6. Wi Flashcards
Calculation of the energy yield within a period T: For each wind speed individually calculate the power output, add it up and weight it according to frequency.
But: For project planning of a wind turbine complex procedure
Therefore: ??
The frequency distribution of the individual wind speeds is estimated using the Weibull distribution.
Formular to calculate the power output of a wind turbine for a specific wind speed?
Pi = roh/2 * A * v_i^3 * c_p,i
roh: density of air
vi: Wind speed
A: rotor area (also nicht wie A bei hi-Berechnung (Weibull)!)
cp,i: Power coefficient (different between turbines and also depending on the wind speed vi)
Why is the mean value of all measured wind speeds not sutable to determine the energy production of a wind turbine?p
Does not provide information about the distribution of wind speeds, which is crucial for energy generation.
Also small changes of high wind speeds already have a strong impact on the energy yield (because of v^3)
1) Why can’t wind speeds be discribed by a normal distribution?
2) Which distribution reprents the wind speeds better?
1) Wind speeds are not symmetrically distributed. They have an asymmetrical distribution with a longer right tail (more low values, fewer high values).
2) The Weibull distribution
Why do small changes in high wind speeds have a strong impact on energy production?
Wind power is proportional to the cube of wind speed (v^3).
A small increase in wind speed results in a much larger increase in power output.
Higher wind speeds have a greater influence on energy yield than lower wind speeds.
Weibull distribution density function? (Formula)
hi = (k/A) * (v_i/A)^(k-1) * e^-(v_i/A)^k
A: Scaling parameter - Measure of characteristic wind speed [m/s]
k: Form factor, varies between 1 and 4
Weibull distribution density function
1) A small k-value means what?
2) What results in a larger k-value?
1) Very variable winds
2) constant winds
Weibull distribution density function
A is proportional to what?
the mean value of the wind speed
Weibull parameters and mean wind speeds at 10m height for different locations in Germany.
-> see exerc.
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Form factors (k) for different climate regions.
Artic region: ??
Central Europe: ??
Trade wind regions: ??
Artic region: 1
Central Europe: 2
Trade wind regions: 3-4
For Central Europe, form factor 2 applies.
Simplification through what possible?
Rayleigh distribution (instead of Weibull distribution)
h_R = f(v_Strich)
v_Strich: Mean wind speed (= A * k.te_Wurzel_aus(0,287 * k^-1 + 0,688 * k^-0,1))
1) Formula Rayleigh distribution?
2) Formula to calculate the mean wind speed for Rayleigh distribution?
1) h_R = f(v_Strich) mit v_Strich: Mean wind speed
2) v_Strich = A * k.te_Wurzel_aus(0,287 * k^-1 + 0,688 * k^-0,1)
What are the advantages of using Rayleigh distribution? (3)
(instead of Weibull distribution)
Mean wind speed only
Known for many locations
The yield calculation of wind turbines of many manufacturers is based on the assumptions of a Rayleigh distributed wind.
Yield Calculation:
The yield in period T provided by the turbine with given power curve Pi(v) is obtained from?
From the yields of the individual wind speeds
Yield Calculation:
The yield in period T provided by the turbine with given power curve Pi(v) can be calculated how? (Formula)
E_i = h_i * P_i * T
With:
Pi = (roh/2) * A * v_i^3 * c_p,i (A here rotor area)
T = 8.760h
Yield calculation (E)
Formula for calculating E_total?
E_total = Sum(E_i) = Sum(h_iP_iT)
insert P_i:
E_total = Sum(h_i * (roh/2) * A * v_i^3 * c_p,i * T)
(
h_i: How often (%) does wind speed i occur compared to others (power weighting)
P_i: Power output of the wind turbine at wind speed i
c_p,i: Optimum power coefficient (refers to rotor speed)
Energy Yield of a Wind Turbine - Conclusion
-> See corresponding slide!!
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Do Energy Yield Task!!! (Durchlesen!! (vor allem d und e), ab und zu rechnen!)
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Sketch the a typical curve for each the power output of P_Wind, P_Betz and P_Turbine! Explain!
See exercise solution or sum.!
P_Wind ohne c_p, darum voll von v^3 abhängig (klar kubische Form)
P_Betz mit c_p,Betz=16/27 (bzw. around 0,593)
-> v^3 > c_p,Betz*v^3
-> theoretisches Optimum (aber nicht technisch gesehen); also noch deutlich höher als bei Turbine
-> höherer power output bei gleichem wind speed als bei
P_Turbine mit c_p,Turbine
-> c_p,Turbine < c_p,Betz
-> geringerer power output bei gleichem wind speed
-> außerdem muss ab bestimmtem wind speed abgeregelt werden (Protection against generator overload for example)
What is the reason for c_p,Turbine < c_p,Betz?
Aerodynamic, mechanical and electrical losses !!
