Financial Leverage and Capital Restructuring
Explore the concepts of financial leverage, capital restructuring, and choosing a capital structure to maximize stockholder wealth. Learn how changes in leverage can impact firm value and the importance of finding the optimal capital structure. Dive into examples and break-even analysis to understand the effects of leverage on stockholders.
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Presentation Transcript
New Approaches to Power Curve Adjustments in WakeBlaster Philip Bradstock Power Curve Working Group 1 February 2018, Hamburg 1 (c) ProPlanEn, 2018
WakeBlaster Measurements (SCADA or virtual) 3D RANS Wake Model Power Capture 2
Power Capture Corrections Turbulence Air Density Rotor Equivalent Wind Speed (REWS) Shear Wake profile Yaw Misalignment 3
Turbulence Zero Turbulence Power Curve 0.5 6000 Current IEC method assumes constant Cp below rated: Not true at low wind-speeds. Not true just below rated 5000 0.4 4000 Power Coefficient 0.3 Power (kW) 3000 0.2 2000 0.1 1000 0 0 3 8 13 18 Wind Speed (m/s) Cp (NREL) Cp (IEC) Power (NREL) Power (IEC) 4
Zero TI power curve - Alternative Method Taylor Series Expansion of TI=0 power curve about mean wind speed (assume cubic relationship). ??=0? ? = ??=0 ? +???=0 ?? 2 ? = power ? = wind speed ? = 10-min mean wind speed ? = turbulence intensity ?2??=0 ? ??2 ?3??=0 ? ??2 ? ? +1 ? ?2+1 ? ?3+ ?(4) 6 10-minute mean by Gaussian Integration: 2 1 ? ? ? ? 1 ?(?) = ??=0? ? ? ? ? 2 ? 2? ?2??=0 ? ??2 = ??=0 ? +1 ?2 2 => expression relating zero turbulence and 10-minute mean power curves. 5
Zero TI power curve - Alternative Method (0)? = ?(?) ??=0 Apex Correction: Find point of inflection and extrapolate with cubic Rated Power Correction: Limit all values to rated power (?)? ?2??=0 ??2 (?+1)? = ?(?) 1 Finite difference method: ?2 ??=0 2 (?)? ?2??=0 ??2 =? ? ? 2? ? + ? ? + ? ?2 ??=0? Warning: Iteration has a tendency to diverge limit number of iterations 6
Alternative Method Testing Measured 10-min power curve TI=0 power curve Error Simulated 10-min power curve GE 1.5MW Vestas 2MW 40 40 30 20 Power Error (kW) Power Error (kW) 20 0 10 3 4 5 6 7 8 9 10 11 12 13 14 -20 0 4 5 6 7 8 9 10 11 12 13 14 15 -40 -10 -20 -60 Wind Speed (m/s) Wind Speed (m/s) IEC WakeBlaster IEC WakeBlaster 7
Alternative Method Testing Measured 10-min power curve TI=0 power curve Error Simulated 10-min power curve Nordex 2.5MW Offshore 6MW 60 300 250 Power Error (kW) Power Error (kW) 40 200 150 20 100 50 0 0 4 5 6 7 8 9 10 11 12 13 14 15 3 4 5 6 7 Wind Speed (m/s) 8 9 10 11 12 13 14 -50 -20 Wind Speed (m/s) IEC WakeBlaster IEC WakeBlaster 8
Air Density Adjustment IEC method: ???? = reference air density (1.225) ? = alternate air density ? = wind speed dependent exponent 1 3 ???? ? ?? = ????? Svenningsen method (semi-empirical): ? 1 3 ? ????? 2 ???? ? ?? = ????? 3 because ?? varies with ? 9
Air Density Adjustment ? = tip-speed ratio ?? ?(?,?) directly, ??= ?(?) Cp-TSR curve ?? curve is designed to be quite flat at the peak. Optimal TSR in centre of flat region. Controller keeps turbine at optimal TSR by managing torque-speed relationship 0.5 0.4 0.3 Cp 0.2 0.1 0 2 4 6 8 10 12 14 TSR 10
Air Density Adjustment ? = tip-speed ratio ?? ?(?,?) directly, ??= ?(?) Cp-TSR curve ?? curve is designed to be quite flat at the peak. Optimal TSR in centre of flat region. Controller keeps turbine at optimal TSR by managing torque-speed relationship 0.5 Rated 5m/s 0.4 4m/s Optimal TSR 6-10m/s 0.3 Cp 0.2 0.1 0 2 4 6 8 10 12 14 TSR 11
Air Density Adjustment If the controller is not modified, a change in air density moves the TSR values. Flat top => no change in Cp for small changes in density. => 1/3 power rule is valid. Density=1.225 Density=1.2 Density=1.0 0.5 0.5 0.5 5m/s Rated Rated Rated 5m/s 5m/s 0.4 0.4 0.4 4m/s 4m/s 4m/s Optimal TSR Optimal TSR Optimal TSR 6-10m/s 6-10m/s 6-10m/s 0.3 0.3 0.3 STALL!! Cp Cp Cp Large change in air density: => change in Cp Operates in stall at rated 1/3 power rule no longer valid Controllers for high-altitude sites modified to correct TSR 0.2 0.2 0.2 0.1 0.1 0.1 0 0 0 2 2 2 4 4 4 6 6 6 8 8 8 10 10 10 12 12 12 14 14 14 TSR TSR TSR 12
Air Density Adjustment 1/3 power rule should be implemented on TI=0 power curve. Due to large uncertainty in ??=0(?), may cause more problems than it solves. Nordex 2.5MW Adjusting from 1.225kg/m3 to 1kg/m3 Nordex 2.5MW Adjusting from 1.225kg/m3 to 1kg/m3 2600 2600 2400 2400 POWER OUTPUT (KW) POWER OUTPUT (KW) 2200 2200 2000 2000 1800 1800 1600 1600 1400 1400 1200 1200 1000 1000 8 8 9 9 10 10 11 11 12 12 13 13 14 14 WIND SPEED (M/S) WIND SPEED (M/S) rho=1.225 rho=1.0 rho=1.225 IEC rho=1.0 WakeBlaster Svenningsen 13
Rotor Equivalent Wind Speed Current practice based on ? ?3: 31 ?3?? ?????= ? ? ? = rotor area ? = aerodynamic torque = rotor speed BUT: is fixed for whole rotor ? ?2 ? = ? and rotor will operate at varying non-optimal angles of attack. WakeBlaster calibration & validation shows linear average to be more representative: ?????=1 ? ??? ? 14
Any Questions? philip.bradstock@proplanen.com wakeblaster@proplanen.com 15
