The kinetic energy contained in wind can be converted into the electrical energy through the WECSs where the kinetic energy captured by the wind turbine blades rotates the drive-train shaft and consequently the coupled generator shaft that produces the electri-cal energy. In this Chapter, the wind energy conversion system components are ex-plained as the following: the aerodynamic characteristic of the wind turbine is shortly discussed, the model of the two-mass drive-train system is presented, and the three-phase model of the PMSG in the abc-frame is expressed, respectively. The model of grid-connected inverter, the transformer and the grid are out of the scope of this thesis since the focus is on modelling and control of the machine-side converter.
2.1 Wind turbine characteristics
Generally, the wind turbine operation can be characterized by the extracted mechanical power from the wind power πw which can be given as
πw= 12ΟΟR2π£w3, (2.1) where Ο denotes the air density, R represents the radius of the rotor blade and π£w is the
wind speed. Equation (2.1) reveals that the wind power is significantly sensitive to the wind speed. Moreover, as the wind power is proportional to the square of the wind turbine blade radius, one concludes that by doubling the radius the wind power can be four times that is the main reason why the use of large wind turbines are growing. The wind power expressed in (2.1) represents an ideal power that a WECS can capture. However, in practice, the power coefficient πΆp determines the amount of the mechanical power that can be obtained from the wind power. The power coefficient can be given as
πΆp(π, π½) = 1 πm
2ΟΟR2π£w3 , (2.2) where πm represents the mechanical power and power coefficient is shown as a function
of the tip speed ratio π and pitch angle π½. Therefore, the actual mechanical power that can be extracted by the wind turbine is expressed as
πm = 1
2πΆp(π, π½)ΟΟR2π£3. (2.3) The wind turbines characteristics can be differentiated by the power coefficient as
πΆp(π, π½) = π0(π1
πiβ π2π½ β π3) π
π4
πi , (2.4) 1
πi= 1
π+c5π½2β c6
π½3+1 , (2.5) where it depends on the aerodynamic characteristics of blades defined by constant
pa-rameters π0, β¦ β¦ π6 [21]. It can be seen from the power coefficient equation that the tip speed ratio is an important factor in determining the efficiency of wind turbine. The tip speed ratio can be defined to be the ratio between the speed of the turbine blades and the wind speed as
π = Rππ‘
π£w , (2.6) where πt is the turbine angular speed.
The blade pitch angle is basically used in mechanical controllers to regulate the captured power by the wind turbine through changing the alignment of the turbine blades with the wind. The pitch angle controllers are mostly utilized in the large wind turbine applications where their main responsibility is to keep the wind turbine during the gust condition. In this thesis, the operation of the pitch controller will not be discussed since the focus is on the operating points below the rated wind speed thus the pitch angle can be assumed to be zero.
By substituting (2.6) into (2.4) and (2.5), the coefficient power can be obtained as a func-tion of the turbine angular speed. Hence, the πmβ πt curve in which the characteristic of the turbine power is as a function of the turbine angular speed can be derived for the different wind speeds as illustrated in Figure 2.
Figure 2. π·πβ ππ curve for different wind speed [22]
Figure 3. ππβ ππ‘ curve for different wind speed [22]
Using the mechanical power presented in (2.3), the turbine shaft torque of a wind turbine can be expressed as
πm=πm
πt = 0.5πΆp(π,π½)ΟΟR
2π£w3
πt . (2.7) Like the πmβ πt curve, the turbine torque can be depicted as a function of the turbine
angular speed as shown in Figure 3 where it can be observed that the optimum torque at which the MPPT occurs is located in the right-hand side of the curve. It will be dis-cussed later that the stable region for the WT operation is the right-hand side of the πmβ πt curve.
2.2 Drive-train model
The different types of the drive-train system of WECSs can be described based on the speed of the generator that is used in the transmission system. The most common types are a high-speed generator coupled to a three-stage gearbox, a medium-speed genera-tor connected to a one/two-stage gearbox and a low-speed generagenera-tor directly coupled to the turbine shaft known as a gearless construction. Due to the downtime issue caused by the gearbox components and the high demand for maintenance, the direct-driven technology has become popular during the recent decade.
