HVDC Insulation Boundary Conditions for Modeling and ... - IEEE Xplore

IEEE Transactions on Dielectrics and Electrical Insulation

Vol. 22, No. 1; February 2015

35

HVDC Insulation Boundary Conditions for Modeling and Simulation Thomas Christen ABB Switzerland Ltd. Segelhofstr. 1K, CH-5405 Baden-Dättwil, Switzerland

ABSTRACT Insulator boundaries like electrode contacts, interfaces, and surfaces play decisive roles for the electric field distributions in high-voltage direct-current (HVDC) insulation. Well-known examples are charging effects at injecting and blocking electrode contacts, and at interfaces or surfaces. Reliable modeling of boundaries and interfaces poses a challenge and can be crucial for the quality of the resulting field distributions in HVDC equipment, but is in practice often put in second place of priority in simulation models. This paper emphasizes the macroscopic boundary conditions and related issues for electrical transport and illustrates their relevance with simple examples. Importance is attached to an appropriate compromise between physical generality and practical usability for simulations in the development process of HVDC insulation systems. Index Terms - HVDC insulation, Electric transport, Charge injection, Electrode contacts, Interface phenomena, Electric field simulations, Finite element method (FEM)

1 INTRODUCTION DIMENSIONING and designing of electrical insulation for HV devices and equipment [1] nowadays is inconceivable without the help of computer simulations. In the research and development process of insulators, modeling and simulation of the electrical and electro-thermal behavior are needed for two purposes: prediction of the behavior of a product prototype prior to realization (for cost reduction of the development process), and understanding and evaluation of the experimental results used to characterize the insulation materials. While the latter is usually not a problem for AC materials, which are in many cases sufficiently characterized by the bulk permittivity and the loss angle at 50 or 60 Hz, the characterization of conduction in HVDC insulation poses a huge challenge [2]. Insulation boundaries like contacts (conductive electrode / insulation material), surfaces (condensed matter insulation / gas insulation), and insulator interfaces (between condensed mater insulations) often play a dominant role in HVDC applications. As a consequence, for a reliable prediction of the electric behavior, boundary effects must be modeled and simulated in a realistic manner. The purpose of the present paper is to highlight some effects that illustrate the distinct relevance of insulation material boundaries for HVDC insulation, to recommend an approach how to formulate boundary conditions in numerical simulations, e.g., with the finite element method (FEM), and to mention some consequences for material characterization. The paper is organized as follows. The challenges of HVDC insulation design [3]-[5] will be briefly reviewed in Secion 2. Section 3 recalls the general electric conduction model for Manuscript received on 16 January 2013, in final form 30 May 2014, accepted 3 June 2014.

insulation material [6]-[10], and subsequently discusses the boundary conditions [9]-[13]. A few illustrative examples are investigated in Section 4.

2 DESIGNING HVDC INSULATION 2.1 CHALLENGES Design development of HVDC insulation is considerably more challenging than for AC, because i. Resistive electric fields are governed by complex conduction processes, which are much less understood than the permittivity that governs capacitive (AC) fields. Often, neither the charge carrier types, nor their origin and process of motion are reliably known. Experiments are difficult because of tiny currents and long relaxation times, and results are sometimes contradictive. Furthermore, the often intrinsically heterogeneous or amorphous structures of real insulation materials hamper an exact theoretical description in the manner as one is used from, e.g., semiconductor physics. ii. The low-frequency conduction behavior shows a very sensitive dependence on various conditions and on parameters, which can lead to a lack of robustness of the resistive field distribution. Highly resistive insulation naturally tends to have large relative resistance variations. Noteworthy is the thermally activated behavior of conduction and the consequent strong sensitivity on temperature. iii. HVDC devices need to withstand not only resistive but also capacitive and mixed capacitive–resistive field distributions. Type tests involve switching surge and sometimes lightning impulse conditions (frequently with

DOI 10.1109/TDEI.2014.004559

36

iv.

T. Christen: HVDC Insulation Boundary Conditions for Modeling and Simulation

DC pre-stress), and the voltage switching-on process consists of capacitive-resistive transient fields. As compared to AC, there is poor experience with possible HVDC failure modes as well as with special DC aging mechanisms, like electro-chemical material changes or low-mobility ion extraction. Consequently there are much less reliable empirical design rules and lifetime estimates for DC than for AC.

