electronics

Article Characteristic Impedance Analysis of Medium-Voltage Underground Cables with Grounded Shields and Armors for Power Line Communication

Hongshan Zhao *, Weitao Zhang and Yan Wang School of Electrical and Electronic Engineering, North China Electric Power University, Baoding 071003, China; [email protected] (W.Z.); [email protected] (Y.W.) * Correspondence: [email protected]; Tel.: +86-135-0313-5810



Received: 25 March 2019; Accepted: 18 May 2019; Published: 23 May 2019


Abstract: The characteristic impedance of a power line is an important parameter in power line communication (PLC) technologies. This parameter is helpful for understanding power line impedance characteristics and achieving impedance matching. In this study, we focused on the characteristic impedance matrices (CIMs) of the medium-voltage (MV) cables. The calculation and characteristics of the CIMs were investigated with special consideration of the grounded shields and armors, which are often neglected in current research. The calculation results were validated through the experimental measurements. The results show that the MV underground cables with multiple grounding points have forward and backward CIMs, which are generally not equal unless the whole cable structure is longitudinally symmetrical. Then, the resonance phenomenon in the CIMs was analyzed. We found that the grounding of the shields and armors not only affected their own characteristic impedances but also those of the cores, and the resonance present in the CIMs should be of concern in the impedance matching of the PLC systems. Finally, the effects of the grounding resistances, cable lengths, grounding point numbers, and cable branch numbers on the CIMs of the MV underground cables were discussed through control experiments.

Keywords: power line communication; medium-voltage underground cable; impedance matching; characteristic impedance matrix; partial conductor grounding

1. Introduction Smart grid (SG) technologies are attracting growing attention given their capacity to sustainably manage power using intelligent grids. SG infrastructures require a bi-directional communication to enhance the energy efficiency, reliability and safety of the electricity systems [1]. When exploring SG communication systems, a universal data and network communications protocol is essential to achieve an interoperable environment [2]. In this respect, IEC 61850, an object-oriented standard, is considered the foundation of the SG communications systems [3,4]. The communication medium used in the communication system is another pivotal issue. Given the wide range of costs, latency, and bandwidth requirements of smart grid communications, a communication system that integrates multiple communication mediums is desirable [5,6]. Therefore, many communication technologies have since been evaluated and compared [6,7]. Among them, power line communication (PLC) is considered a feasible alternative that has attracted significant interest, and it has been gradually applied in advanced measurement systems and in remote fault detection systems [8,9]. This is due to its low cost, convenient deployment, and wide coverage compared with the high data capability communication technologies (e.g., fiber-optic ethernet) [6,10].

Electronics 2019, 8, 571; doi:10.3390/electronics8050571 www.mdpi.com/journal/electronics Electronics 2019, 8, 571 2 of 18

However, power line networks have numerous branches, especially in the medium-voltage (MV) and low-voltage (LV) networks, and their network typologies and loads may change unpredictably and extremely, which makes the impedance characteristics of power lines complicated and changeable, thereby increasing the difficulty of impedance matching in PLC systems. The methods of impedance matching mainly include characteristic impedance matching and complex conjugate matching. The purpose of characteristic impedance matching is to reduce or eliminate the reflected components of the carrier signal to increase the signal voltage at the receivers and to improve the signal-to-noise ratio (SNR) [11–13]. This method requires impedance matching between the internal impedances of the receivers and the characteristic impedances of the power lines. The purpose of complex conjugate matching is to adapt the internal impedance of the transceivers to the changeable access impedance of the power lines to achieve maximum power transmission [14–16]. Additionally, some optimization methods based on access impedance have been proposed to increase the data rate and channel capacity of the PLC systems [17,18]. Here, the access impedances of the power lines are closely related to their characteristic impedances [19,20]. Therefore, the characteristic impedances of the power lines are important in the analysis of the impedance characteristics and in the realization of impedance matching. The power lines of interest in this paper are MV underground cables, which are employed in almost all new urban and suburban installations. The current studies on their characteristic impedances and impedance matching are mainly based on the two-conductor transmission line model [19,21]. However, the MV underground cables have an important multiconductor nature, meaning that the signal coupling between the conductors cannot be ignored, and thus the impedance matching at the terminals of the power lines requires impedance matching at any port comprised of two power lines [13]. At this point, it is necessary to analyze the characteristic impedance matrices (CIMs) of the MV underground cables. In the MV underground cable networks, the shields and armors are grounded at the ring main units (RMUs), where the cable branches exist. The grounding points may considerably affect the PLC channel characteristics [22–24]. Similar issues have been studied for overhead lines when the shield wires are grounded at towers, and different treatments have been adopted. A typical approach involves assuming that the shield wires are continuously grounded and that the shield wires have zero potential along their entire length [25–29]. In previous studies [30–32], this assumption was proven to be unsuitable when the electrical length between towers is odd multiples of a one-half wavelength of signal. In addition, these studies mainly investigated the signal attenuation based on transmission and admittance matrices. For the characteristic impedances of power lines, a simplified method involves ignoring the mutual inductance between the ungrounded and grounded conductors, so that their characteristic impedances are considered separately, despite this method being applicable in certain special cases [33,34]. In Reference [35], for the first time, the CIMs of overhead lines with shield wires that were periodically grounded were calculated under a lossless assumption. In other studies [36,37], the CIMs of typical high-voltage (HV) lossy overhead lines with periodically grounded and sectionalized shield wires were analyzed under an ideal grounding assumption (i.e., grounding points had zero potential), and the properties of the CIMs have been further investigated [37,38]. However, in current studies on the CIM of the MV underground cable, the grounded partial conductors were not considered. The related investigations are confined to the CIM of the uniform MV underground cable [39,40], and the results only represent the CIM of one cable section rather than that of the whole cable structure between the PLC transceivers, which is often nonuniform owing to the grounded shields and armors. The latter is more important in reducing or eliminating the reflected signal at the receivers or transmitters [36]. In this study, we focused on MV underground cables with grounded shields and armors, where the CIMs were calculated and analyzed after considering the multiconductor and non-uniformity characteristics of the cables. Simultaneously, the high-frequency loss and non-ideal grounding conditions (i.e., the shields and armors were grounded only at the RMUs, and the grounding points potential was non-zero) were considered. Electronics 2019, 8, 571 3 of 18

The remainder of this paper is organized as follows. Section2 details the parameter matrices and multiconductor transmission line representations of the MV underground cables. Section3 is Electronics 2019, 8, x FOR PEER REVIEW 3 of 19 dedicated to calculating the CIMs of the MV underground cables with grounded shields and armors, andand thethe results results are are validated validated through through experimental experime measurementsntal measurements in Section 4in. Then,Section the characteristics4. Then, the characteristicsof the CIMs and of the the CIMs effects and of the the effects network of the structures network and structures parameters and parameters are discussed are discussed in Section in5. SectionThe conclusions 5. The conclusions are provided are inprovided Section 6in. Section 6. 2. Multiconductor Transmission Lines Equations 2. Multiconductor Transmission Lines Equations In this paper, the investigation of the CIM of the MV underground cable is based on multi-conductor In this paper, the investigation of the CIM of the MV underground cable is based on multi- transmission lines theory. Therefore, this section presents the geometry of MV underground cable and conductor transmission lines theory. Therefore, this section presents the geometry of MV the basic multi-conductor transmission line representations. underground cable and the basic multi-conductor transmission line representations. At present, the MV (10kV–35kV) underground cable that is used widely in China is a three-phase At present, the MV (10kV–35kV) underground cable that is used widely in China is a three- cable, which has a cyclic-symmetric structure. The 10 kV underground cables that were examined in phase cable, which has a cyclic-symmetric structure. The 10 kV underground cables that were this paper were the YJV22-type cables with a sectional area of 3 240 mm2. The layout of this type of examined in this paper were the YJV22-type cables with a sectional× area of 3 × 240 mm2. The layout cable is depicted in Figure1a. The original geometric and electromagnetic parameters of the cable are of this type of cable is depicted in Figure 1a. The original geometric and electromagnetic parameters provided in AppendixA. The cable was located 3 m below the earth surface, where the soil resistivity of the cable are provided in Appendix A. The cable was located 3 m below the earth surface, where is 100 Ω m. the soil resistivity· is 100 Ω∙m.

