Frequency-Dependent Pi Model of a Three-Core Submarine ... - MDPI

energies Article

Frequency-Dependent Pi Model of a Three-Core Submarine Cable for Time and Frequency Domain Analysis Carlos Ruiz 1, * , Gonzalo Abad 1 , Markel Zubiaga 2 , Danel Madariaga 2 and Joseba Arza 2 1 2

*

Electronics and Computing Department, Mondragon University, 20500 Mondragon, Spain; [email protected] Ingeteam R&D Europe S. L., 48170 Zamudio, Spain; [email protected] (M.Z.); [email protected] (D.M.); [email protected] (J.A.) Correspondence: [email protected]

Received: 24 September 2018; Accepted: 14 October 2018; Published: 16 October 2018







Abstract: In this paper, a Frequency-Dependent Pi Model (FDPi) of a three-core submarine cable is presented. The model is intended to be used for the representation of submarine cables in an Offshore Wind Power Plant (OWPP) scenario for both time and frequency domain analysis. The frequency-dependent variation of each conductive layer is modeled by a Foster equivalent network whose parameters are tuned by means of Vector Fitting (VF) algorithm. The complete formulation for the parameterization of the model is presented in detail, which allows an easy reproduction of the presented model. The validation of the model is performed via a comparison with a well-established reference model, the Universal Line Model (ULM) from PSCAD/EMTDC software. Two cable system case studies are presented. The first case study shows the response of the FDPi Model for a three-core submarine cable. On the other hand, the second case study depicts the response of three single-core underground cables laying in trefoil formation. This last case shows the applicability of the FDPi Model to other types of cable systems and indirectly validates the response of the aforementioned model with experimental results. Additionally, potential applications of the FDPi model are presented. Keywords: cable model; pi sections; power systems; time and frequency domain; vector fitting

1. Introduction Increasing construction of Offshore Wind Power Plants (OWPPs) has been accompanied by a progressive installation of submarine cables in the power system [1]. Since these type of cable networks are affected by many transient phenomena (frequent load changes, switching operations, etc.) and harmonics that can be amplified by cable resonances, proper modeling of submarine cables is crucial in order to perform an adequate analysis of their behavior and their effects on the power system. State of the art time domain models of power cables take into account the frequency dependency of the physical parameters in order to enable accurate transient simulations [2]. Due to their formulation, these models cannot be directly converted into the Laplace domain or in a state-space form, in order to analyze the stability and the frequency response of the power system under study. Thus, submarine cables are commonly represented in power system stability studies by cascaded traditional Pi sections that neglect the frequency-dependent effects [3]. This way of modeling is unable to accurately represent the damping characteristic of the cable and can lead to incorrect results and analysis [4]. Regarding this issue, several contributions have been done in order to improve the damping characteristic of the traditional Pi model. In general terms, these contributions consist of including Energies 2018, 11, 2778; doi:10.3390/en11102778

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the inductive coupling between the cable layers (core and screen) [5] or by including equivalent networks that represent the variation in frequency of the cable parameters [4,6]. The model proposed in reference [4] eliminates the contribution of the cable screens by assuming them to be at ground potential (even though they might not be grounded at both ends). On the contrary, references [6,7] represent a cable system with a Pi equivalent circuit including the frequency dependence of the cable longitudinal impedance in the modal domain. For this model approach, transformations between both domains, phase and modal, are required. This paper presents an alternative Frequency-Dependent Pi (FDPi) Model that represent reasonably well the steady state and transients comprising frequencies up to 10 kHz but can be extended to higher frequencies if required. The model consists of cascaded Pi sections, which incorporate both inductive coupling and the frequency dependent characteristic of some series impedance terms of the conductive layers. The strengths of the presented model in comparison to other published model contributions lie in the following points:

•

• •

•

•

The model represents each cable conductive layer, thus not assuming the elimination of the cable screens and external wire armour, which allows the visualization of the voltages and currents of these conductors and the representation of different grounding configurations. Furthermore, it is possible to account for the frequency dependency of the earth-related terms by explicitly representing the earth as another conductor [8,9]. The model is explicitly represented in the phase domain, thus avoiding the transformation between phase domain to modal domain and vice versa. The model can be directly formulated in the Laplace domain or in a state-space representation. Thus allowing different studies oriented to frequency analysis, stability, sensitivity to parameter deviation, participation factors of the system’s variables, and others. Backward Euler method, or other techniques presented in reference [10], can be used in order to compute the voltages and currents along the cable in the time domain. In this sense, both time and frequency domain analysis can be performed. The model can be easily implemented in Matlab/Simulink or any other circuit simulation tool by means of discrete RLC components. Coupled resistances and inductances are directly implemented by means of a generalized mutual inductance component. Time domain simulations of the submarine cable model can be performed directly by selecting a solver option and a sample time. Typical solver methods available in simulation tools are Backward Euler, Trapezoidal Rule and Runge-Kutta. The modeling approach is not only applicable to three-core submarine cables. It can be extended to other types of cable systems, such as single-core underground and submarine cables or a set of single-core cables arranged in flat or trefoil formation.

In this sense, the FDPi model presented in this paper is intended to be in use with the Wind Turbine Harmonic Model presented in reference [11] and other power component models (such as transformers, filters, etc.) for the analysis and study of an OWPP. The studies can be oriented towards design aspects, harmonic assessment, stability studies, frequency analysis, and others. The paper is structured as follows: Section 2 introduces the electrical schematic and the general equations that describe the behavior of the FDPi model. Section 3 presents the parameterization of the model. A detailed description of the formulation is presented in this section, which allows an easy reproduction of the model and the case studies without the need of resorting for additional information. Section 4 treats the validation of two case studies, a three-core submarine cable and three single-core underground cables laying in trefoil formation. The validation of the FDPi model is performed via a comparison of the proposed model with the well-established Universal Line Model (ULM) from PSCAD/EMTDC software. Section 5 provides a methodology for the determination of a suitable order of the proposed model. The order of the FDPi model is highly dependent on the accuracy and

Figure 1 shows the electrical schematic of the FDPi Model. The model physically represents the seven conductors of a three-core submarine cable system: three core conductors, three screen conductors and a pipe representing the external wire armour of the cable. As depicted in Figure 1a, the model consists of N cascaded FDPi sections. Each FDPi section Energies 2018, 11, 2778 3 of 21 consists of a coupled resistance and inductance matrix computed at the nominal frequency, 50 Hz or 60 Hz. The capacitive coupling between conductors is computed at the nominal frequency as well. The frequency dependent behavior of the series impedance terms each conductive layer is frequency range requirements and the intrinsic characteristics of the cableofsystem. Section 6 introduces represented by means of of a Foster equivalent network depicted in Figureabout 1b. the presented model the potential applications the proposed FDPi model.as Finally, conclusions The order ofresults the FDPi is given by the number of cascaded sections (N) together with the and the obtained are model detailed. order of the Foster equivalent networks (M). The order of the model is chosen depending on the cable 2.length, Frequency-Dependent Pi Model accuracy and frequency range requirements. The general equations that theof behavior of Model. the model expressed in therepresents Laplace Figure 1 shows the electricaldescribe schematic the FDPi Theare model physically domain s =jω ). The terminal voltagessubmarine and currents of system: the FDPithree model areconductors, given by Equation (1). the seven( conductors of a three-core cable core three screen Being [ I ] , and the identity matrix of order conductors a pipe representing the seven. external wire armour of the cable.

