CHAPTER 6 MODELLING AND STUDYING
6.2 Digital Simulation Methods
Since analytical solution methods are not capable of predicting non-periodic ferroresonance occurrences and even sometimes sub-harmonic ferroresonance, researchers have used different digital simulation techniques including EMT analysis and chaos theory to analyse ferroresonant circuits.
It was Kieny [30
],
[44] who first suggested applying chaos to the study of ferroresonance in electric power systemsby studying the possibility of ferroresonance in power transformers, particularly in the presence of long capacitive lines as highlighted by occurrences in France in 1982, and produced a bifurcation diagram indicating stable and unstable areas of operation. Kieny [53] was also able to present a non-periodic (pseudo-periodic), but not chaotic waveform, using Lyapunov exponents and was able to prove the stability of this waveform and postulated that the bifurcation theory was the right mathematical framework to study ferroresonance phenomenon.
A fourth-order Runge-Kutta formula was introduced [33] to solve numerically the differential equation of a basic ferroresonant circuit consisting of a capacitor in series with a nonlinear inductor. The same circuit was later used [54] in a different software toolkit package [55] to solve the nonlinear equations of the circuit and produced bifurcation diagrams with source voltage E being the bifurcation parameter for different exponent of the nonlinearity.
Nonlinear dynamical techniques were used [56] in an attempt to establish a methodical approach for identifying all possible initial conditions and, consequently, the different types of ferroresonant oscillations that can occur in a capacitor voltage transformer.
EMT type program was used [57] to model the behaviour of an 8.33 MVA power transformer with construction of Poincaré maps, phase plane trajectories, frequency spectrum and bifurcation diagrams. An add-on computer simulation was developed [58] which was subsequently incorporated into an EMT type software to study the problem of ferroresonance in a three-phase five-legged grounded wye-wye power transformer to reveal different types of ferroresonant wave-forms. EMT type software was also used in [15] to simulate the conditions present during a voltage transformer failure concluding that the method provided an accurate and inexpensive means to simulate the potential for ferroresonance.
Modelling nonlinear dynamical systems generally leads to a set of differential equations where some control parameters influence explicitly the solution type. That can be the voltage supply of an electric circuit, the flux circulating in the iron core of the transformer, the temperature of a chemical reaction or the gain of a feedback system. When critical values of the parameters induce an abrupt change in the type of solution, there is a bifurcation for the system.
Two main aspects have to be addressed when using such approaches, linked to the study of dynamic systems: a) Simplification of the electrical system, from a real multiphase network, to a set of ODEs, describing the main
characteristics of the network, especially the source, the resonant circuit, including its non linearity, and the dissipative elements.
b) Searching for solutions of the system (harmonics, pseudoperiodic, chaotic) using adaptive numerical methods when varying the main state variables.
The following sections briefly describe the different mathematical tools used in the study of dynamical systems.
Phase Space
The phase space of a dynamical system is a mathematical space in which the instantaneous state of the system is represented by the movement of a point representing the state variables of the system. As time evolves, the initial state point follows a trajectory which closes onto itself if the response is periodic and is called a cycle.
For a chaotic solution, the phase space will have a very complex trajectory never closing onto itself and is called a strange attractor.
Poincaré Section
A Poincaré section is a tool invented by Henri Poincaré as a means of simplifying phase space diagrams of complicated systems. It is constructed by recording the phase space trajectory as a sequence of discrete points at constant time intervals. If this sampling is done at intervals corresponding to the system’s forcing frequency (in the case of ferroresonance the power frequency) then, for a periodic waveform, with the same frequency as the forcing function, the Poincaré section will show only one point. Likewise a sub-third harmonic waveform will produce three points. However a chaotic waveform will produce a Poincaré section with a random set of points confined to a particular region of the plane as can be seen in Figure 6-1.
For a dynamical system such any ferroresonance configuration, the Poincaré section provides a simplification of the phase space diagram while retaining the essential features of the dynamics.
Bifurcation Diagram
Phase space diagrams and Poincaré section provide information about the dynamics of the system for specific parameter values. The dynamic behaviour may also be viewed more globally over a range of parameter values, thereby allowing simultaneous comparison of regions of periodic and chaotic behaviour.
A change in the type of solution to a set of ordinary differential equations when a parameter is varied, is called a bifurcation. A bifurcation diagram provides a summary of the essential dynamics and is therefore a useful method of acquiring this overview. It is an important tool for discovering interesting parameter regimes for a dynamic system. A bifurcation diagram is actually a collection of many Poincaré sections each calculated for a different value of a particular parameter in the system. A typical bifurcation diagram for a ferroresonance circuit is shown in Figure 6-2.
Figure 6-1 Poincaré section of a chaotic waveform
0 0.2 0.4 0.6 0.8 1 1.2 -2.7 -2.65 -2.6 -2.55 -2.5 -2.45 -2.4 -2.35 -2.3 -2.25 -2.2 -2.15 de riv at iv e of st at e va ri ab le state variable
Figure 6-2 Bifurcation diagram for ferroresonance circuit