corresponding to the two considered bonding designs). Such approach also does not agree with the guidelines of IEC 60287-1-1 [1]. Furthermore, when comparing the ampacities of cables having conductor cross- sections of 800 mm2 and 1000 mm2 from Table 4 to the corresponding ampacities from Tables 5 and 6, it is noticed that the ampacities of 810 A and 855 A are significantly higher than those of Table 5, and significantly less than those of Table 6. If it is taken into account that Equation (2) is correct, then it follows that the error occurs when applying Equation (7). This was an alternative way to show that the ampacities of 810 A and 855 A correspond to the case of cables with metallic screens and armourings bonded and earthed at one end. Thus, the given data can be used to validate the accuracy of the proposed formula for eddy-current loss factor, that is, Equation (14). The validation process consists of the following five steps: (i) In the FEM-based steady-state thermal model used to generate the corresponding results from Table 4, it should be specified W'se,v=0 W/m3, while all other parameters remain the same. (ii) The value of the equivalent volume power of heat sources located in the equivalent metallic screen W'se,v needs to be increased from simulation to simulation until a value corresponding to the continuously permissible temperature of 90 °C is identified. (iii) The losses due to eddy currents in an enlarged metallic screen per phase and unit length of a cable Wse,l need to be calculated using Equation (20). (iv) The losses in a conductor per phase and unit length of a cable WC,l need to be calculated using Equation (18). (v) The eddy-current loss factor needs to be calculated as 1 = Wse,l . WC,l. The results of this validation process are shown in Table 7 and Fig. 3. Table 7 Volume powers of heat sources, maximum conductor temperatures and eddy- current loss factor obtained for the tabulated ampacity values taken from [20], and metallic screens and armourings bonded and earthed at one end SC,n/Ss,n IC,T WC,v Wd,v W'se,v C,max * C,max ** "1 *** (mm2)/(mm2) (A) (W/m3) (W/m3) (W/m3) (°C) (°C) (–) 800/50 810 22964.1 155.8 14198.3 90.000 90.080 0.543899 1000/50 855 18306.0 156.1 15039.1 90.000 90.082 0.656629 * Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the constant temperature boundary condition ** Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the convection boundary condition *** Values obtained using Equations (18) and (20) in combination with the following definition: lClse WW ,,1 /" = According to Tables 6 and 7, each eddy-current loss factor from Table 7 is approximately twice as large as the corresponding loss factor from Table 6. The ratio between the eddy-current loss factors from Tables 7 and 6 is obviously comparable to the cross-section ratio )/( ,,, effAeffseffC SSS + that appears in Equation (14). It can therefore be considered that, in this manner, the accuracy of Equation (14) is validated. In addition, temperature distributions over the part of the computational domain in Fig. 2(a) generated using the data from Table 7 for the considered cables with 800 mm2 and 1000 mm2 conductor cross-sections are shown in Figs. 3(a) and 3(b), respectively. These two temperature distributions correspond with the cases where the equivalent metallic screen has the thermal conductivity of copper and where the earth surface is represented by the Correction of the IEC Formula for Eddy-Current Loss Factor 405 constant temperature boundary condition. In order to show how the temperature distributions could be affected by the assumption that the equivalent metallic screens are made of aluminium instead of copper, two additional simulations were performed. The results of these additional simulations are shown in Figs. 3(c) and 3(d). Fig. 3 Temperature distributions over the part of the computational domain in Fig. 2(a) generated using the data from Table 7 for the Cu/XLPE/CTS/PVC/AWA/PVC 1/C 19/33 kV (BS 6622) power cables with (a) 800 mm2 conductor cross-section and equivalent copper screen; (b) 1000 mm2 conductor cross-section and equivalent copper screen; (c) 800 mm2 conductor cross-section and equivalent aluminium screen; (d) 1000 mm2 conductor cross-section and equivalent aluminium screen 406 M. ŠUĆUROVIĆ, D. KLIMENTA, D. TASIĆ According to Figs. 3(a) and 3(c), as well as Figs. 3(b) and 3(d), the assumption that the equivalent metallic screens are made of aluminium instead of copper would lead to increases in the temperatures of the conductors by approximately 0.03 °C. Accordingly, it follows that the assumption used for equivalent metallic screens is justified by these comparisons. Furthermore, there is no need to show additional temperature distributions because of their similarity with those of Fig. 3. Table 8 outlines the eddy-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the ampacities calculated in accordance with IEC 60287-1-1 [1] using the correct formula for 1, other service conditions taken from [19,20], and metallic screens and armourings bonded and earthed at one end. Table 8 Eddy-current loss factor, volume powers of heat sources and maximum conductor temperatures obtained for the ampacities calculated in accordance with IEC 60287-1- 1 [1] using the correct formula for 1, and metallic screens and armourings bonded and earthed at one end SC,n/Ss,n IC "1 WC,v Wd,v Wse,v C,max * C,max ** (mm2)/(mm2) (A) (–) (W/m3) (W/m3) (W/m3) (°C) (°C) 70/16 268.024 0.004658 325711.1 147.7 320.7 89.951 90.010 95/16 319.498 0.008646 242743.5 149.4 581.4 90.034 90.095 120/16 361.663 0.013312 199230.4 150.3 875.0 90.005 90.067 150/25 405.628 0.020640 162888.4 151.3 1103.1 90.011 90.075 185/25 456.540 0.031134 133314.3 152.1 1613.1 89.990 90.055 240/25 524.264 0.053211 100677.6 153.0 2663.6 89.957 90.024 300/25 584.204 0.081535 80964.3 153.8 3748.3 89.972 90.041 400/35 651.874 0.128467 62524.7 154.4 5428.0 89.930 90.001 500/35 719.131 0.201665 47577.9 154.9 7458.1 89.932 90.005 630/35 779.400 0.324354 33613.8 155.5 10604.6 89.829 89.906 800/50 819.562 0.501903 23509.5 155.8 13413.1 89.858 89.938 1000/50 846.368 0.689442 17938.2 156.1 15473.4 89.846 89.928 * Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the constant temperature boundary condition. ** Values obtained using the FEM in COMSOL 4.3 for the earth surface represented by the convection boundary condition. According to Table 8, it is clear that the maximum conductor temperatures are again close to 90 °C. This means that the IEC-based ampacity calculation using the correct formula for 1 was carried out properly. Except for the case of a conductor cross-section of 1000 mm2, it is obtained that the ampacities are higher than those provided by the manufacturer in [20] (Table 4), and that the eddy-current loss factors are lower than the corresponding circulating-current loss factors. Again, on the basis of Tables 4, 5 and 8, it follows that the ampacities of the manufacturer for conductor cross-sections from 70 mm2 to 630 mm2 are obtained as approximate mean values of the ampacities from Tables 5 and 8. As in previous discussion, the results again confirmed that the ampacities of the manufacturer (from Table 4) related to 800 mm2 and 1000 mm2 conductor cross-sections correspond with the case where metallic screens and armourings are bonded and earthed at one end in accordance with Fig. 1(b). Correction of the IEC Formula for Eddy-Current Loss Factor 407 5. CONCLUSION Based on the obtained results and their discussion, the following conclusions are reached: (i) It was shown that the ampacities obtained using the proposed formula for the eddy-current loss factor are closer to the tabulated ampacities provided by the manufacturer than those obtained using the current IEC formula. (ii) It was also shown that the tabulated ampacities provided by the manufacturer do not correspond with the design according to which metallic screens and armourings are bonded and earthed at both ends. (iii) It was found that the manufacturer switched from the case of metallic screens and armourings bonded and earthed at both ends to the case of metallic screens and armourings bonded and earthed at one end for conductor cross-sections of 800 mm2 and 1000 mm2, where the circulating-current loss factor is higher than 0.8. (iv) For conductor cross-sections of 800 mm2 and 1000 mm2, it was shown that the current IEC formula gives approximately half the value of that obtained by applying the correct formula. Finally, FEM-based calculations of the circulating- and eddy-current loss factors for underground power cables with metallic screens and armourings bonded and earthed at both ends, using the IEC 60287-1-1 standard and assuming non-uniform current densities across the cross-sections of conductors, metallic screens and armourings, might be the subject of a future study. 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