+ +

Vc1g _ s

Vs1g _ s

+

FDPi − N

FDPi − 2

FDPi − 1

I c1_ s

I c1_ r

c1_ s1

c1_ r1

c1_ s 2

c1_ r 2

c1_ sN

c1_ rN

s1_ s1

s1_ r1

s1_ s 2

s1_ r 2

s1_ sN

s1_ rN

c 2 _ rN

c 2 _ s1

c 2 _ r1

c2 _ s2

c2 _ r 2

c 2 _ sN

s 2 _ s1

s 2 _ r1

s2 _ s2

s2 _ r 2

s 2 _ sN

s 2 _ rN

c3 _ s1

c3 _ r1

c3 _ s 2

c3 _ r 2

c3 _ sN

c3 _ rN

s3 _ s1

s3 _ r1

s3 _ s 2

s3 _ r 2

s3 _ sN

s3 _ rN

p _ s1

p _ r1

p _ s2

p _ r2

p _ sN

p _ rN

g

g

g

g

g

Vc1g _ r

Vs1g _ r

Ip_r

Ip_s

V pg _ s

+ +

+ V pg _ r

g

(a) Coupled  Rˆ  ⋅ lsection  

Coupled  Lˆ  ⋅ lsection  

Rˆ ⋅ l Rˆ ⋅ l Rˆ c 0 ⋅ lsection Lˆc 0 ⋅ lsection c1 section cM section

c1_ sn

Cˆ c1s1 ⋅ lsection 2

s1_ sn

s1_ rn Lˆs1 ⋅ lsection LˆsM ⋅ lsection

Cˆ c 2 s 2 ⋅ lsection 2

Cˆ s1s 2 ⋅ lsection 2

c 2 _ sn

c 2 _ rn

Cˆ s 2 s 3 ⋅ lsection 2

s 2 _ sn

s 2 _ rn

Cˆ s1s 3 ⋅ lsection 2

c3 _ sn

Cˆ s 3 p 2

c3 _ rn

⋅ lsection

s3 _ sn

Cˆ s1 p 2 p _ sn

Rˆ s 0 ⋅ lsection Lˆs 0 ⋅ lsection

c1_ rn

Lˆc1 ⋅ lsection LˆcM ⋅ lsection Rˆ s1 ⋅ lsection Rˆ sM ⋅ lsection

Cˆ s 2 p 2

Cˆ c 3s 3 ⋅ lsection 2

⋅ lsection

s3 _ rn

Rˆ ⋅ l Rˆ ⋅ l Rˆ p 0 ⋅ lsection Lˆ p 0 ⋅ lsection p1 section pM section p _ rn

⋅ lsection

Cˆ pg 2

Lˆ p1 ⋅ lsection Lˆ pM ⋅ lsection

⋅ lsection

g

g

(b) Figure1.1.Electrical Electrical schematic of the Frequency-Dependent Pi (FDPi) (a) Cascaded FDPi Figure schematic of the Frequency-Dependent Pi (FDPi) Model.Model. (a) Cascaded FDPi sections. sections. (b)schematic Detailed of schematic an FDPi section (Capacitances at the side right-hand side to (b) Detailed an FDPi of section (Capacitances at the right-hand are equal to are the equal left-hand the left-hand butare their names are omitted for simplicity). side, but their side, names omitted for simplicity).

As depicted in Figure 1a, the model consists of N cascaded FDPi sections. Each FDPi section consists of a coupled resistance and inductance matrix computed at the nominal frequency, 50 Hz or 60 Hz. The capacitive coupling between conductors is computed at the nominal frequency as well. The frequency dependent behavior of the series impedance terms of each conductive layer is represented by means of a Foster equivalent network as depicted in Figure 1b. The order of the FDPi model is given by the number of cascaded sections (N) together with the order of the Foster equivalent networks (M). The order of the model is chosen depending on the cable length, accuracy and frequency range requirements.

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The general equations that describe the behavior of the model are expressed in the Laplace domain (s = jω). The terminal voltages and currents of the FDPi model are given by Equation (1). Being [ I ], the identity matrix of order seven. "

[Vs ] [ Is ]

#

"

=

Z Y

[ I ] + [ ][2 ] Y Z Y [Y ] + [ ][ 4 ][ ]

[Z] Y Z [ I ] + [ ][2 ]

#N "

[Vr ] [ Ir ]

·

# (1)

The voltages and currents at the sending and receiving ends are vectors defined by Equation (2). Each voltage is referenced to ground potential as depicted in Figure 1:       [Vs ] =     

Vc1g_s Vs1g_s Vc2g_s Vs2g_s Vc3g_s Vs3g_s Vpg_s





     ;    

     [Vr ] =     

Vc1g_r Vs1g_r Vc2g_r Vs2g_r Vc3g_r Vs3g_r Vpg_r





     ;    

     [ Is ] =     

Ic1_s Is1_s Ic2_s Is2_s Ic3_s Is3_s I p_s





     ;    

     [ Ir ] =     

Ic1_r Is1_r Ic2_r Is2_r Ic3_r Is3_r I p_r

          

(2)

The series impedance matrix [ Z ] and the shunt admittance matrix [Y ] are given by Equations (3) and (4), respectively:   (3) [ Z ] = Zˆ · lsection   (4) [Y ] = Yˆ · lsection The length of a Pi section is given by Equation (5), where lcable is the total length of the cable: lcable N   The per-length unit series impedance matrix Zˆ is given by Equation (6): lsection =

(5)

      Zˆ = Zˆ Fn + Zˆ Foster

(6)

  Being Zˆ Fn matrix defined by Equation (7):       ˆ R =  

Rˆ 11 Rˆ 21 .. . ˆ R71

Rˆ 12 Rˆ 22 .. . ˆ R72

··· ··· .. . ···

     Zˆ Fn = Rˆ + s Lˆ

Rˆ 17 Rˆ 27 .. . ˆ R77

(7)





    

    ˆL =   

;

Lˆ 11 Lˆ 21 .. . ˆL71

Lˆ 12 Lˆ 22 .. . ˆL72

··· ··· .. . ···

Lˆ 17 Lˆ 27 .. . ˆL77

     

(8)

The Zˆ Foster matrix includes the contribution of the Foster equivalent networks of each conductive layer (cores, screens and armour) and is given by Equations (9)–(13): 



   Zˆ Foster =  

 Zˆ F1 [ 0 ] 2×2 [ 0 ] 2×2 [ 0 ] 1×2



 Zˆ Fj =

  

[ 0 ] 2×2   Zˆ F2 [ 0 ] 2×2 [ 0 ] 1×2 "

Zˆ Fc 0

[ 0 ] 2×2 [ 0 ] 2×2   Zˆ F3 [ 0 ] 1×2 0 ˆ ZFs

[ 0 ] 2×1 [ 0 ] 2×1 [ 0 ] 2×1 Zˆ Fp

    

(9)

# (10)

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M

∑

Zˆ Fc = Rˆ c0 + s Lˆ c0 +

m =1 M

∑

Zˆ Fs = Rˆ s0 + s Lˆ s0 +

m =1 M

Zˆ Fp = Rˆ p0 + s Lˆ p0 +

∑

m =1

s Rˆ cm Lˆ cm Rˆ cm + s Lˆ cm

(11)

s Rˆ sm Lˆ sm ˆ Rsm + s Lˆ sm

(12)

s Rˆ pm Lˆ pm ˆ R pm + s Lˆ pm

(13)

  The per-length unit shunt admittance matrix Yˆ is defined by Equation (14):     Yˆ = s Cˆ

(14)

being:      ˆ C =  

Cˆ 11 Cˆ 21 .. . Cˆ 71

Cˆ 12 Cˆ 22 .. . Cˆ 72



· · · Cˆ 17 · · · Cˆ 27 .. .. . . · · · Cˆ 77

    

(15)

The per-length unit capacitances between conductors (depicted in Figure 1b) are given by the following equations: Cˆ c1s1 = −Cˆ 21 = −Cˆ 12

;

Cˆ c2s2 = −Cˆ 43 = −Cˆ 34

;

Cˆ c3s3 = −Cˆ 65 = −Cˆ 56

(16)

Cˆ s1s2 = −Cˆ 42 = −Cˆ 24

;

Cˆ s1s3 = −Cˆ 62 = −Cˆ 26

;

Cˆ s2s3 = −Cˆ 64 = −Cˆ 46

(17)

Cˆ s1p = −Cˆ 72 = −Cˆ 27

;

Cˆ s2p = −Cˆ 74 = −Cˆ 47

;

Cˆ s3p = −Cˆ 76 = −Cˆ 67

(18)

The capacitances to ground for each cable conductor (cores, screens and armour) are computed according to the following equations: Cˆ c1g = Cˆ 11 + Cˆ 12

;

Cˆ c2g = Cˆ 33 + Cˆ 34

Cˆ s1g = Cˆ 22 + Cˆ 21 + Cˆ 24 + Cˆ 26 + Cˆ 27

;

;

Cˆ c3g = Cˆ 55 + Cˆ 56

Cˆ s2g = Cˆ 44 + Cˆ 42 + Cˆ 43 + Cˆ 46 + Cˆ 47

(19) (20)

Cˆ s3g = Cˆ 66 + Cˆ 62 + Cˆ 64 + Cˆ 65 + Cˆ 67

(21)

Cˆ pg = Cˆ 77 + Cˆ 72 + Cˆ 74 + Cˆ 76

(22)

