Protection studies may be performed using either a steady-state program or a time-domain tool (e.g.
EMTP-like tools). When using a time-domain simulation tool, the representation of power system com- ponents is usually made taking into account guidelines recommended for low-frequency and switching transients [24]. This section provides a summary of modelling guidelines for those power system com- ponents that can be affected by a fault and whose behaviour is critical for the behaviour of the protection system [10, 11, 25].
11.2.1 Line Models
Transmission line parameters are evenly distributed along the line length, and some of them are also func- tions of frequency. For steady-state studies, such as short-circuit calculations, positive- and zero-sequence parameters calculated at the power frequency from tables and simple handbook formulas may suffice. For electromagnetic transient studies, line parameters are generally computed using auxiliary subroutines available in most EMTP-like tools [13]. These tools contain two major categories of transmission line models: constant- and frequency-dependent models. In both cases, the models may use either a lumped- or a distributed-parameter representation. In the constant-parameter category, there are a variety of options such as positive- and zero-sequence lumped-parameter representation, pi-section representation, or distributed-parameter transposed and untransposed line representation. In the frequency-dependent category, the distributed-parameter approach may be considered for either transposed or untransposed lines.
Line models for steady-state studies:There are steady-state studies for which modelling transmission lines at only one particular frequency may suffice. Software tools have a number of models that could be used for this purpose:
rExact-pi circuit model: This lumped-parameter model can represent the line accurately at one specific frequency. This model is based on hyperbolic equations, may take into account skin effect and ground return corrections and may be multiphase in the phase domain with constant parameters. The model is correct for steady-state solutions and frequency scans, but is not adequate for transient studies.
rNominal-pi circuit model: This model is derived from the exact-pi model and used when the frequency of transients is low. The line is generally assumed untransposed and can be used to model particular transposition schemes in great detail by cascaded connection of pi sections. This model has the same limitations as the exact-pi model in addition to being limited for short lines. It cannot represent frequency-dependent parameter lines and should not be used for ‘electrically long’ lines.
Line models for transient studies: Distributed- and frequency-dependent parameter models are the most adequate for transient studies. They use travelling wave solutions, and can be valid over a very wide frequency range:
rConstant distributed-parameter line model: This model assumes that the line parameters are constant.
The line inductance and capacitance are distributed, and losses are lumped. The above conditions are met for positive-sequence parameters to approximately 1–2 kHz, but not for zero-sequence parameters, so the model is good only where the zero-sequence currents are very small, or oscillate with a frequency close that at which the parameters were calculated. This frequency should not be very high, to meet the conditionRl≪Zsurge, otherwise the line must be split into smaller sections.
rFrequency-dependent distributed-parameter line model: The line parameters are not constant but a function of frequency. Most frequency-dependent models are based on the modal theory where multiphase line equations are decoupled through modal transformation matrices, so that each mode can be studied separately as a single-phase line. The transformation matrices for untransposed, or
unbalanced lines are complex and frequency-dependent. However, it is possible to obtain good accu- racy by using real and constant transformation matrices. Some programs may provide the option of using frequency-dependent transformation matrix. Recent frequency-dependent line models are based on a frequency-dependent phase-domain model, which does not use the modal transformation matrix.
The exact pi-model can be used when a steady-state analysis suffices. For transient analysis the frequency-dependent distributed-parameter model should be used for the lines of main interest, and the constant distributed-parameter model used for lines of secondary interest. Although the pi-circuit model is not a good choice for transient studies, it has been used for transient studies by cascading a number of nominal-pi sections.
11.2.2 Insulated Cables
The application of insulated cable models for protection studies follows the same guidelines as for overhead lines. Cable models may also use either a lumped- or a distributed-parameter representation, and their parameters can also be computed using auxiliary subroutines available in most EMTP-like tools [13].
