Vertical Axis Wind Turbine
1. Aerodynamics
For verticalaxis wind turbines, the blade element/moment (BEM) method is implemented as a doublemultiple streamtube (DMS) calculation. The conservation of momentum and local aerodynamic coefficients on the airfoils are considered on the upwind and downwind parts of the rotor.
The conservation of momentum is used to find the induced velocity at the swept surface. For a DMS momentum balance, each swept surface element on the upwind side is associated with its mirror element across the (\(Y_r,Z_r\)) plane. The flow which leaves the upwind element is assumed to impinge on the downstream element. That means that the incoming flow at the downstream element (\(V_m\)) is taken to be
where \(V_0\) is the undisturbed incoming flow at the upwind element, and \(V_i\) is the induced velocity at the upwind element.
In practice, ghost elements are introduced in order to calculate the appropriate force at each surface element. In this way, the force associated with each surface element is updated smoothly, and the dynamic stall forces are updated for the real blade elements.
Momentum balance calculations for horizontal axis wind turbines include a correction for the fact that the lift on a blade of finite span goes to zero at the tips. This correction is known as the Prandtl correction. In the present implementation, the user can choose between allowing the code to compute the Prandtl factor (recommended for Htype rotors) or inputing a Prandtl factor (recommended 1.0 for Darrieus rotors).
Dynamic inflow acts as a time lag in the induced velocity computed by the described DMSBEM method. In the implemented aerodynamic model, the Stig Øye dynamic inflow model is applied. This is the same model as implemented for horizontal axis wind turbines in SIMO.
Airfoil forces are computed by first looking up coefficients from an input table, as a function of Reynolds number \(Re\) and the instantaneous angleofattack \(\alpha\), and then correcting the coefficients for dynamic stall. Interpolating coefficients from the input table is the most computationallyintensive part of the DMSBEM calculation.
The basic features of dynamic stall can be understood in terms of a timedelay on the motion of the separation point with respect to the instantaneous angleofattack. As a verticalaxis wind turbine rotates, the blade retreats from the wind over part of the cycle. When the windspeed is high, the angleofattack shifts rapidly from a large negative value to a large positive value, over a timescale that is comparable to the time lag in the dynamic stall equation; therefore dynamic stall is relevant. However, the dynamic stall equations are not welldefined when the angle of attack and separation point position parameter are of opposite sign. Special logic is included to handle this situation.
2. Requirements for vertical axis wind turbine modelling

Support body and rotor (present) body with 6 degrees of freedom.

Moment coupling attached to the rotor body in direction parallel with body fixed Zaxis, ie. moment around the Zaxis
3. Recommendations for vertical axis wind turbine modelling
The rotor body and the support body may be connected by use of

2 docking cone couplings to transfer radial forces.

Each docking cone coupling has its contact point located in the rotor body Zaxis and its cone attached to the support body. A distance between the two contact points on the rotor axis is necessary in order to transfer moments about x and yaxes. The orientation of the direction vector of the docking cones should be coincident with the rotor body Zaxis.

1 axial force coupling between the rotor body and the support body to transfer axial rotor force.

The axial force coupling should be aligned with the rotor body Zaxis.

Note that the applied electrical torque is handled by MOMENT COUPLING. Thus, no constant moment should be specified for the coupling. Any specified torsional stiffness will be set to zero during dynamic analysis.
4. Blade geometrical input
The blade input is given for one blade, which is assumed to initially be in the body +x direction. Additional blades are equally distributed around the body +z axis.
5. Description of internal control system for VAWT generator torque and electrical power
The implemented generator torque control system includes startup logic as well as operational logic. A PI controller for the generator torque is applied to achieve the desired rotor speed.
During the startup period, the reference rotor speed is based on the rotor speed and the current time. The integral control parameter starts with a large value and is relaxed to the operational value after the startup period.
After the startup period, the reference rotor speed is based on the lowpass filtered wind speed.
5.1. Control measurement filter
The generator speed is the sole feedback input. A recursive, singlepole lowpass filter exponential smoothing to reduce the high frequency excitation of the control systems is provided. The discrete time recursion equation for this filter is
\( \mathrm{ \omega_{f.k} = \alpha \omega_{f.k1} + (1  \alpha) \omega_K }\)
where
\( \mathrm{ \alpha = exp((\Delta t)/(TC))}\)
where \( \mathrm{ \Delta t}\) is the discrete time step, \( \mathrm{ TC}\) is the filter time constant, \( \mathrm{ \alpha}\) is the lowpass filter coefficient \( \mathrm{ \omega_f}\) is low pass filtered generator speed and \( \mathrm{k }\) indicates the time step. The relation between the filter time constant and the cut off (corner) frequency \( \mathrm{ f_C}\) is given by:
\( \mathrm{ TC = \frac{1}{2 \pi f_C}}\)
5.2. Generator torque controller
The generator torque controller uses PI control to try to attain a specified reference speed. The reference speed is determined based on a lookup wind speed/rotor speed table, which is given as input or taken from the default table shown below. The lowpass filtered wind speed is used to determine the reference rotor speed at each time step. Linear interpolation is used in between the reference values.
The default rotor speed/wind speed table
Wind speed  Reference rotor speed 

[m/s] 
[rad/s] 
3.0 
0.200 
8.0 
0.544 
23.0 
0.544 
35.0 
0.200 
PI control
The PI controller determines the generator torque based on the proportional and integrated errors in rotor speed. The simple regulator algorithm is given by
\( \mathrm{R(t + \Delta t) = R (t) + \Delta \omega \Delta t }\)
\( \mathrm{ Q = K_P \Delta \omega + K_I R (t + \Delta t) }\)
where \( \mathrm{\Delta t }\) is the regulator time step, \( \mathrm{ \Delta \omega}\) is the rotor speed error, i.e. the difference between filtered rotor speed and rated rotor speed. \( \mathrm{ R}\) is the accumulated time integrated speed error. \( \mathrm{ Q}\) is the instructed/required generator torque.
The factor \( \mathrm{\tau_i}\) is used to relate the proportional and integral gains.
\( \mathrm{K_I = K_P/\tau_i}\)
During startup, the value of \( \mathrm{\tau_i}\) should be small. Therefore, an initial value, final value, and a time period for relaxing \( \mathrm{\tau_i}\) are introduced. The relaxation period begins after the defined startup period.
Startup
During the specified startup period, the reference rotor speed is first calculated based on the filtered wind speed and then a linear ramp factor is applied. The ramp factor is zero at the start of the simulation and increases to one at the end of the startup period.
Gain scheduling
The optimal proportional and integrator gains may vary with rotor speed. At each step the gain will be corrected based on the present rotor speed. The user may specify a gain scheduling law or choose to apply the default law. For intervening generator speeds, linear interpolation is used.
The default gain scheduling law
Rotor speed  Correction factor 

[rad/s] 
[] 
0.00 
1.00 
0.55 
1.00 
0.60 
1.50 
1.00 
1.50 