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TEMPUS ENERGY: GENERATORS IN WIND TURBINES: EXERCISES 1: ASYNCHRONOUS GENERATOR WITH CAGE ROTOR Consider a wind turbine where the rotor blades (1), using a gearbox (3), drive an asynchronous generator (4) with cage rotor. Using a transformer (6), the generated power is injected into a high voltage grid (7). Notice the capacitors (5) generate reactive power which avoids the grid has to supply the reactive power required by the generator (or reduces the reactive power supplied by the grid). The generator generates a 50 Hz voltage having a line voltage of 690 V. The generated active power equals 750 kW and the power factor equals 0.86. Questions part 1: 1) Calculate the apparent power of the generator. 2) Calculate the RMS value of the current provided by the generator. 3) Calculate the capacitor values C needed to obtain a power factor equal to 1. Suppose the gearbox has 1.2 % losses. Suppose the generator has an efficiency π = 95 % and the transformer has an efficiency π = 99 %. Questions part 2: 1) Calculate the active power injected into the grid. 2) Calculate the overall efficiency of the entire installation. Solutions part 1: 1) Since π = 750 ππ and πππ π = 0.86, π = 872 πππ΄. 2) Since π = β3 πππππ πΌππππ and πππππ = 690 π, πΌππππ = 730 π΄. 3) Since π = π π πππ, π = 445 πππ΄π . One single capacitor, having a phase voltage of 400 π, has to generator 148 πππ΄π . Since π = π 2 π πΆ, a πΆ = 3000 ππΉ is needed. Solutions part 2: 1) The generator generates 750 ππ, since ππ‘ππππ ππππππ = 0.99 an active power of 742.5 ππ is injected into the grid. 2) The mechanical power at the output of the gearbox equals 750 ππβππππ = 789.5 ππ. The mechanical power at the input of the gearbox provided by the rotor blades equals 789.5 ππβππππππππ₯ = 800 ππ. This implies the overall efficiency equals ππ‘ππ‘ = 742.5 ππβ800 ππ = 93 %. Notice ππ‘ππ‘ = ππππππππ₯ ππππ ππ‘ππππ ππππππ . 2: DOUBLY-FED INDUCTION MOTOR Consider a wind turbine equipped with a doubly-fed induction machine. The turbine has a nominal power of 3 MW. The speed of rotation of the rotor blades varies between 13 revolutions per minute and 20 revolutions per minute. The gearbox has a speed ration of 1:90. The generator has four poles and the stator is connected with the 50 Hz public grid. The nominal power of 3 MW is supplied in cased the speed of the rotor blades equals 20 revolutions per minute. All losses are neglected. Questions: 1) 2) 3) 4) 5) 6) 7) 8) 9) Calculate the synchronous speed of the generator. Calculate the speed of rotation of the rotor of the generator. Calculate the slip speed. Calculate the slip. Calculate the power supplied by the stator (in MW). Calculate the power supplied by the rotor (in MW). Calculate the power injected into the grid by the frequency converter. Calculate the rotor frequency. Suppose the rotor frequency equals 7.6 Hz and the generator has a sub synchronous speed. Calculate the speed of rotation of the rotor blades. Solutions: 1) The stator of the generator, having four poles, is connected with a 50 Hz grid implying a synchronous speed of 1500 πππ. 2) The speed of rotation of the rotor blades equals 20 πππ, due to the speed ratio 1β90, the rotor of the generator has a speed of rotation which equals 1800 πππ. An super synchronous speed is obtained. 3) The slip speed equals 300 πππ. 4) The slip equals β0.2. 5) Since ππ β π ππ and π < 0 in combination with ππ < 0, ππ > 0. This means active power is extracted from the rotor windings and injected into the grid by the frequency converter. The total power injected into the grid equals βππ + ππ = β(1 β π ) ππ = 3 ππ. This implies ππ = β2.5 ππ (the stator injects 2.5 ππ into the grid). 6) The rotor (using the frequency converter) injects ππ = π ππ = 0.5 ππ in the grid. 7) The losses in the frequency converter are neglected implying the frequency converter injects 0.5 ππ into the grid. 8) The frequency of the rotor currents and voltages equals π2 = π π1 = 10 π»π§ . 9) Since π2 = π π1 = 7.6 π»π§ with π1 = 50 π»π§, the slip π = 0.152 implying a speed of rotation of the rotor which equals ππ = (1 β π )ππ = 1272 πππ. Due to the speed ratio 1β90 of the gearbox, the rotor blades have a speed of 14.1 πππ.