The capacitances of Equations (19)–(21) are zero because of the physical structure of the cable. Therefore, they are not represented in the schematic of Figure 1b. The rest of the capacitance elements presented in Equation (15) and not named previously are zero as well. Supporting information about the capacitance matrix of Equation (15) is presented in [12]. 3. Model Parameterization The frequency-dependent variation of the electrical parameters of a cable system is due to skin effect and proximity effect. There are several methods proposed in the technical literature for determining the electrical parameters of a cable system and their variations with frequency. The most remarkable techniques are based on analytical formulas [13–15], numerical tools based on Finite Element Method (FEM) [16,17], conductor partitioning [18,19] and the Method of Moments (MoM-SO) proposed in [20–22]. Each technique has its advantages and disadvantages in terms of complexity, accuracy, computational time, phenomena that represent and other aspects. In this paper, the analytical equations, presented in reference [13], are used for the parameterization of the model. The main reason for its selection is that most Electromagnetic Transient (EMT) tools use this formulation to compute the electrical parameters of the cable. Therefore,

determining the electrical parameters of a cable system and their variations with frequency. The most remarkable techniques are based on analytical formulas [13–15], numerical tools based on Finite Element Method (FEM) [16,17], conductor partitioning [18,19] and the Method of Moments (MoMSO) proposed in [20–22]. Each technique has its advantages and disadvantages in terms of complexity, accuracy, computational time, phenomena that represent and other aspects. Energies 2018, 11, 2778 6 of 21 In this paper, the analytical equations, presented in reference [13], are used for the parameterization of the model. The main reason for its selection is that most Electromagnetic Transient (EMT) tools use is this formulation to compute thethe electrical parameters of model the cable. this way of parameterization followed in order to compare behavior of the FDPi with Therefore,model, this way of parameterization followed in order to compare the behaviorItofisthe FDPito a reference Universal Line Modelis(ULM) from PSCAD/EMTDC software. worth model with a reference model, Universal Line Model (ULM) from software. is mention, that the analytical equations, addressed in reference [13], PSCAD/EMTDC only take into account theItskin worth to mention, that the analytical equations, addressed in reference [13], only take into account effect and do not consider the proximity effect. If it is required to consider the proximity effect, skin effect and do not consider the proximity effect. If it is required to consider the proximity thethe methodology presented in reference [14] and other special methods already mentioned above such effect, the methodology presented in reference [14] and other special methods already mentioned as FEM, conductor partitioning and MoM-SO can be used for model parameterization. above such as FEM, conductor partitioning and MoM-SO can be used for model parameterization. In order to perform the cable parameterization according to the analytical equations, it is necessary In order to perform the cable parameterization according to the analytical equations, it is to perform a data conversion of the original cable parameters into a new set of parameters. This new necessary to perform a data conversion of the original cable parameters into a new set of parameters. set of parameters consists in the simplification of the cable layers and in the correction of the materials This new set of parameters consists in the simplification of the cable layers and in the correction of properties. Details of the conversion procedure can be found in references [23,24]. Figure 2 shows the materials properties. Details of the conversion procedure can be found in references [23,24]. theFigure simplified cross-section of the cable under Table 1 gives information about the required set 2 shows the simplified cross-section of study. the cable under study. Table 1 gives information about of the parameters. required set of parameters. The determination model involves involvesthree threesteps, steps,which whichare are The determinationofofthe theelectrical electricalparameters parametersof of the the FDPi FDPi model described in in the next subsections of the the model. model. described the next subsectionsininorder ordertotoallow allowan aneasy easy reproduction reproduction of

r5

dk

r6

θ jk dj

r7 r2

r3 r4

r1

Layer 1 Layer 2 Layer 3 Layer 4 Layer 5 Layer 6 Layer 7

Figure 2. Simplified cross-section of a three-core submarine cable. Figure 2. Simplified cross-section of a three-core submarine cable. Table 1. Simplified layers of a three-core submarine cable. Layer Number

Layer Name

Parameters

1 2 3 4 5 6 7

Core Conductor Core Insulation Screen Conductor Screen Insulation Armour Inner Insulation Armour Conductor Armour Outer Insulation

r1 , ρc , µrc r2 , µrins1 , ε rins1 r3 , ρs , µrs r4 , µrins2 , ε rins2 r5 , µrins3 , ε rins3 r6 , ρ p , µrp r7 , µrins4 , ε rins4

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    3.1. Computation of Rˆ and Lˆ Matrices For the determination of the resistance and inductance matrices, the per-length unit series   impedance matrix Zˆ Fn is computed at the nominal frequency and is given by Equation (23): 

   Zˆ Fn = [ Zi ] + Z p + [ Zc ] + [ Ze ]

(23)

The cable internal impedance matrix [ Zi ] is given by Equation (24):    [ Zi ] =  

[ Zi1 ] [ 0 ] 2×2 [ 0 ] 2×2 [ 0 ] 1×2

[ 0 ] 2×2 [ Zi2 ] [ 0 ] 2×2 [ 0 ] 1×2

[ 0 ] 2×2 [ 0 ] 2×2 [ Zi3 ] [ 0 ] 1×2

"

Zcsj Zssj

[ 0 ] 2×1 [ 0 ] 2×1 [ 0 ] 2×1 0

    

(24)

being: 



Zij =

Zccj Zcsj

# (25)

Zccj = ZCSins + ZSinner + ZSouter + ZSins − 2ZSmutual Zcsj = ZSouter + ZSins − ZSmutual ;

Zssj = ZSins

(26) (27)

The series impedance of the core insulation is given by Equation (28): ZCSins =

  sµins1 r2 ln 2π r1

(28)

The screen inner and outer series impedance are given by Equations (29) and (30), respectively: ZSinner =

ρs ms J0 ( x2 )K1 ( x3 ) + K0 ( x2 ) J1 ( x3 ) · 2πr2 J1 ( x3 )K1 ( x2 ) − J1 ( x2 )K1 ( x3 )

(29)

ZSouter =

ρs ms J0 ( x3 )K1 ( x2 ) + K0 ( x3 ) J1 ( x2 ) · 2πr3 J1 ( x3 )K1 ( x2 ) − J1 ( x2 )K1 ( x3 )

(30)

being: r x2 = m s r2 ;

x3 = m s r3 ;

ms =

sµs ρs

(31)

Jn ( x ) : Bessel function of x, of first kind and order n. Kn ( x ) : Bessel function of x, of second kind and order n. The series impedance of the screen insulation is given by Equation (32): ZSins

  sµins2 r = ln 4 2π r3

(32)

The screen mutual series impedance is given by Equation (33): ZSmutual =

ρs 1 · 2πr2 r3 J1 ( x3 )K1 ( x2 ) − J1 ( x2 )K1 ( x3 )

(33)

  The pipe internal impedance matrix Z p is defined by Equation (34):    Z p11        Z p21  Zp =   Z p31 [ 0 ] 1×2

  Z  p12  Z  p22  Z p32 [ 0 ] 1×2

  Z  p13  Z  p23  Z p33 [ 0 ] 1×2

[ 0 ] 2×1 [ 0 ] 2×1 [ 0 ] 2×1 0

    

(34)

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being: " 

Z pjj

# Z pjj ; Z pjj #

Z pjj Z pjj



Z pjj =

" sµ0 µrp D pnum · + Q jj + 2µrp y jj ; = 2π x5 D pden

h

"

i

Z pjk Z pjk

Z pjk =

Z pjk

#

Z pjk Z pjk

(35)

" # sµ0 µrp D pnum · + Q jk + 2µrp y jk = 2π x5 D pden

D pden = K1 ( x5 ) J1 ( x6 ) − J1 ( x5 )K1 ( x6 )   ∞ r 5  − ∑ cn Q jk = ln q n d2 + d2 − 2d d cos θ n =1

D pnum = J0 ( x5 )K1 ( x6 ) + K0 ( x5 ) J1 ( x6 ); ( Q jj = ln

y jj =

r5 r4

" 1−

∞

d j dk

n =1

r52

∑

cn =



dj r5

2 #) ;

j

!n

n 1 + µrp +

d j dk r52

!n



x 5 K n −1 ( x 5 K n ( x5 )

(38)

jk

y jk =

; )

cn
n=1 n 1 + µrp +

∑

x 5 K n −1 ( x 5 ) K n ( x5 )

s



x5 = m p r5 ;

cos nθ jk ;

The mutual impedance matrix [ Zc ] Equation (41):  [ Zc1 ]2×2  Z  [ c1 ]2×2 [ Zc ] =   [ Zc1 ]2×2 [ Zc2 ]1×2

(37)

∞

1


j k

k

(36)

x6 = m p r6 ;

mp =

sµ p ρp

(39)

(40)

between pipe inner and outer surfaces is given by

[ Zc1 ]2×2 [ Zc1 ]2×2 [ Zc1 ]2×2 [ Zc2 ]1×2

[ Zc1 ]2×2 [ Zc1 ]2×2 [ Zc1 ]2×2 [ Zc2 ]1×2

[ Zc2 ]2×1 [ Zc2 ]2×1 [ Zc2 ]2×1 Zc3

    

(41)

being: Zc1 = ZPouter + ZPGinsul − 2ZPmutual ;

Zc2 = ZPouter + ZPGinsul − ZPmutual ;

Zc3 = ZPGinsul

(42)

The pipe outer series impedance is given by Equation (43): ZPouter =

ρ p m p J0 ( x6 )K1 ( x5 ) + K0 ( x6 ) J1 ( x5 ) · 2πr6 J1 ( x6 )K1 ( x5 ) − J1 ( x5 )K1 ( x6 )

(43)

The series impedance of the pipe outer insulation is given by Equation (44): ZPGins =

  r7 sµins4 ln 2π r6

(44)

