The Effect of the Axial Heat Transfer on Space Charge Accumulation

energies

Article The Eﬀect of the Axial Heat Transfer on Space Charge Accumulation Phenomena in HVDC Cables

Giuseppe Rizzo * , Pietro Romano , Antonino Imburgia , Fabio Viola and Guido Ala

L.E.PR.E. HV Laboratory, Dipartimento di Ingegneria–Università di Palermo, Viale delle Scienze, Ediﬁcio 9, 90135 Palermo, Italy; [email protected] (P.R.); [email protected] (A.I.); [email protected] (F.V.); [email protected] (G.A.) * Correspondence: [email protected]

Received: 5 August 2020; Accepted: 11 September 2020; Published: 15 September 2020

Abstract: To date, it has been widespread accepted that the presence of space charge within the dielectric of high voltage direct current (HVDC) cables is one of the most relevant issues that limits the growing diffusion of this technology and its use at higher voltages. One of the reasons that leads to the establishment of space charge within the insulation of cables is the temperature dependence of its conductivity. Many researchers have demonstrated that high temperature drop over the insulation layer can lead to the reversal of the electric field profile. In certain conditions, this can over-stress the insulation during polarity reversal (PR) and transient over voltages (TOV) events accelerating the ageing of the dielectric material. However, the reference standards for the thermal rating of cables are mainly thought for alternating current (AC) cables and do not adequately take into account the effects related to high thermal drops over the insulation. In particular, the difference in temperature between the inner and the outer surfaces of the dielectric can be amplified during load transients or near sections with axially varying external thermal conditions. For the reasons above, this research aims to demonstrate how much the existence of “hot points” in terms of temperature drop can weaken the tightness of an HVDC transmission line. In order to investigate these phenomena, a two-dimensional numerical model has been implemented in time domain. The results obtained for some case studies demonstrate that the maximum electric field within the dielectric of an HVDC cable can be significantly increased in correspondence with variations along the axis of the external heat exchange conditions and/or during load transients. This study can be further developed in order to take into account the combined effect of the described phenomena with other sources of introduction, forming, and accumulation of space charge inside the dielectric layer of HVDC cables.

Keywords: space charge; thermal rating; HVDC cables

1. Introduction To date, the demand of high voltage direct current (HVDC) cables is growing due to the tendency to increase the amount of electric energy exchanged over long distances. Old HVDC cables with mass impregnated insulation are being replaced with extruded cables resistant to higher temperatures and with better mechanical properties. The achievement of higher temperatures leads to an increase in the ampacity of a cable, reducing the specific costs of transmission lines, but, at the same time, some collateral effects are catalyzed. In particular, the rise of Joule losses inside the conductor causes an increase in the temperature drop over the dielectric layer. As a result, this leads to an increase in the secondary mechanical stresses and to a radial variation in the electrical conductivity of the insulation. Indeed, in polymeric materials, this quantity grows exponentially with the temperature. As widely demonstrated by many studies, in the insulation of a cable, the simultaneous existence of an electric field and a gradient in conductivity leads to a non-zero divergence of the current and

Energies 2020, 13, 4827; doi:10.3390/en13184827 www.mdpi.com/journal/energies Energies 2020, 13, 4827 2 of 18 therefore to the presence of space charge [1–3]. For the main commercial materials used as insulation for HVDC cables, it can be asserted that as the radial thermal gradient increases, the higher the electric field is close to the interface with the outer semicon, the lower the one near the interface with the inner semicon. In some cases, the electric field at the external boundary of the insulation layer of a loaded cable can exceed its value at the inner boundary reached in no-load conditions. This scenario is more likely to occur in large cables loaded with rated current or under emergency load conditions. However, in any case, the inversion of the electric field profile can lead to higher electric stresses than the rated ones in case of transient over voltages (TOV) or polarity reversal (PR). This can cause the occurrence of partial discharges near the external surface of the dielectric layer and the accelerated ageing of the insulation [4–6]. Many researchers have focused their attention on the amount of space charge accumulated within the insulation of HVDC cables in presence of thermal gradient, coming to the conclusion that the importance of this phenomenon can be generally considered as proportional to the size of the cable [7–10]. This is because the thicker the dielectric of a cable, the greater its thermal resistance and therefore the drop in temperature between the internal and the external boundary. In addition, recent studies have focused their attention to the effect of environmental issues on the temperature drop over the dielectric in cables [11]. Other researchers have deepened the issues related to the role of the insulating material [12]. In this paper, attention is focused on the conditions that can lead to local and/or temporary peaks of thermal gradient in the dielectric layer of HVDC cables and the possibility to reach high levels of electric field near its external surface is verified. A discontinuity along the cable axis of the external heat exchange parameters can lead to the achievement of a local peak of temperature drop. In certain conditions, the axial thermal conduction can prevail on the radial one due to the high thermal conductivity of metals in comparison to that of polymeric materials. The area near a joint, the crossing by the cable of an interface between two different means or local external heat or cold sources are potentially cases in which axial heat transfer can play an important role in defining the highest electric field within the insulation of a cable. In Figure1, the e ffect on the temperature drop of a hot spot is illustrated. Nevertheless, the large difference in the heat capacities of conductor and dielectric can temporarily increase the temperature drop during load transients. This can be amplified near geometric variations such as near joints. The standards IEC 60853-2 and IEC 60287 propose procedures for the calculation of the temperature inside cables and for the current rating based on a one dimensional approach that neglect the axial heat transfer [13,14]. This is because these standards are applicable to AC cables or DC cables under less than 5 kV of voltage. In AC cables, the main parameter to be controlled is the conductor temperature and this is certainly achieved in the 2D section where the worst external heat exchange occurs. In this case, the axial heat conduction cannot further increase the conductor temperature in a hot spot. To date, no dedicated standard of the International Electrotechnical Commission (IEC) provides a guide for the thermal design of HVDC cables at rated voltages greater than 5 kV. This lack is covered by the Cigrè brochure 640 [15] that takes into account the need to limit the temperature drop over the dielectric layer in HVDC cables. This document pays attention to the role of the axial heat transfer in the establishment of local variations in the radial temperature distribution. Some researchers have simulated the behavior of axial discontinuities by the thermal point of view by means of finite element analysis (FEA) in three-dimensional geometries [16–18]. However, this method requires high calculation powers. To date, no standards have been dedicated to the calculation of the temperature profile and the current rating for HVDC cables. In the DC case, since the thermal gradient plays an important role in limiting the rated current, a 2D approach for modeling a laying section is not suitable to describe cautiously the operative conditions and a 3D modeling is necessary to take into account axial heat transfer phenomena. Energies 2020, 13, 4827 3 of 18 Energies 2020, 13, x FOR PEER REVIEW 3 of 20

Figure 1. SchematicFigure 1. Schematic of prevalent of prevalent axial axial conduction conduction closeclose to to an an interface interface between between two external two externalmeans. means.

