Energy2green Wind And Solar Power System
4.4.1 Construction and Properties
Asynchronous or induction machines operating as motors are the most widely used electromechanical converters. In an induction machine the stator is identical to the one for synchronous machines shown in Figure 4.3 in which three-phase currents supplied to the stator produce a rotating magnetic field (RMF). The rotor, however, is radically different and it has neither an external magnetizing source nor permanent magnets. Instead, alternating currents are injected in the rotor from the stator through induction or transformer action - hence the useful parallel with the operation of a transformer. It is the interaction between these induced rotor currents and the stator RMF that results in torque production.
In its most common form, the rotor consists of axial conductors shorted at the ends by circular rings to form a squirrel-cage orjust cage- as shown in Figure 4.12. Although for the purposes of renewable energy sources there is interest in the generation mode, it is easier initially to understand the operation of the induction machine from the motoring perspective.
As the stator RMF moves at os (given by Equation (4.4)) past the stationary rotor conductors, three-phase electromotive forces (EMFs) are induced in the spatially shifted rotor conductors by a flux cutting action. The resulting rotor currents, according to Lenz's law [1], are of such magnitude and direction as to generate a torque that speeds up the rotor. If the rotor were to achieve speed os, there would be no change in flux linkage, no induced voltage, no current in the rotor conductors and therefore no torque. For EMFs to be induced in the rotor conductors they should possess some relative speed with respect to the stator RMF. For motoring, the rotor therefore turns at a lower speed (
It can be shown that the rotor currents produce an RMF whose speed depends on the frequency of these currents. For a constant torque interaction to take place, the rotor RMF must rotate in synchronism with the stator RMF, as in the case of the synchronous machine. How is this accomplished if the rotor rotates at a lower speed than os?
The difference between cos and cot is expressed as a ratio with respect to cos and is known as slip s where s = (4.15)
Therefore
The relative motion between the stator's and rotor's RMFs determines how frequently the stator RMF cuts the rotating rotor conductors, so the frequency of the rotor induced voltages and currents fr is f = sf (4.17)
where f is the mains frequency. The frequency of the rotor currents determines the speed of the rotor RMF with respect to the rotor:
The speed of the rotor RMF with respect to the stationary stator is the rotor speed plus the rotor RMF's speed with respect to the rotor:
It can be concluded that the rotor and stator RMFs rotate together at synchronous speed as required for a uniform torque to be developed as in the synchronous machine. However, in contrast to the synchronous machine, the rotor RMF is produced through induction from the stator. The larger the applied braking torque the higher the slip, the larger are the induced EMFs and resulting rotor currents, and the stronger the interaction between the two RMFs to produce an electrical torque equal and opposite to the braking torque. The induction motor therefore exhibits a small decrease in speed with increments in braking torque.
An ideal induction machine could be imagined to operate at zero slip. This is equivalent to the 'idling' state of the fixed speed synchronous machine. The vital difference, however, in the induction machine is that motoring or generating torques will be accompanied by a decrease or increase of speed below or above synchronous respectively. For generating, Equation (4.16) now gives a negative slip. The induction machine will move seamlessly from the motoring into the generation mode as the external torque changes from a decelerating to an accelerating type. Indeed, in small wind turbines, it is very common to find that the induction generator was originally designed as a motor and has been employed as a generator without any modification.
A six-pole 50 Hz induction motor runs at 4% slip at a certain load. Calculate the synchronous speed, the rotor speed, the frequency of the rotor currents, the speed of the rotor RMF with respect to the rotor and the speed of the rotor RMF with respect to the stator.
Model answer
The synchronous speed from Equation (4.4) is Ns = f/p = 50/3 rev/s = 50 x 60/3 = 1000rev/min
The rotor speed from Equation (4.16) is (1 - s)Ns = (1 - 0.04) x 1000 = 960rev/s The frequency of the rotor currents are: fr = sf = 0.04 x 50 = 2 Hz The speed of the rotor RMF with respect to the rotor:
The speed of the rotor RMF with respect to the stator:
(i.e. the rotor and stator RMFs rotate together)
The induction machine can be viewed as a transformer with a rotating secondary. Imagine an induction machine with its rotor mechanically locked, i.e. at standstill. The stator RMF will be rotating at cos with respect to the rotor and inducing in each phase the voltage E2 at mains frequency f. The current that flows in each phase will be h =-E-= —E (4.19)
where R2 and L2 are the effective per-phase resistance and inductance of the rotor winding and X2 is the rotor reactance at mains frequency. At standstill the slip s = 1 and the rotor voltages and currents are of the stator frequency f. At any other rotor speed, the slip is s, the
I2 R2 jX2
a e2
Figure 4.13 Equivalent circuit of induction machine rotor
R1 X1 Air gap
Ideal transformer Figure 4.14 Induction machine stator-rotor equivalent circuit induced voltage is sE2. the rotor frequency is sf and the rotor reactance is sX2. For the rotor current at slip s we can write the more general expression:
Equation (4.20) provides the rational for the equivalent circuit shown in Figure 4.13. This circuit resembles that of the secondary winding of the transformer (Figure 4.10) but with a variable resistive load connected to its output. The energy conservation principle indicates that the electrical power transferred to the rotor is the real part of E21*. The power lost irreversibly in the rotor ohmic resistance is R2 /| and the remainder, i.e. /| R2[(1 - s)/s], must be and indeed is the electrical power converted into mechanical power.