I Filler XPLE Insulation c,A Core + Steel Armor Shield Is,A + Copper Shield Vc,A Armor + PVC Sheath Core Ir Vs,A Vr Ground - -- x

(a) (b)

FigureFigure 1. ((aa)) Structure Structure of of the the medium-voltage underground cable. XPLE XPLE stands stands for for cross-linked cross-linked polyethylene,polyethylene, PVC PVC stands stands for for polyvinyl polyvinyl chloride; chloride; ( (bb)) Voltage Voltage and and current current definitions definitions of of the the A-phase A-phase core,core, A A phase phase shield, shield, and and the the armor. armor.

TheThe MV MV underground cable cable includes 7 7 conductors with three-phase cores, three-phase shields andand the outermostoutermost armor.armor. They They can can be be treated treated as theas the uniform uniform multiconductor multiconductor transmission transmission lines, basedlines, basedon the on telegraph the telegraph equations equations in the in frequency the frequency domain: domain: ( dd ( ) = ( ) dx V𝑽x(𝑥) =−𝒁𝑰ZI x(𝑥, ), dd𝑥 − (1) I(x) = YV(x). dxd − (1) 𝑰(𝑥) =−𝒀𝑽(𝑥). where x denotes the position at the conductors.d𝑥 Z and Y are the per unit length (p.u.l.) impedance wherematrix x and denotes admittance the position matrix at of the the conductors. MV underground 𝒁 and 𝒀 cable, are the respectively. per unit length The 7 (p.u.l.)1 vectors impedanceV and I × matrixrepresent and the admittance voltage and matrix current of the in theMV phasor underground form, respectively. cable, respectively. In this work, The 7V ×and 1 vectorsI are expressed 𝑽 and 𝑰 as follows: 𝑽 𝑰 represent the voltage and currenth in the phasor form, respectively. In this work,iT and are expressed as follows: V = Vc,A Vs,A Vc,B Vs,B Vc,C Vs,C Vr , (2)

h iT 𝑽=[𝑉, 𝑉, 𝑉, 𝑉, 𝑉, 𝑉, 𝑉] , (2) I = Ic,A Is,A Ic,B Is,B Ic,C Is,C Ir . (3)

T where [ ] denotes the matrix transpose.𝑰=[𝐼, V𝐼, , V𝐼, and𝐼, V𝐼,are𝐼, the voltages𝐼] . of the A, B and C phase(3) · c,A c,B c,C cores, respectively. Vs,A, Vs,B and Vs,C are the voltages of the A, B and C phase shields, respectively. Vr where [∙] denotes the matrix transpose. 𝑉,, 𝑉, and 𝑉, are the voltages of the A, B and C phase is the voltage of the armor. Ic,A, Ic,B and Ic,C are the currents in the A, B and C phase cores respectively. cores, respectively. 𝑉,, 𝑉, and 𝑉, are the voltages of the A, B and C phase shields, respectively. Is,A, Is,B and Is,C are the currents in the A, B and C phase shields, respectively. Ir is the current in the 𝑉 is the voltage of the armor. 𝐼,, 𝐼, and 𝐼, are the currents in the A, B and C phase cores armor. The positive directions of voltages and currents are shown in Figure1b taking the A phase and respectively. 𝐼,, 𝐼, and 𝐼, are the currents in the A, B and C phase shields, respectively. 𝐼 is thethe current armor as in examples. the armor. The positive directions of voltages and currents are shown in Figure 1b taking the A phase and the armor as examples. Apart from the filler, armor and sheath, the MV underground cable can be considered as a system that contains three subsystems, and each subsystem is comprised of the core, insulation and shield. Therefore, the p.u.l. impedance matrix 𝒁 and the admittance matrix 𝒀 can be expressed in the forms:

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Apart from the filler, armor and sheath, the MV underground cable can be considered as a system that contains three subsystems, and each subsystem is comprised of the core, insulation and shield. Therefore, the p.u.l. impedance matrix Z and the admittance matrix Y can be expressed in the forms:

 T   ZA ZAB ZAC ZAr     Z Z Z ZT   AB B BC Br   T , (4)  ZAC ZBC ZC ZCr    ZAr ZBr ZCr Zr

 T   YA YAB YAC YAr   T   Y Y Y Y   AB B BC Br   T . (5)  YAC YBC YC YCr    YAr YBr YCr Yr where both the rows and columns of Z and Y are arranged in the order of the A-phase subsystem, the B-phase subsystem, the C-phase subsystem, and the armor. There are identical meanings for the subscripts of submatrices in Z and Y. Taking the submatrices in Z as examples, the 2 2 submatrices Z , × A ZB and ZC are the self-impedance matrices of A, B and C phase subsystem, respectively. The element Zr is the self-impedance of the armor. The 2 2 submatrices Z , Z and Z are the mutual impedance × AB AC BC matrices of two different subsystems. The 2 2 submatrices Z , ZBr and Z are the mutual impedance × Ar Cr matrices between the one-phase subsystem and the armor. Given the structural symmetry, the above submatrices have the relationships as follows   ZA = ZB = ZC,   ZAB = ZAC = ZBC , (6)   ZAr = ZBr = ZCr.   YA = YB = YC,   YAB = YAC = YBC , (7)   YAr = YBr = YCr. The impedances and admittances matrices in Equations (6) and (7) were analyzed in References [41–45]. Each conductor has its own self-impedance, and there are mutual impedances between any two conductors. The mutual admittances exist only between directly adjacent conductors. For example, there is a mutual admittance between the core and the shield, but no mutual admittance between the core and the armor because of the shield. Furthermore, the aforementioned submatrices (taking partial submatrices for examples) can be expressed as: " # " # Zc Zcs Zu Zu h i ZA = , ZAB = , ZAr = Zur Zur . (8) Zcs Zs Zu Zu " # " # Yc Ycs 0 0 h i YA = − , YAB = , YAr = 0 Yur . (9) Ycs Ys 0 Yu − − where both the rows and columns of the matrices in Equations (8) and (9) are arranged in the order of the core, the shield. Zc and Zs are the self-impedances of the core and the shield, respectively. Zcs is the mutual impedance between the core and the shield. Zu is the mutual impedance between the cores or shields of two different subsystems. Zur is the mutual impedance between the core (or the shield) and the armor. Yc and Ys are the self-admittances of the core and the shield, respectively. Ycs is the mutual admittance between the core and the shield. Yu is the mutual admittance between the shields of two different subsystems. Yur is the mutual admittance between the shield and the armor. The ATP-EMTP (Version 6.0, NTNU/SINTFE, Trondheim, Norway, 2015) software provides a routine named Cable Constants to calculate the p.u.l. parameters of cables [42]. This routine was Electronics 2019, 8, 571 5 of 18 adopted in this work. For example, the p.u.l. impedance and admittance matrices of the YJV22-type cable at 300 kHz are shown in AppendixB. The first-order phasor equations in Equation (1) can be expressed in the form of second-order ordinary differential equations:  2  d V(x) = ZYV(x),  dx2  2 (10)  d I(x) = YZI(x). dx2 To solve the differential equations, the actual phasor line voltages and currents are transformed to modal quantities, as described in Reference [13]: ( V(x) = QVm(x), (11) I(x) = SIm(x). where Vm and Im are the modal voltage vector and the modal current vector, respectively. The complex matrices Q and S are the corresponding modal transformation matrices, which define changes in the variables between the actual phasor line voltages and currents, and the modal voltages and currents. Substituting Equation (11) into Equation (10) gives:

 2  d Vm(x) = Q 1ZYQVm(x) = H2Vm(x),  dx2 −  2 (12)  d Im(x) = S 1YZSIm(x) = H2Im(x). dx2 − where Q and S are the matrices chosen to diagonalize ZY and YZ, respectively, via similarity transformations. H = diag(γ , γ , , γ ), where the diagonal elements are the eigenvalues of 1 2 ··· 7 the matrices ZY and YZ, or are known as the propagation constants of the modal quantities. Then, Equation (12) has a simple solution, transforming the solution back to the actual line voltages and currents via Equation (11) gives:     ( ) = m Hx + m Hx  V x ZcS I f e− Ib e ,    (13)   I(x) = S Ime Hx ImeHx . f − − b where subscripts f and b represent the forward and backward modal quantities, respectively. 1 1 Zc = ZSH− S− is the CIM of the uniform MV underground cable. The MV underground cable can be viewed as having 2 7 ports, with seven ports at the sending × end (x = 0) and seven ports at the receiving end (x = l), where l is the length of the cable. According to Equation (13), when one end of the underground cable is connected to the CIM, the access impedance matrix of the other end is equal to the CIM. In addition, the phasor voltages and currents at the sending end and the receiving end have the relationship [13] as: " # " # V(l) V(0) = T . (14) I(l) I(0) where T is the transmission matrix, and can be expressed as:

" T T # Qcos h(Hl)S Qsin h(Hl)S Zc T = 1 T −1 T . (15) Z− Qsin h(Hl)S Z− Qcos h(Hl)S Zc − c c 3. Characteristic Impedance Matrices with Grounding Points For safe operation, the shields and armors of the MV underground cables are grounded at the RMUs, resulting in the whole cable structure being non-uniform, such that the above uniform transmission line model is no longer valid. In this section, the CIMs of the entire underground cable containing the grounding points are provided based on an eigen analysis of the overall transmission Electronics 2019, 8, 571 6 of 18 matrix of this non-uniform structure. The shields and armors are assumed to be grounded with grounding resistances only at the RMUs, of which the positions are arbitrary. As described, the whole cable structure between the transceivers is divided into k sections by the grounding points, an example being when k = 3, as shown in Figure2. Here, the three phase subsystems of the MV underground cables are drawn vertically to show the grounding structure Electronicsmore clearly. 2019, 8, x FOR PEER REVIEW 6 of 19

l1 RMUl2 RMU l3

x Rg Rg Armor Shield Core

Figure 2. Structure of medium-voltage underground cables with shields and armors grounding at the Figure 2. Structure of medium-voltage underground cables with shields and armors grounding at the ring main units (RMUs). ring main units (RMUs). Assuming that the type and length of each section is arbitrary, and the total length is L, the voltage Assuming that the type and length of each section is arbitrary, and the total length is L, the and current vectors at the sending end (x = 0) and the receiving end (x = L) have the following relations: voltage and current vectors at the sending end (𝑥=0) and the receiving end (𝑥=𝐿) have the following relations: " # " # k 1 " # V(L) V(0) Y− V(0) = T = (TjTg)Tk , (16) I(L𝑽)(𝐿) I𝑽((00)) 𝑽I((00) =𝑻 =(𝑻j=1 𝑻 )𝑻 , (16) 𝑰(𝐿) 𝑰(0) 𝑰(0) where: " # where: AB T = . (17) CD𝑨 𝑩 𝑻 = . (17) 𝑪𝑫 where T is the overall transmission matrix, and A, B, C and D are the sub-blocks. Tj is the transmission wherematrix 𝑻 of theis thejth cableoverall section transmission(j = 1, 2, matrix,, k), whichand A can, B, be C calculatedand D are from the Equationsub-blocks. (15). 𝑻T𝑗 gisis the the ··· transmissiontransmission matrix matrix of of the the jth grounding cable section structure, (𝑗=1,2,⋯,𝑘 which can), which be written can be as: calculated from Equation (15). 𝑻 is the transmission matrix of the grounding structure, which can be written as: 𝑔 " # E 0 Tg = 𝑬𝟎. (18) 𝑻 =Y E . (18) −−𝒀in𝒊𝒏 𝑬   where 𝒀 = diag(0,0,0, 𝐺,𝐺,𝐺,𝐺) is the input admittance matrix of the grounding structure, where Yin = diag 0, 0, 0, Gg, Gg, Gg, Gg is the input admittance matrix of the grounding structure, 𝐺 =1/𝑅; 𝑅 is the grounding resistance; and 𝑬 is the 7 × 7 identity matrix. Gg = 1/Rg; Rg is the grounding resistance; and E is the 7 7 identity matrix. × IfIf the the positive positive direction direction of of the the current current is is the the di directionrection flowing flowing into into the the port, port, then then the the impedance impedance matrixmatrix 𝒁Z of of thethe entireentire cablecable structurestructure cancan bebe expressedexpressed as:as: 𝑽(0) 𝑰(0) 𝒁 𝒁 𝑰(0) " # =𝒁" #=" 𝟏𝟏 𝟏𝟐#" , # (19) V(0) ( ) I(0)( ) Z11 Z12 (I(0)) 𝑽 𝐿= Z −𝑰 𝐿 = 𝒁𝟐𝟏 𝒁𝟐𝟐 −𝑰 𝐿 , (19) V(L) I(L) Z21 Z22 I(L) where: − − where:  𝒁 =−𝑪𝟏𝑫,  ⎧ Z 𝟏𝟏= C 1D,  ⎪ 11 − 𝟏  𝒁𝟏𝟐 =−𝑪− 1 ,  Z12 = C− ,𝟏 (20) ⎨𝒁𝟐𝟏 =𝑩−− 𝑨𝑪 𝑫, (20)  = 1  ⎪Z21 B AC−𝟏D,  ⎩ 𝒁𝟐𝟐 =−− 𝑨𝑪1 .  Z22 = AC− . The MV underground cable considered here− is passive, thereby satisfying the condition of The MV underground cable considered here is passive, thereby satisfying the condition of reciprocity, such that the impedance matrix is symmetrical: reciprocity, such that the impedance matrix is symmetrical: 𝒁𝟏𝟏 =𝒁𝟏𝟏, 𝒁𝟏𝟐 =𝒁𝟐𝟏, (21) 𝒁𝟐𝟐 =𝒁𝟐𝟐. The eigen analysis of the overall transmission matrix 𝑻 is based on the equation: 𝑨 −𝝀𝑬 𝑩 𝑱 𝟎 = . (22) 𝑪𝑫−𝝀𝑬𝑲 𝟎 where 𝜆 is the eigenvalue of 𝑻 and [𝑱𝑲] is the corresponding eigenvector. The elimination of 𝑲 using the first equation in Equation (22) gives:

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 T  Z = Z ,  11 11  T  Z12 = Z , (21)  21  T  Z22 = Z22. The eigen analysis of the overall transmission matrix T is based on the equation: " #" # " # A λEB J 0 − = . (22) CD λE K 0 − h iT where λ is the eigenvalue of T and JK is the corresponding eigenvector. The elimination of K using the first equation in Equation (22) gives:

h 2 1  1 1 1 i λ C− λ C− D + AC− + AC− D B J = 0. (23) − − Then, combining Equation (20) with Equation (23), it follows that:

h 2   i λ Z12 + λ Z11 + Z22 + Z21 J = 0. (24)

Equation (24) is then divided by λ2 and transposed; so, it follows that:

h 2 1  i λ− Z12 + λ− Z11 + Z22 + Z21 J = 0. (25)

1 Equations (24) and (25) indicate that both λ and λ− are the eigenvalues of T, so the eigenvalue matrix Λ of the overall transmission matrix T can be denoted as:

   1 1 1 Λ = diag Λ , Λ = diag λ , λ , , λn, λ− , λ− , , λ− . (26) f b 1 2 ··· 1 2 ··· n 1 where Λ and Λ are the 7 7 diagonal matrices, Λ = Λ− . f b × b f From the diagonalization of the overall transmission matrix T, it follows that: " # " # 1 Λ f 0 Q f Qb P− TP = , P = . (27) 0 Λb S f Sb where P is the nonsingular matrix, and Q f , Qb, S f and Sb are the sub-blocks. Substituting Equation (27) into Equation (16) gives: " # " # " # V(L) Λ f 0 1 V(0) = P P− . (28) I(L) 0 Λb I(0) or in the form of: " # " #" # V(x) Q Q M (x) = f b f , (29) I(x) S f Sb Mb(x) " # " #" # M (L) Λ 0 M (0) f = f f . (30) Mb(L) 0 Λb Mb(0) where x = 0 or x = L. Compared to Equation (13), when x = 0 or x = L, M f and Mb are the equivalent forward and backward modal quantities, respectively. Λ f and Λb represent the relationships between them at both ends, so Λ f and Λb can be denoted as:   Λ = e Hx,  f − (31)  Hx  Λb = e . Electronics 2019, 8, 571 8 of 18

  where H = diag γ , γ , , γ is the equivalent propagation constant matrix. The voltage and current 1 2 ··· 7 vectors at both ports can be represented by the equivalent modal currents, such that:   M (x) = Im(x)  f f ,  (32)  M (x) = Im(x). b − b m m where I f and Ib are the equivalent forward and backward modal current vectors, respectively. By substituting Equation (32) into Equations (29) and (30), the voltage and current vectors are:   V(x) = Z S Ime Hx + Z S ImeHx,  c, f f f − c,b b b  (33)  I(x) = S Ime Hx S ImeHx. f f − − b b where x = 0 or x = L, and:   = 1  Zc, f Q f S−f ,  1 (34)  Z = Q S− . c,b − b b Equation (33) has the same form as Equation (13). S f and Sb are the transformation matrices of the forward and backward modal currents, respectively; Q f and Qb are the transformation matrices of the forward and backward modal voltages, respectively; and Zc, f and Zc,b are the forward characteristic impedance matrix (FCIM) and the backward characteristic impedance matrix (BCIM) of the whole cable structure, respectively. This indicates that the carrier signals at both ends of the MV underground cables with grounded shields and armor can still be regarded as the superposition of a series of modal quantities, as can the uniform MV underground cables without grounding points. Equation (33) only applies to x = 0 or x = L. There is no definite equivalence relation between S f and Sb, or between Zc, f and Zc,b, which indicates that the characteristic impedance matrices of the whole cable structure are different from those of one cable section. The former has two different characteristic impedance matrices, being the forward and backward impedance matrices, whereas the latter only has one unique characteristic impedance matrix. Additionally, the latter can be regarded as a special case of the former when the transformation matrices of the forward and backward modal quantities are equal. Equation (33) shows that the access impedance matrix at the sending end is equal to Zc, f only if the receiving end is connected to Zc, f . Similarly, the access impedance matrix of the receiving end is equal to Zc,b only if the sending end is connected to Zc,b.