11.2.3 Source Models
Source models used in protection studies are represented by means of detailed machine models or as ideal sinusoidal sources behind subtransient reactances or the equivalent Thevenin impedances of the system [24]. The choice of a specific model depends on system configuration, the location of the fault and the objectives of the study:
rModel 1: A detailed model of the machines involved in a disturbance is mostly used for representing small generating stations in non-integrated systems where the system disturbance is likely to cause change in frequency and the relays are slow in responding to that disturbance. The model requires complete machine data, including the mechanical part and the control systems, depending upon the time frame of study and their response time.
rModel 2: A representation based on an ideal source with subtransient reactance is used for representing large generating stations. The assumption is that the system inertia is infinite and the disturbance under study does not cause the system frequency to change. The time frame of interest is small (approximately 10 cycles) and the machine controls, such as excitation system and governor, will not respond to the disturbance. Large systems can be divided into subsystems, and each subsystem can be then reduced to an ideal three-phase source in series with equivalent positive- and zero-sequence Thevenin impedances. The main advantage of this model is that the computation requirements are significantly reduced; its main disadvantage is that the Thevenin impedance represents the system equivalence at power frequency only, so the transient response is not as accurate as when the complete system is represented.
11.2.4 Transformer Models
Transformer modelling over a wide frequency range still presents substantial difficulties: the transformer inductances are nonlinear and frequency dependent, the distributed capacitances between turns, between winding segments and between windings and ground produce resonances that can affect the terminal and internal transformer voltages. Models of varying complexity can be developed for power transformers using supporting routines or built-in models available in EMTP-like tools. Although none of the existing models can portray the physical layout of the transformer, or the high-frequency characteristics introduced by interwinding capacitance effects, most EMTP-like tools have capabilities that can be used to model any transformer type over a particular frequency range.
Here is a summary of built-in capabilities available in EMTP-like tools for representing power transformers [24, 25]:
rIdeal transformer model: It ignores all leakages, by assuming that all the flux is confined in the magnetic core, and neglects magnetization currents by assuming no reluctance in the magnetic material. This capability can be used together with other linear and nonlinear components to represent more complex transformer models not available in EMTP-like tools.
rSaturable transformer model: It considers that around each individual coil there is a magnetic leakage path and a magnetic reluctance path. This model uses a star-circuit representation for single-phase transformers with multiple windings. This model requires, as a minimum, the following information:
(1) the voltage rating of each winding; (2) the leakage impedance of each winding, (3) the core satu- ration characteristic; (4) the transformer connectivity information. The leakage impedances are fixed inductances and resistances, separated into individual elements for each winding. The representation of the magnetizing branch is optional. This model is good for low-frequency transients. Since the winding resistances are frequency dependent, they need to be modified to reflect proper behaviour at higher frequencies. In general, the turns ratio cannot be dynamically changed during the simulation to reflect tap changer operation, although some tools include this option.
rMatrix model: In a transformer bank of single-phase units, the individual phases are not magnetically coupled, and their modelling is balanced, assuming that all three phases have equal parameters. In three-phase core transformers, there is magnetic coupling between windings. In addition, they may have asymmetry of magnetic path lengths, which results in asymmetrical flux densities in the individual legs of the transformer core. The core asymmetry effects are more noticeable for unbalanced operation.
An accurate representation requires the use of a model that takes into consideration the coupling of every phase winding with all other phase windings. Models based on matrices of mutually coupled coils can represent quite complex coil arrangements. The matrix elements for transformers with any number of windings can be derived from the short-circuit impedances between pairs of windings. The calculations are rather complex. Support routines available in some EMTP-like tools can be used to produce the branch matrices from the positive- and zero-sequence short-circuit and excitation test data.
The resulting models are good up to 2 kHz. They can take into account excitation losses, but nonlinear behaviour is not represented and must be added externally. Two matrix representations are possible for transformer modelling: the admittance and impedance matrix representations. The impedance matrix representation is only possible if the exciting current is non-zero; otherwise the matrix is singular.
