3/2/20141The University of Iowa Intelligent Systems LaboratoryEnergy OutputAndrew KusiakIntelligent Systems Laboratory2139 Seamans CenterThe University of Iowa Iowa City, Iowa 52242 - [email protected]: 319-335-5934 Fax: 319-335-5669http://www.icaen.uiowa.edu/~ankusiakThe University of Iowa Intelligent Systems LaboratoryOutline The wind variation Distribution plotting The average bottle fallacy Mean power of the wind Betz' law Power density Power curves The power coefficient Calculator guide The power calculator Annual energy outputThe University of Iowa Intelligent Systems LaboratoryCharacterizing Wind Variability  Characterization of the wind speed variability is of importance to the wind industry Turbine designers use it to optimize the design of theturbines, e.g., by minimizing the energy generation cost  Wind farm designers use it to locate wind farms and selectturbines Investors use it to estimate the income from electricity generationThe University of Iowa Intelligent Systems LaboratoryWind Variations: WeibullDistribution Weibull distribution describesthe wind variation for a typical site The site has the mean wind speed of 7 m/s  The shape parameter of the functionis k = 2 The data has been collected over ayear periodThe General Pattern of Wind Speed Variation3/2/20142The University of Iowa Intelligent Systems LaboratoryWeibull Distributionwhere:λ > 0 is the scale parameterk > 0 is the shape parameter of the distributionhttp://en.wikipedia.orgFor k = 3.4, the Weibull distribution appears similar to the normal distributionFor k = 1, the Weibull distribution becomes the exponential distributionThe University of Iowa Intelligent Systems LaboratoryWeibull Distributionhttp://en.wikipedia.orgMedianModeMeanwhere the gamma function isScale Shape The University of Iowa Intelligent Systems LaboratoryCumulative Distribution FunctionThe University of Iowa Intelligent Systems LaboratoryPDF and CDFE. Hau (2006)3/2/20143The University of Iowa Intelligent Systems LaboratoryCDF: Mean vs Median Wind SpeedE. Hau (2006)The University of Iowa Intelligent Systems LaboratoryWind Speed Histogram0.4001.6152.8304.0455.2606.4757.6908.90510.12011.33512.55013.76514.98016.195Wind speed (m /s)0102030405060708090Number of observationsThe University of Iowa Intelligent Systems LaboratoryDescription of Wind Speed  The area under the pdf curve is always exactly 1 (the probability that the wind is blowing at some wind speed including 0 is 100 %)  Median = 6.6 m/s (Half of the blue area is to the left)  This means that 50% of the time the wind speed is less than 6.6 m/s, the other 50% of the time it is greater than 6.6 m/s Note: The median 6.6 m/s is not equal the mean 7m/s due to asymmetry of the pdfThe University of Iowa Intelligent Systems Laboratory Mode = 5.5 m/s (The most common wind speed)  The statistical distribution of wind speeds depends on locationclimate conditions, the landscape, and its surface  The Weibull distribution may thus vary in its shape,determined by the pdf parametersDescription of Wind Speed3/2/20144The University of Iowa Intelligent Systems LaboratoryThe Average Bottle Fallacy  The average energy content of the wind at a turbine site can not be determined from the average wind speed, rather the Weibull distribution is needed  How large (in volume) is the average bottle, one is 0.24m and the other is 0.76m tall and both are of same shape? Though one is only only 3.17 taller than the other, its volume is actually 3.173= 32 times larger than the small bottle (bottle V = cube of its size)The University of Iowa Intelligent Systems LaboratoryBetz' Law The Ideal Braking of the Wind  The more kinetic energy a wind turbine extracts from the wind, the more the wind will be slowed down (as it leaves the left side of the turbine in the tunnel)  An attempt to extract all the energy from the wind, would reducethe speed to zero, i.e., the air could not leave the turbine  In that case we would not extract any energy at all, as the new air would obviously be prevented from entering the rotor of the turbine  Passing air without speed change would lead to zero extracted energyThe University of Iowa Intelligent Systems LaboratoryBetz' Law The optimal point is: An ideal wind turbine slows down the wind by 2/3 of its original speed (v2= 1/3v1) Betz' law (Year 1919) says that one can only convert not more than 16/27 (or 59%) of the kinetic energy in the wind to mechanical energy using a wind turbine. (Albert Betz, German Physicist) Modern rotors achieve values of the coefficient of performance Cp= 0.4 - 0.5, which is 70% to 80% of the theoretically possible value of Cpmax= 0.59The University of Iowa Intelligent Systems LaboratoryPower Density Function The area under the blue curve showstheoretical power that can be extracted(Betz' law says, 16/27 of the total power in the wind)  The total area under the red curve represents the electrical power a certain wind turbine could produce  Turbine’s power curve determines the actual power produced 35.0 vAP The Power of the Wind3/2/20145The University of Iowa Intelligent Systems LaboratoryMore Detailed Power Density Function E. Hau (2006)where: E = Wind Turbine System Efficiency= ERotor x EGearbox x EGenerator x EPowerConverterE = Erotx EgerxEgenx E p-convThe University of Iowa Intelligent Systems LaboratoryMechanical Power: Summary 131()2pPAC VrVP Mechanical power produced by the rotorAir density A Rotor swept areaV Wind speedTip speed ratioCpPower coefficient r Rotor radiusRotor speed0 2 4 6 8 1000.050.10.150.20.250.30.350.4Tip speed ratio Power coefficient Cp()The University of Iowa Intelligent Systems LaboratoryMechanical Power: Summary 20204060801000246810-2000200400600800Rotor power P(W)Rotor speed (rad/s)Wind speed V(m/s)The University of Iowa Intelligent Systems LaboratoryPower CurveA: Insufficient power to overcome frictionand initial torqueB: Turbine operates to maximizeefficiencyC: Fixed (rated) power operation0204060801000246810-2000200400600800Rotor power P (W)Rotor speed (rad/s)Wind speed V(m/s)31()2pPACVrV3/2/20146The University of Iowa Intelligent Systems LaboratoryActual Power CurveThe University of Iowa Intelligent Systems LaboratoryActual Power CurveThe University of Iowa Intelligent Systems LaboratoryValidating Power Curves