Although the system is well described in abc coordinates for the stator, in the flux equations (2.35) it can be noted that the inductance terms are time-varying, since some variables change along with θr, which is a function of time. Using a transformation on stator variables in a rotating reference comes as a solution to the problem. It reduces the complexity of the natural form abc. With this background, the transform will be presented and then applied to the established voltage and flux linkage equations.
2.3.1 Park’s dq0 Transformation
This subsection aims to introduce Park’s transformation [20]. The work ofRobert Park related to the dq0 coordinate system transform was well discussed and established as a reference in the modeling of electric machines [21,22,23]. However, due to the relevance
of this reference-frame changing in subsequent equations, some basis for this transformation and the author’s interpretation will be presented below.
For the matter of interpretation, let’s consider, at first, a three-phase sinusoidal vector fs. The following equivalences can be assured for balanced and equilibrated systems [24].
Just as can be noticed, fd q0 is a rotating reference-frame depending directly on its angular frequency. The magnitude of d and q axis components are obtained directly
Re n
|fabc|e−jωto=fd; (2.37) Imn|fabc|e−jωto=fq (2.38) while the zero-sequence component dimension is given by
1
3 fa+fb+fc=f0. (2.39)
The transformation matrix that makes this change in the coordinate system has one of its variants, as will be used later in the reference reflection from the stator to the rotor, as follows
Chapter 2. Synchronous Generator Modeling 36
and the inverse transformation is given by
For the application of the Td q0 transform in the machine system, the angular component θ=p2f dθr. Naturally, the dq0 conversion can be applied to any three-phase abc coordinate system, whereas there is particular interest in the use related to the stator voltage equation (2.3). Nevertheless, without regard to the resistive element of the circuit, the use of Td q0transform must be cautious, especially when applied to a derivative element.
To deal with this issue, [15] presents a matrix approach to derivate over the transform attending the trigonometric identities presented in Appendix A as follows:
fqd=fq −fd 0Ö (2.44)
Although it is well described and commonly used to solve this topic, a different mathematical perspective can be used in the deduction to endorse the already known results.
From (2.36) the following similarities present the issue, where the αβ reference-frame will aid the deduction.
d
dtfs= d
dt |fabc|e−jωt= d
dt (fα+j·fβ)·e−jωt (2.46) The above equalities can be expanded by the product rule, resulting in
d
By inspection and making the appropriate correlations it possible to verify the next equivalences.
For this reason, beyond the coordinate system, both axes (d and q) are influenced by each other, and it is said that they are dynamically coupled. The zero-axis maintains the coupling independence as can be verified in (2.45).
2.3.2 Flux linkage with dq0 components
Aiming to have the variables and inductance quantities in the rotor reference frame, the coming matrix equivalence applies this changing through Park’s transformation. In the matrix reference conversion, I4 is a 4x4 identity matrix.
2
From three-phase trigonometric identities of Aapplied to the inductance matrix in (2.50), it can be verified that the position varying variables are dependent on the difference between the rotor angle and an arbitrary angle (θr−θ) from theTd q0 transformation. At this point, it is convenient to establish another transformation between the two angles to adjust the coordinates to the same reference.
For this, formal consideration must be made. Although it may seem like a math-ematical refinement (θ could have been set to θr), the arbitrary angle θ could take a different value in another application of the transform. Therefore, not to confuse the readers of this thesis, it will be assumed that the angles are different. Therefore, Figure 2.2 presents both coordinate references. The first aligns with the machine’s field of rotation, while the other is the angular position of the transformation of (2.41).
fq
Figure 2.2 – Twodq reference frames transformation
As can be seen in Figure2.2, a transformation between two dq orientated frames might be dependent on the angular position and the rotation frequency. However, for ω=ωr the dependence of the rotation frequency turns out to be constant so that the frames
Chapter 2. Synchronous Generator Modeling 38
transformation is only relative to the angular difference. The following complex/space-phasor relationship exemplifies this dependence.
frd+jfrq= fd+jfqej(θr−θ−π/2)
,→∴ frd q=fd qej(θr−θ−π/2) (2.51) The transformation that leads to the angular difference (θr−θ−π/2)cancellation is given by the matrix transform in (2.52). It is interesting to verify that the transformation matrix Kr is orthogonal, that is, the matrix transpose is equal to its inverse KrÖ=Kr-1.
Thus, the complete inductance matrix with the angular difference transformation is achieve applying Kr to (2.50) in a similar way Td q0 was applied, as follows
2
where ther superscript denotes an variable in rotor reference frame. The following matrix structure presents the inductance terms on the complete flux linkage system.
ψrd q0
This section aims to get an expression of the electromagnetic torque (Te). The expression is deduced from the instantaneous power assumptions at the stator terminals, as well considering the new reference frame equivalences stated above. Initially, the stator three-phase instantaneous power must be declared as:
Pt =vÖabc siabc s, (2.55)
which can be also expressed in dq0 components that, under balance (v0=i0=0), results in the following
Pt=3
2 vdid+vqii+2 *0
v0i0 (2.56)
According to [16], the torque can be solved by considering the winding forces as the result of the cross product of the currents by the flux. In fact, from the stator voltage