4. Method Validation In this section, two examples are provided to validate the calculation of the characteristic impedance matrices of the whole cable structure between the PLC transceivers. Both examples possess a structure similar to that shown in Figure2. Example 1 has three sections of cables, l1 = 100 m , l2 = 300 m and l3 = 500 m, whereas example 2 has two sections of cables, l1 = l2 = 300 m. All the cables are of the YJV22-type MV underground cables described in Section2, and the grounding resistances ( Rg) are 1 Ω. As shown in Equation (33), if and only if the receiving end of the whole cable structure is connected to the FCIM, then the access impedance matrix of the sending end is equal to the FCIM. Similarly, if and only if the sending end of the whole cable structure is connected to the BCIM, then the access impedance matrix of the receiving end is equal to the BCIM. These characteristics were used to validate the calculation results of the FCIM and BCIM. For instance, take the validation of the FCIM in example 1. Firstly, the receiving end of the whole cable in example 1 was connected to the calculated FCIM. Then, the access impedance matrix of the sending end was measured. The calculated FCIM was validated if it was equal to the measured access impedance matrix. The principle for the validation of the BCIM was similar, the sending end of the whole cable was connected to the calculated BCIM, and the access impedance matrix of the receiving end was measured. Electronics 2019, 8, 571 9 of 18

The measurement setup is shown in Figure3. Each cable section in examples 1 and 2 was simulated using 50 cascaded π circuits, which were welded using lumped parameter components in the laboratory. The parameters of the π circuits could be easily calculated using the p.u.l. parameters and the cable lengths. The receiving or sending end was connected to the lumped electrical networks where the impedance matrix was equal to the calculated FCIM or BCIM, respectively. These electrical networks are welded in the laboratory, and are labelled the FCIM network and BCIM network in this paper, respectively. Finally, the access impedance matrix of the sending or receiving end of the cables was measured using a HIOKI 3532-50 LCR meter. Given the multiport characteristics of the cables, the measurements were recorded for each port of the sending or receiving end. Each measurement Electronics 2019, 8, x FOR PEER REVIEW 9 of 19 was repeated three times, where the average value was taken as the measurement value.

1 1 Sending end: 2 Cable model: 2 Receiving end: ...... LCR meter 7 Cascaded π circuits 7 FCIM network

x = 0 x = L (a) 1 1 Sending end: 2 Cable model: 2 Receiving end: ...... BCIM network 7 Cascaded π circuits 7 LCR meter

x = 0 x = L (b) Figure 3. Measurement setup. The measurement of (a) the sending port’s access impedance matrix Figureand (b) 3. the Measurement receiving port’s setup. access The impedance measurement matrix. of (a) the sending port’s access impedance matrix and (b) the receiving port’s access impedance matrix. In this study, the FCIMs and BCIMs of two examples were calculated over the frequency of 10 kHz–500In this study, kHz, whichthe FCIMs was sampledand BCIMs into of 246 two frequency examples points were withcalculated a sample over intervalthe frequency of 2 kHz. of 10kHz–500kHz,For measurements, which the abovewas sampled procedures into were246 frequency also carried points out at with some a sample typical frequenciesinterval of 2 within kHz. For the measurements,same frequency the range. above The procedures selected frequencies were also included carried thoseout at at some which typical the real frequencies or imaginary within parts the of samethe calculated frequency characteristic range. The selected impedances frequencies reached included the local those maximums at which or minimumthe real or values,imaginary the otherparts offrequency the calculated points werecharacteristic distributed impedances evenly within reached the frequencythe local maximums ranges that or did minimum not contain values, the local the othermaximum frequency or minimum points were values. distributed evenly within the frequency ranges that did not contain the local Inmaximum Figures4 or and minimum5, the solid values. lines represent the numerical results of the FCIM and BCIM for exampleIn Figures 1. The crosses4 and 5, show the thesolid measurement lines represent results the ofnumerical the access results impedance of the matrices FCIM and of the BCIM sending for exampleand receiving 1. The ports. crosses Specifically, show theZc (measurement1, 1), Zc(4, 4) and resultsZc(7, of 7) theare theaccess characteristic impedance impedances matrices of of the sendingcore, the shieldand receiving and the armor,ports. respectively.Specifically, Z𝒁c𝒄((1,1)2, 4) is, the𝒁𝒄(4,4) off-diagonal and 𝒁𝒄(7,7) element are in the the characteristiccharacteristic impedancesimpedance matrices, of the core, which the shield results and from the the armor, mutual respectively. impedance 𝒁𝒄 between(2,4) is the the off-diagonal shields of the element A-phase in theand characteristic B-phase. impedance matrices, which results from the mutual impedance between the shields of theFigure A-phase4 shows and B-phase. that when the receiving port of the cable was connected to the calculated FCIM, 600 600 300 300 the measured500 access impedance500 matrix of the sending200 port matched well with200 the calculated FCIM, Ω Ω () () Ω Ω () indicating() 400 the credibility of400 the FCIM numerical results.100 Similar conclusions100 were obtained when 300 300 0 0 Real comparingReal the numerical results and the measured results of the BCIM, as shown in Figure5. At high 200 200 -100 -100 Imaginary Imaginary frequencies,100 the differences100 between the numerical results-200 and the measured-200 results were larger than 0 0 -300 -300 the differences0 100 200 at 300 low 400 frequencies. 500 0 100 This 200 300 happened 400 500 because0 the 100 approximation 200 300 400 500 degree0 100 of 200 the 300 cascaded 400 500 π Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) circuits decreased with increasing frequency.Z (4,4) Zc (1,1) c Zc (1,1) Zc (4,4) 200 250 300 300 For example 2, the numerical results are shown in200 Figure6. The FCIM and200 BCIM in example 2 Ω Ω () 150 200 () 100 100 Ω Ω

were() the same. Then, the same measurements were recorded, and Figure7 provides a comparation () 150 100 0 0 100 Real of the numerical results andReal the measured results, which-100 also demonstrates-100 the credibility of the Imaginary 50 50 Imaginary numerical results of the characteristic impedance matrices.-200 -200 0 0 -300 -300 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) Z (7,7) c Zc (2,4) Zc (7,7) Zc (2,4) Numerical results Measured results Numerical results Measured results (a) (b)

Figure 4. The validation of the forward characteristic impedances of medium-voltage underground cables with grounded shields and armor in example 1: (a) The real parts and (b) the imaginary parts of the characteristic impedances.

Electronics 2019, 8, x FOR PEER REVIEW 9 of 19

1 1 Sending end: 2 Cable model: 2 Receiving end: ...... LCR meter 7 Cascaded π circuits 7 FCIM network

x = 0 x = L (a) 1 1 Sending end: 2 Cable model: 2 Receiving end: ...... BCIM network 7 Cascaded π circuits 7 LCR meter

x = 0 x = L (b)

Figure 3. Measurement setup. The measurement of (a) the sending port’s access impedance matrix and (b) the receiving port’s access impedance matrix.