The widespreadThe widespread ways ways to measure to measure the the radial radial patternpattern of ofspace space in cables in cables are the are pulsed the electro pulsed electro acoustic (PEA) method and the thermal step method (TSM) [19–24]. In particular, the former is based acoustic (PEA) method and the thermal step method (TSM) [19–24]. In particular, the former is based on the detection of an acoustic impulse generated by the interaction between charges and a pulsed on the detectionelectric field of an externally acoustic generated. impulse Due generated to the attenuation by the of interactionacoustic signals between when they charges pass through and a pulsed electric fielda solid externally material, generated. the larger is the Due cable to size, the attenuationthe lower is the of measurable acoustic charge signals density. when TSM they is passbased through a solid material,on the the measurements larger is the of cablea current size, of the order lower of ispA the proportional measurable to the charge amount density. of space TSM charge is based on density. Due to the low levels of currents that are to be measured, this method is not suitable for the measurements of a current of the order of pA proportional to the amount of space charge density. measuring small space charge densities. Since the phenomena described above are more likely to Due to theoccur low levelsin large ofHVDC currents cables, that at the are state to be of measured,the art, experimental this method methods is notare suitablenot suitable for for measuring small spaceunderstanding charge densities. the impact Since of axial the discontinuities phenomena on described the distribution above of the are electric more field. likely to occur in large HVDC cables,The at establishment the state of of the a temperature art, experimental gradient in methodsthe dielectric are of HVDC not suitable cables plays for an understanding important the role during programmed or accidental voltage transients [25]. impact of axial discontinuities on the distribution of the electric field. In this paper, a 2D model in the radial and axial coordinates as well as in time domain is used to The establishmentsimultaneously calculate of a temperature the distributions gradient of temperature, in the dielectric space charge, of HVDC and electric cables field. plays an important role during programmedUnlike the bipolar or accidental charge transp voltageort models transients used recently [25]. by other researchers, a conductivity In thismodel paper, has a been 2D used model in this in theresearch radial to evaluate and axial the charge coordinates distribution as well in the as dielectric in time [26]. domain Even if is used to more approximate, the conductive model provides sufficiently reliable information for the evaluation simultaneously calculate the distributions of temperature, space charge, and electric field. of the electric field profile during a voltage transient [27]. Unlike theIn bipolarthe second charge paragraph, transport the model models implemented used in recently Matlab® is by described, other researchers, whilst in the third, a conductivity a model hascase been study used is inpresented. this research Finally, to the evaluate results ar thee discussed charge distributionin the fourth paragraph in the dielectric and further [26 ]. Even if more approximate,development the ideas conductive for future research model are provides offered in su theffi cientlyconclusions. reliable information for the evaluation of the electric2. Mathematical field profile Model during a voltage transient [27]. In the second paragraph, the model implemented in Matlab® is described, whilst in the third, As mentioned above, a time domain 2D mathematical model has been developed to study the a case studydistribution is presented. of the electric Finally, field within the results the insulation are discussed of an HVDC incable the close fourth to a discontinuity paragraph along and further developmentthe axis ideas of forthe heat future exchange research parameters are off withered the in external. the conclusions. The equations of the thermal and the electrical models are expressed in cylindrical geometry assuming axial symmetry. In this work, the 2. Mathematicalangular heat Model conduction inside the cable is neglected in order to reduce the computational effort. The geometry of a typical HVDC cable is discretized into radial and axial nodes in order to model a As mentionedmultilayer stratigraphy above, a time such as domain the one shown 2D mathematical in Figure 2. All modelthe materials has of been the cable developed are modeled to study the distributionas ofcontinuous the electric and homogeneous, field within therefore the insulation equivalent of parameters an HVDC are cable used closefor the to stranded a discontinuity along the axisconductor of the and heat the wire exchange screen in parameters order to take into with account the external. their filling The factor. equations of the thermal and the electrical models are expressed in cylindrical geometry assuming axial symmetry. In this work, the angular heat conduction inside the cable is neglected in order to reduce the computational effort. The geometry of a typical HVDC cable is discretized into radial and axial nodes in order to model a multilayer stratigraphy such as the one shown in Figure2. All the materials of the cable are modeled as continuous and homogeneous, therefore equivalent parameters are used for the stranded conductor and the wire screen in order to take into account their filling factor. For each layer, thermal and electrical properties are assigned based on the materials as well as the tables of the standards IEC 60287 and IEC 60853-2. The geometry is discretized in NR radial nodes and NZ axial nodes. For each of these, thermal conductivity, mass density, and specific heat characterize the material in the thermal model as well as permittivity and electrical conductivity at 20 ◦C are used in the electric model. All the parameters used Energies 2020, 13, 4827 4 of 18 in the thermal model are assumed constant and homogeneous for each material, whilst simplified equations are used for the calculation of electrical conductivity on dependence of temperature and electricEnergies field. 2020, 13, x FOR PEER REVIEW 4 of 20

FigureFigure 2. Typical 2. Typical high high voltage voltage directdirect current current (HVDC) (HVDC) large large size sizecable. cable.