The pipe mutual series impedance is given by Equation (45): ZPmutual =

ρp 1 · 2πr5 r6 K1 ( x5 ) J1 ( x6 ) − J1 ( x5 )K1 ( x6 )

(45)

Finally, [ Ze ] is a square matrix of order seven that represents the earth-return impedance. This impedance term can be defined by Saad’s approximation, Equation (46): " # ρseabed m2e 2e−2hme Ze = K0 ( x 7 ) + 2π 4 + m2e r72

(46)

being: r x7 = m e r7

;

me =

sµseabed ρseabed

(47)

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  Once all previous terms are calculated, the Zˆ Fn matrix is computed at nominal frequency and     the Rˆ and Lˆ matrices can be obtained by splitting real and imaginary parts, as given by Equations (7) and (8). 3.2. Computation of Foster Equivalent Networks For the computation of the Foster equivalent networks shown in Figure 1b, a frequency sweep is done to ZCouter , ZSouter and ZPouter terms. The series impedance of the core conductor ZCouter is given by Equation (48): ρc mc J0 ( x1 ) (48) ZCouter = 2πr1 J1 ( x1 ) being: r x1 = m c r1 ;

mc =

sµc ρc

(49)

ZSouter and ZPouter terms are defined in Equations (30) and (43), respectively. Table 2 shows the general data that is needed to obtain the Foster equivalent networks. The frequency vector is mainly defined in dependence of the frequency range of interest. Table 2. Required general data for the determination of the Foster equivalent networks. ω1

ω2

ω3

···

ωt

ZCouter (ω1 ) ZSouter (ω1 ) ZPouter (ω1 )

ZCouter (ω2 ) ZSouter (ω2 ) ZPouter (ω2 )

ZCouter (ω3 ) ZSouter (ω3 ) ZPouter (ω3 )

··· ··· ···

ZCouter (ωt ) ZSouter (ωt ) ZPouter (ωt )

Energies 2018, 11, x FOR PEER REVIEWbehavior The frequency-dependent

10 of of 22 for each series impedance term is approximated by means Vector Fitting (VF) algorithm [25–27]. VF provides an adequate representation of the series elements in elements in the frequency domain by means of a rational function on pole residue form, as given by the frequency domain by means of a rational function on pole residue form, as given by Equation (50): Equation (50): M M cmc f (s)f = se =  m ++ d + + se ( s )∑ s − p pm m=1m =1 s − m

(50) (50)

Foster equivalent equivalent The rational function, obtained from the fitting process, is represented by a Foster network of order M, as depicted in Figure 3. The order of the equivalent network depends mainly on and accuracy accuracy requirements. requirements. the frequency range and

Rˆ0

Rˆ1

Lˆ0 Lˆ1

Rˆ 2 Lˆ2

Rˆ M LˆM

Figure Foster equivalent equivalent network Figure 3. 3. Foster network of of order order M. M.

The The electrical electrical parameters parameters of of the the Foster Foster equivalent equivalent network network are are given given by by Equation Equation (51): (51): M M cm c d −  m; Rˆ 0 = dR垐 − 0 =∑ p =1 p m m =1 m m

cm Rˆ Rˆ m = m ;; LLˆm m= = − −m ;Lˆ 0 L= e e ; ; RˆR垐 mm = 0 = pmm pm pm

(51)

ˆ and ˆLˆ terms, calculated for each conductive layer (core, It is is possible It possible to to incorporate incorporate the the R Rˆ 00 and L00 terms, calculated for each conductive layer (core,     ˆ ˆ screen and pipe), into the main diagonal of the In this this way, way, screen and pipe), into the main diagonal of the  RˆR and and LLˆ  matrices matricesof ofEquation Equation (7). (7). In a reduction in the number of components shown in Figure 1b can be performed, leading to a more a reduction in the number of components shown in Figure 1b can be performed, leading to a more practical model and improving the computational load due to a lower number of components. practical model and improving the computational load due to a lower number of components.

3.3. Computation of  Cˆ  Matrix The per-length unit shunt capacitance matrix  Cˆ  is computed at the nominal frequency from the potential coefficient matrix P , as given by Equations (52) and (53):

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  3.3. Computation of Cˆ Matrix   The per-length unit shunt capacitance matrix Cˆ is computed at the nominal frequency from the potential coefficient matrix [ P], as given by Equations (52) and (53):   Cˆ = [ P]−1

(52)

  [ P] = [ Pi ] + Pp + [ Pc ]

(53)

The cable internal coefficient matrix [ Pi ] is given by Equation (54): 

[ Pi1 ] [ 0 ] 2×2 [ 0 ] 2×2 [ 0 ] 1×2

  [ Pi ] =  

[0 ] 2×2 [ Pi2 ] [ 0 ] 2×2 [ 0 ] 1×2

[ 0 ] 2×2 [0 ] 2×2 [ Pi3 ] [ 0 ] 1×2



[ 0 ] 2×1 [ 0 ] 2×1 [0 ] 2×1 0

   

(54)

being: "

# Pcj + Psj Psj Pij = Psj Psj     r2 1 r 1 ln ; Psj = ln 4 Pcj = 2πε ins1 r1 2πε ins2 r3   The pipe internal coefficient matrix Pp is given by Equation (57): 



   Ppjj i  h     Ppjk Pp =  h i  P  pjk [ 0 ] 1×2

h

i Ppjk   Ppjj i h Ppjk

h h

[ 0 ] 1×2

[ 0 ] 1×2

Ppjk

i

i Ppjk   Ppjj

[ 0 ] 2×1

(55) (56)



 [ 0 ] 2×1    [ 0 ] 2×1   0

(57)

being: " 



Ppjj =

Ppjj Ppjj

Ppjj Ppjj

Ppjj =

# ;

Q jj ; 2πε ins3

h

i

"

Ppjk =

Ppjk =

Ppjk Ppjk

Q jk 2πε ins3

Ppjk Ppjk

# (58)

(59)

The potential coefficient matrix between pipe inner and outer surfaces [ Pc ] is a square matrix of order seven with all its elements given by Equation (60):   1 r7 Pc = ln 2πε ins4 r6

(60)

As already shown, the above procedure details the steps and the formulation required for model parameterization. 4. Validation Case Studies This section introduces two numerical case studies in order to validate the model of Figure 1. The FDPi model is directly implemented in Simscape Electrical (Matlab/Simulink) and the Backward Euler technique is chosen as solver method for time domain computations. 4.1. Case Study #1: Three-Core Submarine Cable This case study consists of a buried three-core submarine cable. Figure 4 shows the simplified cross-section of the submarine cable under study and gives information about the surrounding medium

The FDPi model is directly implemented in Simscape Electrical (Matlab/Simulink) and the Backward Euler technique is chosen as solver method for time domain computations. 4.1. Case Study #1: Three-Core Submarine Cable This case study consists of a buried three-core submarine cable. Figure 4 shows the simplified Energies 2018, 11, 2778 11 of 21 cross-section of the submarine cable under study and gives information about the surrounding medium properties and burial depth of the cable. The required geometrical and material properties properties depth of the3.cable. The required geometrical and4 material offrom the cable of the and cableburial are listed in Table The information presented in Figure and Tableproperties 3 is adopted [22,28]. are listed in Table 3. The information presented in Figure 4 and Table 3 is adopted from [22,28]. Sea Seabed

h =1m

ρ seabed = 20 Ωm μ rseabed = 1

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Table 3. Geometrical and material parameters for Case Study #1. Layer

Parameters

Figure 4. Cable cross-section for = Case Study #1= 1.7241 and (laying depth and and Figure 4. Cable cross-section and installation conditions (laying depth ⋅installation 10−8 Ωm , μconditions 11.6Study mm , ρc#1 1 for r1Case rc = 1 surrounding medium properties). surrounding medium properties). r2 = 26.6 mm , μ rins1 = 1 , ε rins1 = 2.3 2 r3 = 26.82 mm , ρs = 2.20 ⋅10−7 Ωm , μrs = 1 3 Table 3. Geometrical and material parameters for Case Study #1. 4

Layer

5

r4 = 30.82 mm , μrins 2 = 1 , ε rins 2 = 2.3 r5 = 72 mm , μrins Parameters 3 = 1 , ε rins 3 = 2.3

r = 80 mm , ρ = 2.86 ⋅10−8 Ωm , μ = 1 r16 = 11.6 mm,p ρc = 1.7241 · 10−rp8 Ωm, µrc = 1 rr7 == 83mm μ rins 4 µ= 1 , ε rins = 2.3 26.6,mm, =4 1, ε = 2.3

