Research Papers

Development of an Analytical Unsteady Model for Wind Turbine Aerodynamic Response to Linear Pitch Changes

[+] Author and Article Information
Mohamed M. Hammam

Department of Mechanical and
Manufacturing Engineering,
University of Calgary,
Calgary, AB T2N 1N4, Canada
e-mail: mmahamma@ucalgary.ca

David H. Wood

Department of Mechanical and
Manufacturing Engineering,
University of Calgary,
Calgary, AB T2N 1N4, Canada
e-mail: dhwood@ucalgary.ca

Curran Crawford

Associate Professor
Department of Mechanical Engineering,
University of Victoria,
Victoria, BC V8W 2Y2, Canada
e-mail: curranc@uvic.ca

1Corresponding author.

Contributed by the Solar Energy Division of ASME for publication in the JOURNAL OF SOLAR ENERGY ENGINEERING: INCLUDING WIND ENERGY AND BUILDING ENERGY CONSERVATION. Manuscript received June 5, 2015; final manuscript received April 28, 2016; published online June 14, 2016. Assoc. Editor: Yves Gagnon.

J. Sol. Energy Eng 138(5), 051001 (Jun 14, 2016) (9 pages) Paper No: SOL-15-1171; doi: 10.1115/1.4033592 History: Received June 05, 2015; Revised April 28, 2016

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.

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Fig. 1

Trailing wake of mixed vorticity as a result of unsteady operation (the wake is assumed of constant pitch and diameter)

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Fig. 2

The velocity and force diagram of a blade element

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Fig. 3

The contribution of each term in Eq. (7a) and the total rate of change of induction for the NREL Ph VI at induction value of a = 0.5, nonlinear term α̂rar2, linear term βrar, and the free term γr

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Fig. 4

The relation between thrust and induction for a linear change in pitch angle from 0 deg to 10 deg at four different pitch rates and tip speed ratio of 7.5 for a blade element at the mid-radius of the NREL phase VI wind turbine

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Fig. 5

Changes in thrust coefficient for a linear change in pitch angle at the same initial conditions, and λ=7.5 for a blade element at the midradius of the NREL phase VI wind turbine. (a) With pitch rates at constant time, and (b) with time for four different pitch rates.

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Fig. 6

Change in thrust coefficient with tip speed ratio at two different instants for θ˙p=10 deg/s for a blade element at mid-radius of the NREL phase VI wind turbine

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Fig. 7

Changes in thrust coefficient with initial conditions of a mid-radius blade element at the switching time of t=1 s and conditions of θ˙p=10 deg/s, and λ=7.5

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Fig. 8

Tangential induction factor distribution with time for blade elements from r/R = 0.6–0.97 at tip speed ratio of 7.5 and pitch rate of 10 deg/s of the NREL VI wind turbine

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Fig. 9

Calculated thrust coefficient of the NREL VI wind turbine at pitch rate of 10 deg/s for three different tip speed ratios of (a)7.5, (b) 5, and (c) 3

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Fig. 10

Change in thrust coefficient overshoot with pitch rate and tip speed ratio

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Fig. 11

The measured and calculated torque of case (ii.4) of the Tjæreborg wind turbine, measurements were digitized from Ref. [8]




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