For metals,For each a linear layer, relationthermal and between electrical electrical properties conductivity are assigned andbased temperature on the materials is used as well considering as the coeffithecient tables proposed of the standards by the IEC standards 60287 and IEC above. 60853-2. In polymers, the electrical conductivity can be expressed asThe an geometry exponential is discretized function in ofNRtemperature radial nodes and and NZ electric axial nodes. field. For The each model of these, described thermal in this conductivity, mass density, and specific heat characterize the material in the thermal model as well article uses the following equation for the electrical conductivity of the dielectric: as permittivity and electrical conductivity at 20 °C are used in the electric model. All the parameters used in the thermal model are assumed constant and homogeneous for each material, whilst a(T T0)+bE simplified equations are used for σthe(T ,calculationE) = σ0e of− electrical, conductivity on dependence of (1) temperature and electric field. where σ0 is theFor metals, electrical a linear conductivity relation between at the electrical temperature conductivityT0 and and in temperature absence of is an used external considering electric field, σ(T,E) isthe its coefficient value at theproposed temperature by the standardsT and under above. the In electric polymers, field theE electricalexpressed conductivity in V/m, a canand beb are the positiveexpressed coefficient as an respectively exponential forfunction the temperature of temperature and and theelectric electric field. field. The model The referencedescribed in temperature this article uses the following equation for the electrical conductivity of the dielectric: T0 is equal to 20 ◦C. The coefficient b is also variable with the temperature and it can be evaluated by () means of the following: (, ) = , (1) 2 where σ0 is the electrical conductivityb = cat( Tthe temperatureT ) + d(T T0 andT ) in+ absencee, of an external electric field, (2) − 0 − 0 σ(T,E) is its value at the temperature T and under the electric field E expressed in V/m, a and b are the where cpositive, d, and coefficiente are parameters respectively depending for the temperatur on thee material. and the electric The Equationfield. The reference (2) and temperature its coefficients are obtainedT0 by is equal fitting to the20 °C. values The coefficient reported b in is thealso reference variable with [28 ].the The temperature coefficient anda isit can weakly be evaluated variable by with the electric stressmeans butof the it following: can be considered as constant within the electric field ranges obtained in this work. The distortion of the radial electric field profile in a loaded cable in comparison to the Laplacian =(−) +(−) +, (2) field is mainly due to the temperature gradient. Indeed, the coefficient a is about 2–3 times higher than where c, d, and e are parameters depending on the material. The Equation (2) and its coefficients are b in mostobtained of the by commercial fitting the values dielectric reported materials. in the reference [28]. The coefficient a is weakly variable with Duethe to electric the strong stress prevalence but it can be of considered the radial as component constant within of the the electric electric field field over ranges the obtained others, thein this variable E is herework. referred to the radial component of the electric field vector. In addition, due to its direct proportionalityThe todistortion the electrical of the radial field, electric the current field profile density in Ja isloaded analogously cable in comparison referred to to the the radial Laplacian component of the currentfield is densitymainly due vector. to the temperature gradient. Indeed, the coefficient a is about 2–3 times higher than b in most of the commercial dielectric materials. The volumetric heat source inside the conductor is calculated from the axial current density as Due to the strong prevalence of the radial component of the electric field over the others, the functionvariable of time E is and here the referred electric to the resistivity radial component depending of the on electric the temperature. field vector. In Ataddition, the axial due boundaries,to its an adiabaticdirect conditionproportionality is imposed to the electrical whilst field, a convective the current heat density transfer J is analogously is considered referred at theto the radial radial external limit. Thiscomponent heat exchange of the current mode density is defined vector. by an external temperature and a convective coefficient for each axial divisionThe volumetric of the heat geometry. source inside The temperaturethe conductor is distribution calculated from vs. the time axial is calculatedcurrent density by as means of the thermalfunction conduction of time and law the in electric an axial-symmetric resistivity depending cylindrical on the temperature. coordinate At system: the axial boundaries, an adiabatic condition is imposed whilst a convective heat transfer is considered at the radial external limit. This heat exchange mode is defined by an external temperature and a convective coefficient for 1 ∂ ∂T(r, z, t) ∂ ∂T(r, z, t) ∂T(r, z, t) k(r)r + k(r) + q000 (r, z, t) = rho(r)C(r) , (3) r ∂r ∂r ∂z ∂z ∂t Energies 2020, 13, 4827 5 of 18 where r, z, and t are respectively the radial, the axial, and the time coordinates, k(r) is the thermal conductivity, q000(r,z,t) is the thermal power density, rho(r) is the mass density, and C(r) is the specific heat. Equation (3) has been discretized in a linear system of NR*NZ equations through the finite differences method and iteratively solved for each time step. Whilst the thermal model is applied to the entire cable segment, the domain of the electric model includes only the layers between the inner and the outer semicon and assumes that voltage is applied between these two layers. The electrical conductivity of the semicon layers is much greater than that of the dielectric, therefore the charge is accumulated at the interfaces in much faster times than those necessary for accumulation of the space charge within the insulation. This can lead to numerical instabilities during the transients, therefore the approach followed in this work is different for the dielectric and the semicon layers. It has been assumed that the space charge accumulated within the semicon layers reaches immediately a stationary value, therefore a stationary model has been applied to this portion of domain. This is complementary to the transient model applied to the dielectric layer. By defining ND, the number of radial nodes that divide the insulation thickness, the electric model provides a set of linear systems each composed by ND*NZ equations in ND*NZ unknown values of radial components of the electric field E and current density J as well as space charge density ρ. The electric field is calculated through the Gauss’s law expressed in cylindrical coordinates and assuming axial symmetry and neglecting variations along the axis:

1 ∂ ρ(r, z, t) [rεr(r)E(r, z, t)] = , (4) r ∂r ε0 where εr(r) is the relative permittivity of the material, ε0 is the vacuum dielectric constant and ρ(r, z, t) is the space charge density. Since at least two values of E are required to obtain a single value of ρ, the differential Equation (4) can be discretized in a linear system of (ND 1)*NZ equations in ND*NZ − unknowns, where ND is the number of radial nodes that discretize the dielectric thickness. As an initial condition, the space charge is considered equal to zero in the whole the domain. Consequently, the first radial distribution of the electrical field is a Laplacian pattern. Once known, the temperature and the electric field in each node, the conductivity distribution can be calculated from Equations (1) and (2) in order to obtain the following:

σ(r, z, t) = σ(T, E). (5)

The radial component of the current density through the dielectric can be therefore calculated from the Ohm’s law that can be simply applied to each node:

J(r, z, t) = σ(r, z, t) E(r, z, t). (6) ∗ As it can be noted from (6), the current field depends on the electrical conductivity distribution and therefore on the thermal and electrical field. For most commercially used dielectric materials, the electrical conductivity is more dependent on the temperature than on the electric field. By considering the assumption above, the current continuity equation expressed in cylindrical coordinates can be written as follows:

1 ∂ ∂ρ(r, z, t) rJ(r, z, t) = . (7) r ·∂r − ∂t As it can be noted from the Equation (7), two values of J are needed for each discretized equation in ρ, therefore, the linear system is closed, imposing a radial current density equal to zero in the dummy nodes outside the domain of the dielectric layer. This linear system of equations provides the distribution of the space charge inside the dielectric that can be introduced in the system obtained from the (4) at the subsequent time step. Energies 2020, 13, 4827 6 of 18

As introduced above, a diﬀerent approach has been followed to calculate the charge accumulated within the semicon layers. Since the time constants of this phenomena are proportional to the product between the electrical conductivity and permittivity, the charges accumulate in the semicon, near the interfaces with the dielectric, in a practically inﬁnitesimal time compared to that required for accumulation inside the bulk of the dielectric. For this reason, a semi-stationary model has been developed to extend the linear systems of the equation deriving from the (4)–(7) to a domain containing the semicon layers. By setting the time variation of space charge equal to zero and introducing the (6) in the (7), the following equation is obtained:

∂ [rσ(r, z, t) (r, z, t)] = 0. (8) ∂r × The linear equations obtained from the Equation (8) can be added to those extrapolated from the (4) to forming a system of (NDL 1) NZ equations in NDL NZ unknowns where NDL is the − × × number of the radial nodes in which the layers between the external surface of the conductor and the inner surface of the shield are divided. In order to complete the linear system for the calculation of the electric ﬁeld, the equation relating the electric ﬁeld to the potential has to be used:

Z rout E(r, z, t)dr = U(t), (9) rin where rin and rout are respectively the inner and the outer radius of the dielectric layer and U(t) is the applied voltage. Equation (9) is discretized for each axial division and provides NZ equations to the linear system for the calculation of the electric ﬁeld distribution. Finally, the space charge distribution in the semicon layers is calculated from the (4) after obtaining theEnergies electric 2020 ﬁeld, 13, x distribution. FOR PEER REVIEW In Figure 3, the calculation model is schematized. 7 of 20

FigureFigure 3. 3.Scheme Scheme ofof thethe calculationcalculation model.