6 1 7 2 2 rins1 rins1 3 r3 = 26.82 mm, ρs = 2.20 · 10−7 Ωm, µrs = 1 The requirement is to represent steady state and transients comprising 4 of the validation r4 = 30.82 mm, µthe rins2 = 1, ε rins2 = 2.3 frequencies up to 105kHz, which is the rfrequency range of interest for the studies mentioned at the 5 = 72 mm, µrins3 = 1, ε rins3 = 2.3 end of Section 2 and6 in Section 6. To fulfill this objective, the FDPi modelµis built with 32 cascaded − 8 r6 = 80 mm, ρ p = 2.86 · 10 Ωm, rp = 1 FDPi sections (N = 32), equivalent networks of order three (M = 3) for each 7 each one with rFoster = 83 mm, µ = 1, ε = 2.3 7 rins4 rins4

conductive layer term. The series impedance terms were fitted for a frequency range from 0.1 Hz up to 10 kHz by means of VF algorithm. In order toof validate the FDPi model in the time domain, the test configuration of Figure 5 has The requirement the validation is to represent the steady state and transients comprising been used considering a cable length of 50 km. The test is as proposed in references [29,30]. The test frequencies up to 10 kHz, which is the frequency range of interest for the studies mentioned at the end consists of energizing one-phase of the cable with an AC voltage source at its sending end and with of Section the 2 and in Section 6. To fulfill objective, FDPi model built with 32 cascaded FDPi receiving ends open. The input this source signal is a the cosine function withis a nominal voltage of 100 sections (NkVrms. = 32),The each one withtakes Foster equivalent ofinput order threein(M = 3) for eachtheconductive energization place at the peak networks voltage of the source order to produce transient current. layer term.highest The series impedance terms were fitted for a frequency range from 0.1 Hz up to 10 kHz by Figure 6 shows the time domain response of the receiving end signals depicted in Figure 5. Each means of VF algorithm. graphic shows information about the steady state and transient state (top one) and a zoom in for the In order validate the in FDPi timeresponse domain, the test configuration of Figure 5 has first to 10 ms (down one) ordermodel to showin thethe transient in detail. been used considering cable length of 50 km. is as proposed in references [29,30]. As depictedain Figure 6, the results of theThe FDPitest model show a very good agreement with the The test obtained one-phase from the reference shape the time are veryend similar consists ofones energizing of themodel. cableThe with anand ACamplitude voltage of source atsignals its sending and with the and a good representation of the wave travel time is performed by the FDPi model. receiving ends open. The input source signal is a cosine function with a nominal voltage of 100 kVrms. Figure 7 shows the magnitude and phase of the cable admittance. As indicated for the signals in The energization takes place at the peak of the input source in order the highest the time domain, the admittance of thevoltage cable (magnitude and phase) is very similar to for produce both models transient current. up to a frequency value of 10 kHz.

50 km 1 mH

t = 0 ms

I s1_ r

Vc1g _ r

Vin = 100 kVrms Ip_r 1 mΩ

5. Setup timeresponse response validation—Case StudyStudy #1. FigureFigure 5. Setup for for time validation—Case #1.

1 mΩ

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Figure 6 shows the time domain response of the receiving end signals depicted in Figure 5. Each graphic shows information about the steady state and transient state (top one) and a zoom in for the first 10 ms (down one) in order to show the transient response in detail. As depicted in Figure 6, the results of the FDPi model show a very good agreement with the ones obtained from the reference model. The shape and amplitude of the time signals are very similar and a good representation of the wave travel time is performed by the FDPi model. Figure 7 shows the magnitude and phase of the cable admittance. As indicated for the signals in the time Energies domain, of the cable (magnitude and phase) is very similar for both models 2018,the 11, x admittance FOR PEER REVIEW 13 of 22 Energies 2018, 11, x FOR of PEER 13 of 22 up to a frequency value 10REVIEW kHz. 3 2

10

5

3000

32FDPi

Voltage (V) Voltage (V)

ULM

32FDPi

1 0 0 -1

-1 -2 0

0.01

32FDPi

3000 2000

ULM

Vc1g _ r

2 1

ULM

ULM

Vc1g _ r

5

Current (A) Current (A)

10

3

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

I s1_ r

2000 1000

1000

I s1_ r

0

0 -1000

0.1

0

-2

32FDPi

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

100.01 5

10 3 2

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

0 3000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

Vc1g _ r

Transient detail

32FDPi

3000 2000

ULM

Vc1g _ r

Current (A) Current (A)

Voltage (V) Voltage (V)

0.1

32FDPi

1 0 -1

-1 -2 0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0

0.001

0.002

0.003

0.004

0.005

1000

I s1_ r

0

0 -1000 0

0.001

0.002

0.003

0.004

0.005

(a) (a)

0.007

0.008

0.009

0

0.01

0.006

0.007

0.008

0.009

0.01

Time (s)

-1000 0.006

32FDPi

I s1_ r

Transient detail

2000 1000

0.01

Time (s)

-2

0.1

32FDPi

Transient detail

ULM

Transient detail

0.09 ULM

ULM

5

2 1

0

0.09

-1000 0 3

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

(b) (b) Time (s)

Time (s)

1000

1000

Current (A) Current (A)

0

0 -1000

Ip_r

-1000 -2000

ULM

Ip_r

32FDPi ULM

-2000

0

0 1000

0.01

0.01

0.02

0.02

0.03

0.03

0.04

0.04

0.05

0.05

0.06

0.06

0.07

0.08

0.07

0.08

0.09 32FDPi 0.09

0.1

0.1

1000

Current (A) Current (A)

0

0 -1000

Ip_r

Transient detail

Ip_r

Transient detail

-1000 -2000

ULM 32FDPi ULM

0

-2000

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009 32FDPi

0.01

Time (s) 0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

(c) (c) Time (s)

Figure 6. Time response comparison for Case Study #1. (a) Vc1g_r. (b) Is1_r. (c) Ip_r.

Figure Time response comparison CaseStudy Study #1. c1g_r. (b). I(b) Ip_r. .(c) Ip_r . Figure 6. Time6.response comparison forfor Case #1.(a) (a)VV Is1_r c1g_r s1_r. (c)

Magnitude (S) Magnitude (S)

0.06

ULM 32FDPi ULM

0.06 0.04

32FDPi

0.04 0.02 0.02 0 10

0

10

1

10

1

10

2

10

2

10

3

3

10 10

4

4

100

Phase (Deg) Phase (Deg)

100 50 50 0

ULM 32FDPi ULM

0

32FDPi

10 10

1

1

10 10

2

2

10

Frequency (Hz)

10

3

3

10 10

4

4

Frequency (Hz)

7. Cable admittancecomparison comparison for Study #1. #1. FigureFigure 7. Cable admittance forCase Case Study Figure 7. Cable admittance comparison for Case Study #1.

Thedeviation short deviation the FDPi model withrespect respect to model is due four The short of theofFDPi model with to the thereference reference model is to due tomain four main The short of the FDPi model respect to model is due to four main reasons. First,deviation the parameterization that iswith performed by the thereference ULM model from PSCAD/EMTDC reasons. First, the parameterization that is performed by the ULM model from PSCAD/EMTDC reasons. parameterization is performed by the ULM from of PSCAD/EMTDC softwareFirst, is notthe exactly the same as that in reference [13]. Therefore, the model comparison both models is softwaresoftware is not exactly the same as in reference [13]. Therefore, the comparison of models both models is is not exactly the same as in reference [13]. Therefore, the comparison of both is performed with slightly different parameter values. performed with slightly different parameter values. performed with different parameter values. Second, theslightly representation of the FDPi model is built in order to consider frequencies up to 10 theof representation FDPi model is built inaorder tofrequency consider frequencies up to kHz.Second, The error the model canofbethe reduced by considering higher range. In order to10 do kHz. error of by considering higher frequency range. order to do this, The the order ofthe themodel modelcan hasbe toreduced be increased but at the aexpense of increasing the In computational this, cost.the order of the model has to be increased but at the expense of increasing the computational cost.

