Presentation of global behaviour of nonlinear AC power systems using bifurcation diagrams

International Journal of Electrical Engineering Education, Jul 1997 by Bodger, P S, Irwin, G D, Woodford, D A, Gole, A M

Abstract Operational voltage levels can force a.c. power systems into non-linear behaviour. The system global characteristics are presented in bifurcation diagrams, generated using an electromagnetic-transients program. They reveal the onset of ferroresonance, period-doubling and chaos when series capacitance is present between the supply and saturating transformers

1 INTRODUCTION

Operational voltage levels can force a.c. systems into non-linear behaviour. Ferroresonance, period doubling and chaos can be the result. Ferroresonance can occur with sinusoidal excitation where an incremental change in the amplitude of the input or in the magnitude of one of the parameters of the system causes a sudden change in signal amplitude somewhere elset. The appearance of ferroresonance may be random2, with jumps and hysteresis, subharmonics, quasi-periodic oscillations, intermittency and chaotic behaviour3.

In power systems, ferroresonant circuits consist of iron-cored (and therefore non-linear) inductances in series with capacitance4. Typically, transformers provide a non-linear magnetizing inductance through operation into the saturation region. Capacitance may be intentional components such as grading capacitors across circuit breakers, series compensation capacitors, within CVTs, or capacitive coupling between transmission/distribution lines or cables3.

Field observation of ferroresonance in power systems is reported, for example, in Refs. [2, 4, 5]. Experiments in the field have been conducted and reported in Refs. [2, 7-9]. Many authors have studied the phenomenon in the controlled space of a laboratory3,s,lo-2 . A number of these and others have attempted analytical expressions for ferroresonant behaviour13,16.20-26 and, where possible, aligned this analysis with measurements on real ferroresonant systems.

Unfortunately, obtaining analytical expressions of non-linear networks is difficult and a number of authors have used either network analyzers or digital simulation techniques to solve models of ferroresonant systems5-7,15'17'26-29. It has been typical of researchers to reduce the complexity of a real circuit to a much more simple form to facilitate experiment, analysis or simulation.

In a previous paper30, the usefulness of chaos tools in predicting the behaviour of a ferroresonant circuit was established. Bifurcation diagrams removed the guesswork as to where changes in the steady state behaviour of the system occurred. At selected points, specific behaviour was studied. The paper used a simple, single non-linear component model27, as representative of a circuit to which more complicated systems have been reduced.

In this paper, bifurcation diagrams are used to compare the global steadystate behaviour of more complicated systems. The characteristics of single phase, multiple transformer; three phase single and multiple transformer; through to radial and interconnected series capacitor compensated a.c. power systems are studied. Use is made of a digital electromagnetic transients simulation program31 to model the systems as a network of individual components without reduction of the systems to simple equivalents. Information on specific variables of interest is thus gained by metering of variables, such as voltages and currents.

2 CHAOS AND BIFURCATION DIAGRAMS

Deterministic non-linear systems can generate apparently random behaviour. They are referred to as being chaotic32. Chaotic systems are characterized by extreme sensitivity to tiny perturbations33,34. The results can be affected such that time and frequency domain presentation become selective at best. Important behavioural transitions can be missed since two different initial states, arbitrarily close to each other, give rise to completely different system behaviour2,35.

Instead, the global behavioural characteristics of a system can be presented in a bifurcation diagram33,34. This is a multiple run representation of an output variable relative to some driving variable. When critical values of the driving parameter induce an abrupt change in the behaviour of the system, there is a bifurcation3. The system changes state as it goes from periodic behaviour, through quasi-periodicity, to chaos".

Bifurcation diagrams presented in this paper are derived from Poincare sampling of a variable, i.e. sampling is undertaken at the source frequency, once per cycle. The samples become points on the bifurcation diagram. All values of the samples are plotted. For a periodic waveform, there is one point per driving parameter value. For chaos there are many.

3 SYSTEM MODELLING AND INFORMATION PROCESSING

The systems of this paper are modelled using an electromagnetic transients program31. Fundamental frequency sinusoidal sources excite each system. All systems include a capacitor in series with one or more non-linear inductances as represented by transformers in saturation.

In the previous paper3, a single phase source excited a capacitor and open circuit transformer rated at 8.33 MVA, with voltages at 63.5 kV and 25.4 kV phase to neutral. The leakage reactance has been set to 0.1 p.u. The series capacitor was 0.48344 (mu)F, core loss resistance was 27.06 kQ and winding resistance was set to 0.1 4. To provide a comparison, these values, or their three phase equivalents, were used in the initial systems of this paper.