3. Boundary, Initial Conditions and Model Assumption of the Simulated Case Study With the aim of evaluating the effect of an axial thermal flux in a HVDC cable, several simulations have been carried out varying the external temperature drop and the convective heat flux coefficient at the interface. A typical large size HVDC cable is reproduced in order to study the phenomenon under investigation. The sample cable has the geometrical and thermal characteristics shown in Table 1 [29,30]. The specific heats in Table 1, are obtained from the thermal capacities indicated in the IEC 60853-2 and the mass density of each material. This is with the aim of evaluating the linear weight of the sample cable. In Table 2, the electrical properties of the layers are shown. The values in Table 2 are referred to a temperature of 20 °C and an electric field of 5 kV/mm. With reference to Equation (1), the temperature coefficient a has been calculated by means of a linear interpolation between its values of 0.104 °C-1 and 0.114 °C-1 respectively under the electric fields of 20 and 30 kV/mm. The values of b have been calculated in accordance to Equation (2) and adopting the following coefficients: c = −4.13 × 10−11 m/(V × K2), d = 2.83 × 10−9 m/(V × K), e = 1.25 × 10−8 m/V. These parameters are compatible with those of a thermoplastic dielectric material. The study has been carried out on a section of 6 m of HVDC cables exchanging heat with the external by means of convection. Since the thermal model is two-dimensional, the different external temperature and convective heat exchange coefficient can be set along the axial coordinate of the cable. Different levels of variation are simulated in order to evaluate the dependence between the maximum electric field and the axial variation of the outer thermal conditions.

Table 1. Geometrical and physical properties of the cable taken as the case study.

Thermal External Mass Density Specific Heat Layer Description Conductivity Radius [mm] [kg/m3] [J/kgK] [W/mK]

Energies 2020, 13, 4827 7 of 18

In spite of a calculation through a commercial software using FEA, the procedure described in this article minimizes the computational effort. In order to take into account the axial heat transfer that plays an important role in proximity of axial discontinuities, a two dimensional approach is necessary for the thermal modeling. On the other hand, due to a negligible axial electric field and thermal gradient compared to their radial components, a one-dimensional approach is sufficient to model the charge and electric field distribution.

3. Boundary, Initial Conditions and Model Assumption of the Simulated Case Study With the aim of evaluating the effect of an axial thermal flux in a HVDC cable, several simulations have been carried out varying the external temperature drop and the convective heat flux coefficient at the interface. A typical large size HVDC cable is reproduced in order to study the phenomenon under investigation. The sample cable has the geometrical and thermal characteristics shown in Table1[ 29,30]. The specific heats in Table1, are obtained from the thermal capacities indicated in the IEC 60853-2 and the mass density of each material. This is with the aim of evaluating the linear weight of the sample cable. In Table2, the electrical properties of the layers are shown. The values in Table2 are referred to a temperature of 20 ◦C and an electric field of 5 kV/mm. With reference to Equation (1), the temperature coefficient a has been calculated by means of a linear interpolation between its values of 0.104 ◦C-1 and 0.114 ◦C-1 respectively under the electric fields of 20 and 30 kV/mm. The values of b have been calculated in accordance to Equation (2) and adopting the following coefficients: c = 4.13 10 11 m/(V K2), d = 2.83 10 9 m/(V K), e = 1.25 10 8 m/V.These parameters − × − × × − × × − are compatible with those of a thermoplastic dielectric material. The study has been carried out on a section of 6 m of HVDC cables exchanging heat with the external by means of convection. Since the thermal model is two-dimensional, the different external temperature and convective heat exchange coefficient can be set along the axial coordinate of the cable. Different levels of variation are simulated in order to evaluate the dependence between the maximum electric field and the axial variation of the outer thermal conditions.

Table 1. Geometrical and physical properties of the cable taken as the case study.

External Mass Density Speciﬁc Heat Thermal Conductivity Layer Description Radius [mm] [kg/m3] [J/kgK] [W/mK] 1 Conductor 1 20 8960 385 390 2 Inner semicon 21 1180 1900 0.6 3 dielectric 51 1400 1714 0.2857 4 outer semicon 52 1180 1900 0.6 5 swelling tape 53 930 1900 0.6 metallic screen 6 1 55 8960 385 390 7 swelling tape 56 930 1900 0.6 aluminum 8 58 2700 896 290 laminate PVC outer 9 62 1400 1214 0.167 covering 1 Filling factors are taken into account by means of eﬀective lumped parameters.

Table 2. Electrical properties of the cable taken as the case study.

Layer Description Permittivity Reference Resistivity (@20 C) [Ohm m] ◦ × 2 Inner semicon 5000 2.18 3 dielectric 2.5 1.4 1014 × 4 outer semicon 5000 2.18 5 swelling tape 5000 2.18 Energies 2020, 13, 4827 8 of 18

The radial electric field distribution over the radius and the axis of the modeled cable section has been calculated in steady state conditions and during a PR event. The transient has a duration of 24 h with the electrical boundary conditions shown in Table3. This scenario is compatible with a slow PR event [31]. In order to investigate the behavior of the phenomenon with different thermal boundary conditions, five different scenarios have been simulated. In Table4, the external temperature and the convective heat transfer coe fficients are shown for the simulated scenarios. These parameters have been assumed to vary linearly in an axial length of 0.3 m in proximity of the center of the 6 m long modeled cable portion. The values of the external temperature and convective coefficients are compatible with an external cooling in water and air. In particular, the portion of the cable with an axial coordinate of less than 2.7 m is immersed in water, whilst the remaining part exchanges heat with the air. As can be seen from Table4, only the heat exchange parameters in the air-cooled region are varied in the di fferent scenarios. This choice has been made with the aim of comparing the maximum values of the electric field reached in proximity of the water–air interface with reference values relating to the portion of cable immersed in water and sufficiently distant from the liquid surface.

Table 3. Electrical boundary conditions during the simulated transient.

Time [s] Conductor Voltage [kV] Screen Voltage [kV] Current [A] 0–60480.0 510 0 1700 60480.0–60481.0 From 510 to 0 0 0 60481.0–60661.0 0 0 0 60661.0–60662.0 From 0 to 510 0 0 − 60662.0–61262.0 510 0 0 − 61262.0–86400.0 510 0 1700 −

Table 4. Thermal boundary conditions during the simulated transient.