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Second, the representation of the FDPi model is built in order to consider frequencies up to 10 kHz. The error of the model can be reduced by considering a higher frequency range. In order to do this, the order of the model has to be increased but at the expense of increasing the computational cost. Third, the FDPi model only takes into account the frequency dependency of the series impedance Energies 11, REVIEW 14 terms ZCouter ZSouter andPEER ZPouter . The frequency dependent behavior of other terms is not Energies, 2018, 2018, 11,xxFOR FOR PEER REVIEW 14of of22 22 taken into account. Third, into the Third, the the FDPi FDPi model model only only takes into account account the the frequency frequency dependency dependency of the series Finally, an additional oscillation is takes presented in the results of the FDPi model of due toseries the nature impedance , Z Souter and . The frequency dependent behavior of other terms is impedance terms terms ZZCouter and ZZPouter Couter , ZSouter Pouter . The frequency dependent behavior of other terms is that lumped parameters models introduce an unreal resonance, as explained in reference [31]. Is in fact, not not taken taken into into account. account. a resonance that is not present in the realiscable but that appears because of the model characteristics. Finally, presented Finally, an an additional additional oscillation oscillation is presented in in the the results results of of the the FDPi FDPi model model due due to to the the nature nature In orderthat to show this unreal resonance, Figure 8 depicts the cable admittance for a frequency lumped parameters models introduce an unreal resonance, as explained in reference [31]. that lumped parameters models introduce an unreal resonance, as explained in reference [31]. Is Is in in range fact, a resonance that is not present in the real cable but that appears because of the model of 100 kHz. be seen that FDPi 3 and N = 32) considered previously, fact, Ita can resonance that is for not the present in model the realorder cable (M but=that appears because of the model characteristics. In order show 88 depicts cable admittance characteristics. Inlocated order to toaround show this this unreal resonance, Figure depicts the cable admittance for for aa the unreal resonance is 40unreal kHz resonance, and with aFigure magnitude ofthe 0.01 Siemens. frequency range of 100 kHz. It can be seen that for the FDPi model order (M = 3 and N = 32) considered frequency of 100 kHz. It can be seenofthat for the FDPi model(N) order = 3 and N FDPi = 32) considered Figure 8 alsorange performs an evaluation different number of(M cascaded sections while previously, previously, the the unreal unreal resonance resonance is is located located around around 40 40 kHz kHz and and with with aa magnitude magnitude of of 0.01 0.01 Siemens. Siemens. keeping the same order of Foster equivalent networks (M = 3). Two importantFDPi aspects canwhile be inferred Figure Figure 88 also also performs performs an an evaluation evaluation of of different different number number (N) (N) of of cascaded cascaded FDPi sections sections while from Figure 8: keeping keepingthe thesame sameorder orderof ofFoster Fosterequivalent equivalentnetworks networks(M (M==3). 3).Two Twoimportant importantaspects aspectscan canbe beinferred inferred

• •

from Figure 8:

from Figure An increase in 8: the model order allows extending the frequency range of the model, which gives a •• deviation An the order extending the An increase increase in the model model order allows allows extending thefrequency frequency range range of of the the model, model, which which gives gives lower inin time and frequency domain results. aa lower deviation in time and frequency domain results. lower deviation in time and frequency domain results. The unreal resonance is shifted to the right in frequency and decreased in magnitude by increasing •• The The unreal unreal resonance resonance is is shifted shifted to to the the right right in in frequency frequency and and decreased decreased in in magnitude magnitude by by the model order. the model order. increasing increasing the model order.

Figure 9Figure shows the impact of the model order timesignals. signals. It can be seen that for lower 99 shows the of model order on the ItIt can be that lower Figure shows the impact impact of the the model orderon onthe the time time signals. can be seen seen that for for lower model orders there is a higher oscillation magnitude. model orders there is a higher oscillation magnitude. model orders there is a higher oscillation magnitude. 0.08 0.08 ULM ULM 12FDPi 12FDPi

0.07 0.07

22FDPi 22FDPi 32FDPi 32FDPi

0.06 0.06

Magnitude(S) (S) Magnitude

0.05 0.05

0.04 0.04

0.03 0.03

0.02 0.02

0.01 0.01

00 1 1 10 10

2 2 10 10

4 4 10 10

3 3 10 10

5 5 10 10

Frequency Frequency(Hz) (Hz)

8. admittance comparison for #1 different number (N) Figure 8.Cable Cable admittance comparisonfor forCase Case Study #1with with different numbernumber (N)of ofcascaded cascaded FDPi Figure 8.Figure Cable admittance comparison CaseStudy Study #1 with different (N) ofFDPi cascaded sections. sections. FDPi sections. 5

3

10 5 10

3

3000 3000

VVc1g _ r c1 g _ r

2.5 2.5

2000 2000

2

IIs1_ r s1_ r

1500 1500

1.5 1.5

1

ULM ULM 12FDPi 12FDPi 22FDPi 22FDPi 32FDPi 32FDPi

2500 2500

Current (A) Current (A)

Voltage (V) Voltage (V)

2

ULM ULM 12FDPi 12FDPi 22FDPi 22FDPi 32FDPi 32FDPi

1000 1000

500 500

1

0 0 0.5 0.5

0

0 0

-500 -500

-1000 -1000 0

0.5 0.5

1

1

1.5 1.5

Time (s) Time (s)

(a) (a)

2

2

2.5 2.5

3 -3

3

10 -3 10

0 0

0.5 0.5

1 1

1.5 1.5

Time (s) Time (s)

2 2

2.5 2.5

3 3 -3

10 -3 10

(b) (b)

Time comparison with different number cascaded FDPi sections for Figure 9.Figure Time9.9.response comparison with different (N)of cascaded FDPi sections for Case Figure Time response response comparison with differentnumber number (N) (N) ofof cascaded FDPi sections for Case Case c1g_r I.Is1_r (a) VV(b) c1g_r.I. (b) (b) s1_r.. Study #1. (a) Study #1.Study (a) V#1. . c1g_r s1_r

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4.2. Case Study #2: Single Core Underground Cable The cable system Energies 2018, 11, 2778

for this second case study consists of three single core cables (one for14each of 21 phase) as depicted in Figure 10. This figure shows the simplified cross-section of each single core cable and gives information about the surrounding medium properties, burial depth and laying 4.2. Case Study #2:cable Single Core Underground Cable formation of the system. Eachcable single core underground operates with of a nominal voltage of 150 (one kV. The cablephase) has a The system for this secondcable case study consists three single core cables for each 2 core cross-section of 1200 mm figure and its insulation material is XLPE (cross-linked polyethylene). as depicted in Figure 10. This shows the simplified cross-section of each single core cable and The required geometrical and material properties of the cable are listed in Table 4.ofThe gives information about the surrounding medium properties, burial depth and laying formation the information cable system.presented in Figure 10 and Table 4 is adopted from [31]. Air Earth

ρ earth = 100 Ωm μrearth = 1

1.22224 m

1.30324 m

Figure 10. Cable cross-section cross-section for for Case Case Study #2 and installation conditions (laying depth and 10. Cable surrounding surrounding medium medium properties). properties). Source Source [31]. [31].

Each singleTable core 4. underground cable operates with a nominal voltage of 150 [31]. kV. The cable has a Geometrical and material parameters for Case Study #2. Source core cross-section of 1200 mm2 and its insulation material is XLPE (cross-linked polyethylene). Layer Parameters The required geometrical and material properties of the cable are listed in Table 4. The information −8 ρ = 3.19 ⋅ 10 Ωm , μrc = 1 r = 20.75 mm , 1 c 1 presented in Figure 10 and Table 4 is adopted from [31]. r = 40.25 mm , μ = 1.08 , ε rins1 = 2.68 2 2 rins1 −7 Table 4. Geometrical and material parameters Case Study = 1.19 ⋅10 Ωm#2., Source r3 = 42.76 mm , ρ s for μrs = 1 [31]. 3

4 Layer 1

r4 = 46.76 mm , μ rins 2 = 1 , ε rins 2 = 2.5 Parameters r1 = 20.75 mm, ρc = 3.19 · 10−8 Ωm, µrc = 1

As in the previous case, theµvalidation is to represent the steady state and 2 the requirement r2 = 40.25of mm, rins1 = 1.08, ε rins1 = 2.68 − 7 transients comprising frequencies up rto 1042.76 kHz. Toρfulfill the objective the model is built with 3 mm, · 10 Ωm, µrs = FDPi 1 s = 1.19 3 = = 46.76 µrins2 = 1, ε rins2 networks = 2.5 35 cascaded FDPi sections4(N = 35), eachr4one withmm, Foster equivalent of order three (M = 3) for each conductive layer term. The series impedance terms were fitted for a frequency range from 0.1 Hz 10previous kHz by means of VF algorithm.of the validation is to represent the steady state and Asup in to the case, the requirement In order to validate the response FDPi To model inthe theobjective time domain, the test configuration of transients comprising frequencies up of to the 10 kHz. fulfill the FDPi model is built with Figure 11 has beensections used considering a cable length of 50equivalent km. Similar to the previous case, the test 35 cascaded FDPi (N = 35), each one with Foster networks of order three (M = 3) consists of energizing one-phase of the cable with an AC voltage source at its sending end and with for each conductive layer term. The series impedance terms were fitted for a frequency range from the Hz receiving ends The input signal is a cosine function with a voltage magnitude of 150 0.1 up to 10 kHzopen. by means of VFsource algorithm. kVrms voltage the given by the cable energization place at the peak In (nominal order to validate response of themanufacturer). FDPi model in The the time domain,takes the test configuration voltage of the input source in order to produce the highest transient current. of Figure 11 has been used considering a cable length of 50 km. Similar to the previous case, the test Figure 12 shows the time domain the receiving signals in Figure 11) consists of energizing one-phase of the response cable withofan AC voltage end source at its(depicted sending end and with for the Case Study #2. AsThe in the previous each shows information about the steady and the receiving ends open. input sourcecase, signal is aplot cosine function with a voltage magnitude of transient state (top one) and a zoom in for the first 10 ms (down one) in order to show the transient 150 kVrms (nominal voltage given by the cable manufacturer). The energization takes place at the peak response detail. voltage ofin the input source in order to produce the highest transient current. Figure 12 shows the time domain response of the receiving end signals (depicted in Figure 11) for the Case Study #2. As in the previous case, each plot shows information about the steady and transient state (top one) and a zoom in for the first 10 ms (down one) in order to show the transient response in detail.

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t = 0 ms t = 0 ms

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Vc1g _ r Vc1g _ r

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Figure 11. Setup for time response validation – Case Study #2. Figure 11. Setup for time response validation—Case validation – Case Study #2. 10

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Figure 12.12. Time response comparison forfor Case Study #2.#2. (a)(a) Vc1g_r (c) IIs2_r Figure Time response comparison Case Study Vc1g_r. .(b) (b)IIs1_r s1_r.. (c) .. s2_r Figure 12. Time response comparison for Case Study #2. (a) Vc1g_r. (b) Is1_r. (c) Is2_r.

Figure and phase of the cable admittance for afor frequency rangerange of 10 of kHz. Figure13 13shows showsthe themagnitude magnitude and phase of the cable admittance a frequency 10 Figure 13 shows the magnitude and phase of the cable admittance for a frequency range of 10 kHz.As depicted in the previous figures, time and frequency domain results obtained from the FDPi kHz. Asshow model a very agreement with the from the reference model. The shape and depicted ingood the previous figures, timeones andobtained frequency domain results obtained from the FDPi As depicted in the previous figures, time and frequency domain results obtained from the FDPi amplitude of the time signals are very similar and a good representation of the wave travel time is model show a very good agreement with the ones obtained from the reference model. The shape and model show a very good agreement with the ones obtained from the reference model. The shape and obtained byofthe it can be aseen that the results obtained for this casetime study amplitude theFDPi timemodel. signalsFurthermore, are very similar and good representation of the wave travel is amplitude of the time signals are very similar and a good representation of the wave travel time is present a lower deviation in comparison with the previous one. obtained by the FDPi model. Furthermore, it can be seen that the results obtained for this case study obtained the deviation FDPi model. Furthermore, itthe can be seen that the results obtained for this case(single study Thisaby lower is because formulation used for cable parameterization present lower deviation inmainly comparison with the previous one. present a lower deviation in comparison with the previous one. core cables) is thedeviation same as the one used by thethe ULM model from PSCAD/EMTDC software. Therefore, This lower is mainly because formulation used for cable parameterization (single This lower deviation is mainly because the formulation used for cable parameterization (single the same values for the series impedance terms and shunt admittance terms are obtained and the core cables) is the same as the one used by the ULM model from PSCAD/EMTDC software. Therefore, core cables) is the same as the one used by the ULM model from PSCAD/EMTDC software. Therefore, differences in the for results mainly due to the characteristics of both models. the same values the are series impedance terms and shunt admittance terms are obtained and the the same values for the series impedance terms and shunt admittance terms differences in the results are mainly due to the characteristics of both models. are obtained and the differences in the results are mainly due to the characteristics of both models.

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As presented in the previous case study, some changes are done to the number (N) of cascaded Aspresented presentedininthe theprevious previouscase casestudy, study, somechanges changes aredone donetotothe thenumber number(N) (N) of cascaded As FDPi sections in order to show its effects onsome the frequencyare range and unreal resonanceofofcascaded the cable FDPisections sectionsininorder ordertotoshow showitsitseffects effectson onthe thefrequency frequencyrange rangeand andunreal unrealresonance resonanceofofthe thecable cable FDPi (Figure 14). Similar conclusions are obtained as in the previous example. (Figure 14). Similar conclusions are obtained as in the previous example. (FigureAs 14). obtained that as inall thethe previous example. lastSimilar point, conclusions it is worth toare emphasize parameters information for this cable is taken Aslast lastpoint, point, it it is worth to emphasize that allall thethe parameters information for this cablecable is taken As is worth to emphasize that parameters for this from reference [31]. In this reference, field measurements were done information to a single section of a 150 is kV fromfrom reference [31]. In thisInreference, field measurements were done to a single sectionsection of a 150 kV taken reference [31]. this reference, field measurements were done to a single of a underground cable and the results were compared with the Universal Line Model (ULM) from underground cable and the results were compared with the Universal Line Model (ULM) from 150 kV underground cable and theconclusions results wereobtained comparedinwith the Universal Line Model (ULM) from PSCAD/EMTDC software. The reference [31] describes a good agreement PSCAD/EMTDC software. software. The The conclusions conclusions obtained obtained in inreference reference[31] [31]describes describesaagood goodagreement agreement PSCAD/EMTDC between measurements and ULM model for the coaxial mode of the real cable. In this sense, it can be betweenmeasurements measurementsand andULM ULMmodel model forthe thecoaxial coaxialmode modeofofthe thereal realcable. cable.InInthis thissense, sense,ititcan canbebe between stated that the results presented in this for example indirectly validate the response of the FDPi model statedthat thatthe theresults resultspresented presentedininthis thisexample exampleindirectly indirectlyvalidate validatethe theresponse responseofofthe theFDPi FDPimodel model stated with experimental measurements. withexperimental experimentalmeasurements. measurements. with 0.2

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Figure 13. Cable admittance comparison for Case Study #2. Figure Figure13. 13.Cable Cableadmittance admittancecomparison comparisonfor forCase CaseStudy Study#2. #2. 0.16 0.16 0.14 0.14

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Figure Figure14.14.Cable Cableadmittance admittancecomparison comparisonfor forCase CaseStudy Study#2#2with withdifferent differentnumber number(N) (N)ofofcascaded cascaded Figure 14. Cable admittance comparison for Case Study #2 with different number (N) of cascaded FDPi FDPisections. sections. FDPi sections.

5. Methodology for the Selection of the FDPi Model Order 5. Methodology for the Selection of the FDPi Model Order 5. Methodology for the Selection the FDPi Model Order As it can be inferred from the of results shown in the previous section, the choice of an optimal As it can be inferred from the results shown in the previous section, the choice of an optimal modelAs order, selection of M and N, does not obey deterministic rules and isthe highly on it can be inferred from the results shown in the previous section, choicedependent of an optimal model order, selection of M and N, does not obey deterministic rules and is highly dependent on several factors. This characteristic difficult the possibility of presenting guidelines that are valid for model order, selection of M and N, does not obey deterministic rules and is highly dependentall on several factors. This characteristic difficult the possibility of presenting guidelines that are valid for the scenarios. several factors. This characteristic difficult the possibility of presenting guidelines that are valid for all the scenarios. all the scenarios.

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This is is capable ofof selecting a Thissection sectionisisintended intendedtotoserve serveasasaabasis basisfor forcreating creatinga asoftware softwaretool toolthat that capable selecting suitable model order for representing a given cable under specific performance requirements. Figure 15 a suitable model order for representing a given cable under specific performance requirements. shows should bethat implemented the software in toolthe forsoftware the determination of Figurethe 15 general shows procedure the generalthat procedure should be in implemented tool for the the optimal order of the FDPi model. determination of the optimal order of the FDPi model.

Cable System Definition

Definition of Performance Requirements

Evaluation of Model Order

Selection of Optimal Model Order

Figure Figure 15. 15. General General procedure procedure for for the the determination determination of of the the optimal optimal order orderof ofthe theFDPi FDPimodel. model.

The to to take intointo account is theisdefinition of the cable system study. This definition Thefirst firstpoint point take account the definition of the cableunder system under study. This comes from the characteristics of the cable system, which are described next: definition comes from the characteristics of the cable system, which are described next:

•• •• •• •• •• •• ••

Type Typeof ofcable: cable:Underground Undergroundor orsubmarine submarinecables. cables.Single-core Single-coreor orthree-core three-corecables. cables. Cable length. Cable length. Cable disposition: disposition: Tight Tightor orspaced spaceddisposition dispositionin inflat flator ortrefoil trefoilformation. formation. Cable Surrounding medium properties: resistivity and relative permeability of the surrounding medium. Surrounding medium properties: resistivity and Burial depth. depth. Burial Grounding configuration: Single-end Single-endbonding, bonding,both-ends both-endsbonding bondingor orcross crossbonding. bonding. Grounding configuration: Load connected to the cable system: The cable impedance depends on the termination the Load connected to the cable system: The cable impedance depends on the termination of the of cable. cable. This impedance vary continuously (infinite possibilities). cases can for be This impedance can vary can continuously (infinite possibilities). Extreme casesExtreme can be considered considered for the cable system, short-circuited cable and open-ended cable. the cable system, short-circuited cable and open-ended cable. • Couplings between other cable systems. • Couplings between other cable systems. All these aspects change the time response and frequency response of the cable; hence need to All these aspects change the time response and frequency response of the cable; hence need to be be evaluated in order to select an optimal model order. evaluated in order to select an optimal model order. The second step consists of defining the performance requirements that the FDPi model must The second step consists of defining the performance requirements that the FDPi model must fulfill. The most relevant requirements are: fulfill. The most relevant requirements are: • Frequency range: The definition of the frequency range is crucial for the intended application of • Frequency The definition of the frequency range is crucial for theallintended application of the model.range: An adequate value should be assigned in order to include the relevant dynamics the model. An adequate value should be assigned in order to include all the relevant dynamics that can have an effect on the studies that are going to be performed. that can have effect onto the studies that are going to belimit performed. • Accuracy: It isan necessary define a minimum accuracy and quantify the accuracy. In order • Accuracy: It is necessary to define a minimum accuracy limit and quantify the accuracy. order to quantify the accuracy, a measure of the error associated with the FDPi model must be In defined. to quantify the can accuracy, measure of the errorofassociated withasthe FDPi model must be Several criteria be useda for the quantification the error such averaged error, weighted defined. Several criteria can be used for the quantification of the error such as averaged error, error and absolute error. Additionally, the error can be computed in the time domain, frequency weighted error and absolute error. Additionally, the error can be computed in the time domain, domain or taking into account both domains. The authors suggest assessing the accuracy of the frequency domain taking into account both domains. The authors suggest assessing FDPi model in the or frequency domain by computing the absolute error point by point over the the accuracy of the FDPi model in the frequency domain by computing the absolute error point by frequency range of interest. Absolute error is chosen over other methods mainly to avoid a high point over the frequency range of interest. Absolute error is chosen over other methods mainly error in a single frequency (for example in a resonance frequency). Furthermore, the accuracy of to error in a single frequencyby (forevaluating example inthe a resonance frequency). theavoid FDPia high model should be performed amplitude error andFurthermore, phase error the accuracy of the FDPi model should be performed by evaluating the amplitude and phase separately. ULM can be considered as the reference model if measurements are error not available. error separately. ULM can be considered as the reference model if measurements are not available. The third block of Figure 15 consists of evaluating the order of the FDPi model. A sweep of M third should block ofbeFigure 15 consists ofcable evaluating order of theAccuracy FDPi model. sweep of M and The N values performed for the systemthe under study. and A computational and N values should be performed for the cable system under study. Accuracy and computational time must be recorded for each M and N combination. time must be step recorded for each and Nmodel. combination. The last is to select theM optimal The selection of the optimal model order is a balanced The last step isthe to select the optimal model. The selection of the optimal model order is a balanced trade-off between following two conditions: trade-off between the following two conditions: • Accuracy level within the allowable limit. The lowest computational time. •• Accuracy level within the allowable limit. • The lowest computational time. In general, higher values of M and N will lead to a better accuracy. However, beyond a certain

model order, the improvements in accuracy with further increase of the model order tend to become almost negligible.

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In general, higher values of M and N will lead to a better accuracy. However, beyond a certain model order, the improvements in accuracy with further increase of the model order tend to become almost negligible. Another important aspect to take into account for the accuracy of the FDPi model is the way VF algorithm performs the approximation of the frequency-dependent variation of the electrical parameters of the cable system. Vector fitting algorithm may be fine-tuned by applying a weighting function [25]. This weighting function can improve the accuracy in a specific frequency region. 6. Potential Applications of the FDPi Model The model presented in this paper is intended to be used in any computer-based simulation tool (such as Matlab/Simulink) with the aim of representing different cable-based scenarios. It can help in representing and analyzing the behavior of the cable for the steady state and for transient responses in both time domain and frequency domain. The main features of the model are:

• • •

•

•

The capability of representing different types of cables (underground and submarine single core and three-core cables) and overhead lines. The capability of representing different cable dispositions or laying formations (flat formation, trefoil formation). The capability of representing different grounding configurations due to the fact of including the different conductive layers of the cable. In this sense, grounding configurations such as single end bonding, both ends bonding and cross bonding can be represented with the model. The possibility to derive simplified analytical models with different levels of abstraction from the complete model. This means that the model can be easily reconfigured with the aim of obtaining a model with a higher abstraction level that allows deducing dominant behaviors and parameters of the system under study, allowing to do many other subsequent analysis. All by obtaining a reduced computational cost model with reduced accuracy for higher frequencies. The possibility to configure a very accurate model at expense of the computational cost in order to study a specific or special phenomenon.

These previous features allow the correct characterization of the cable system: resonance frequencies, impedance spectrum, sequence impedances and impedance-admittance matrices. As already presented in this paper, the FDPi model allows the formulation of a mathematical description, Laplace domain or state-space form, of the behavior of the cable. This feature allows the inclusion of the cable model with the mathematical description of other power components (such as transformers, filters, power converters including their control loops and harmonics, and others). Thus allowing the creation of a global analytical model of scenarios such as onshore and offshore wind power plants [32,33], electrically propelled vessels [34,35], electric railways, electrical grids [36] and many more, in where different types of cables and elements are presented. With this modeling approach, different valuable studies can be performed in order to obtain a higher understanding degree and contributing to the analysis and results presented in the previous references, which consider cable models at power frequency and neglecting their frequency dependent behavior. Focussing the attention on an OWPP scenario, some of the valuable studies (in which the FDPi model can be used and contribute to them) can be oriented towards aspects in terms of system stability and power quality, as main industry concerns. System Stability:

• •

The development of a more complete analytical model of an OWPP scenario is required in order to perform this kind of study. The application of techniques (Nyquist and/or Lyapunov techniques) in order to analyze and evaluate the stability of the system.

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The design of solutions oriented towards the improvement of the stability of the system (design of controls and active damping strategies). Power Quality:

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The design of passive and/or active filters in order to damp the resonances presented in the OWPP. The improvement of the total harmonic distortion (THD). The evaluation of harmonics (harmonic assessment) based on the OWPP topology [37]. Possibility of this cable model to be used with the wind turbine harmonic model proposed in [11] and be able to test different modulation strategies, evaluate the fulfillment of BDEW [38] and IEEE 519 [39] grid codes, evaluate the propagation (magnitude and phase angle) of harmonics in the point of common coupling (PCC) and other points of the OWPP, and other studies.

This cable model can be used as well in the early design stages of the OWPP in order to contribute to the fulfillment of the design requirements. Some of these requirements can be the topology of the OWPP, the magnitude and frequency of the resonances, the maximum number of installable wind turbines (based on the topology), the fulfillment of low-voltage ride through (LVRT) and high-voltage ride through (HVRT) capability and many more. 7. Conclusions In this paper, a frequency-dependent Pi model has been developed in order to reproduce the behavior of a three-core submarine cable. The model consists of N cascaded FDPi sections. Each FDPi section incorporates the following: (i) the magnetic coupling which is represented by an inductance matrix, (ii) the electrical coupling between conductors and represented by means of capacitances, (iii) the frequency dependent variation of the conductive layers by means of Foster equivalent networks and iv) the loss mechanism of the cable represented with each resistive element. All these four points allow an easy implementation of the model by means of discrete RLC components. Parameterization of the model is described in detail and is performed according to analytical equations. However, other parameterization methods can be used to account for proximity effect and other medium characteristics. Regarding the validation of the FDPi model, two cable system case studies are presented and a comparison is performed with a reference model, the ULM model from PSCAD/EMTDC software. The first case presents the results of the FDPi model for the cable under study, a three-core submarine cable. The second case study presents the results of three single-core underground cables laid in trefoil formation. This last case confirms two important points in order to give more weight to the validity of the FDPi model. The first point is the model applicability to other types of cable systems and the second point is an indirect validation with experimental measurements that were performed on a real cable system for the coaxial mode. The results for both case studies show a very good agreement of the FDPi model in representing the behavior of a power cable, submarine and underground cable, for frequencies up to 10 kHz. If a higher frequency range needs to be represented, the order of the model has to be increased in order to fulfill the accuracy requirements but at the expense of increasing the computational burden. Additionally, the effects of model order on the time response and the unreal resonance of the cable model are presented for a different number of cascaded FDPi sections. Finally, it is important to emphasize that the choice of a suitable FDPi model order does not obey deterministic rules. It is highly dependent on the requirements such as desired accuracy and frequency range of interest on the one hand, and characteristics of the cable under study such as type of cable, cable configuration and cable length on the other hand.

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Author Contributions: C.R.: conceptualization, formal analysis, investigation, software, writing—original draft preparation. G.A.: conceptualization, formal analysis, investigation, writing—review and editing. M.Z.: conceptualization, formal analysis, investigation, writing—review and editing. D.M.: conceptualization, formal analysis, investigation, writing—review and editing. J.A.: conceptualization, formal analysis, investigation. Funding: This research received no external funding. Conflicts of Interest: The authors declare no conflict of interest.

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