In contrast to AC, for DC boundary and interface effects play a decisive role for all four items. Although this paper focuses mainly on (i) some general remarks on (ii)-(iv) are in order. The lack of robustness of the electrical behavior (ii) is important because the insulation resistivity can easily vary by orders of magnitudes in space, time, for different samples due to variations in the production process parameters, and for other differences in history. Pragmatic solutions of the robustness problem may involve means for resistive field grading, fault tolerance, and redundancy [5], [14]-[16]. The enlarged set of requirements (iii) narrows down the feasible set of solutions for the design optimum, which makes the product design process more difficult for DC as compared to AC. Reliable design rules (iv) are not only less established for DC because there is less field experience, but also because it is unclear how to derive design rules for real devices by temporally and spatially upscaling short-term lab-scale experience. HVDC failure modes and breakdown are also strongly affected by boundaries, however a discussion of differences between DC and AC failure modes goes beyond the purpose of this paper. Upscaling is a problem also for HVDC material characterization of electrical conduction properties. It has its origin partly in the main subject of this paper: boundary and finite size effects. The conductance cannot be simply derived from a volume independent bulk conductivity via multiplication by a simple geometrical factor. Moreover, as we will see, boundary and finite size effects can lead to a nonlinear system behavior. 2.2 APPLICATIONS A selected list of relevant boundaries in HVDC devices is shown in Table 1. For physical modeling reasons it is convenient to distinguish between electrode-insulator contacts, insulatorinsulator interfaces between condensed phases, and surfaces where one phase is a gas. Important for applications are the geometrical configuration and topological arrangement of the materials in composite insulation systems. Obviously, the effect of an interface in a series arrangement (e.g., cable and joint insulations in a cable-joint [14]) strongly differs from a parallel arrangement (e.g., epoxy-spacer and SF6 in a gas insulated switchgear (GIS) or line (GIL) [17], [18]) in the influence on the system behavior. Strategies for increasing the robustness of field distributions (like resistive stress grading etc [5], [16]) depend strongly on the topology and the geometry of the insulators and conductors in an application. In practice the configuration also determines the temperature distribution which affects the electrical behavior [17]; however, thermal effects will not be discussed in this rather conceptual paper, as it is methodologically straightforward to include the heat balance equation in a numerical simulation of the electric field distribution.

Table 1. HVDC application devices [1],[3] [4] and relevant boundaries. DEVICE Cables

Cable accessories (prefabricated and factory joints, termination)

Spacers, suspension and support insulators (in GIS, GIL, power lines) Capacitors Bushings (similar to termination) Transformer

RELEVANT BOUNDARIES Polymer cable: Insulation / semicon electrodes MIP (Mass impregnated paper): Insulation / carbon paper semicon Impregnated paper / paper layers Press contacts (accessory / cable interface): Accessory insulation / cable insulation Field grading material / cable insulation Interface between different termination insulation materials (solid, liquid, gas) Air / solid at termination surface Chemical solid/solid interfaces: Accessory insulation / conductive rubber Accessory insulation / field grading material Vulcanized cable-joint insulation contact Metal-electrodes / solid/gaseous insulation Solid insulation surface / gas (air, SF6) (outdoor: water film, dirt layer, …) Press contact: electrode foil / insulation film Deposited interface: insulation / metallization Bushing surface / gas Insulation / metal-foil windings of bushing body Oil / pressboard, … Oil / cellulose, pressboard, epoxy resin, …

2.3 MATERIAL CHARACTERIZATION Electric transport in technical HVDC insulation [19] is modeled with equations for drift, diffusion, generation, recombination, and trapping, etc., of charge carriers (cf. Section 3). These partial differential equations need to be completed by specifying boundary and interface conditions, which should model the contact and interface physics. The generic structure of the bulk equations is in principle well-known, and the main work is material characterization, i.e., to determine experimentally or theoretically the model parameters like carrier types, their mobility, kinetic rates for trapping, generation and recombination, etc.. A similar statement holds for the boundary conditions (BC): although it may require larger effort to determine BC model parameters due to the inherent microscopic character of lowerdimensional structures like contacts and interfaces, BC can be mathematically formulated in a generic way (cf. Section 3.2). Table 2 lists a selection of experimental techniques for material characterization (there exist other methods not listed here, particularly for surface studies like the Kelvin Probe Microscopy [20]). The dominance of boundary effects in HVDC insulation makes it obvious that always the “system” consisting of insulation material with boundaries, interfaces and electrode contacts, must be characterized rather than a bulk material alone [21]. At an interface, bulk material parameters can change in their value (e.g., the permittivity or the carrier mobility etc.), or parameters can have localized specific values (like densities for interface traps or surface recombination centers etc.). Electrode contacts confine the simulation region for electric transport and can act as carrier sources and sinks. To determine the model parameters related to such effects is difficult, because experiments often measure integral quantities (like the total current or the voltage). It may require special effort to separate