Depending on the focus of the research, the complexity of the drive-train modelling can be determined. It has been suggested in the literature [10] that the two-mass model of the drive-train is necessary for the WECSs stability studies in which the effect of shaft flexibility and consequently the drive-train torsional modes accurately captured. Moreo-ver, higher-order models have been proposed in the literature [23]. HoweMoreo-ver, as they are
commonly utilized for the mechanical fatigue of the drive-train studies which is out of scope of this thesis, a two mass model drive-train is considered in this work that can be presented as
π½tππt
ππ‘ = πt β πsπsβ π΅sπt+ π΅sπg , (2.8) π½gππg
ππ‘ = πsπs β πg+ π΅sπtβ π΅sπg , (2.9) ππs
ππ‘ = πtβ πg , (2.10) where, π½t and π½g are the inertia of the turbine and the generator, respectively. πg is the PMSG rotor speed, and πt is the speed of shaft. πs denotes the shaft angle, πΎs and π΅s respectively represent the shaft stiffness and damping coefficient. πt is the input mechan-ical torque applied on the wind turbine rotor and the electromagnetic torque developed inside the PMSG is shown by πg. Furthermore, the rotational frequency of the drive-train torsional mode in the two-mass model can be given as
πosc = βπΎs(π½t+ π½g)
π½tπ½g . (2.11)
2.3 Permanent-magnet synchronous generator model
An equivalent wye scheme of a PMSM model in abc-frame is derived according to [24]
where all voltages and currents are instantaneous values. Using the stator instantaneous voltage equations, the PMSG model may be expressed in matrix form as
π’abcs= π
ππ‘πabcsβ πsπabcs , (2.12) where πs is the stator winding resistance. The terminal stator voltage and phase-current vectors are π’abcs and πabcs, respectively. πabcs represent the stator flux linkages and
π
ππ‘ denotes derivate. As the wind energy is the study case in this thesis, the generator-mode convention is applied here; therefore, the direction of the stator current is chosen to be positive. The stator flux linkages can be given as
πabcs = πΏs(πr)πabcs+ πm,abcs(πr), (2.13) where
πm,abcs(πr) = [πmcosπr πmcos (πrβ 2Ο
3) πmcos (πr+ 2Ο
3)]T. (2.14) Here, πm,abcs(πr) is the flux linkage vector of the permanent magnets and πm is the maxi-mum flux linkage produced by the rotor magnet. πr is the rotor angle which defines the
angle between magnetic axis of the rotor and the stator phase π winding. πΏs(πr) denotes the inductance matrix of the generator stator windings and can be expressed as
πΏs(πr) = [
πΏs πs πs
πs πΏs πs
πs πs πΏs
] β πΏre [
cos2(πr) cos2 (πrβ Ο3) cos2 (πrβ 2Ο3) cos2 (πrβ Ο
3) cos2 (πrβ 2Ο
3) cos2 (πrβ 3Ο
3) cos2 (πrβ 2Ο
3) cos2 (πrβ 3Ο
3) cos2 (πrβ 4Ο
3)]
. (2.15)
The stator self-inductance and the mutual inductance are denoted by πΏs and πs, respec-tively. πΏre represents the reluctance effect which depends on the rotor position. Substi-tuting (2.15) to (2.12), the abc-frame model of the PMSG can be presented as
π’abcs= π
ππ‘πm,abcs(πr) β πsπabcsβ π
ππ‘(πΏs(πr)πabcs), (2.16) where the first term represents the back-electromotive force (EMF) vector, induced by the magnet flux in the stator winding. The abc-frame three-phase model of the PMSG derived in this section will be used in the next Chapter for the small-signal modelling of the PMSG where it is connected to the DC-link through the AC/DC converter.