Scenario Axial Position < 2.7 m Axial Position > 3 m 2 2 Text [◦C] h [W/(m K)] Text [◦C] h [W/(m K)] A 15 100 20 8 B 15 100 15 8 C 15 100 20 16 D 15 100 15 16 E 15 100 25 8

4. Results and Discussion In this section, the results of the simulations of the above described scenarios are shown. In Figure4, an example of the axial distribution of temperature drop over the dielectric layer is shown. The illustrated graph refers to Scenario E, in which the maximum local peak of the thermal drop is reached. With the aim of illustrating the role of the interface on the steady state temperature distribution over the radius and the axis of the modeled cable portion, a 3D graph is illustrated in Figure5. As it can be observed, the temperature drop reaches its peak just above the liquid level in accordance with the initial hypothesis illustrated in Figure1. The hottest part of the cable exchanging heat with still air heats up the coldest part by means of conduction through the cable core. Therefore, the conductor of the cable just below the liquid level has almost the same temperature of the portion cooled in air whilst the external surface in the same axial position is cooled with a high convection coeﬃcient by the water. Consequently, a thermal gradient peak is expected at the interface between the two diﬀerent means by which the cable exchanges heat. In stationary conditions for Scenario E, it has been found that the thermal drop at the interface is 1.15 times greater than the portion cooled in water and 2.7 m below the liquid level. Energies 2020, 13, x FOR PEER REVIEW 10 of 20

Energies 2020, 13, x FOR PEER REVIEW 10 of 20 air heats up the coldest part by means of conduction through the cable core. Therefore, the conductor ofair the heats cable up just the coldestbelow the part liquid by means level ofhas conducti almoston the through same temperature the cable core. of theTherefore, portion the cooled conductor in air whilstof the thecable external just below surface the inliquid the same level axialhas almost position the is same cooled temperature with a high of convection the portion coefficient cooled in byair thewhilst water. the externalConsequently, surface a in thermal the same gradient axial position peak is is expectedcooled with at thea high interface convection between coefficient the two by differentthe water. means Consequently, by which the a cablethermal exchanges gradient heat. peak In stationaryis expected conditions at the interface for Scenario between E, it has the been two found that the thermal drop at the interface is 1.15 times greater than the portion cooled in water and Energiesdifferent2020 ,means13, 4827 by which the cable exchanges heat. In stationary conditions for Scenario E, it has9 ofbeen 18 2.7found m below that the the thermal liquid level. drop at the interface is 1.15 times greater than the portion cooled in water and 2.7 mIn below Figure the 5, itliquid can be level. seen that the temperature at the external boundary of the cable in the portion immersedInIn Figure Figure in 5air, 5, it isit can canmore be be seen thanseen that that15 the°C the higher temperature temperature than the at at the environmentalthe external external boundary boundary temperature of of the the cable cableset for in in the Scenario the portion portion E. immersedThisimmersed shows in inthat air air the is is more axialmore thanthermal than 15 15 ◦transmission C°C higher higher than than thro the theugh environmental environmental the core depends temperature temperature more on theset set forvariation forScenario Scenario of the E.E. ThisconvectiveThis shows shows thatcoefficient that the the axial axial at thermalthe thermal interface transmission transmission than on the through thro variationugh the the core ofcore temperature. depends depends more more on on the the variation variation of of the the convectiveconvectiveAs specified coe coefficientﬃcient above, at at the the the interface interface presence than than of on ona the radialthe variation variation thermal of of temperature. gradienttemperature. causes the onset of a high conductivityAsAs speciﬁed specified gradient above, above, due the presencetothe the presence exponential of a radial of a thermallawradial of dependencethermal gradient causesgradient of the the causeselectric onset of theconductivity a high onset conductivity of ona high the gradienttemperatureconductivity due toand gradient the this exponential leads due toto thethe law accumulationexponential of dependence law of of spaceof the dependence electriccharge conductivitywithin of the the electric dielectric. on the conductivity temperature on and the thistemperature leads to the and accumulation this leads to of the space accumulation charge within of space the dielectric.charge within the dielectric.

Figure 4. Axial distribution of thermal drop—Scenario E. Figure 4. Axial distribution of thermal drop—Scenario E. Figure 4. Axial distribution of thermal drop—Scenario E.

Figure 5. Temperature vs. radius and axis in steady state conditions—Scenario E. Figure 5. Temperature vs. radius and axis in steady state conditions—Scenario E. In Figure6Figure, the steady 5. Temperature state space vs. chargeradius and distribution axis in steady within state the conditions—Scenario insulation layer is E. illustrated by means of a 3D graph. As can be noted, peak value is reached just below the liquid level in proximity of the outer boundary, following the same trend of the axial distribution of the thermal drop shown in

Figure4. In the stationary conditions for Scenario E, the maximum space charge density at the liquid level has been found to be 1.27 times greater than that in the portion cooled in water and 2.7 m below the interface. As it can be expected, the accumulation of space charge leads to inversion of the radial electric ﬁeld proﬁle with the maximum absolute values reached in proximity of the external surface of the dielectric layer. Energies 2020, 13, x FOR PEER REVIEW 11 of 20

EnergiesIn 2020 Figure, 13, x 6, FOR the PEER steady REVIEW state space charge distribution within the insulation layer is illustrated11 of 20 by means of a 3D graph. As can be noted, peak value is reached just below the liquid level in In Figure 6, the steady state space charge distribution within the insulation layer is illustrated proximity of the outer boundary, following the same trend of the axial distribution of the thermal by means of a 3D graph. As can be noted, peak value is reached just below the liquid level in drop shown in Figure 4. In the stationary conditions for Scenario E, the maximum space charge proximity of the outer boundary, following the same trend of the axial distribution of the thermal density at the liquid level has been found to be 1.27 times greater than that in the portion cooled in drop shown in Figure 4. In the stationary conditions for Scenario E, the maximum space charge water and 2.7 m below the interface. As it can be expected, the accumulation of space charge leads to density at the liquid level has been found to be 1.27 times greater than that in the portion cooled in inversion of the radial electric field profile with the maximum absolute values reached in proximity water and 2.7 m below the interface. As it can be expected, the accumulation of space charge leads to Energiesof the external2020, 13, 4827 surface of the dielectric layer. 10 of 18 inversion of the radial electric field profile with the maximum absolute values reached in proximity In Figure 7, the steady state radial electric field distribution over the radius and the axis of the of the external surface of the dielectric layer. cable is shown. In the stationary conditions for Scenario E, the maximum electric field at the liquid InIn FigureFigure7 7,, the the steady steady state state radial radial electric electric field field distribution distribution over over the the radius radius and and the the axis axis of of the the level is 1.11 times greater than the portion cooled in water and 2.7 m below the liquid level. It can be cablecable isis shown.shown. InIn thethe stationarystationary conditionsconditions forfor ScenarioScenario E,E, thethe maximummaximum electricelectric fieldfield atat thethe liquidliquid stated that, in stationary conditions, the local increase in the electric field caused by the effect of an levellevel isis 1.111.11 timestimes greatergreater thanthan thethe portionportion cooledcooled inin waterwater andand 2.72.7 mm belowbelow thethe liquidliquid level.level. ItIt cancan bebe interface between two means in which an HVDC cable is immersed depends on the local increase in statedstated that,that, inin stationarystationary conditions,conditions, the locallocal increaseincrease in thethe electricelectric fieldfield causedcaused byby thethe eeffectffect ofof anan the thermal gradient. Nevertheless, the maximum value of the steady state electric field depends on interfaceinterface betweenbetween twotwo meansmeans inin whichwhich anan HVDCHVDC cablecable isis immersedimmersed dependsdepends onon thethe locallocal increaseincrease inin the local temperature gradient close to the external surface of the dielectric layer. thethe thermalthermal gradient.gradient. Nevertheless,Nevertheless, thethe maximummaximum valuevalue ofof thethe steadysteady statestate electricelectric fieldfield dependsdepends onon thethe locallocal temperaturetemperature gradientgradient closeclose toto thethe externalexternal surfacesurface ofof thethe dielectricdielectric layer. layer.

