Tld Mlds

where

Lld Rld the rotor time constants for the direct-axis, cr = 1 —

Lf L

f Lld the leakage factors, and

LdLld

MldMr LldMf f

Furthermore, it is clear from Eq. (3.97) that the direct-axis stator flux is frequency dependent, and that its DC-gain is equal to Ld (for Uf =0), or equal to kMf /Rf (for id = 0). In the sequel, these DC-gains are referred to as , and -GfdC.

However, Eq. (3.97) is valid only for the situation that the (fictitious) damper winding can be adequately represented by one winding located on the direct-axis. Recall that the number of damper windings depends on both the physical construction of the rotor and the accuracy required by the model. More general, the direct-axis stator flux equation is described by

where Ld(s) and Gfd(s) transfer functions. The order of the numerator and denominator polynomials of Ld(s) are equal to the number of damper windings located on the d-axis plus the field winding, while Gfd(s) has the same denominator as Ld(s), but a different numerator. The order of the numerator of Gfd(s) is one less than the denominator. Both transfer functions depend on the design of the synchronous generator and Ld(s) is usually referred to as "operational inductance".

In addition, it is shown in Appendix D that

and that

where Ldo(s), Lfdo(s) = Ldfrj(s), and Lfo(s) are proper transfer functions, which depend on the design of the synchronous generator.

Quadrature-axis. The equation for the voltage across the quadrature-axis damper is given by

0 = -RiqIiq - s^1q The flux linkage equations on the quadrature-axis are given by

Rearranging the latter two expressions and substituting the results in the voltage equation allows elimination of I1q and ^1q, resulting in the following equation for the quadrature-axis stator flux

where

R1q the rotor time constant for the quadrature-axis, and k 2M2q a1q =1 - TT1

LqL1q the leakage factor. Furthermore, it is clear from Eq. (3.101) that the quadrature-axis stator flux is frequency dependent, and that its DC-gain is equal to Lq. In the sequel, the DC-gain is referred to as L.

However, Eq. (3.101) is valid only for the situation that the (fictitious) damper winding can be adequately represented by one winding located on the quadrature-axis. More general, the quadrature-axis stator flux equation is described by

where Lq(s) is a proper transfer function, which function depends on the design of the synchronous generator. The order of the numerator and denominator polynomials of Lq(s) are equal to the number of damper windings located on the q-axis.

The voltage equations in the dq reference frame of the electromagnetic part of a synchronous generator have been given in matrix form in Eq. (3.93). Removing the zero-sequence equation gives the following set of equations

Ud f

with ud the direct-axis voltage, Rs the stator-winding resistance, id the direct-axis current, pum the generator speed, *q the quadrature-axis winding flux, t time, *d the direct-axis winding flux, uq the quadrature-axis voltage, iq the quadrature-axis current, U' the field-winding voltage, R' the field-winding resistance, f the field-winding current, and the field-winding flux.

A few observations can be made, the most important one being that equations (3.103) are coupled via the fluxes. In addition, they depend on the generator speed pwm, thereby introducing non-linearities. For time-domain simulation purposes, it is convenient to rewrite the this set of equations in the following form

*d = -J (Ud + Rsid + P^m*q ) dt *q = -J (Uq + Rsiq - P^m*d) dt = jiuf - R'i' ) dt

with the fluxes as state variables.

The fluxes, in turn, have been given by

with s the Laplace operator and Ldo(s), Lfdo(s) = Ldfa(s), Lq(s), L'o(s) proper transfer functions which can, for a finite number of damper windings, be expressed as a ratio of polynomials in s [147]. The direct-axis flux equations can be conveniently expressed in matrix form

It can be easily shown that the inverse transformation is given by

" *d(s) '

. (s) ,

" Id(s) '

It is shown in Appendix D that the denominators of Ldo(s), Ldfrj(s) = Lfdo(s), and Lfo(s) are all the same. Consequently, the denominators in the above matrix equation are identical.

Finally, the inverse of the quadrature-axis flux equation is given by

The dynamic behavior of an ideal synchronous generator is thus fully described by the sets of equations (3.104), (3.107) and (3.108) expressed in the dq reference frame. These equations are implemented in DAWIDUM's Elec2 module (see Appendix I.2.4). The resulting block diagram is depicted in Fig. 3.26.

Figure 3.26: Block diagram of an ideal synchronous machine.

Obviously, for simulation as well as control design purposes, accurate information about the transfer functions Ldo(s), Lfdo(s), Lq(s) and Lfo(s), as well as the resistances Rs and Rf, is required. In Section 4.3 a new procedure is developed for identifying the transfer functions Yd(s) and Yq(s) (see Fig. 3.26) of Park's dq-axis model of a synchronous generator from time-domain standstill test data. To complete the analysis, relations for power and torque are needed.

Electromagnetic torque

The instantaneous power output of a three-phase synchronous generator is simply the sum of the stator ui products

It can be shown that by applying Park's power-invariant transformation to Eq. (3.109), the following power output equation expressed in odq coordinates can be obtained

Assuming that the generator is star connected with the star point not used, this equation reduces to

Substitution of ud and uq from Eq. (3.93) in Eq. (3.111) gives

(~RS • id - P^m^q - d?) • id + (—RS • iq + P^m^d - ^^ • iq

- Rs {ij + iq) - ^ddtd id + ddtL iq^j + P^m (^d • iq - ^q • id) (3.112)

The first term in Eq. (3.112) describes the power dissipated in the stator windings (the so-called stator copper losses). The second term corresponds to the time rate of change of the magnetic energy stored in the inductances of the generator, and the third term reflects the power transferred across the airgap [211]. This power is equal to the electromechanical power developed, hence

Dividing the electromechanical power by the mechanical speed of the generator shaft (wm), the following expression is obtained for the instantaneous electromechanical torque developed by a synchronous generator with P pole-pairs

The value of Tem from the above expression is negative for motoring, and positive for generator operation.

Furthermore, it is assumed that the generated electrical power is equal to the generator power output minus the converter losses, the electric excitation losses (the so-called rotor copper losses) and the cable losses

Pelec Pg Pconv Pcu Pcable (3.115)

The converter losses include the switching losses in the power electronic interface and are, here, divided into no-load losses (which are constant) and losses which are proportional to the stator current squared

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Renewable Energy 101

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