6.3.1 Extent of the Network Model
Ferroresonance is a localised phenomenon and, as a general rule, large network models are not necessary. Only the main elements directly involved in to ferroresonant circuit (i.e. non-linear reactance, capacitances and voltage source) need to be represented in detail. Therefore both in ferroresonance and resonance studies the feeding network can be represented as a Thevenin source equivalent calculated at power frequency.
6.3.2 Overhead Line Model
An accurate representation of the line parameters at resonant and near-resonant frequencies is essential. The most critical elements that need to be reproduced accurately are the circuit capacitances, therefore the model has to be based on accurate geometrical configuration of conductors at the towers and along the spans. Circuit phase transpositions, if present, need to be represented explicitly. The frequency dependency of the model parameters is not critical since the phenomena of interest is resonance at (or near) power frequency. A multi-phase distributed parameter model, such as Bergeron, calculated at power frequency normally yields reasonable results. A frequency dependant model like JMARTI can also yield accurate results when the transformation matrix is calculated at (or near) power frequency.
Corona losses can reduce the amplitude of resonant overvoltages when the critical corona onset voltage is exceeded. These losses are dependent on a large number of random variables, atmospheric conditions among them. Although there is significant literature dealing with corona losses under normal voltage operating conditions, there is very limited published experimental data on corona at power frequency above critical voltage [72]. In practical terms, corona will only contribute to the attenuation of Temporary OverVoltages and most studies tend to ignore these losses to add a safety margin to the computed results.
-1.5 -1 -0.5 0 0.5 1 1.5
0 20 40 60 80 100 120 140 160 180
statevariable
bifurcation parameter
Depending on the ferroresonant topology, it may be necessary to represent overhead lines in detail, for example in cases where energy is coupled from an energized parallel circuit or, alternatively, to assess the detuning effect of switching a long circuit.
Given that ferroresonance is a low frequency phenomenon, frequency dependency is not a critical feature of the line model. Therefore, a multi-phase distributed parameter line model like Bergeron is sufficient for most ferroresonant studies. The exception is when ferroresonance is initiated by a line switching transient where a frequency dependant model of the circuit under study is recommended. Other circuits not involved in the switching can be represented with lumped or distributed parameter models. If inter-circuit capacitances are part of the ferroresonant circuit, both parallel circuits must be represented in detail using accurate tower geometry. Phase-transposition, if employed, must be explicitly modelled. If the overhead line employs any series capacitors, these can be modelled as lumped capacitive elements connected in series between two line sections.
Special attention must be paid if frequency dependant line models are used in order to avoid non-passivity at low frequency (from 0 Hz to 100 Hz). This problem has been reported in [66]. When excited at low frequencies, these models “create” power due to apparent negative resistivity. The situation can be dangerous when dealing with resonance and ferroresonance because non-passive line models can give incorrect results that can be confused with regular ferroresonant shapes. A general procedure to check passivity in a frequency dependent line model is to calculate the admittance matrix [Y] and check that the eigenvalues of the real part of [Y] are positive.
6.3.3 Transformers
Power transformers must be represented with a three-phase model in order to reproduce correctly the coupling between phases. Voltage transformers on the other hand can be represented with single-phase models, with the secondary and tertiary winding connections represented externally to the model. Transformer stray, bushing and inter-winding capacitances can be represented as lumped elements in parallel with the appropriate windings.
The nonlinear behaviour of the magnetic core of the transformer is the most critical aspect of the model and therefore correct representation of the saturation effects along with losses is a key factor for the accuracy of the simulation results. Some transformer models available in commercially available EMT type software packages do not support inclusion of this data. In such cases the magnetic core data has to be represented externally. Some models include the hysteretic behaviour of the magnetic core including losses where this is normally the area inside the hysteresis loop. Normally, it is difficult to implement this type of model due to unavailability of data to the user.
To overcome this, some models employ a single-valued representation of a nonlinear inductor with a damping resistor added externally to account for the losses. The nonlinear inductor provides a smooth computation since flux is the integral of voltage and performs satisfactorily provided that the curve is not defined by too many segments.
In some stand alone cases, a Preisach type mode [59] or a Preisach-Biorci-Pescetti hysteresis model [60] has been utilised to include the hysteresis effect in the study of ferroresonance. It should be mentioned that although this type of saturation modelling provides very good results it has the drawback of being extremely difficult to implement due to the unavailability of data without specialised equipment testing.
