Qualitative analysis of mathematical arc models using Lyapunovtheory R Genesio and A Vieino Istituto di Elettrotecnica Generale, Politecnico di Torino, Italy M Tartaglia Istituto Elettrotecnico Nazionale 'Galileo Ferraris', Torino, Italy fin approach is proposed to the study of the electric arc near current zero by means of mathematical models. The approach is based on L yapunov's stability theory and allows a qualitative analysis of the nonlinear differential equations describing the phenomenon. The main results concern the determination of the set of conditions leading to arc extinction and their dependence on the physical parameters involved. The classical Mayr and Cassie models are studied using the method, and some numerical results are given. Keywords: mathematical models, electric arcs, Lyapunov method, electric power system disturbances I. Introduction Much effort has been devoted to the study of electric arc behaviour in circuit breakers near current zero by means of mathematical models. Such models allow arc evolution to be simulated in quite different conditions once a few para- meters have been experimentally evaluated. Analysis of the model sensitivity with respect to the variables taken into account may provide useful suggestions for improving the design of circuit breakers. The dynamics of arc phenomena have been described both in terms of ordinary differential equations H9 and by means of partial differential equations 2°-2s. Because of the nonlinear features of such models, their analytical solu- tion generally appears impossible, except for simple cases in which some physical variables are imposed ~'2,3,19 Numerical techniques have often also been used to con- sider the electric circuit connected to the arc and to study their interaction s, 6, 8-1s, 22, 24, 26. However, this approach can be rather difficult when analysing general system properties because of the many computations needed and the wealth of results that must be interpreted. Received: 3 March 1982 The solution of a set of differential equations requires a knowledge of the model's initial conditions, which deter- mine the transient evolution of the phenomenon and there- fore the final state reached, i.e. an extinguished or perma- nent arc. Actually, these initial conditions are difficult to evaluate because they depend on the complex phenomena preceding the current zero, and generally the simulation is performed just for a single initial system condition. In the present paper, a new approach 27 is proposed for investigating the structural properties of a model with regard to any possible initial condition. In particular, the procedure enables one to associate the final state reached by a transient with the corresponding initial state;that is, to detect the extinction or the reignition of the arc depend- ing on the initial conditions. The approach is based on the Lyapunov theory of stability, which allows both local and global stability characteristics of an equilibrium point of a dynamical system to be investigated without integrating the corresponding equations. So, the determination of the initial conditions leading to a particular final state is reduced to the estimation of a region of asymptotic stability. In the present paper, the approach is developed for ordinary differential-equation arc models. Section II introduces the well known Mayr model and derives the equations of an RLC circuit connected to the arc. In Section III the main idea of the paper is presented for the analysis of arc models by means of the Lyapunov stability theory. Section IV is concerned with some structural properties of the above Mayr model, while, in Section V, the main results are re- ported and discussed in terms of their links with the classi- cal experiences of electric arc behaviour. Finally, Section VI contains a similar stability analysis of the Cassie arc model, showing some basic structural differences between it and the Mayr model. II. The Mayr are model The well known Mayr modeP represents the electric arc as a time-varying conductance ga of the form Vol 4 No 4 October 1982 0142-0615/82/040245-08 $03.00 © 1982 Butterworth & Co (Publishers) Ltd 245