IEEE Transactions on Dielectrics and Electrical Insulation

Vol. 22, No. 1; February 2015

bulk from boundary effects. One way is by investigating the lengthscaling of the conduction behavior. Also other finite-size effects, like transit-time peaks (Section 4.3) or certain nonlinearities (Section 4.4) exhibit specific size-scaling properties and can provide important information on the electrical behavior of boundaries and their effects on conduction. For instance, with sizescaling one may separate bulk-inherent nonlinearities (e.g., PooleFrenkel effect, barrier hopping, etc.) from finite-size induced nonlinearities (e.g., Schottky injection, space charge limited conduction, etc.). However, it must be reemphasized and it will also become clearer below, that the DC conduction properties often refer to a global insulation-system behavior including bulk, interfaces, and contacts, and it does not allow a decomposition in series resistances for, e.g., contacts and bulk. This holds particularly if the majority carriers origin from current injection. Table 2. Experimental methods for electric characterization of HVDC insulation, quantities of interest, and influencing boundary effects. EXPERIMENT

QUANTITIES

BOUNDARY EFFECTS

Isothermal de-/polarization currents Steady state currents

Dielectric relaxation time Time-of-flight I-V-characteristics Activation energy Trap energy levels and densities of states Bulk conductivity Dielectric polarization strength Space charge mobility charge polarity Carrier mobility Recombination centers Void sizes and number Breakdown strength

Charge injection Contact resistance Current noise by PD in interface voids Interface trap effects T-dependence of injection may mask trap release peaks Injection / blocking (nonlinear effects see Sect. 4.4) Hetero-and homocharge formation, injection/blocking Surface trapping Surface recombination Contact voids differ from bulk voids Interfaces/contacts are often weakest parts

different temperatures Thermally stimulated currents (incl. x-ray induced) Impedance spectroscopy (low-frequency or broad-band) Space charge measurements Surface potential decay Electroluminescence Partial discharge Breakdown measurements

This is also related to a downside of the nontrivial scaling of boundary effects: the above mentioned upscaling problem of lab-experiments. The insulation behavior of (larger) real applications might strongly differ from those of smaller lab samples. As is well-known, while thin insulation samples may show space charge limited current (SCLC) behavior with 1/L3scaling, sufficiently thick insulators show ohmic conduction with 1/L-scaling. A list of different scaling behavior for different underlying physics is discussed in [2]. In small samples (which are usually the objects of material characterization) finite size effects competing with bulk or volume effects frequently preponderate. Generally, in HVDC insulators electronic and ionic carrier types may contribute to conduction at the same time (mixed conduction). Mixed conduction HVDC polymer insulation is poorly understood, and it can occur in solids and liquids [22]. A part of the complexity is due to the possible different behavior of different carrier types at boundaries. For instance, a contact can be blocking for ions but ohmic for electrons. In general a variety of different scenarios may occur, which aggravates the system characterization, the interpretation of measurements, and the modeling of the electric properties.

37

3 CONDUCTION MODEL The main task of an HVDC conduction model in industrial R&D is to predict the electric field distribution for type test conditions [3], [4]. A model should also be able to describe small lab samples in different experiments used for characterization of the insulation, in order to determine the needed model parameters and their values. 3.1 BULK EQUATIONS The relevant electric equation is Ampere’s law

 jtot  0

(1)

for the total current density

jtot  j  D/ t

(2)

where j is the conduction current and D=0E+P is the electric displacement field with polarization P. The displacement satisfies the first of Maxwell’s equations (Gauss’s law, or Poisson’s equation for the electric potential): D   (3) The space charge density  can be expressed by the differences of the densities nk (k=1,...,N) of N charge carrier types, and their charge-neutral equilibrium densities nk(0)    q k ( n k  n k( 0 ) ) (4) k

where qk are the charges of the carriers. The material equations consist of a polarization model for P and a conduction model, which provides the nk and the current. Although the polarization dynamics is of practical importance, it is not relevant for the purpose of this paper and will not be discussed in the following (see [9]). A constant relative static permittivity s of the materials will be assumed, and we will use D=E=s0 E in the low-frequency regime. The conduction model consists of N partial differential equations of the form nk / t  q k1  jk  f k ( n1 , n2 ,..., E , T ) (5) with current densities j k  q k  k n k E  q k Dk  n k

(6)

where the k and Dk are the mobility and the diffusion constant of species k. In local thermodynamic equilibrium, the Einstein relation qkDk = kBTk holds at relevant temperatures. The set of equations (5) are of drift-diffusion type with source and sink terms as used in semiconductor physics [23]. Note that equations (5) and (6) include immobile traps, acceptors, and donors, as well as neutral diffusing molecules (immobility refers to k=Dk=0, and neutrality to qk=k=0). The functions fk, which describe generation and recombination, trapping by and emission from traps, etc., are typically polynomials in the nk, which can be constructed from the order of the kinetic process [23]. For instance, recombination of species l with species j is a two-particle process associated with a term kljnlnj with a kinetic coefficient klj that generally depends on T and E. Note also that charge conservation implies kqkfk = 0. Electric transport can then be simulated by solving the equation (5) with equation (6), where equations (3) and (4) can be used for the determination of the electric potential , which

