Abstract:

This paper describes the experience of the author in teaching nonlinear system stability analysis to master and undergraduate students of electrical engineering programmes.

The author has more than one decade 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 [1]. A relevant aspect that has emerged as a common issue is the difficulty that students encounter when they get in touch for the first time with theoretical aspects of power system stability analysis.

The inherent complexity of power systems is due to two key factors:
(i) power systems are composed of several different devices; and
(ii) the equations that describes power system dynamics are nonlinear. This basically means that the behaviour of an interconnected power system is not just the combination of the behaviours of each device alone.

In mathematics, it is well-known that nonlinear dynamic systems are way more complex and intrinsically more difficult to understand than linear ones.

These are some relevant reasons:
(i) the stability of linear systems can be defined univocally and is a global property, whereas, in the best case, only local stability properties can be determined for nonlinear systems;
(ii) most control techniques can guarantee to work properly only for linear systems;
(iii) bifurcations, e.g., structural changes of the dynamic response of a dynamic systems, only occur in nonlinear systems.

As a result, the behaviour of nonlinear systems is, to a large extent, unpredictable. This can be overcome only with experience, which in the case of power systems, implies dedicating several hours to solve computer simulations. The lack of experience can be only partially covered by lectures and tutorials. In fact, paraphrasing a famous sentence by Leo Tolstoy, all linear systems are alike; each nonlinear system is nonlinear in its own way.

The learning approach designed by the author in his modules is as follows:
(i) to assign to the students several lab activities that require solving a large number of simulations and studying the dynamic response of several test systems; and
(ii) to provide, during the lectures, a variety of examples of the idiosyncrasies of the behaviour of nonlinear systems and ask the students questions on how they would expect the system to respond following given disturbances.

The contributions of the final paper will be as follows:
1. A discussion on the didactic challenges to teach nonlinear systems to engineering students. 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 peculiar and unexpected behaviours of nonlinear systems. These examples are also presented during the lectures, hence, typical answers provided by the students and the rationales why such answers are typically incorrect are duly discussed.
3. An overview of the lab activities that are proposed for the module “Power System Stability Analysis” offered by the author at UCD as part of the ME Programme in Electric Energy Systems, as well as a discussion of difficulties encountered by the students to prepare lab reports.

References:
[1] F. Milano, Power System Software Tools, available at: http://faraday1.ucd.ie/software.html

Keywords:

Nonlinear systems, power system dynamics, power system stability analysis, electrical energy systems.