A simple unsteady blade element analysis is used to account for the effect of the trailing wake on the induced velocity of a wind turbine rotor undergoing fast changes in pitch angle. At sufficiently high tip speed ratio, the equation describing the thrust of the element reduces to a first order, nonlinear Riccti's equation which is solved in a closed form for a ramp change in pitch followed by a constant pitch. Finite tip speed ratio results in a first order, nonlinear Abel's equation. The unsteady aerodynamic forces on the NREL VI wind turbine are analyzed at different pitch rates and tip speed ratio, and it is found that the overshoot in the forces increases as the tip speed ratio and/or the pitch angle increase. The analytical solution of the Riccati's equation and numerical solution of Abel's equation gave very similar results at high tip speed ratio but the solutions differ as the tip speed ratio reduces, partly because the Abel's equation was found to magnify the error of assuming linear lift at low tip speed ratio. The unsteady tangential induction factor is expressed in the form of first order differential equation with the time constant estimated using Jowkowsky's vortex model and it was found that it is negligible for large tip speed ratio operation.

References

References
1.
Leishman
,
J.
,
2002
, “
Challenges in Modeling the Unsteady Aerodynamics of Wind Turbines
,”
Wind Energy
,
5
(
2
), pp.
85
132
.
2.
Schepers
,
J.
,
2012
, “
Engineering Models in Wind Energy Aerodynamics: Development, Implementation and Analysis Using Dedicated Aerodynamic Measurements
,”
Ph.D. thesis
, TU Delft, Delft University of Technology,
Delft
,
The Netherlands
.
3.
Øye
,
S.
,
1986
, “
Unsteady Wake Effects Caused by Pitch Angle Changes
,”
IEA R&D WECS Joint Action on Aerodynamics of Wind Turbines, 1st Symposium
, London, Oct. 15, pp.
58
79
.
4.
Jonkman
,
J.
,
Butterfield
,
S.
,
Musial
,
W.
, and
Scott
,
G.
,
2009
, “
Definition of a 5-MW Reference Wind Turbine for Offshore System Development
,” National Renewable Energy Laboratory, Golden, CO, Technical Report No. NREL/TP-500-38060.
5.
Burton
,
T.
,
Sharpe
,
D.
,
Jenkins
,
N.
, and
Bossanyi
,
E.
,
2001
,
Wind Energy Handbook
,
Wiley
,
West Sussex, UK
.
6.
Fingersh
,
L.
, and
Johnson
,
K.
,
2002
, “
Controls Advanced Research Turbine (CART) Commissioning and Baseline Data Collection
,” National Renewable Energy Laboratory, Golden, CO,
Technical Report No. NREL/TP-500-32879
.
7.
Hand
,
M.
,
Simms
,
D.
,
Fingersh
,
L.
,
Jager
,
D.
,
Cotrell
,
J.
,
Schreck
,
S.
, and
Larwood
,
S.
,
2001
, “
Unsteady Aerodynamics Experiment Phase VI: Wind Tunnel Test Configurations and Available Data Campaigns
,” National Renewable Energy Laboratory, Golden, CO, Technical Report No. NREL/TP-500-29955.
8.
Snel
,
H.
, and
Schepers
,
J.
,
1995
, “
Joint Investigation of Dynamic Inflow Effects and Implementation of an Engineering Method
,” Netherlands Energy Research Foundation (ECN), Petten, The Netherlands,
Technical Report No. ECN-C-94-107
.
9.
Wood
,
D.
,
2011
,
Small Wind Turbines
,
Springer
,
Berlin
.
10.
Hecht
,
Y.
, and
Rand
,
O.
,
1993
, “
Response of Helicopter Blades to a Sharp Collective Increase
,”
J. Aircr.
,
30
(
6
), pp.
889
896
.
11.
Carpenter
,
P.
, and
Fridovich
,
B.
,
1953
, “
Effect of a Rapid Blade-Pitch Increase on the Thrust and Induced-Velocity Response of a Full-Scale Helicopter Rotor
,” National Advisory Committee for Aeronautics, Langley Field, VA,
Technical Report No. NACA-TN-3044
.
12.
Rebont
,
J.
,
Valensi
,
J.
, and
Soulez-Larivire
,
J.
,
1960
, “
Response of Rotor Lift to an Increase in Collective Pitch in the Case of Descending Flight, the Regime of the Rotor Being Near Autorotation
,” National Advisory Committee for Aeronautics, Washington, DC,
Technical Report No. NASA-TT-F-18, L-455
.
13.
Duraisamy
,
K.
, and
Brown
,
R.
,
2008
, “
Aerodynamic Response of a Hovering Rotor to Ramp Changes in Pitch Input
,”
American Helicopter Society 64th Annual Forum
, Montreal, Canada, Apr. 29–May 1.
14.
Bossanyi
,
E.
,
1996
,
GH Bladed Theory and User Manuals
,
Garrad Hassan and Partners Limited
,
Bristol, UK
.
15.
Murphy
,
G.
,
1960
,
Ordinary Differential Equations and Their Solutions
,
Van Nostrand
,
New York
.
16.
Sheng
,
W.
,
Galbraith
,
R. A. M.
, and
Coton
,
F. N.
,
2009
, “
On the s809 Airfoil's Unsteady Aerodynamic Characteristics
,”
Wind Energy
,
12
(
8
), pp.
752
767
.
17.
Gupta
,
S.
, and
Leishman
,
J. G.
,
2006
, “
Dynamic Stall Modeling of the s809 Aerofoil and Comparison With Experiments
,”
Wind Energy
,
9
(
6
), pp.
521
547
.
18.
Polyanin
,
A.
, and
Zaitsev
,
V.
,
2003
,
Handbook of Exact Solutions for Ordinary Differential Equations
,
Chapman & Hall/CRC Press
,
Boca Raton, FL
.
19.
McWilliam
,
M.
, and
Crawford
,
C.
,
2011
, “
The Behavior of Fixed Point Iteration and Newton–Raphson Methods in Solving the Blade Element Momentum Equations
,”
Wind Eng.
,
35
(
1
), pp.
17
32
.
20.
Ning
,
A.
,
2014
, “
A Simple Solution Method for the Blade Element Momentum Equations With Guaranteed Convergence
,”
Wind Energy
,
17
(
9
), pp.
1327
1345
.
21.
Spera
,
D.
,
1994
,
Wind Turbine Technology
,
ASME Press
,
New York
.
22.
Panayotounakos
,
D.
, and
Zarmpoutis
,
T.
,
2011
, “
Construction of Exact Parametric or Closed Form Solutions of Some Unsolvable Classes of Nonlinear ODEs (Abel's Nonlinear ODEs of the First Kind and Relative Degenerate Equations)
,”
Int. J. Math. Math. Sci.
,
17
(
9
), pp.
1327
1345
.
23.
Henriksen
,
L.
,
Hansen
,
M.
, and
Poulsen
,
N.
,
2013
, “
A Simplified Dynamic Inflow Model and Its Effect on the Performance of Free Mean Wind Speed Estimation
,”
Wind Energy
,
16
(
8
), pp.
1213
1224
.
24.
Chen
,
X.
, and
Agarwal
,
R.
,
2014
, “
Inclusion of a Simple Dynamic Inflow Model in the Blade Element Momentum Theory for Wind Turbine Application
,”
AIAA
Paper No. 2014-2849.
25.
Okulov
,
V.
,
Sørensen
,
J.
, and
Wood
,
D.
,
2015
, “
The Rotor Theories by Professor Joukowsky: Vortex Theories
,”
Prog. Aerosp. Sci.
,
73
, pp.
19
46
.
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