The complex Hurwitz test for the stability analysis of induction
generators
Marc Bodson, Fellow, IEEE
Abstract— The paper proposes a new approach for the
analysis of stability of induction generators. A PI-controlled
doubly-fed induction generator is considered. The new method
is based on a Hurwitz test for polynomials with complex
coefficients. Using the test, stability of the PI controller is found
to be guaranteed if a simple quadratic inequality relating the
proportional and integral gains of the controller is satisfied. The
application of the complex Hurwitz test is also demonstrated on
another example. The method provides a new, systematic pro-
cedure for stability assessment and control design in induction
machine problems.
I. I NTRODUCTION
Wound-rotor induction generators find increasing uses,
especially in the field of renewable energy [6], [11], and wind
energy in particular [8], [10]. Their ability to produce power
at fixed frequency and for a range of rotor speeds makes them
particulary attractive. Wound-rotor induction generators may
be connected directly to the grid, with the stator currents con-
trolled indirectly through the rotor currents. Such generators
are usually referred to as doubly-fed induction generators.
Their advantage is that power electronic conversion is needed
only at the lower power level of the rotor, instead of the
stator. Methods have been proposed to control the active and
reactive power delivered to the grid [1], [2], [7], [12], [13].
Typically, these methods are based on the theory of field-
oriented control for induction motors, although some are
based on a model expressed in the reference frame of the
grid, or stator voltages [5], [9]. Assuming an infinite bus,
control of the currents in the reference frame of the grid
directly translates into the control of the active and reactive
powers absorbed or delivered by the generator. Control of
the reactive power is useful to maintain the voltage at the
end of a long transmission line, which often happens with
renewable energy sources [2].
In this paper, we consider the proportional-integral (PI)
control algorithm of [5]. This elegant algorithm of low com-
plexity was derived by applying passivity-based nonlinear
control techniques. It is a feedback linearization algorithm
that results in a linear closed-loop system. The authors of
[5] did not prove the stability of their algorithm, which
is described by a 6
order characteristic polynomial. Yet,
analysis for special cases and simulations demonstrated that
a wide range of parameter values would yield stability. This
paper answers the stability question exactly by reducing the
6
order characteristic polynomial with real coefficients to
Manuscript received September 15, 2009.
M. Bodson is with the Department of Electrical and Computer Engineer-
ing, University of Utah, 50 S Central Campus Dr Rm 3280, Salt Lake City,
UT 84112, U.S.A (e-mail: bodson@eng.utah.edu).
a cubic polynomial with complex coefficients. Interestingly,
while a Routh-Hurwitz test for the 6
order polynomial was
found to be intractable, application of a little-known Hurwitz
test for complex polynomials yields a simple stability test,
requiring that a single quadratic inequality be satisfied by the
PI gains.
The method proposed in the paper is also applicable to a
limited, but not insignificant class of control problems with
certain symmetry properties. In [3], the Hurwitz test was used
to find analytic conditions for spontaneous self-excitation in
induction generators. This paper shows how the approach can
be used to solve the stability problem for another example
[9] involving doubly-fed induction generators. Although a
stability proof existed in that case, it was based on a
Lyapunov function. The method of this paper provides a
systematic approach that does not rely on the skill of the
control engineer to find such a Lyapunov function.
II. MODELLING AND PI CONTROL OF A DOUBLY- FED
INDUCTION GENERATOR
A. Induction generator model
Consider the model of a two-phase wound-rotor induction
generator
(1)
where
,
are the stator voltages,
,
are the
stator currents,
,
are the rotor voltages,
,
are
the rotor currents, and is the speed of the generator. The
parameters of the generator are
, the stator inductance,
, the rotor inductance, , the mutual inductance between
the stator and rotor windings,
, the stator resistance,
,
the rotor resistance,
the synchronous frequency, and
,
the number of pole pairs. The and subscripts indicate
that the phase variables have been transformed into a frame
of reference rotating at frequency
. For the stator voltages
2010 American Control Conference
Marriott Waterfront, Baltimore, MD, USA
June 30-July 02, 2010
ThA05.4
978-1-4244-7425-7/10/$26.00 ©2010 AACC 2539