The transformer equivalent circuit analogy is extended in Figure 4.14 to include the stator parameters. Here R1 and X1 represent the stator winding resistance and inductance while Xm represents the magnetizing reactance drawing the current necessary to establish the RMF. The dashed line corresponds to the air gap interface across which energy is transferred from the stator to the rotor.
In a further simplification the ideal transformer can be omitted by transferring elements from secondary to primary using the transformation ratio. In Figure 4.15 Rs and Xs are the stator winding resistance and reactance respectively. The elements Rr and Xr represent the rotor resistance and reactance respectively referred to the stator using the rotor-stator transformation ratio. The product Rr[(1 - s)/s]/s2 = Rem represents the electrical power per-phase converted into mechanical power. This equivalent circuit tells us that when the rotor is locked, s = 1, Rr[(1 - s)/s] = 0, so all the input to the rotor is converted into heat in Rr. When s < 1, the energy into the rotor is partly converted into heat in the winding resistances and partly into mechanical form.
R2 X2
Rs gap
Rr jXr
Figure 4.15 Induction machine equivalent circuit referred to the stator
With the induction machine generating, the slip is negative and the notional resistance Rr[(1 - s)/s] is also negative. This is perfectly consistent with circuit analysis. A positive RI2 implies irreversible conversion of electrical energy into thermal energy. A negative RI2 implies the conversion of some other type of energy (in this case mechanical) into electrical.
I rrespective of whether the machine is motoring or generating the mains to which the machine is connected supplies the reactive voltamperes absorbed by all the inductive components of the equivalent circuit. This must be so as only positive or negative real power is associated with the mechanical/electrical energy conversion in the resistance Rr(1 - s)/s. The consequence is that induction generators always absorb reactive power from the mains.
If the total electrical input power per phase fed into the stator is Ps, the power crossing the motor's air gap i.e. the power per phase transferred from the stator to the rotor is Pr = Ps - RsIs2. All of Pr is dissipated in {Rr + Rr[(1 - s)/ s]} = Rr/s so, Pr = (Rj s)Ir2. Hence the rotor copper loss is:
Subtracting the rotor copper loss from Pr gives the average per-phase mechanical power Pm = Pr - Rr Ir2 which through substitution from Equation (4.21) gives
The developed torque Qm of the motor is its total mechanical power 3Pm divided by the motor shaft speed ®r. Therefore
Substituting Equations (4.16) and (4.22) into (4.23) gives
(1 - s)(0s ®s where Pag = 3Pt is the total three-phase power crossing the air gap.
Neglecting the stator copper losses and rotating mechanical losses, the efficiency of an induction motor is given approximately by:
Pin Pr Pr
For an induction generator (s negative) the power flow is in the reverse direction hence approximately
Equations (4.25) and (4.26) indicate that for the conversion efficiency to be high, s at full load must be as small as possible.
Real induction generators have losses that have not been taken into account in this simplified analysis. The mechanical power available to produce electricity is reduced by windage and other mechanical frictional losses within the generator. Additionally, electrical and magnetic losses within the rotor reduce the power that is transferred from the rotor across the air gap to the stator. Finally, in the stator there are more losses associated with the winding resistance and the setting up of the magnetic excitation in the shunt branch of the equivalent circuit. As a consequence, large induction generators have efficiencies in the region of 90%. These extra losses will be referred to in a later section.
4.4.4 The Induction Machine Speed-Torque Characteristic
An important characteristic of any electromechanical converter is its speed-torque relationship. For the induction machine the developed torque Qm from Equations (4.23), (4.22) and (4.16) is
Substituting for
and using the equivalent circuit of Figure 4.15 we get
This relationship is nonlinear and can be generalized to describe typical performances of an induction machine by normalizing it in terms of torque and speed. Taking as normal torque the rated torque and as normal speed the synchronous speed, the normalized relationship of Equation (4.27) for a typical induction generator is plotted in Figure 4.16 with Rr as a parameter.
Note that, as for the synchronous machine, there is a maximum or pullout torque beyond which the generator will accelerate uncontrollably. However this condition is far away from the normal operating regime. The curve for rotor resistance Rr' represents the performance of a typical induction machine with low rotor resistance and shows that the variation in speed from zero input torque to rated torque varies by about 3-4%. For s small, Rr/s is large compared to Rr and Xs + Xr and to a good approximation Equation (4.27) can be written as
Equation (4.28) indicates that in the normal operating range (zero to rated torque), torque is directly proportional to slip and therefore speed and is inversely proportional to Rr. Curves for Rr' > Rr and Rr» > Rr are also plotted on Figure 4.16. By selecting the value of the rotor resistance, a designer has the ability to change the slope of the torque-speed characteristic. If a substantial variation of speed with torque is required, the rotor can be designed to have a large resistance. The downside of such an arrangement is the unacceptable reduction in efficiency.
A method to access the rotor windings and therefore exploit the property of speed change is to arrange a rotor that has coils rather than short circuited bars, with the coil terminals connected to slip rings and brushes so that additional external resistance can be connected in series with the windings. In such a rotor wound induction machine, the rotor winding is similar to that on the stator. The disadvantages of the wound rotor induction generator include a higher capital cost and a higher maintenance cost.
A wind turbine rated at 450 kW has the following induction generator parameters in ohms: Rs = 0.01, Xs = Xr = 0.15, Rr = 0.01 and Xm = 6. Ata time when it is supplying its rated output the slip is 0.01. Calculate the mains voltage and the power factor at which the induction generator is supplying power to the grid using the simplified equivalent circuit.
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Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable.