Sensitivity to Parameters

6.4.1 Effect of Magnetising Curve

The saturation curve is one of the key parameters for the occurrence of ferroresonance, whether the non linearity is pronounced or not. One of the key components of the saturation curve is the slope in the fully saturated region

normally referred to as Lsat. This is equal to Lair, the air core inductance of the transformer, minus the inductance

corresponding to stray losses. Value of Lair can be estimated with a very good accuracy either by analytical

formulas [64] or by 3D electromagnetic calculations [65], assuming that the relative permeability of the iron core is equal to 1.

Figure 6-3 shows three measured magnetizing curves for three different VTs all used in a 400 kV network. They are presented in the same scale, to demonstrate differences between them. Figure 6-4 shows voltage - frequency dependencies for the above VTs for capacitances of 1 and 10 nF for a voltage range of 0 to 1600 kV based on EMT type software simulation. VT3, with the highest magnetisation knee point, connected to a capacitor of more than 5 nF is not able to oscillate on the fundamental power frequency for realistic levels of voltages, so fundamental frequency ferroresonance is impossible for this particular VT. On the contrary VT1, with its easy saturability, can exhibit ferroresonance for a wide range of frequencies.

Figure 6-3 Magnetizing curves (Wb - I) for three different 400 kV VTs

Figure 6-4 Frequency of oscillations as a function of initial voltage for three types of VTs

Ferroresonance is a highly nonlinear phenomenon which is very sensitive to the circuit parameters and initial conditions for the transformer and the power system. The choice of representation for the magnetising curve was investigated using EMT type software and is illustrated below. For this analysis, two approaches were compared: (i) piecewise linear representation and (ii) two – term polynomial curve representation. The data used in this analysis is shown in Figure 6-5. Both saturation curves were tested on the same network model for the two typical ferroresonant configurations described below. In both cases, the simulation results were dependent on the selected representation for the magnetising curve. These two examples illustrate the high sensitivity of ferroresonance to small variations in the circuit parameters or initial conditions.

VT1 0 500 1000 1500 2000 2500 0 0,2 0,4 0,6 0,8 1 VT2 0 500 1000 1500 2000 2500 3000 0 0,2 0,4 0,6 0,8 1 VT3 0 500 1000 1500 2000 2500 3000 3500 4000 4500 0 0,2 0,4 0,6 0,8 1

Current [A] Current [A] Current [A]

C=1 nF 0 20 40 60 80 100 120 140 160 180 0 200 400 600 800 1000 1200 1400 1600 Voltage (kV) F re q u e n c y (H z ) VT3 VT2 VT1 C=10 nF 0 20 40 60 80 100 120 140 0 200 400 600 800 1000 1200 1400 1600 Voltage (kV) F re q u e n c y (H z ) VT3 VT2 VT1

Figure 6-5 Current-Flux magnetising curve of a transformer approximated with (a) piecewise linear model and (b) polynomial curve model

Case 1: Transformer energized through grading capacitance of circuit breaker

The circuit shown in Figure 6-6 is assumed to operate in a steady state no-load condition and a temporary three- phase short circuit fault is applied to the secondary side of the transformer. The simulation results are shown in Figure 6-7 (a) and (b). Based on the piecewise linear magnetization characteristic, the simulated voltage show a normal operating condition for the transformer voltage whereas in case of the polynomial saturation curve, the transient state is followed by a fundamental mode of a ferroresonance oscillation.

Figure 6-6 Fault clearance leaving transformer energized through the grading capacitance of a circuit breaker

Figure 6-7 Simulation of fault clearance leaving transformer energized through the grading capacitance of a circuit breaker (a) normal response using piecewise linear representation of saturation curve (b) ferroresonant

response using a polynomial representation for the saturation curve

Case 2: Transformer connected to a double circuit transmission line

characteristic results in a dangerous ferroresonance overvoltage, whereas the polynomial saturation curve (b) does not show any overvoltage.

Figure 6-8 Line disconnection leaving transformer energized through the coupling with parallel circuit

Figure 6-9 Simulation of line disconnection leaving transformer energized through the coupling with parallel circuit (a) ferroresonant response using piecewise linear representation of saturation curve (b) normal response using a

polynomial representation for the saturation curve

It should be noted that the polynomial function representation has the problem that it has only three parameters and has therefore limited flexibility to accurately represent the core nonlinearity in the linear part around the knee point and in the saturation region. An inaccurate magnetization characteristic can result in erroneous ferroresonance simulations; therefore it is not very useful for ferroresonance analysis.

6.4.2 Influence of Circuit Breaker Closing Times

The closing times of a circuit breaker are a key parameter for the suppression of ferroresonance; the reason is mainly due to the fact that the major state-variable in the electrical circuit is the flux circulating in it, which is constituted mainly by the magnetic flux circulating in the iron core of the transformer. It may also be shown, in the case of discrepancies in the closing times for circuit breaker poles, that the phenomena may be generated due to it, and stopped when re-synchronizing the poles of the circuit breaker which operates.

6.4.3 Influence of the Damping in the Circuit

Damping factors are of major importance. In other parts of physics, dealing with non linear systems, especially in chemistry, dissipative aspects are of major importance, linked to thermodynamic aspects, entropy [17], which characterises the balance between the energy brought into the system (source as an input), and the dissipative parts (losses of the system). In electrical network, when the energy is injected in the electrical system from one side, through the up-stream network, the resonance phenomena may be damped through the dissipative components.