The location of the saturation curve is also important in three phase power transformers. To obtain reasonable results, the saturation curve must be represented in parallel with the closest winding to the magnetic core (this is normally the LV winding). This approximation gives accurate results for frequencies below 1 kHz.
The magnetic core losses are critical in any simulation involving saturation. Various representations can be found in the literature: hysteresis loop, non-linear resistor and linear resistor. The advantages and limitations of each representation are discussed in [14], [61], [62]. Those technical publications conclude that the most accurate representation is the hysteresis reactor; however the parameters required for its developments are normally not available to the user and involve special testing. The non-linear resistor representation, on the other hand, can have serious limitations since hysteresis losses depend on flux and not voltage. A linear resistance is the most common representation for the magnetic core losses. It is reported in [14] that this core loss representation, if it
For more detailed information on the modelling of transformers for this type of studies the reader is advised to refer to CIGRE TB on Transformer Energization – A Study Guide.
6.3.4 Shunt Reactors
The core construction has a large effect on the reactors’ behaviour during unbalanced open-phase conditions. The only type of reactor that does not have direct magnetic coupling between phases is the single-phase unit. Three-phase reactors present different levels of zero sequence coupling depending on the core design. The shell-type and four/five-legged core-type reactors provide a magnetic path for the zero sequence flux, hence, the coupling between phases is very small and can normally be neglected. The three-legged core construction, however, presents a strong magnetic coupling between phases, which must be accurately represented when this reactor type is used for line shunt compensation.
Shunt reactors are normally specified to remain linear up to a knee point of around 125% to 175% of nominal voltage. For a preliminary analysis of line resonance, the shunt reactors can be represented as linear lumped elements. However, operation near a resonant peak may drive the reactor into saturation and initiate a ferroresonant oscillation. Therefore, a detailed analysis of the circuit must include reactors’ saturation.
Reactor losses affect the amplitude of near-resonant overvoltages. Typical quality factors for modern reactors built with low loss materials are in the order of 1000. This can be represented by a lumped resistor connected in series with the reactor.
6.3.5 Other Substation Equipment
The main aspect of the substation equipment that needs to be modelled accurately is the capacitive component.
The capacitances of all pieces of plant contributing to the ferroresonant circuit need to be represented accurately.
These elements include the busbars, bay conductors, disconnectors, current transformers, voltage transformers, surge arresters, circuit breakers, power transformers, shunt capacitors, etc.
Series capacitances are of major importance when the capacitance is connected to the target transformer; i.e. the case of series ferroresonance when the series capacitance is formed by the capacitances between the lines of a double circuit line, with the first line being energized, and the second one is de-energized with the transformer connected to it.
A distinction is made between capacitances to ground of conductors and stray capacitances of plant apparatus:
Capacitances to Ground of Busbars and Bay Conductors
An accurate model of the busbars and bay conductors is required. These conductors must be modelled based on the geometrical layout of the station. Lumped parameter line models are normally adequate for busbars or typical conductor lengths within the station unless very long busbars are involved, in which case distributed models may be employed. However, this approach restricts the size of the integration time-step and increases the computation time. While this restriction may not be important for one single simulation, it may be impractical when carrying out a large number of parametric analysis simulations. In practice, for normal lengths of conductors within a substation, a lumped parameter model (i.e. multi-phase “pi” representation) calculated at power frequency yields adequate results.
Stray Capacitances of Apparatus
The stray capacitance of each apparatus involved in the ferroresonant circuit can be represented as lumped elements connected between the phase conductors and ground. Accurate values of these capacitances must be obtained from the equipment manufacturers and test reports.
In addition to the above, certain equipment may need to be modelled explicitly, as follows:
Surge arresters may need to be represented if there is a concern about their energy absorption capabilities being exceeded during the ferroresonant oscillations. Surge arresters can also influence the ferroresonant oscillation mode [85].
Current transformers and PLC line traps do not need to be represented.
Circuit breakers can be represented as ideal switches. If the circuit breakers are equipped with grading capacitors, these need to be represented in parallel with the switch.
Capacitor banks can be represented as lumped capacitive elements with the appropriate connections (i.e.
wye or delta).
Circuit Breakers
Circuit Breakers can be represented as ideal time-controlled switches. Circuit breaker grading capacitors should be represented explicitly as a parallel capacitance across the ideal time-controlled switch. Stray capacitances of the circuit breaker are of major importance when the phenomena appearing at the transformer de-energization are of importance. In particular, they have an impact on the final value of the residual flux remaining in the iron core of the transformer following the opening of the circuit breaker poles.