Transformer saturationshould be modelled if the flux will exceed the linear region. In the saturated region, it may make a difference where it is placed. EMTP-like tools have auxiliary routines to calculate the magnetization branch saturation parameters. The supporting saturation routine generates the data for the piecewise linear inductance by converting the rms voltage–current data into peak flux–peak current data. The resulting curve is single-valued (without hysteresis). Although the nonlinear inductor model works well for a number of cases, it also exhibits several limitations: it is frequency independent and does not represent hysteresis effects, which means that remanent flux in the core cannot be represented.
EMTP-like tools also provide a pseudo-nonlinear hysteretic reactor, which could overcome some of the above limitations. Data for this model can be obtained using another supporting routine. The user needs to supply the scaling – that is the location of the positive saturation point (the point of the first quadrant where the hysteresis loop changes from being multivalued to being single-valued). Hysteretic models have some limitations and numerical problems [25].
Low-frequency transformer models: They can be based on the built-in capabilities summarized above.
Supporting routines may be used for modelling transformer windings as mutually coupled branches.
When matrix models are used, the magnetic core of the transformer is typically represented with a nonlinear or a hysteretic reactance branch, connected externally to the terminals of the windings. The built-in saturable transformer component is simpler to use than the matrix models, but if zero-sequence behaviour of three-phase core-type transformers needs to be modelled, then the matrix approach must
be used. The transformer models discussed here are valid only at moderate frequencies. In general, these models are accurate enough for overcurrent protection studies. Eddy currents in the transformer core introduce losses and they delay flux penetration into the core. Modelling of eddy currents is not an easy task since data is not usually available. No-load losses include hysteresis and eddy current losses and can be represented by a resistor in parallel with the magnetizing inductance branch.
High-frequency transformer models: The above models are used in studies at frequencies below 2 kHz.
At frequencies above 2 kHz, capacitances and capacitive coupling between windings can be important or very important. For frequencies of up to 30 kHz, the simple addition of total capacitances of windings to ground and between windings is sufficient for many purposes. For frequencies above 30 kHz, a more detailed representation of the internal winding arrangement is required, and capacitances between winding and among winding segments must be modelled. The values of terminal-to-ground capacitance including bushing capacitance vary considerably, with typical values in the range of 1–10 nF. This is due mainly to the physical arrangement of the transformer windings and the overall transformer design.
High-frequency transformer models may be derived from its terminal behaviour. In relaying studies, the interest may be in internal faults [26]. If explicit representation of transformers is not required, the user can model transformer effects without modelling the transformer itself; for example, a Thevenin equivalent representation in the sequence domain can be used in these situations. New high-frequency transformer models have been developed in recent years [27, 28].
11.2.5 Circuit Breaker Models
Circuit breakers are usually represented as ideal switches; that is, the switch opens at a current zero and there is no representation of arc dynamics and losses. Custom-made circuit breaker models can be employed for detailed arc modelling [29, 30]. The types of switches that are applicable for protection studies are presented below [13, 25].
rTime-controlled switch: In this type of switch both the time at which it is to close and the time at which it is to open are specified. The actual switch opening time will occur at the next current zero after the time at which it is required to open. To simulate current chopping, a current margin is also specified and the switch actually opens at the instant the current magnitude falls below the current margin and the time is greater than the time at which it is to open.
rStatistical switch: This type of switch is used to open or close the circuit breaker randomly with predetermined distribution functions, such as Gaussian or uniform. The user needs to specify the mean time and the standard deviation, in addition to the type of distribution, and the switch performs the same close or open operation repetitively according to the specified distribution characteristics.
This switch can be employed to determine the maximum peak currents that can flow through a relay when closing into a fault. Usually a few hundred simulations are run to determine the statistical distribution of interest such as the maximum relay currents.
rControlled switch: Most EMTP-like tools have a module that can provide the control signal to open or close a switch following a given strategy. In such cases, the switch would behave similarly to the time-controlled switch.