In this study, the FCIMs and BCIMs of two examples were calculated over the frequency of 10kHz–500kHz, which was sampled into 246 frequency points with a sample interval of 2 kHz. For measurements, the above procedures were also carried out at some typical frequencies within the same frequency range. The selected frequencies included those at which the real or imaginary parts of the calculated characteristic impedances reached the local maximums or minimum values, the other frequency points were distributed evenly within the frequency ranges that did not contain the local maximum or minimum values. In Figures 4 and 5, the solid lines represent the numerical results of the FCIM and BCIM for example 1. The crosses show the measurement results of the access impedance matrices of the sending and receiving ports. Specifically, 𝒁𝒄(1,1) , 𝒁𝒄(4,4) and 𝒁𝒄(7,7) are the characteristic impedances of the core, the shield and the armor, respectively. 𝒁𝒄(2,4) is the off-diagonal element in Electronics 2019, 8, 571 10 of 18 the characteristic impedance matrices, which results from the mutual impedance between the shields of the A-phase and B-phase. 600 600 300 300 500 500 200 200 Ω Ω () () Ω Ω () () 400 400 100 100 300 300 0 0 Real Real 200 200 -100 -100 Imaginary Imaginary 100 100 -200 -200 0 0 -300 -300 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) Z (4,4) Zc (1,1) c Zc (1,1) Zc (4,4) 200 250 300 300 200 200 Ω Ω () 150 200 () 100 100 Ω Ω () () 150 100 0 0 100 Real Real -100 -100 Imaginary 50 50 Imaginary -200 -200 0 0 -300 -300 0 100Electronics 200 300 2019, 400 8, x 500FOR PEER0 REVIEW 100 200 300 400 500 0 100 200 300 400 500 0 100 20010 of 30019 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) Z (7,7) 350c 350 Zc (2,4) 150 Zc (7,7) 150 Zc (2,4) 100 100 300 Numerical results Measured300 results Numerical results Measured results Ω Ω

250 250 () 50 () 50 Ω Ω () () 200 200 0 0 -50 Real -50 Real 150 (a) 150 (b) 100 100 -100 -100 Imaginary Imaginary 50 50 -150 -150 Figure 4. The0 validation of the 0 forward characteristic -200 impedances of medium-voltage-200 underground 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 cables with groundedFrequency (kHz) shieldsshields andand armor armorFrequency in in(kHz) example example 1: 1: ( a()a) The TheFrequency real real parts (kHz) parts and and (b ()b the) Frequencythe imaginary imaginary (kHz) parts parts of Z (1,1) Z (4,4) Z (4,4) Electronics 2019, 8, x FOR PEERc REVIEW c Zc (1,1) c 10 of 19 100 150 60 100 ofthe the characteristic characteristic impedances. impedances. 40 80 Ω Ω 50 () 100 () 20 Ω Ω

60 () () 1500 150 350 350 0 40 -20100 100 Real 300 Real 300 50 -40 Ω Ω

Imaginary -50 Imaginary

250 20 250 () 50 () 50 Ω Ω -60 () () 200 0 200 0 -80 0 -100 0 0 100 200 300 400 500 0 100 200 300 400 500 -500 100 200 300 400 500 0 100 200 300 400 500 Real -50 Real 150 150 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) 100 100 -100 -100 Zc (7,7) Zc (2,4) Imaginary Zc (7,7) Imaginary Zc (2,4) 50 50 -150 -150 Numerical results Measured results Numerical results Measured results 0 0 -200 -200 0 100 200 300 400 500 (0a) 100 200 300 400 500 0 100 200 300(b 400) 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) Z (1,1) Z (4,4) Z (4,4) Figurec 5. The validation of the backwardc characteristic impedances of medium-voltageZc (1,1) underground c 100 150 60 100 40 80 cables with grounded shields and armor in example 1: (a) The real parts and (b) the imaginary parts Ω Ω 50 () of the characteristic100 impedances. () 20 Ω Ω

60 () () 0 0 40 -20 Real Real 50 Figure 4 shows that when the receiving port of the cable-40 was connected to the calculated FCIM,

Imaginary -50 20 Imaginary the measured access impedance matrix of the sending port-60 matched well with the calculated FCIM, 0 0 -80 -100 0 100indicating 200 300 the 400 credibility 500 0 of 100 the 200 FCIM 300 numerical 400 500 results. Similar0 100 200conclusions 300 400 were 500 obtained0 100 200when 300 400 500 comparingFrequency (kHz) the numerical resultsFrequency and the(kHz) measured results of the BCIM,Frequency as (kHz) shown in Figure 5. FrequencyAt high (kHz) Z (7,7) Z (2,4) Z (7,7) Z (2,4) frequencies,c the differences betweenc the numerical results and the measuredc results were larger thanc Numerical results Measured results Numerical results Measured results the differences at low frequencies. This happened because the approximation degree of the cascaded π circuits decreased(a) with increasing frequency. (b) For example 2, the numerical results are shown in Figure 6. The FCIM and BCIM in example 2 Figure 5. Figurewere 5. The the validation same. Then, of the the same backward measurements characteristic were recorded, impedances and Figure of medium-voltage 7 provides a comparation underground cablesof withwith the groundednumericalgrounded results shieldsshield ands and and the armor armor measured in in example example results, 1: which1: (a ()a The) Thealso real realdemonstrates parts parts and and (theb ()b the)credibility the imaginary imaginary of the parts parts of theof the characteristic numericalcharacteristic results impedances. impedances. of the characteristic impedance matrices.

600 600 Figure 4 shows500 that when the receiving port of the500 cable was connected to the calculated FCIM,

Ω 400 Ω 400 () () the measured access300 impedance matrix of the sending 300port matched well with the calculated FCIM, Real indicating the 200credibility of the FCIM numerical results.Real 200 Similar conclusions were obtained when 100 100 comparing the numerical0 results and the measured results0 of the BCIM, as shown in Figure 5. At high frequencies, the300 differences between the numerical results300 and the measured results were larger than 200 200 Ω Ω () the differences() 100at low frequencies. This happened because100 the approximation degree of the cascaded π circuits decreased0 with increasing frequency. 0 -100 -100 Imaginary For exampleImaginary -200 2, the numerical results are shown in-200 Figure 6. The FCIM and BCIM in example 2 -300 -300 were the same. Then,0 the 100 same 200 measurements 300 400 500 were recorded,0 100and Figure 200 3007 provides 400 a 500 comparation of the numerical results andFrequency the (kHz)measured results, which also demonstratesFrequency (kHz) the credibility of the Z (1,1) Z (4,4) Z (7,7) Z (2,4) Z (1,1) Z (4,4) Z (7,7) Z (2,4) numerical results ofc the characteristicc c impedancec matrices. c c c c (a) (b) 600 600 Figure 6. TheFigure characteristic 6. The characteristic impedances impedances of medium-voltage of medium-voltage underground underground cablescables with with grounded grounded shields 500 500 and armorshields in example and armor 2: in (a )example The numerical 2: (a) The numerical resultsand results (b )and the (b numerical) the numerical results results of of the the backward

Ω 400 Ω 400

() backward characteristic impedances. () characteristic300 impedances. 300 Real 200 Real 200 100 100 0 0 300 300

200 200 Ω Ω () () 100 100 0 0 -100 -100 Imaginary Imaginary -200 -200 -300 -300 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Z (1,1) Z (4,4) Z (7,7) Z (2,4) Z (1,1) Z (4,4) Z (7,7) Z (2,4) c c c c c c c c (a) (b)

Figure 6. The characteristic impedances of medium-voltage underground cables with grounded shields and armor in example 2: (a) The numerical results and (b) the numerical results of the backward characteristic impedances.

Electronics 2019, 8, 571 11 of 18 Electronics 2019, 8, x FOR PEER REVIEW 11 of 19

600 600 300 300 500 500 200 200 Ω Ω () () Ω Ω 100 100 () () 400 400 300 300 0 0 Real Real 200 200 -100 -100 100 100 Imaginary -200 Imaginary -200 0 0 -300 -300 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz)

Zc (1,1) Zc (4,4) Zc (1,1) Zc (4,4) 400 400 150 150 100 100 Ω 300 300 Ω () () 50 50 Ω Ω () () 200 200 0 0 Real Real -50 -50

100 100 Imaginary Imaginary -100 -100 0 0 -150 -150 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz)

Zc (7,7) Zc (2,4) Zc (7,7) Zc (2,4) Numerical results Measured results Numerical results Measured results (a) (b)

Figure 7. The validation of the forwardforward characteristiccharacteristic impedances of medium-voltagemedium-voltage underground cables with groundedgrounded shieldsshields and and armor armor in in example example 2: 2: ( a()a The) The real real parts parts and and (b ()b the) the imaginary imaginary parts parts of theof the characteristic characteristic impedances. impedances.

