On the state space representation of synchronous generators Emmanuel Delaleau Universit´e Paris-sud Laboratoire des signaux et syst`emes CNRS – Sup´elec 3 rue Joliot-Curie 91 192 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 resulting 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, stability 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 electromechanical 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
(1b)
= h(x, u, $)
where u is the control vector, $ the disturbance vector, x the state vector and y is the output vector. In some cases, f and h can explicitly depend on time and, usually, the output equation (1b) is only a function of the state and does not depend on u nor on $. The input in (1), which consist of u and $, can be thought as the “cause” acting on the system. Mathematically the vector functions t 7→ u(t) and t 7→ $(t) must be 1
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 output y can be thought as the “effect” of the operation of the system. It is usually 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 more general state-variable representation in which input derivatives may appear: x˙ = f (x, u, $, u, ˙ $, ˙ . . .) y = h(x, u, $, u, ˙ $, ˙ . . .)
(2a) (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 of cause and effect about electrical machines leads to the conclusion that the motor operation can be represented 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 synchronous 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 understand. 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 DC machine are: J Ω˙ = Kt I − Kf Ω − Tex dI U = L + RI + Kb Ω dt
(3a) (3b)
where = +1 in motor operation and = −1 in generator operation. The variables are Ω (angular speed), I (current), U (voltage) and Tex (external torque applied to the shaft). The parameter are J (momentum of inertia), Kt (torque constant), Kf
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(friction coefficient), L (coefficient of auto-inductance), R (resistance) and K b (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 Ω, T ex , I and U . So one needs to provide 2 time functions to be able to express its solution or to 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 is u = (U ) and the disturbance input is $ = (Tex ). Note that in this case, Tex represents the load torque undergone by the schaft. Knowing the time functions t 7→ U (t) and t 7→ T ex (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 thus x = (Ω I)t . The output is e.g. y = (Ω), leading to the classical or Kalman state representation: −Kf /J Kt /J 0 −1/J x˙ = x+ u+ $ −Kb /L R/L 1/L 0 y
= (1 0) x
Generator operation. In this case, the control input is the torque applied to the motor u = (Tex ), which here represents the torque applied to drive the generator, and the disturbance input is the electrical load of the generator represented by the current $ = (I). (Remember that in this case = −1.) For a given behavior of t 7→ T ex (t) and t 7→ I(t), one needs only one initial condition to express the solution of (3). There is only one state variable to pick and a natural choice is x = (Ω). The output is in this mode of operation clearly the voltage produced by the generator y = (U ). The corresponding representation reads: 1 Kt Kf x− u+ $ J J J = Kb x + R$ + L$ ˙
x˙ = − y
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 = L d = Lq whose model in the DQ frame reads: θ˙ = Ω ˙ = np Km iq − Kf Ω − Tex , JΩ did L = −Rid + np LΩiq + vd dt diq L = −Riq − np LΩid − np Km Ω + vq dt 3
(4a) (4b) (4c) (4d)
where = ±1 whether one considers motor or generator operation, i d and iq are called respectively the “direct” and “quadrature” currents, v d and vq are the direct and quadrature voltages. As this model contains 4 equations relating 7 variables, this in a 3 input system. Motor operation. In this case, the input is u = (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 = (θ Ω id iq )t . The output can be for instance y = (θ) or y = (Ω) according to the type of control task is to be achieved. So, the state representation is ( = +1): x˙ 1 x˙ 2 x˙ 3 x˙ 4 y
= θ˙ = Ω np Km Kf = Ω˙ = iq − ω− J J R did = − id + np ωiq + = dt L R diq = − iq − np ωid − = dt L = θ
(5a) 1 Tex J 1 vd L np Km 1 ω + vq L L
(5b) (5c) (5d) (5e)
which is of type (1). Generator operation. In this mode of operation, the control input is the torque applied 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 = (θ Ω). Obviously, the output is y = (vd vq ). The state variable representation is then ( = −1): x˙ 1 x˙ 2 y1 y2
= θ˙ = ω
(6a)
Kf 1 np Km = ω˙ = − ω + Tex − iq J J J did = vd = Rid − np Lωiq + L dt = vq = Riq + np Lωid − np Km ω + L
(6b) (6c) diq 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 synchronous generator (with field excitations and damping windings) and its consequence for simulation and control. 4
A model of such a electrical generator can be expressed in an appropriate frame by: vd
=
vq
=
vF
=
0
=
0
=
J ω˙
=
diF diD did − kMF − kMD dt dt dt diq diQ −ωLd id − ωkMF iF − ωkMD iD + ri q − Lq − kMQ dt dt did diF diD −rF iF + kMF + LF + MR dt dt dt diF diD did − MR − LD rD iD − kMD dt dt dt diQ diQ rQ iQ − kMQ − LQ dt dt Cm − Cem − Dω
Cem
=
1/3 [(Lq − Lq )id iq + kMF iq iF + kMD iq iD − kMQ id iQ ]
rid + ωLq iq + ωkMQ iQ − Ld
(7a) (7b) (7c) (7d) (7e) (7f) (7g)
For the details we refer the reader to [1]. The natural choice of control input is u = (Cm vF )t and the disturbance is in this case $ = (id iq )t . The behavior of the generator is modeled by a 4th order dynamics and a natural choice of state is thus x = (ω iF iD iQ )t . The output is obviously the to-be-regulated windings voltages y = (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 electrical load, 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 dynamics. IEEE Trans. Automat. Control, 35:994–1001, 1990. [3] M. Fliess. Some basic structural properties of linear generalized systems. Systems Control Lett., 15:391–396, 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, 58:277–283, 1991.
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