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Most textbooks covering induction machines cover only the motor use-case scenario. When the machine is used as a motor, the slip factor is determined by the frequency of the stator current by the equation

\$ \Huge{s=f_s - \frac{\frac{p}{2} \cdot \frac{n_r}{60}}{f_s}} = \frac{n_s - n_r}{n_s} \$

where

\$ s \$ is the slip factor,

\$ f_s \$ is the stator frequency

\$ p \$ is the number of poles

\$ n_s \$ is the motor synchronous speed in rpm

\$ n_r \$ is the actual rotor speed in rpm

This all makes perfect sense when the machine is being operated as a motor and/or is connected directly to the transmission network which provides magnetizing current and a stable frequency to the stator windings.

My question: What happens if the machine is being operated as a generator and is on an island network? I understand that we need a source of reactive power -- provided by soft-start capacitor bank, for example -- but if the speed of the machine is variable, how in heaven's name is the slip factor determined? There's no alternating current magnetization, and hence no frequency on the stator winding. How is the frequency being set?

And how can I create a simple model of this that I can use in Simulink or in Scilab xcos?

I want to model a wind generating system on an island network, or a variable-speed system coupled to the transmission network via a rectifier and then a DC-DC converter. As usual I bit off more than I could chew and now I understand why everybody else does these models with permanent magnet synchronous machines: it's way easier.

Perhaps you can prove me wrong?

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Do you mean "alternator on an island network"? If so, When you say "Island network" do you mean the alternator is the sole member of the network? If so (and this is NOT a specialist area of mine) I think you may have a "special case" and have to provide 'signal' / excitation to suit. You may have to assume that there is a remnant of permanent magnetisation present to start the system - or provide some sort of excitation. Ive seen people use 3 phase motors with capacitor arrangements to allow self excitation and use as an alternator. – Russell McMahon Sep 22 '12 at 13:21
@RussellMcMahon: yes, that's basically what I mean. I get the part about a reactive power source for excitation, but that's normally done with a capacitor bank. As the bank discharges, the stator windings are magnetised, but this stator current doesn't alternate initially. The rotor turns through the stator's field and a current flows, which creates a magnetic field. Eventually, the induced field in the rotor will drive an alternating current in the stator... at least, that's my understanding of it. – Stephen Bosch Sep 22 '12 at 14:08
@RussellMcMahon: my challenge is this: the frequency of the stator current is what determines the synchronous speed of the machine. The slip factor (see above) is a function of the synchronous speed and the actual rotor speed, except that, without an external ac source, there is no fixed frequency; it would have to be coming entirely from the induction generator. My goal is to model this mathematically so that I can simulate it, but this is a closed loop. – Stephen Bosch Sep 22 '12 at 14:23
I just realised that I had written "motor" when I meant "generator". Sorry for the confusion. I've edited the original question. – Stephen Bosch Sep 22 '12 at 15:42
For an 3 phase induction machine to work as a generator, you need to feed it first with an rotating field. Then wind will make the rotor turn faster than the rotating field in stator and only then slip can become negative and the machine will start to run in generator mode. Therefore, with too little wind, a windmill will run in motor mode and usually be turned off all together. – jippie Sep 22 '12 at 18:30
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3 Answers

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According to this wikipedia article, if the motor is used as a stand alone generator:frequency and voltage are complex function of machine parameters, capacitance used for excitation, and load value and type.

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This link may provide information in regards to using an induction motor as a generator.
Motors as Generator
Another possibility would be to use a sine wave inverter for excitation of your generator.
Or use this method to excite the field windings of a DC motor/generator,
this should provide a constant frequency independent of rotor speed.
I hope this helps.

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The concept of slip is a bit misleading for variable speed wind turbines. The type which uses an induction machine are called doubly fed induction generators (DFIGs) and their speeds are allowed to vary through the use of an AC-AC converter applied to their rotor windings. Because there is no actual induction in these machines, the term slip might be a bit odd to use, but the value for s can still be calculated, and is negative.

For these type of generators no capacitor banks are needed to supply reactive power, as the converters can be used to control the production of reactive power from the induction machine.

A DFIG must have an AC source to run, but this does not prohibit it from being run on an island system. Island systems are often AC systems.

I'm not really sure why permanent magnet synchronous machines would be easier to model. For wind turbine applications these machines also use power electronics, and their rotation speed is also variable through a wide range.

If you have the SimPowerSystems toolbox then these types of generators can be modelled in Simulink

http://www.mathworks.se/help/physmod/powersys/ref/windturbinedoublyfedinductiongeneratorphasortype.html

but there are also free models which are available

http://www.ece.umn.edu/users/riaz/

Another free MATLAB tool which can be used for modelling power systems is

http://www.ece.umn.edu/users/riaz/

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Your statement "Because there is no actual induction in these machines" doesn't make any sense to me; there is induction by definition, without induction the machine cannot develop stator voltage (in an island system). Also, DFIG are not the only induction machines used in windpower applications; there are others (the Danish concept comes to mind). Further, all the examples I have seen assume a reactive power source on the stator windings, so I don't see how the presence of an AC-AC inverter on the rotor will help here. – Stephen Bosch Nov 21 '12 at 20:33
Also - even if there is an AC-AC inverter on the rotor windings: in a black-start condition, it needs to get its power from somewhere. – Stephen Bosch Nov 21 '12 at 20:35
There is no induction induced in the rotor windings. The voltage there is directly applied from the network through the converter to the rotor windings. The stator is also connected to the network, and therefore has a voltage. By Danish concept I assume you mean a squirrel cage induction generator, but these are not variable speed turbines, which was what was asked for in the question. Yes, reactive power sources on the stator side are used for these turbines, but they do not have power electronics, so reactive compensation is required. – Katt Nov 22 '12 at 15:58
But maybe I see that what the question was asking for is how should a variable speed wind turbine be used for black start, not actually island operation? – Katt Nov 22 '12 at 16:42
Shouldn't "island operation" imply that it must also work in black-start conditions, since in either case the system has to be started somehow? I know this is almost never done in practice, but that is why I want to simulate it. This site shows a lab bench example: arthropodsystems.com/AsynchronousGenerator1/… I have since discovered this variable speed condition in an island is usually modeled with space vectors, which is something I am still learning and is apparently fairly challenging to do. – Stephen Bosch Dec 2 '12 at 13:38

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