5. Analysis of the Characteristic Impedance Matrices In this section, the characteristics of the FCIM and BCIM of the MV underground cables with grounded shields shields and and armors armors are are analyzed analyzed based based on on the the results results presented presented in Section in Section 4, 4and, and on onthe theinvestigation investigation of Equations of Equations (33) and (33) (34). and To (34). determin To determinee the effects the of e differentffects of difactorsfferent (e.g., factors the high- (e.g., thefrequency high-frequency loss and lossgrounding and grounding resistances), resistances), more experiments more experiments under underdifferent diff scenarioserent scenarios have havebeen beenperformed. performed. 5.1. Characteristics of the Forward and Backward Characteristic Impedance Matrices 5.1. Characteristics of the Forward and Backward Characteristic Impedance Matrices As shown in Figures4–7, the MV underground cables with grounded shields and armor have As shown in Figures 4–7, the MV underground cables with grounded shields and armor have FCIM and BCIM that are generally not equal, as shown in Equation (34). However, if the cable is FCIM and BCIM that are generally not equal, as shown in Equation (34). However, if the cable is longitudinally symmetrical, its FCIM and BCIM are equal, because the sending and receiving ends can longitudinally symmetrical, its FCIM and BCIM are equal, because the sending and receiving ends be interchanged when the cable is longitudinally symmetrical. can be interchanged when the cable is longitudinally symmetrical. Using the experimental results, we simultaneously checked that if the receiving end or sending end Using the experimental results, we simultaneously checked that if the receiving end or sending of the whole cable structure was connected to the FCIM or BCIM, the access impedance matrix at the end of the whole cable structure was connected to the FCIM or BCIM, the access impedance matrix other end was equal to the FCIM or BCIM. Combining these results with Equation (33), we concluded at the other end was equal to the FCIM or BCIM. Combining these results with Equation (33), we that if the receiving or sending port of the whole cable structure was connected with the FCIM or concluded that if the receiving or sending port of the whole cable structure was connected with the BCIM, the backward or forward modal quantities would be zero, and thus the signal power would be FCIM or BCIM, the backward or forward modal quantities would be zero, and thus the signal power completely absorbed by the transceiver. Therefore, the characteristic impedance matrices of the whole would be completely absorbed by the transceiver. Therefore, the characteristic impedance matrices cable structure between the transceivers should be considered to reduce or eliminate the reflected of the whole cable structure between the transceivers should be considered to reduce or eliminate the components of the carriers at the receivers or transmitters. reflected components of the carriers at the receivers or transmitters. The characteristic impedances of the cores and shields were almost the same, indicating that the The characteristic impedances of the cores and shields were almost the same, indicating that the grounding of the shields not only affected their own characteristic impedances, but they also had a grounding of the shields not only affected their own characteristic impedances, but they also had a similar influence on the characteristic impedances of the cores covered by them owing to the mutual similar influence on the characteristic impedances of the cores covered by them owing to the mutual inductance between the cores and the shields. inductance between the cores and the shields. Lastly, the results in Section4 demonstrated that resonance occurred when the shields and armor of Lastly, the results in Section 4 demonstrated that resonance occurred when the shields and MV underground cable were grounded. At each resonant frequency, the real parts of the characteristic armor of MV underground cable were grounded. At each resonant frequency, the real parts of the impedances showed high peaks, and the imaginary parts (phase angles) were zero, whereas the whole characteristic impedances showed high peaks, and the imaginary parts (phase angles) were zero, cable structure was pure resistive, as shown in Figures4–7. These typical resonance phenomena whereas the whole cable structure was pure resistive, as shown in Figures 4–7. These typical signified that the electromagnetic waves at the resonant frequencies were standing waves, and the resonance phenomena signified that the electromagnetic waves at the resonant frequencies were electromagnetic energy in the cables could not be transmitted, which hindered the transmission of standing waves, and the electromagnetic energy in the cables could not be transmitted, which carrier signals and adversely affected the impedance matching. hindered the transmission of carrier signals and adversely affected the impedance matching. To further study the above resonance characteristics, a control experiment was performed in which the characteristic impedance matrices from example 2 were calculated without the high-

Electronics 2019, 8, 571 12 of 18

Electronics 2019, 8, x FOR PEER REVIEW 12 of 19 To further study the above resonance characteristics, a control experiment was performed in which thefrequency characteristic loss. The impedance results of matrices the FCIM from are example shown 2in were Figure calculated 8, which without shows the that high-frequency the characteristic loss. Theimpedances results of without the FCIM the are loss shown had ain series Figure of 8resonant, which shows points. that The the resonant characteristic frequencies impedances were consistent without thewith loss the had frequencies a series of in resonant Figures points. 6 and The7, confirming resonant frequencies the above were judgment consistent and withindicating the frequencies that the ingrounding Figures6 andof 7the, confirming shields and the abovearmor judgment would lead and indicatingto resonance, that thereflecti groundingng the of multi-resonance the shields and armorcharacteristics. would lead Figure to resonance, 8 shows reflectingthat the thehigh-frequency multi-resonance loss characteristics. had inhibitory Figure effects8 shows on the that above the high-frequencyresonance, which loss made had the inhibitory curve around effects onthe the reso abovenance resonance, points smoother which made and reduces the curve the around resonant the resonanceamplitudes. points The higher smoother the resonant and reduces frequency, the resonant the larger amplitudes. the high-frequency The higher loss, the resonant and thus, frequency, the more theevident larger the the inhibitory high-frequency effects. loss, and thus, the more evident the inhibitory effects.

1200 1200 1500 1500 1000 1000 1000 1000 800 800 Ω Ω

Ω 500 500 () Ω () () () 600 600 0 0 Real Real 400 400 -500 -500 200 200 Imaginary -1000 Imaginary -1000 0 0 -1500 -1500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) Z (4,4) Zc (1,1) c Zc (1,1) Zc (4,4) 600 1000 1000 1500 500 800 1000 Ω Ω 500 Ω Ω () () () 400 () 600 500 300 0 0 Real Real 400 200 -500 -500 200 Imaginary Imaginary 100 -1000 0 0 -1000 -1500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) Z (7,7) Z (2,4) c c Zc (7,7) Zc (2,4) Lossy Lossless Lossy Lossless (a) (b)

Figure 8. The comparation of the forward characteristic impedances when the cables are lossy and lossless: ( (a)) The The real real parts and ( b) the imaginary parts of the characteristic impedances. 5.2. Effects of the Network Structures and Parameters 5.2. Effects of the Network Structures and Parameters It is fundamental for the PLC system to adapt to various power line networks. To demonstrate It is fundamental for the PLC system to adapt to various power line networks. To demonstrate the effects of different MV network structures and parameters, a series of control experiments were the effects of different MV network structures and parameters, a series of control experiments were conducted and are outlined in the following. The type and parameters of the cables used were the conducted and are outlined in the following. The type and parameters of the cables used were the same as those for the MV underground cables used in Section4. same as those for the MV underground cables used in Section 4. 5.2.1. Effects of Grounding Resistances 5.2.1. Effects of Grounding Resistances To analyze the effects of grounding resistances on the characteristic impedance matrices, To analyze the effects of grounding resistances on the characteristic impedance matrices, the the characteristic impedance matrices from example 1 with different grounding resistances (Rg) characteristic impedance matrices from example 1 with different grounding resistances (𝑅 ) were were calculated. calculated.The elements in the FCIMs are shown in Figure9. When the shields and armor are ungrounded, thenThe the cableselements can in be the regarded FCIMs are as uniformshown in transmission Figure 9. When lines, the and shields thus and there armor is no are resonance ungrounded, in the characteristicthen the cables impedances. can be regarded When as the uniform shields transmis and armorsion arelines, grounded, and thus the there characteristic is no resonance impedance in the matricescharacteristic show impedances. resonance. Furthermore,When the shields the smallerand armor the are grounding grounded, resistance, the characteristic the higher impedance the peaks ofmatrices the characteristic show resonance. impedances Furthermore, at resonant the smaller frequencies. the grounding However, resistance, the differences the higher in the the grounding peaks of resistancesthe characteristic do not impedances affect the resonant at resonant frequencies. frequencies. However, the differences in the grounding resistances do not affect the resonant frequencies. 5.2.2. Effects of Cable Lengths To analyze the effects of the cable lengths on the characteristic impedance matrices, the characteristic impedance matrices from example 2 for different total cable lengths were calculated, assuming that the shields and armor were grounded at the half-length of the whole cables, Rg = 1 Ω. The elements in the FCIMs are shown in Figure 10, which shows that the larger the total lengths of the cables, the more the resonant points shown by the characteristic impedances. Further calculations

Electronics 2019, 8, 571 13 of 18 showed that all the electric lengths of the cables at the resonant frequencies were in the sequence of 0.087, 0.147, 0.211, 0.296, 0.945, 1.607, etc. This indicated that the resonant frequencies under the same structure were determined by the electric length of the whole cables, which was easy to understand considering the resonant characteristics. With increases in the cable lengths, the loss increases, the inhibition effect of high-frequency loss on the resonance gradually strengthens, and thus theElectronics peaks 2019 at, the 8, x resonantFOR PEER frequenciesREVIEW gradually decrease. 13 of 19

600 600 300 300 500 500 200 200 Ω Ω () () Ω 400 Ω 400 100 100 () () 300 300 0 0 Real Real 200 200 -100 -100 Imaginary 100 100 Imaginary -200 -200 0 0 -300 -300 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz)

Zc (1,1) Zc (4,4) Z (1,1) Zc (4,4) 200 250 300 c 300 200 200 200 Ω

150 Ω () ()

Ω Ω 100

() () 100 150 100 0 0

Real Real 100 -100 -100

50 Imaginary 50 -200 Imaginary -200 0 0 -300 -300 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz)

Zc (7,7) Zc (2,4) Zc (7,7) Zc (2,4) R =Ω1 R =Ω5 R =Ω1 R =Ω5 g g Ungrounded g g Ungrounded (a) (b)

Figure 9.9.The The forward forward characteristic characteristic impedances impedances of medium-voltage of medium-voltage underground underground cables with cables diff erentwith Electronics 2019, 8, x FOR PEER REVIEW 14 of 19 groundingdifferent grounding resistances: resistances: (a) The real (a parts) The andreal (bparts) the and imaginary (b) the parts imaginary of the characteristicparts of the characteristic impedances.

1000impedances. 600 600 300 800 500 400 200 Ω Ω Ω Ω () () 100 () 200 () 400 600 5.2.2. Effects of Cable Lengths300 0 0 Real Real 400 200 -200 -100

To200 analyze the effects100 of the cable lengths onImaginary -400 the characteristic impedanceImaginary -200 matrices, the 0 0 -600 -300 characteristic0 100 200impedance 300 400 500 matrices0 100 from 200 300 example 400 500 2 for different0 100 200 total 300 cable 400 500 lengths0 100 were 200 300 calculated, 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) assuming that the shields and armor were grounded at the half-length of the whole cables, 𝑅 =1 Ω. l = 200m l = 600m l = 200m l = 600m 500 400 300 150 The elements in the FCIMs are shown in Figure 10, which shows that the100 larger the total lengths 400 200

300 Ω Ω Ω () () Ω 50 () of the cables, the more () the resonant points shown 100by the characteristic impedances. Further 300 0 200 0 Real calculations showed that allReal the electric lengths of the cables at the resonant frequencies-50 were in the 200 -100 100 -100 sequence100 of 0.087, 0.147, 0.211, 0.296, 0.945, 1.607, etc.Imaginary -200 This indicated that theImaginary -150 resonant frequencies 0 0 -300 -200 under the0 same 100 200 structure 300 400 500 were0 determined 100 200 300 400 by 500 the electric 0length 100 200 of 300the 400whole 500 cables,0 100 which 200 300 was 400 500easy to understandFrequency considering (kHz) the resonantFrequency (kHz) characteristics. With increasesFrequency (kHz) in the cable Frequencylengths, (kHz) the loss l =1000m l =1400m l = 1000m l = 1400m Z (1,1) Z (4,4) Z (7,7) Z (2,4) Z (1,1) Z (4,4) Z (7,7) Z (2,4) increases, the inhibitionc c effect ofc high-frequencyc loss on the resonancec cgraduallyc strengthens,c and thus the peaks at the resonant(a) frequencies gradually decrease. (b)

Figure 10.10.The The forward forward characteristic characteristic impedances impedances of medium-voltage of medium-voltage underground underground cables with cables diff erentwith cabledifferent lengths: cable (lengths:a) The real (a) partsThe real and parts (b) the and imaginary (b) the imaginary parts of theparts characteristic of the characteristic impedances. impedances.

5.2.3. Effects of the Grounding Point Numbers 5.2.3. Effects of the Grounding Point Numbers To analyze the eeffectsffects of the grounding point numbers on the characteristic impedance impedance matrices, matrices, the characteristic impedance matrices of the MV underground cables with ng grounding points were the characteristic impedance matrices of the MV underground cables with 𝑛 grounding points were calculated, assuming that the grounding points were evenly distributed along the cables, where the calculated, assuming that the grounding points were evenly distributed along the cables, where the total length of the cables was 600 m, Rg = 1 Ω. total length of the cables was 600 m, 𝑅 =1 Ω. The elements in the FCIMs are shown in Figure 11, which shows that the larger the grounding The elements in the FCIMs are shown in Figure 11, which shows that the larger the grounding point numbers, the fewer resonant points the characteristic impedances. Similar to the situation under point numbers, the fewer resonant points the characteristic impedances. Similar to the situation different total cable lengths, at each resonant frequency, the electric lengths of the cables between the under different total cable lengths, at each resonant frequency, the electric lengths of the cables two grounding points were in the sequence of 0.046, 0.077, 0.112, 0.160, 0.171, etc. With the decrease in between the two grounding points were in the sequence of 0.046, 0.077, 0.112, 0.160, 0.171, etc. With the decrease in cable length between two grounding points, the high-frequency loss decreases, so that the peaks at the resonant frequencies gradually increase. 200 500 200 600 400 100 400 Ω 150 Ω () () Ω Ω () () 300 200 100 0 0 Real

Real 200 -100

50 Imaginary 100 Imaginary -200 0 0 -200 -400 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) = = = = ng 2 ng 3 ng 2 ng 3 1000 1500 800 1500 600 1200 1000 800 Ω Ω () 400 () Ω Ω () () 600 900 200 500 0 Real Real 400 600 0 -200 Imaginary Imaginary -500 200 300 -400 0 0 -600 -1000 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) n = 4 n = 5 = = g g ng 4 ng 5

Zc (1,1) Zc (4,4) Zc (7,7) Zc (2,4) Z (1,1) Z (4,4) Z (7,7) Z (2,4) c c c c (a) (b)

Figure 11. The forward characteristic impedances of medium-voltage underground cables with different grounding point numbers: (a) The real parts and (b) the imaginary parts of the characteristic impedances.

5.2.4. Effects of Cable Branch Numbers To analyze the effects of the cable branches on the characteristic impedance matrices, the characteristic impedance matrices with 𝑛 cable branches were calculated assuming that the branches were also evenly distributed along the cables. Each cable branch was connected in parallel with one grounding branch, which is common in real-life scenarios. In this situation, the above

Electronics 2019, 8, x FOR PEER REVIEW 14 of 19

1000 600 600 300 800 500 400 200 Ω Ω Ω Ω () () 100 () 200 () 400 600 300 0 0 Real Real 400 200 -200 -100

200 100 Imaginary -400 Imaginary -200 0 0 -600 -300 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) l = 200m l = 600m l = 200m l = 600m 500 400 300 150 100 400 200

300 Ω Ω Ω () () Ω 50 () () 100 300 0 200 0 Real Real -50 200 -100 100 -100 100 Imaginary -200 Imaginary -150 0 0 -300 -200 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) l =1000m l =1400m l = 1000m l = 1400m Z (1,1) Z (4,4) Z (7,7) Z (2,4) Z (1,1) Z (4,4) Z (7,7) Z (2,4) c c c c c c c c (a) (b)

Figure 10. The forward characteristic impedances of medium-voltage underground cables with different cable lengths: (a) The real parts and (b) the imaginary parts of the characteristic impedances.