Figure 6. Space charge vs. radius and axis in steady state conditions—Scenario E. Figure 6. Space charge vs. radius and axis in steady state conditions—Scenario E. Figure 6. Space charge vs. radius and axis in steady state conditions—Scenario E.

Figure 7. Electric ﬁeld vs. radius and axis in steady state conditions—Scenario E. Figure 7. Electric field vs. radius and axis in steady state conditions—Scenario E.

The sameFigure temperature 7. Electric drop field over vs. theradius entire and thickness axis in steady can bestate achieved conditions—Scenario with different E. radial thermal profiles, therefore, contrary to what may appear in a first approximation, there is not a proportional dependence between the increase in thermal drop and the increase in local electric field at the interface. It is more correct to state that the local peak of electric field depends both on the temperature difference between the two means at the interface and of the ratio between the convective coefficients. In Table5, the main results of the simulated scenarios in steady state conditions are shown in terms of increases in thermal drop, space charge and radial electric field respect to the reference values reached 2.7 m below the liquid level. As it can be noted from the Table, the maximum steady state value of the electric field is achieved in Scenario E and it is equal to 33.7 kV/mm. Energies 2020, 13, 4827 11 of 18

Table 5. Local increases in thermal drop, space charge density, and electric ﬁeld.

Scenario Increase in Thermal Drop Increase in Max Space Charge Density Increase in Max Electric Field A 1.1176 1.2015 1.1000 B 1.0866 1.1463 1.0721 C 1.0767 1.1220 1.0656 D 1.0434 1.0742 1.0354 E 1.1529 1.2683 1.1148

Once the steady state conditions have been reached from the thermal and electrical points of view, a PR event has been simulated as specified in Table3. In all simulated scenarios, the maximum electric field was reached at the end of the PR ramp, in proximity of the internal boundary of the dielectric, in the area most distant from the interface and immersed in water. In order to illustrate the temporal trend of the maximum absolute value of the electric field during the simulated transient, the graphs shown in Figures8 and9 have been drawn. These graphs refer to Scenario D, during which the maximum peak of the electric field has been achieved. It should be noted that Scenario D is characterized by the lowest increases in temperature drop and consequently in electric field during the stationary phase. However, the maximum values of the electric field reached at the end of the PR ramp in all the simulated scenarios are all at most 1% lower than the peak found during Scenario D. This is because the thermal distribution 2.7 m below the liquid level is not strongly dependent on the heat exchange conditions of the portion immersed in air. With reference to Figures8 and9, it can be noted an initial reduction of the electric stress due to the inversion of the profile of the electric field after the application of the load and starting from a homogeneous temperature distribution. As described above, due to the establishment of a thermal gradient, the electric field tends to reduce its value in proximity of the internal boundary whilst reaching its maximum at the external. The time necessary to achieve a stationary electric field distribution depends on the time constants of the thermal and electrical phenomena. The former is strictly related to the cable weight, whilst the latter can be expressed as the product between the permittivity and the conductivity of the dielectric material. Since the electrical conductivity of polymers is dependent on the temperature with exponential law, the electrical time constant varies up to 3 orders of magnitude within the operating thermal range in which the cable works. This can play an important role during Energiesrelative 2020 fast, 13 voltage, x FOR PEER transients REVIEW such as PRs and TOVs. 13 of 20

Figure 8. Maximum absolute value of electric ﬁeld in the modeled cable portion—Scenario D. Figure 8. Maximum absolute value of electric field in the modeled cable portion—Scenario D.

Figure 9. Maximum absolute value of electric field in the modeled cable portion—Scenario D, detail.

Approximately 7 min after the inversion, the electric field at the outer surface of the insulation in the air-cooled part exceeds that at the inner boundary in the area immersed in water. This new maximum is in turn overcome, after about 90 min, by the electric field at the external surface of the dielectric close the interface. The latter finally reaches an asymptotic value identical to that reached before the PR. The maximum values of the electric field found at the end of the PR depends not only on the temperature drop over the insulation but also on its average value within the dielectric. Indeed, at higher temperature, the greater charge mobility reduces the time necessary for the relaxation. In example, due to their lower operating temperature, mass impregnated non-draining (MIND) cables could be stressed by higher electric field peaks during a PR [32]. In Figure 10, a comparison is made between profiles of the radial electric field at different times during the transient of the Scenario D in the coldest area of the modeled cable portion. As it can be observed, the maximum value of the electric field achieved in proximity of the external surface of the dielectric before the PR is higher than that reachable in no load conditions at the internal boundary. Once the PR ramp is finished, the electric field achieves its maximum value of 39.27 kV/mm, 19% higher than the maximum steady state value reached on the external surface and near the interface.

Energies 2020, 13, x FOR PEER REVIEW 13 of 20

Energies 2020, 13, 4827 12 of 18 Figure 8. Maximum absolute value of electric field in the modeled cable portion—Scenario D.

FigureFigure 9. 9. MaximumMaximum absolute absolute value value of of electric electric field ﬁeld in in the the modeled modeled cable portion—Scenario D, detail.

ApproximatelyAfter reaching a minimum7 min after value, the inversion, the maximum the electr electricic fieldfield moduleat the outer starts surface to riseagain, of the reaching insulation an inasymptotic the air-cooled value part in approximately exceeds that 15at the h. Once inner the bo steadyundary state in the is achieved,area immersed the electric in water. field This suddenly new maximumcollapses due is in to turn grounding overcome, in 1 after s of the about conductor. 90 min, by the electric field at the external surface of the dielectricAfter close a relaxing the interface. phase of The 180 latter s, the finally maximum reaches electric an asymptotic field suddenly value reaches identical its peakto that at reached the end beforeof the rampthe PR. of the PR. This peak is achieved at the internal boundary of the dielectric layer in the coldestThe region maximum of the values cable. Afterwards,of the electric its valuefield found begins at to the decrease, end of falling the PR below depends the asymptoticnot only on value the temperaturepreviously reached drop over in about the insulation 2 min. but also on its average value within the dielectric. Indeed, at higherApproximately temperature, 7the min greater after thecharge inversion, mobility the reduces electric fieldthe time at the necessary outer surface for the of therelaxation. insulation In example,in the air-cooled due to their part lower exceeds operating that at thetemperature, inner boundary mass impregnated in the area immersed non-draining in water. (MIND) This cables new couldmaximum be stressed is in turn by higher overcome, electr afteric field about peaks 90 min,during by a the PR electric [32]. field at the external surface of the dielectricIn Figure close 10, the a interface.comparison The is lattermade finallybetween reaches profile ans of asymptotic the radial valueelectric identical field at todifferent that reached times duringbefore thethe PR.transient of the Scenario D in the coldest area of the modeled cable portion. As it can be observed,The maximum the maximum values value of theof the electric electric field field found achieved at the in end proximity of the PRof the depends external not surface only onof the dielectrictemperature before drop the over PR the is insulationhigher than but that also reacha on itsble average in no value load withinconditions the dielectric. at the internal Indeed, boundary. at higher Oncetemperature, the PR ramp the greater is finished, charge the mobility electric reduces field achieves the time its necessary maximum for va thelue relaxation. of 39.27 kV/mm, In example, 19% higherdue to than their the lower maximum operating steady temperature, state value mass reached impregnated on the external non-draining surface (MIND) and near cables the interface. could be stressed by higher electric field peaks during a PR [32]. In Figure 10, a comparison is made between profiles of the radial electric field at different times during the transient of the Scenario D in the coldest area of the modeled cable portion. As it can be observed, the maximum value of the electric field achieved in proximity of the external surface of the dielectric before the PR is higher than that reachable in no load conditions at the internal boundary. Once the PR ramp is finished, the electric field achieves its maximum value of 39.27 kV/mm, 19% higher than the maximum steady state value reached on the external surface and near the interface. In this instant, whilst a negative peak is reached at the internal boundary, a residual positive electric field can be observed in proximity of the outer surface. This is due to the lower mobility of the charges at low temperatures. In light of the above, it is evident that a sudden variation along the axis of the thermal conditions outside a cable causes the achievement of local peaks of the electric field in stationary conditions. On the contrary, this scenario has a negligible influence during an event of PR. In order to better explain this behavior, the electric field distribution and the space charge at different times of the Scenario E have been analyzed. In Figure 11, the space charge distribution calculated for Scenario E at the end of the PR ramp is shown. As it can be observed, the minimum absolute value of the space charge density is reached in the area cooled in water and farthest from the surface. Here, the effect of the application of a negative Energies 2020, 13, x FOR PEER REVIEW 14 of 20

