Учебники / 0841558_16EA1_federico_milano_power_system_modelling_and_scripting

The Anderson-Fouad’s model, which, apart from [10], is also reported by the majority of books on power systems, e.g., [14] and [179], is as follows:

and (15.15). The Anderson-Fouad’s model can be considered a simplification of the Sauer-Pai’s model. In fact, the Sauer-Pai’s model leads to the Anderson-Fouad’s one by defining:

The Anderson-Fouad’s model can be also derived from the Marconato’s model by assuming:

Fig. 15.2 Block diagram of stator fluxes for the Marconato’s model of the synchronous machine

15.1.6Simplified Magnetic Equations

There are a variety of possible simplifications for the magnetic equations presented above. Following paragraphs only show some models presented in the literature.

Two d- and One q-Axis Model

Assuming Tq0 ≈ 0 and xq ≈ xq in (15.13) leads to ed ≈ 0. Hence, (15.13) can be rewritten as:

e˙q = (−eq − (xd − xd)(id − γd2ψd − (1 − γd1)id + γd2eq ) + vf )/Td0

A second type of two d- and one q-axis model, can be obtained from (15.16) assuming only one additional circuit on the q-axis (e.g., Tq0 ≈ 0) [184]. The resulting model has three magnetic state variables eq , eq and ed as follows:

Similar equations can be obtained using (15.19) or imposing γd ≈ TAA ≈ 0 in (15.24).

One d- and Two q-Axis Model

Another model presented in [184] assumes:

Similar equations can be obtained using (15.19) or imposing γq ≈ 0 in (15.27).

One d- and One q-Axis Model

In this model, lead-lag transfer functions are used for modelling the d- and q-axis inductances (e.g., Td0 ≈ Tq0 ≈ 0). The resulting magnetic equations have only two state variables, namely eq and ed, as follows:

This model can be obtained indi erently from any of the complete models (15.13), (15.16) and (15.19). This two-axis model is the highest order model on which there is substantial agreement in the literature. Actually, this model is the most commonly used in power system stability analysis because it provides the right compromise between simplicity and accuracy.

A similar fourth order model can be formulated using the sub-transient d- axis voltage ed instead of ed (e.g., Td0 ≈ Tq0 ≈ 0). The di erential equations becomes:

and the algebraic equations:

One d-Axis Model

A common model used in transient stability consists in neglecting all q-axis electro-magnetic circuits, and using a lead-lag transfer function for the d- axis inductance [325, 330]. The only magnetic state variable is eq , with the following di erential equation:

and with the algebraic equations:

337

Fig. 15.4 Comparison of synchronous machine models of di erent types

where the field flux ψf is:

which leads to rewrite (15.39) as:

Clearly, this model has no practical applications since is too detailed and too simplified at the same time. It is too detailed because it includes stator flux dynamics, thus requiring a comparatively small step length in numerical integration, and it is too simplified because it considers only one dynamic on the d-axis, thus resulting inadequate for detailed transient stability analysis.

15.1.8Saturation

As discussed above, the variety of magnetic equations allows defining a huge variety of machine models. Taking into account saturation introduces even more arbitrariness in the formulation of the synchronous machine model.

340 15 Alternate-Current Machines

A general model that accounts for saturation is a generalization of the

0= ψd + xd id − γd1eq − (1 − γd1)ψd − ςd(z)

0= ψq + xq iq + γq1ed − (1 − γq1)ψq − ςq (z)

where z = [eq , ed, ψd , ψq , id, iq ]T and the saturation functions are:

ςf (z): saturation of the dc field winding.

ςd(z) and ςq (z): d- and q-axis saturation of the stator windings. ςq1(z): saturation of current induced in the rotor core.

ςd1(z) and ςq2(z): saturation of the equivalent damper windings.

Instead of focusing on some particular model that includes saturation, following subsections discuss two examples of saturation functions and the data required for defining such saturation. Once the functions are defined, the inclusion of a saturation function is straightforward using (15.42). For simplicity, in the remainder, it is considered a generic saturation function ψ = ς(i) that links a generic current i to a generic flux ψ.

Piece-Wise Saturation Function

A piece-wise saturation model is discussed in [163] and is shown in Figure 15.5. The saturation function consists of three regions, as follows:

where xr = x1/x2. The curve is completely defined by the parameters ψA, ψB , ψC , xr , As and Bs. The main issue of this model is that for iA = ψA/x1, the function is discontinuous, since for i−A , the function returns x1iA and for i+A, the function returns x1iAψA/(ψA + As). This discontinuity is immaterial only if As ψA.