Poster based on research on investigating the non-linear response of a synchronous machine to variations in system parameters (torque and damping), demonstrating the existence of a bifurcation curve within the parameter space. Response was visualized using state space diagrams. This poster was presented at the Power and Energy Conference at the University of Illinois (PECI) in Spring 2017.

1. Research Background and Method
The classical dynamic model of a synchronous machine connected to an infinite
bus can be summarized by the following state equation:
δ = internal machine voltage angle
ω = transient speed
α = mechanical torque term
β = damping coefficient
γ = electrical torque term
The research objective is to verify that a boundary curve known as a homoclinic
bifurcation curve defined as
β =
π
4
α
in terms of the torque and damping coefficient exists for sufficiently small
parameter values within the system parameter space. The proposed research
approach is as follows:
1. System Approximation
First a program would be written to numerically approximate solutions to the
above non-linear system. Supplied with an initial state vector X0 and a value for
the time step, the program would use Euler’s Forward method to calculate the
value of the current state X over a range of discrete time, defining the
corresponding solution trajectory. The trajectory would be represented as a list
of ordered pairs (δ,ω) in a text file.
2. Phase Plane
Next, the results from the program representing the solution trajectories would
be mapped to the 2-dimensional phase space. This would be done through
importing the text file outputs from the program into a command-line based
graphing utility called gnuplot.
3. Experimental Validation
Critical points in which homoclinic bifurcation occurs in the trajectories within
the phase plane would be approximated through monitoring of the solution
behavior in response to small variations in the torque α and damping β around
the proposed line. Collecting a sufficient number of these critical points for
various parameter values would give an experimental representation of the
homoclinic bifurcation curve.
Separatrix Analysis of a Synchronous Machine
Aristotle Boyd-Martin, Peter W. Sauer
Research Significance
It is important for engineers to better understand the non-linear behavior
associated with the machines that they design, and this mathematical tool is a
useful aid in that.
Research Motivation and Objectives
• The non-linear behavior characteristic of most practical dynamical systems
makes it difficult or impossible to analyze the system using traditional
analytical methods. Therefore, it is a challenge to be able to make
predictions about the system under a variety of operating conditions.
• The research objective is to verify that a boundary curve that separates the
state space of a dynamical system modeling a synchronous machine into
two regions of fundamentally different behavior known as a homoclinic
bifurcation curve exists within the parameter space.
• These concepts would then be extended out to a more sophisticated
machine model.
Potential Applications and Future
Research
In order to design a machine for a particular application, it is crucial to have an
understanding for how designer-chosen parameters will affect the machine’s
behavior under a variety of operating conditions in accordance with the
governing differential equations. This mathematical tool provides the connection
between these parameters and the overall separatrix for the system.
Future research will consist of applying these concepts to a more realistic and
widely accepted machine model. If such relationships can be determined, these
can prove to be very useful to engineers in helping them to design more stable
systems.
Research Results
Fig. 1: A comparison between the experimentally obtained homoclinic bifurcation curve (red) and the
theoretical one (black).
The critical points at which bifurcation occurred within the phase space were
found and recorded for a range of parameter values. Note that γ was held
constant throughout. For small parameter values, the resulting separatrix curve
(red) very closes matches the analytical one (black).
Fig. 2: Phase plane showing trajectories for α = 0.3, β = 0.23, and γ = 1. In this case the expected critical
value is βc = 0.2356.
Below the separatrix (β < βc, where βc is the critical value of β for a given α,
marked on the red line), the system exhibits two stable solutions, a sink and a
periodic rotation in the state space. Just past βc (graph below), the stable
periodic solutions disappear entirely through bifurcation.
University of Illinois at Urbana-Champaign
Fig. 3: Showing how the solutions bifurcate when the damping β (which in this
case was 0.24) is nudged just beyond the bifurcation point.