In thisEnergies instant, 2020, 13 whilst, x FOR PEERa negative REVIEW peak is reached at the internal boundary, a residual positive14 electric of 20 field can be observed in proximity of the outer surface. This is due to the lower mobility of the charges at lowIn this temperatures. instant, whilst In a light negative of the peak above, is reached it is evident at the internal that a sudden boundary, variation a residual along positive the axis electric of the thermalfield can conditions be observed outside in proximity a cable of causes the outer the surface. achievement This is due of tolocal the lowerpeaks mobility of the electricof the charges field in stationaryat low temperatures. conditions. On In thelight contrary, of the above, this scenarioit is evident has that a negligible a sudden influence variation duringalong the an axisevent of ofthe PR. thermal conditions outside a cable causes the achievement of local peaks of the electric field in In order to better explain this behavior, the electric field distribution and the space charge at different stationary conditions. On the contrary, this scenario has a negligible influence during an event of PR. times of the Scenario E have been analyzed. In order to better explain this behavior, the electric field distribution and the space charge at different Energies 2020In13 Figure 11, the space charge distribution calculated for Scenario E at the end of the PR ramp is times, of, 4827 the Scenario E have been analyzed. 13 of 18 shown. InAs Figure it can 11,be theobserved, space charge the minimum distribution absolute calculat valueed for of Scenario the space E at charge the end density of the PRis reached ramp is in theshown. area cooled As it canin water be observed, and farthest the minimum from the absolute surface. valueHere, of the the effect space of charge the application density is ofreached a negative in voltagevoltage isthe combined area is combined cooled with in water with a lower anda lower farthest mobility mobility from of theof the the surface. charges charges Here, due due the to to effect the the lower lowerof the temperatureapplicationtemperature of compareda compared negative to to the portionthevoltageof portion the cableis ofcombined the cooled cable with incooled air.a lowerTherefore, in air.mobility Therefore, inof thisthe chargesin area, this during area,due to during the the lower PR the ramp, temperaturePR ramp, thee theﬀ comparedect effect of the of to reducedthe mobilityreducedthe of portion the mobility charges of the of due cablethe tocharges cooled relatively indue air. to lower Therefore, relatively temperatures in lower this area, temperatures appears during the to appears prevailPR ramp, to over theprevail effect that overrelated of the that to the temperaturerelatedreduced to gradient. the mobility temperature of the chargesgradient. due to relatively lower temperatures appears to prevail over that related to the temperature gradient.

FigureFigureFigure 10. 10. 10.Radial Radial Radial electric electric electric ﬁeld fieldfield distributions—Scenario D. D. D.

Figure 11. Space charge distribution at the end of the Polarity Reversal (PR) ramp—Scenario E. Figure 11. Space charge distribution at the end of the Polarity Reversal (PR) ramp—Scenario E. Figure 11. Space charge distribution at the end of the Polarity Reversal (PR) ramp—Scenario E.

In Figure 12, the distribution of the electric field at the end of the PR ramp is shown. As it can be observed, the electric field gradient in the portion of cable immersed in water is higher than that cooled in air. In this colder area, the residual positive electric field in proximity of the external surface can be attributed to a higher electric time constant for the space charge re-distribution. During this phase, the maximum value of the electric field depends strongly on the absolute temperature of the dielectric and, consequently, also on the current flowing through the cable. If on one hand the space charge increases with the thermal gradient and therefore with the load, on the other, the mobility of the charges is reduced with the lowering of the current. It is therefore possible that a load condition can be determined that maximizes the local electric field as a function of the external heat exchange conditions. In Figure 13, the electric field distribution after 10 min from the PR is shown. At this step, the charge re-distribution in the colder area is still not complete, therefore, in this portion, the electric field reaches Energies 2020, 13, x FOR PEER REVIEW 15 of 20

In Figure 12, the distribution of the electric field at the end of the PR ramp is shown. As it can be observed, the electric field gradient in the portion of cable immersed in water is higher than that cooled in air. In this colder area, the residual positive electric field in proximity of the external surface Energies 2020, 13, x FOR PEER REVIEW 15 of 20 can be attributed to a higher electric time constant for the space charge re-distribution. During this phase, theIn maximum Figure 12, the value distribution of the electric of the electric field depefield ndsat the strongly end of the on PR the ramp absolute is shown. temperature As it can be of the dielectricobserved, and, theconsequently, electric field also gradient on the in currentthe portion flowing of cable through immersed the incable. water is higher than that Ifcooled on one in air.hand In thisthe colderspace area,charge the increasesresidual positive with the electric thermal field gradientin proximity and of thereforethe external with surface the load, on thecan other, be attributed the mobility to a higher of the electric charges time is constant reduced for with the spacethe lowering charge re-distribution. of the current. During It is thereforethis possiblephase, that the a maximumload condition value ofcan the be electric determined field depe thatnds maximizes strongly on the the localabsolute electr temperatureic field as of a thefunction dielectric and, consequently, also on the current flowing through the cable. of the external heat exchange conditions. If on one hand the space charge increases with the thermal gradient and therefore with the load, In Figure 13, the electric field distribution after 10 min from the PR is shown. At this step, the on the other, the mobility of the charges is reduced with the lowering of the current. It is therefore Energies 2020, 13, 4827 14 of 18 chargepossible re-distribution that a load incondition the colder can be area determined is still no thatt complete, maximizes therefore, the local electr in thisic field portion, as a function the electric field ofreaches the external its maximum heat exchange in proximity conditions. of the middle of the dielectric thickness. On the other hand, the electricIn fieldFigure distribution 13, the electric in thefield portion distribution immers aftered 10 in min air from has thealready PR is reachedshown. At almost this step, steady the state its maximum in proximity of the middle of the dielectric thickness. On the other hand, the electric conditionscharge which,re-distribution in a specular in the colder way, areatend is to still those not complete,illustrated therefore, in Figure in this7. As portion, it can thebe electricobserved in ﬁeldFigure distributionfield 14, reaches after in approximately its the maximum portion in immersed2 proximity h after the of in PR,the air middle a hasnew already ofequilibrium the dielectric reached condition thickness. almost has On steady beenthe other reached. state hand, conditions which, inthe a electric specular field way, distribution tend to in thosethe portion illustrated immersed in in Figure air has 7already. As it reached can be almost observed steady instate Figure 14, after approximatelyconditions which, 2 h afterin a specular the PR, way, a new tend equilibrium to those illustrated condition in Figure has 7. been As it reached.can be observed in Figure 14, after approximately 2 h after the PR, a new equilibrium condition has been reached.

