1.2 Thesis Outline
2.1.3 Wind Turbine Model
and the stator currents id[p.u.] and iq[p.u.] are obtained via simple nodal analysis of the synchronous generator equivalent circuit (see Fig. 2.4), yielding the following algebraic equations:
0 =vq,j+ra,jiq,j+Xd,j0 id,j−Eq,j0 , 0 =vd,j+ra,jid,j−Xq,jiq,j.
(2.14)
Finally, with the knowledge of the currents and voltages within the armature windings, the active and reactive power can be computed as
0 =Pe,j−vd,jid,j−vq,jiq,j, 0 =Qe,j−vq,jid,j+vd,jiq,j.
(2.15)
2.1.3 Wind Turbine Model
As mentioned earlier, there is an ongoing trend towards more environmentally-driven production of electricity, such as wind and solar generation, in the hope of reducing the CO2emissions and contributing effectively against global warming. Wind turbines are generally the most common wind energy conversion system (WECS) converting the wind kinetic energy into electricity. In contrast to conventional power plants, wind turbines mostly employ asynchronous (induction) generators where the rotor speed is no longer synchronized with the magnetic field of the stator. This makes the induction generators attractive for wind generating stations since they are capable of producing power at varying rotor speeds compared to synchronous generators. Wind turbines are divided into four different type; the fixed speed Type-1, limited variable speed Type-2, or variable speed with either partial or full power electronic conversion, Type-3 and Type-4, correspondingly.
In this thesis we employ the Type-3 wind turbine using the doubly-fed induction generator as the WECS, which is a generating principle widely used in wind turbines [27, 96, 104, 108]. The operating principle is illustrated in Fig. 2.5. The system consists of three main elements; namely, the drive train, the generator, and the back-to-back converter. The drive train has its the low speed shaft facing the wind, and it is
responsible of making the high speed shaft turn approximately between 40 to 50 times faster than the low speed shaft depending on the manufacturer. This in turn rotates the rotor of the induction generator coupled with the high speed shaft. The generating unit is a wound-rotor induction generator, where the stator side is connected directly to the point of common coupling (PCC) prior to the connection with the utility grid, and the rotor side is connected to the PCC via special power converters. The power converter is commonly the back-to-back voltage source converter, which is comprised of two separate bi-directional converters coupled via a DC link. This makes it possible to control the rotor speed where the rotor frequency can freely differ from the frequency of the utility grid. Furthermore, using this topology, one may adjust the rotor currents, which in turn indirectly specifies the active and reactive power fed to the grid from the stator, independently of the rotor turning speed [96, Ch. 20].
Stator Rotor
Gearbox Rotor side
converter Grid side
converter
3 ~
=
Wind
Coupling transformer
3 ~
=
DC-link
Figure 2.5: Schematic diagram of a DFIG-based wind generation system at thei-th bus.
Equivalent circuit
The WECS based on the doubly-fed induction generator can be described as a third order model including dynamics of the drive train, the asynchronous generator, and the power converter [51]. In this modelling framework, the doubly-fed induction generator is equivalent to the electrical circuit shown in Fig. 2.4.
Here, the circuit is supplied via the PCC, and the winding included in the stator and the rotor are aggregated via a constant impedance. The machine parameters and their meanings are described in Table2.2.
One can notice from the equivalent circuit that the rotor voltage depends on a new variablesassociated with the so-called slip ratio defined as
sj= ωs,j−ωr,j ωs,j
, (2.16)
Table 2.2: Machine parameters of the doubly-fed induction generator
Variable Description Unit
Hw Sum of turbine and rotor inertia constant [MWs/MVA]
Xν Magnetizing reactance [p.u.]
Xs Stator reactance [p.u.]
