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On the state space representation
of synchronous generators
Emmanuel Delaleau
Universite Paris-sud
Laboratoire des signaux et systemes
CNRS Supelec
3 rue Joliot-Curie91 192 Gif-sur-Yvette, France
e-mail: [email protected]
Abstract
The paper aims to present a formalism allowing the selection of
state variables
in dymamic models. This is applied to the model of a synchronous
generator result-
ing in a new form of the state equations not encountered in the
existing litterature.
The paper will present the properties of this new model in terms
of simulation, sta-
bility and control, and compares it to more conventional
state-space models. The
main difference relies in the fact that the model presented here
does not include
any load which are externeal to the generator. This should have
consequences for
simulation softwares.
1 State-variable representation of nonlinear systems
In this paper we present a recent formalism for selecting state
variables in dynamic
models of electromechanical systems. We illustrate the procedure
on the example of
a synchronous generator. The procedure is useful in control and
estimation of elec-
tromechanical systems, and if offers an alternative to the
conventional intuition-based
modeling process.
A state-variable representation of a nonlinear system generally
takes the form:
x = f(x,u,) (1a)
y = h(x,u,) (1b)
whereu is thecontrolvector, the disturbancevector,x the
statevector andy is theoutputvector. In some cases, f andh can
explicitly depend on time and, usually, theoutput equation (1b) is
only a function of the state and does not depend on unor on.
The inputin (1), which consist ofu and , can be thought as the
cause actingon the system. Mathematically the vector functions t
u(t)and t (t) must be
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specified or known in order to calculate or simulate a
trajectory of the underdetermined
system of equations (1).State variables (i.e. components of any
state vector) are abstract mathematical
quantities, useful to obtain the first-order representation (1)
when the input is known.
However, in most of the cases state variables are chosen among
the physical variables
representing the plant under study.
Any outputy can be thought as the effect of the operation of the
system. It isusually formed with the to-be-controlled variables.
Most of the time, its components
correspond to measured variables.
A recent framework of nonlinear control [2] resulted in a
moregeneral state-variable
representation in which input derivatives may appear:
x = f(x,u,, u, , . . .) (2a)
y = h(x,u,, u, , . . .) (2b)
In addition to the mathematical foundation of this type of
equations, some examples
of engineering systems have been found to belong to this class
of say generalized
systems (see [4] for the case of an overhead crane).
A careful discussion of choice of input and output in terms
ofcause and effect
about electrical machines leads to the conclusion that the motor
operation can be rep-
resented by (1) whilst the generator operation only admits
state-variable representation
of type (2). More precisely, natural outputs of the electrical
generators let appear input
derivatives. The paper will examine the obtained generalized
state model of a syn-
chronous generator in perspective of stability, simulation, and
control.
2 Introductory examples
The following two examples namely the DC and synchronous
permanent magnets
machines are use to introduce our point as they are quite simple
and easy to under-
stand. Thought this two machines are rarely used as generators,
it is interesting to look
at their state space models in this mode of operation. More
practical and complicated
cases can be analysis with the same tools.
2.1 The permanent magnets DC machine
The basic differential-algebraic equations that models the
behavior of a permanent
magnet DCmachine are:
J = KtI
Kf
Tex (3a)
U = LdI
dt + RI+ Kb (3b)
where = +1in motor operation and = 1in generator operation. The
variablesare (angular speed), I (current), U (voltage) and Tex
(external torque applied tothe shaft). The parameter areJ (momentum
of inertia), Kt (torque constant), Kf
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(friction coefficient),L (coefficient of auto-inductance),R
(resistance) andKb (back
emf constant).Notice this is a linear system and this is a
2-inputs system. The main reason is that
this is a system of 2 independent equations relating 4 variables
namely , Tex, IandU. So one needs to provide 2 time functions to be
able to express its solution orto simulate it. (see [3] for the
details about the notion and size of the input of a linear
system.)
Motor operation. In this situation, the natural control input
isu = (U) and thedisturbance input is = (Tex). Note that in this
case, Tex represents the load torqueundergone by the schaft.
Knowing the time functions t U(t) and t Tex(t)(specification of the
input), one needs 2 initial conditions to express the solutions of
(3).
(See also [3] for an intrinsic determination of the dimension of
the state.) A natural
choice of state is thusx = (I)t. The output is e.g.y = (),
leading to the classicalor Kalman state representation:
x =
Kf/J Kt/JKb/L R/L
x +
01/L
u +
1/J
0
y = (1 0) x
Generator operation. In this case, the controlinput is the
torque applied to the motor
u = (Tex), which here represents the torque applied to drive the
generator, and thedisturbance input is the electrical load of the
generator represented by the current =(I). (Remember that in this
case = 1.) For a given behavior oft Tex(t) andt I(t), one needs
only one initial condition to express the solution of (3). Thereis
only one state variable to pick and a natural choice is x = (). The
output is in
this mode of operation clearly the voltage produced by the
generator y = (U). Thecorresponding representation reads:
x = KfJ
x 1
Ju+
KtJ
y = Kbx+ R+ L
which let appear the disturbance (input) first derivative in the
output equation.
2.2 The permanent magnets synchronous machine
For the sake of simplicity consider a non salient pole machine
such that L= Ld = Lqwhose model in the DQframe reads:
= (4a)
J = npKmiq Kf Tex, (4b)
Ldid
dt = Rid+ npLiq+ vd (4c)
Ldiq
dt = Riq npLid npKm + vq (4d)
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where = 1 whether one considers motor or generator operation, id
and iq are
called respectively the direct and quadrature currents, vd and
vq are the direct andquadrature voltages.As this model contains 4
equations relating 7 variables, this in a 3 input system.
Motor operation. In this case, the input isu = (vd vq)t
(control) and = Tex(disturbance). This choice being done it is not
difficult that ones need 4 initial condition
to express the behavior of (4) in motor mode. The natural choice
of state variable is
x= ( id iq)t. The output can be for instancey = ()or y =
()according to thetype of control task is to be achieved.
So, the state representation is (= +1):
x1 = = (5a)
x2 = =
npKmJ iq
KfJ
1
JTex (5b)
x3 = did
dt =
R
Lid+ npiq+
1
Lvd (5c)
x4 = diq
dt =
R
Liq npid
npKmL
+ 1
Lvq (5d)
y = (5e)
which is of type (1).
Generator operation. In this mode of operation, the control
input is the torque ap-
plied to the machine u = (Tex) and the disturbance is
represented by the currents = (id iq)
t. Consequently one need only 2 initial condition to express the
behavior
of the generator mode, and a natural choice is x = ( ).
Obviously, the output isy= (vd vq).
The state variable representation is then (= 1):
x1 = = (6a)
x2 = = KfJ
+ 1
JTex
npKmJ
iq (6b)
y1 = vd = Rid npLiq+ Ldid
dt(6c)
y2 = vq =Riq+ npLid npKm+ Ldiq
dt(6d)
Note that the (disturbance) input derivatives appears in the
output equations.
3 State variable model of the synchronous generator
The final paper will expose the development of the state-space
model of the syn-
chronous generator (with field excitations and damping windings)
and its consequence
for simulation and control.
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A model of such a electrical generator can be expressed in an
appropriate frame by:
vd = rid+Lqiq+ kMQiQ Lddiddt
kMFdiFdt
kMDdiDdt
(7a)
vq = Ldid kMFiF kMDiD+ riq Lqdiqdt
kMQdiQdt
(7b)
vF = rFiF +kMFdiddt
+LFdiFdt
+MRdiDdt
(7c)
0 = rDiD kMDdiddt
MRdiFdt
LDdiDdt
(7d)
0 = rQiQ kMQdiQdt
LQdiQdt
(7e)
J = Cm CemD (7f)
Cem = 1/3 [(Lq Lq)idiq+ kMFiqiF+kMDiqiD kMQidiQ] (7g)
For the details we refer the reader to [1].
The natural choice of control input isu = (CmvF)t and the
disturbance is in this
case = (id iq)t. The behavior of the generator is modeled by a
4th order dynamicsand a natural choice of state is thus x = ( iF iD
iQ)
t. The output is obviously the
to-be-regulated windings voltagesy = (vd vq)t.The final paper
will discuss precisely the derivation of the state space model
which,
as the simpler previous examples, also involve disturbance
derivatives in the output
equations. Moreover, one will provide a precise discussion of
the consequences in
terms of simulation, control and stability of the generator. In
particular one will proceed
in a careful comparison with more conventional models, in which
the energy-storage
variables are the states.
The main difference of the approach adopted here with more
conventional works is
that we only consider the generator itself in the modeling
process and not the electricalload, which can be of various type.
The load must be of course taken into consideration
in simulation of the behavior of the generator but it is
preferable to include a model of
the load in a separated box. This is important in stand-alone
applications but also
in most of the simulation software that consider every part of a
complex ssytem as an
elementary box. To our point of view, the model of any generatro
does not need to
include any part of the load. The result of the paper should
have consequence in the
development of simulation softwares.
References
[1] P. M. Anderson and A. A. Fouad. Power Systems Control and
Stability. Iowa State
University Press, 1977.
[2] M. Fliess. Generalized controller canonical forms for linear
and nonlinear dynam-
ics. IEEE Trans. Automat. Control, 35:9941001, 1990.
[3] M. Fliess. Some basic structural properties of linear
generalized systems. Systems
Control Lett., 15:391396, 1990.
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[4] M. Fliess, J. L evine, and P. Rouchon. A generalized state
variable representation
for a simplified crane description. Internat. J. Control,
58:277283, 1991.
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