Figure 12. Electric field distribution at the end of the PR ramp—Scenario E. FigureFigure 12. Electric 12. Electric ﬁeld field distribution distribution at th thee end end of ofthe the PR PRramp—Scenario ramp—Scenario E. E.

Energies 2020, 13, x FORFigure PEER 13. ElectricREVIEW field distribution 600 s after the end of the PR ramp—Scenario E. 16 of 20 Figure 13. Electric ﬁeld distribution 600 s after the end of the PR ramp—Scenario E. Figure 13. Electric field distribution 600 s after the end of the PR ramp—Scenario E.

Figure 14. Electric ﬁeld distribution 7200 s after the end of the PR ramp—Scenario E. Figure 14. Electric field distribution 7200 s after the end of the PR ramp—Scenario E.

In light of the above, it can be stated that the time required to reach new stationary conditions after a voltage transient increases with the lowering of the temperature in the dielectric.

Energies 2020, 13, 4827 15 of 18

In light of the above, it can be stated that the time required to reach new stationary conditions after a voltage transient increases with the lowering of the temperature in the dielectric.

5. Conclusions This paper describes the results of a study on the role of the axial heat transfer in HVDC cables on the space charge accumulation phenomena. Because of the accumulation of space charge, the electric field distribution departs from the Laplacian profile and it is conditioned from the thermal field. In addition to the authors’ previous work, the evolution of the space charge and electric field distributions have been investigated under five different scenarios [33]. Moreover, an analysis of the electric field peaks reached during and after a PR event has been carried out. With the aim of performing numerical simulation to carry out this study,a thermal and electric model has been developed in time domain. This model takes into account the accumulation of space charge caused by the dependence of the electrical conductivity of the dielectric on the temperature. It should be stressed that other issues contribute to the space charge phenomena in HVDC cables. As demonstrated in recent studies, these issues, simulated as conductivity discontinuities, play an important role in the increase of the space charge density within the dielectric [34]. Nevertheless, the effect of the temperature drop over the dielectric layer has a weight, between the other space charge phenomena, increasing with the sizes of the cable. In the case study analyzed in this article, a segment of an HVDC cable 6 m long has been modeled by means of a cylindrical axial-symmetric two-dimensional geometry. It has been considered that the insulation of the cable consists of a thermoplastic material whose thermal and electrical characteristics are available in the literature. A portion of a full size HVDC cables has been modeled considering a half in length cooled in water and the other in air. In this way, the effect of the axial variation in the external thermal conditions on the radial temperature gradient has been assessed. This causes the onset of a local peak in the temperature drop over the dielectric layer just below the liquid level. As was found in the previous work of the authors, the results of this research show that the presence of the interface between the two external means plays an important role in the determining of electric field distribution due to a greater amount in this area of space charge. The increase in electric field reached just below the liquid level seems to be almost proportional to the temperature drop growth at the interface. The thermal gradients obtained in this research are comparable to those occurring in operating large size cables. The effect of the sudden axial variation in the thermal conditions can result amplified during emergency loading or fast load transients. The findings of this research show no appreciable dependence between the peak of electric field reached during a slow PR event and the effect of an interface between 2 means out of the cable. On the contrary, at the end of a PR ramp, the maximum value of the electric field is achieved in the coldest area of the modeled cable. This phenomenon can be explained by the lower mobility of the space charge where the dielectric temperature is lower. This effect seems to prevail on the lower thermal drop in the most cooled areas of the cable. The findings of this work are referred to a specific case study but they are extendible to other scenarios in which the radial temperature distribution is conditioned by axial discontinuities by the thermal point of view. Therefore, this study can be extended to other cases in order to assess the actual electric field distribution in proximity of joints, hot and cold points along the line or the proximity to other heat sources. The research offered in this article attempts to provide a contribution for a deeper understanding of the phenomenon of the accumulation of space in large HVDC cables and of its consequences in terms of achieving high levels of electric field. This and other works on this topic may provide elements to take into account in the thermal rating of HVDC cables. Energies 2020, 13, 4827 16 of 18

Author Contributions: Conceptualization, G.R.; Data curation, G.R.; Formal analysis, P.R. and F.V.; Investigation, G.R. and A.I.; Methodology, G.R.; Resources, P.R.; Supervision, G.A.; Writing—original draft, G.R.; Writing—review & editing, P.R. All authors have read and agreed to the published version of the manuscript. Funding: This research received no external funding. Conﬂicts of Interest: The authors declare no conﬂict of interest.

Acronyms AC Alternating Current DC Direct Current FEA Finite Element Analysis HVDC High Voltage Direct Current IEC International Electrotechnical Commission MIND Mass Impregnated Non-Draining PEA Pulsed Electro-Acoustic PR Polarity Reversal TOV Transient Over Voltage TSM Thermal Step Method Variables and Parameters a temperature coefficient for the electrical conductivity b electric field coefficient for the electrical conductivity c first parameter for the evaluation of the coefficient b C(r) specific heat at the radial coordinate r d second parameter for the evaluation of the coefficient b e third parameter for the evaluation of the coefficient b E(r,z,t) radial component of the electric field at the coordinates r, z, t J(r,z,t) radial component of the current density at the coordinates r, z, t k(r) radial thermal conductivity at the radial coordinate r ND radial nodes in which the insulation thickness is discretized NDL radial nodes in which the layers between the external surface of the conductor and the inner surface of the shield are discretized NR Radial nodes in which the cable geometry is discretized NZ Axial nodes in which the cable geometry is discretized q000(r,z,t) thermal power density at the coordinates r, z, t r radial coordinate rho(r) mass density at the radial coordinate r rin inner radius of the dielectric layer rout outer radius of the dielectric layer t time coordinate T(r,z,t) temperature at the coordinates r, z, t T0 reference temperature (20 ◦C) U(t) applied voltage at the time t z axial coordinate ε0 vacuum dielectric constant εr(r) relative permittivity at the radial coordinate r σ(T,E) electrical conductivity at the temperature T and the electric field E σ0 electrical conductivity at the temperature T0 and electric field equal to 0 ρ(r, z, t) space charge density at the coordinates r, z, t

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