Xr Rotor reactance [p.u.]
rs Stator resistance [p.u.]
rr Rotor resistance [p.u.]
withω[p.u.] being the rotational speed. The subscriptss, r, andj corresponds to the stator, the rotor, and the j-th machine, respectively. As stated earlier, the key distinction of asynchronous generators compared to synchronous machines, is that the magnetic field of the rotor is no longer synchronized with that of the stator. In fact, the slip is a very important parameter in the electrical circuit, because it relates how fast the rotor is spinning with the electrical side. Without the slip, the equivalent circuit of the doubly-fed induction generator becomes identical to a transformer circuit, which is a motionless device simply varying the voltage levels from the stator to the rotor.
Point of common coupling (PCC)
Rotor side power converter
Grid power
converter DC-link
Doubly-fed induction generator
3-phase AC grid side
Stator reactance Rotor reactance
Magnetizing reactance
Figure 2.6: Equivalent circuit of the DFIG of the wind energy conversion system.
Differential Equations
Prior to introducing the differential equations governing the mathematical model, some assumptions concerning the doubly-fed induction generator are made. The DC/AC converter on the grid side is assumed to operate loss-less and completely synchronized with the grid, hence the active power flowing in the back-to-back converter is equal and the reactive power of the DC/AC converter is zero. Furthermore, the transient behaviour associated with the stator flux is neglected, i.e. ˙ψs
= 0, due to the fact that the!
wind turbine is connected through the stator to the grid, which is modeled by algebraic variables via the power flow equations (2.8).
With these basic assumptions, the WECS based on the doubly-fed induction generator can be described via the following third order model [96, Ch. 20]
˙
ωr,j= 1 2Hw,j
(Tm,j−Te,j), (2.17)
ψ˙r,d,j =vr,d,j +rrir,d,j+sjωs,jψr,q,j, ψ˙r,q,j=vr,q,j+rrir,q,j−sjωs,jψr,d,j,
(2.18)
whereψr[p.u.] is the rotor flux. The system inputs are the rotor voltagevr[p.u.] and the torqueTm[p.u.].
Here the subscripts j, m, and e are corresponds to the j-th machine, the mechanical and electrical components, respectively, and dandqdenote the d- and q-axes, associated with Park’s transformation.
The differential equation (2.17) considers the electromechanical oscillations of the system via the so-called swing equation and the equations (2.18) handle modelling of the machine magnetic flux on the rotor side.
Algebraic Equations
Similarly to the model of the synchronous generator, the remaining variables are obtained by solving a set of algebraic equations; first the electrical torqueTeis obtained via:
0 =Te,j−Xν,j(ir,q,jis,d,j−ir,d,jis,q,j), (2.19) where i[p.u.] is the current. The stator currents are obtained by applying the nodal analysis on the equivalent circuit (see Fig. 2.6) yielding the following equations
0 =vs,d,j+rs,jis,d,j+ωs,jψs,q,j, 0 =vs,q,j+rs,jis,q,j+ωs,jψs,d,j,
(2.20)
withvs[p.u.] as the stator voltage computed based on the voltages levels at their correspondingh-th bus 0 =vs,d,j−Vhsin(θh),
0 =vs,q,j−Vhcos(θh).
(2.21)
It still remains to compute the rotor currentsir[p.u.] and the stator flux ψs[p.u.]. The aforementioned variables are obtained by solving the following set of equations:
0 =ψr,d,j+ (Xr,jir,d,j+Xν,jis,d,j), 0 =ψr,q,j+ (Xr,jir,q,j+Xν,jis,q,j), 0 =ψs,q,j+ (Xs,jis,q,j+Xν,jir,d,j), 0 =ψs,d,j+ (Xs,jis,q,j+Xν,jir,q,j),
(2.22)
and with the knowledge of the currents and voltages within the windings of the stator and the rotor, the active and reactive power can be computed according to:
0 =Pe,j−vs,d,jis,d,j−vs,q,jis,q,j−vr,d,jir,d,j−vr,q,jir,q,j, 0 =Qe,j−vs,q,jid,j+vr,d,jiq,j.
(2.23)