CHALLENGES IN TEACHING THE DIFFERENCE BETWEEN PHYSICAL AND NUMERICAL INSTABILITY

Abstract

This paper describes a subtle but crucial problem that often arises in the dynamic analysis of complex nonlinear systems, namely the difference between physical and numerical instability. The former is the consequence of the intrinsic behaviour of the system that is simulated, while the latter is an issue of the numerical technique, typical an algorithm, that is used to simulate the system. The author has about fifteen year experience in teaching, in different institutions and countries, power systems modules, including power system modelling, control and stability analysis and in developing open-source software tools for research and educational purposes. A relevant issue that has been observed is the difficulty that students encounter when they have to discuss simulation results of large systems that depends on a large number of parameters and/or of variables. A typical question that often arises is whether the “instability” that a system shows is due to the fact that the system is actually unstable or it is due to some numerical issues, e.g., the numerical scheme shows an erratic behaviour but the system is actually perfectly stable. The first didactic issue to solve is to make the student understand the difference between these two kinds of instability. Explaining the distinction between mathematical models and computer-based implementation is anything but trivial. The second challenge is to find methods, which depend on the analysis to be carried out, to distinguish between the two. Very often, the discrimination can be done introducing advanced mathematical concepts which are not suitable for the student level. Hence, the paper discusses alternative “intuitive” techniques that allow solving the problem, if not rigorously, at least qualitatively. The proposed didactic approach has been tested by the author in his modules “Power System Dynamics and Control” and “Stability Analysis of Nonlinear Systems” taught since 2013 at the UCD School of Electric and Electronic Engineering. The final paper will provide the following contributions: 1. A discussion on the didactic challenges of teaching computer-based modelling of nonlinear dynamic systems to students of engineering modules. This discussion is particularly focused to electric power systems, but main concepts can be applied to any engineering area involving sets of nonlinear differential equations. 2. A variety of examples that illustrate the occurrence of numerical instabilities that can be confused with an actual instability of the simulated system and qualitative approaches that allows discriminating between physical and numerical instabilities. Students' feedback is duly discussed whenever relevant.