38

T. Christen: HVDC Insulation Boundary Conditions for Modeling and Simulation

is related to the electric field E by E = - (magnetic effects are neglected). The total current, measured in an external circuit, is obtained from integration of the total current density of equation (2) over a cross section area of the insulator. 3.2 BOUNDARY CONDITIONS The BC for the electric potential (homogeneous Dirichlet BC for  at conductor surfaces, Neumann for insulators etc.) are well-known and need not to be discussed here. But how to formulate appropriate BC for the transport equations is less well-known. In the literature, one often finds BC which prescribe the current, j=j(0)(E,T), e.g., describing a thermal and/or field emission (Richardson-Schottky, FowlerNordheim, etc [2]) injection current. This BC is a special limit case, but not the mathematically general BC for (5). A general BC relates the {nk}k=1,…,N to the {nk}k=1,…,N normal to the boundary. If one restricts to quasi-linear BC, one can write  a lk n k   b lk sˆ   n  c l for l=1,…,N and where ŝ k

k

k

is the boundary unit normal vector. The parameters alk, blk, and cl of the macroscopic BC model may still depend on other variables like E, T, and {nj}j=1,…,N, and they should, for a fundamental theory, be derived from the microscopic boundary physics. For clarity, we will not consider the most general form of these BC, which requires also a discussion of the properties of the matrices alk and blk. Instead we write the BC in the form j k  j k( 0 )  q k wk n k (7) where jk refers to the boundary normal component of the current density of species k and is defined positive with respect to the inward direction, jk(0) has the meaning of a saturation current, and wk is a characteristic velocity describing the backflow of carriers into the electrode [11]. The form of equation (7) turns out to be general enough for illustrating the relevant physical behavior and simple enough for convenient use in simulations. Most helpful publications on BC in the context of conduction theory in insulators are [6][11], and a general mathematical treatment of BC for a specific but important class of partial differential equations is [24]. Related experimental work on electrode contact physics in the HV engineering community is manifold, we refer to [25] and the literature cited therein. Interesting progress in theory occurs also in the framework of first principle simulations [26]. Equation (7) has a simple interpretation as is illustrated in Figure 1 for the example of a perfect insulator. The latter is defined by the absence of intrinsic charge carriers. At equilibrium (short-circuited electrodes), carriers from the electrodes contacted to the perfect insulator can diffuse into it in order to equilibrate the electro-chemical potential. We assume positive unipolar conduction, i.e., one carrier species (holes) dominates. Hole accumulation layers form and build contact charges, which shifts the potential in the insulator upwards (dashed curve (i) in Figure 1). An externally applied positive voltage U shifts the maximum of the potential to the anode region (solid curve). This maximum defines the potential barrier for current injection, and its location is called virtual electrode (VE). At the VE the electric field, which is

the slope of the potential, vanishes. Hence the current is purely diffusive there. In order to derive the boundary conditions (7) for the macroscopic equations (5) at the VE (x=0), one takes into account for the net current jk the current jk(0) coming from the metal, and a current wkqznk(0) which flows back into the electrode for a given carrier density nk(0). The saturation current jk(0) can, for a simple example case, be calculated by integration of the appropriate current contribution in the Boltzmann tail of the Fermi velocity distribution above the barrier top, in the same manner as the expression for a Richardson current is usually derived. However, as mentioned above, other models are possible (cf. Chapter 6.2 in [2]). The prefactor wk in equation (7) can be derived by assuming a Maxwell velocity distribution at the VE, and deriving the current flow into the electrode by velocity integration only over the appropriate half space [11][12] (this is again only one possible model among a variety). Injection (of electronic or ionic carriers) can also occur in ionic conductors like oil or oil impregnated solids, where the ionic electrode layer must be modeled [22][27]. We will not enter here the interesting area of deriving the macroscopic BC parameters from microscopic models. We will rather adopt a phenomenological approach by assuming that wk and jk(0) are given, and discuss consequences of their values on the electric behavior.

Figure 1. Illustration of the boundary conditions (7) for a single carrier type. (i) Equilibrium potential (normalized with respect to its maximum) in an insulator of thickness L with (positive) majority carrier injection from the electrodes. (ii) Non-equilibrium potential after application of a voltage, with formation of a potential maximum near the cathode (virtual electrode, VE in the inset). Inset (iii): carriers moving in the metal (M) above their Fermi level (EF) towards the VE (with saturation current density j(0)) are partially “backscattered” at the VE (proportional to n at the VE) and partially transmitted (j). The location xVE of the VE can be set as the injecting contact boundary x 0 because xVE