5.2.3. Effects of the Grounding Point Numbers To analyze the effects of the grounding point numbers on the characteristic impedance matrices, the characteristic impedance matrices of the MV underground cables with 𝑛 grounding points were calculated, assuming that the grounding points were evenly distributed along the cables, where the total length of the cables was 600 m, 𝑅 =1 Ω. The elements in the FCIMs are shown in Figure 11, which shows that the larger the grounding Electronicspoint numbers,2019, 8, 571 the fewer resonant points the characteristic impedances. Similar to the situation14 of 18 under different total cable lengths, at each resonant frequency, the electric lengths of the cables cablebetween length the betweentwo grounding two grounding points were points, in the high-frequencysequence of 0.046, loss 0.077, decreases, 0.112, so 0.160, that the0.171, peaks etc. atWith the resonantthe decrease frequencies in cable length gradually between increase. two grounding points, the high-frequency loss decreases, so that the peaks at the resonant frequencies gradually increase. 200 500 200 600 400 100 400 Ω 150 Ω () () Ω Ω () () 300 200 100 0 0 Real

Real 200 -100

50 Imaginary 100 Imaginary -200 0 0 -200 -400 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) = = = = ng 2 ng 3 ng 2 ng 3 1000 1500 800 1500 600 1200 1000 800 Ω Ω () 400 () Ω Ω () () 600 900 200 500 0 Real Real 400 600 0 -200 Imaginary Imaginary -500 200 300 -400 0 0 -600 -1000 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 0 100 200 300 400 500 Frequency (kHz) Frequency (kHz) Frequency (kHz) Frequency (kHz) n = 4 n = 5 = = g g ng 4 ng 5

Zc (1,1) Zc (4,4) Zc (7,7) Zc (2,4) Z (1,1) Z (4,4) Z (7,7) Z (2,4) c c c c (a) (b)

Figure 11.11.The The forward forward characteristic characteristic impedances impedances of medium-voltage of medium-voltage underground underground cables with cables diff erentwith groundingdifferent grounding point numbers: point numbers: (a) The real (a parts) The and real ( bparts) the and imaginary (b) the parts imaginary of the characteristicparts of the characteristic impedances. impedances. 5.2.4. Effects of Cable Branch Numbers 5.2.4.Toanalyze Effects of theCable effects Branch of the Numbers cable branches on the characteristic impedance matrices, the characteristic impedance matrices with n cable branches were calculated assuming that the branches were also To analyze the effectsb of the cable branches on the characteristic impedance matrices, the evenly distributed along the cables. Each cable branch was connected in parallel with one grounding characteristic impedance matrices with 𝑛 cable branches were calculated assuming that the branch, which is common in real-life scenarios. In this situation, the above calculation method was branches were also evenly distributed along the cables. Each cable branch was connected in parallel still valid, because the cable branch and grounding branch could easily be equivalent to one branch. with one grounding branch, which is common in real-life scenarios. In this situation, the above The total length of the main cables was 600 m, and Rg = 1 Ω. The branch cables were the same as the main cables, and the branches were well-matched to their characteristic impedance matrices. The elements in the FCIMs are shown in Figure 12, where the results are similar to that in Figure 11, indicating that the effects of well-matched cable branches on the resonant frequencies are relatively small compared to those of the grounding points. This occurs because the relatively small impedance (i.e., the grounding impedance) plays a major role in the input impedance of the parallel structure. When the cable branches are connected in parallel to the grounding branches, the input impedances at the branch points decrease, and so the peaks in Figure 12 are higher than those in Figure 11, as revealed in Section 5.2.1.

Figure 12. The forward characteristic impedances of medium-voltage underground cables with different branch numbers: (a) The real parts and (b) the imaginary parts of the characteristic impedances. Electronics 2019, 8, 571 15 of 18

6. Conclusions At present, the calculations and analyses of MV underground cables as a power line communication channel are limited to studies under ideal conditions, where the MV underground cables are treated as uniform transmission lines. To consider more realistic situations, in this study, a non-uniform multiconductor transmission line model of MV underground cables was established with special consideration given to the grounded shields and armor. Then, the characteristic impedance matrices of the whole underground cables between the PLC transceivers were calculated through the eigen analysis of the overall transmission matrices. Thereafter, the characteristics of the FCIM and BCIM due to the grounding of the shields and armor were analyzed based on the numerical and measurement results. Lastly, the effects of the grounding resistances, line lengths, ground point numbers, and cable branch numbers on the characteristic impedance matrices were discussed. The main conclusions are as follows:

1. Considering the grounding of the shields and armors, the MV underground cables have FCIM and BCIM that were not equal unless the cables were longitudinally symmetrical. This is different from the case of uniform MV underground cables without grounding points, where the latter only has one unique characteristic impedance matrix. 2. If and only if the receiving end of the MV underground cable is connected to the FCIM, the access impedance matrix at the sending end is equal to the FCIM. Similarly, if and only if the sending end of the MV underground cable is connected to the BCIM, the access impedance matrix at the receiving end is equal to the BCIM. 3. The grounding of partial conductors will lead to resonance, resulting in multiple peaks in the characteristic impedances of the underground cables, which are not only reflected in the shields and armor, but can also be found in the cores. The resonance should be considered in investigations of the impedance matching for PLC to reduce or eliminate the reflected signal power at the transceivers. 4. The smaller the grounding resistance, the larger the amplitudes at the resonant points. The grounding resistance does not affect the resonant frequencies. The longer the cable, the more resonant points in the fixed frequency range, and the smaller the amplitudes at the resonant points. The more grounding points along the cable, the fewer resonant points in the fixed frequency range, and the larger the amplitudes at the resonance points. Compared to the grounding of the shields and armor, the effects of well-matched cable branches on the characteristic impedance matrices are limited.

The above conclusions are useful for understanding the impedance characteristics of MV underground cable and achieving impedance matching in PLC systems. In this study, the ground was modeled as a simple lumped resistance. However, the actual behavior of ground is more complex, and the grounding impedances may be different for different grounding points. The effects of these factors on the CIMs of MV underground cable will be investigated in future works.

Author Contributions: Conceptualization, Methodology and Validation, H.Z. and W.Z.; Investigation and Visualization, W.Z. and Y.W.; Writing—Original Draft Preparation, H.Z. and W.Z.; Writing—Review & Editing, Supervision, H.Z. and Y.W.; Funding acquisition, Y.W. Funding: This work was supported by the National Natural Science Foundation of China (Grant No. 51807063). Acknowledgments: Thanks to the reviewers and editors for their careful review, constructive comments and English editing, which helped improve the quality of the paper. Conflicts of Interest: The authors declare no conflict of interest.

Appendix A The original geometric and electromagnetic parameters of the cable in this paper are provided in Table A1, where XPLE stands for cross-linked polyethylene, PVC stands for polyvinyl chloride. Electronics 2019, 8, 571 16 of 18

Table A1. Data sheet of the examined 10-kV underground cables (YJV22).

Geometric Parameter Radius (mm) Electromagnetic Parameter Value Core conductor 10.125 Core or shield resistivity 1.75 10 8 Ω m × − · Cross-linked polyethylene 15.325 Armor resistivity 9.78 10 8 Ω m (XPLE) insulation × − · Copper shield 15.925 Insulation relative permittivity 2.6 Steel armor 42.050 Sheath relative permittivity 6 Polyvinyl chloride 45.650 Armor relative permeability 1 1000 (PVC) sheath 1 The relative permeability of mediums other than the armor is 1.

Appendix B The per unit length parameter matrices at 300 kHz of the medium-voltage underground cable examined in this paper are as follows:    0.3812 0.3725 0.3539 0.3539 0.3539 0.3539 0.3311     0.3725 0.3626 0.3539 0.3539 0.3539 0.3539 0.3311       0.3539 0.3539 0.3812 0.3725 0.3539 0.3539 0.3311    Real(Z) =  0.3539 0.3539 0.3725 0.3626 0.3539 0.3539 0.3311  Ω/m, (A1)      0.3539 0.3539 0.3539 0.3539 0.3812 0.3725 0.3311     0.3539 0.3539 0.3539 0.3539 0.3725 0.3626 0.3311     0.3311 0.3311 0.3311 0.3311 0.3311 0.3311 0.3311 

   2.7228 2.5627 2.3966 2.3966 2.3966 2.3966 2.2424     2.5627 2.5633 2.3966 2.3966 2.3966 2.3966 2.2424       2.3966 2.3966 2.7228 2.5627 2.3966 2.3966 2.2424    Imag(Z) =  2.3966 2.3966 2.5627 2.5633 2.3966 2.3966 2.2424  Ω/m. (A2)      2.3966 2.3966 2.3966 2.3966 2.7228 2.5627 2.2424     2.3966 2.3966 2.3966 2.3966 2.5627 2.5633 2.2424     2.2424 2.2424 2.2424 2.2424 2.2424 2.2424 2.2424     0.90i 0.90i 0 0 0 0 0   −   0.90i 1.4i 0 0.20i 0 0.20i 0.20i     − − − −   0 0 0.90i 0.90i 0 0 0   −  Y =  0 0.20i 0.90i 1.4i 0 0.20i 0.20i  mS/m. (A3)    − − − −   0 0 0 0 0.90i 0.90i 0   −   0 0.20i 0 0.20i 0.90i 1.4i 0.20i   − − − −   0 0.20i 0 0.20i 0 0.20i 4.90i  − − − References

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