Thrust coefficient correction on yaw misalignment and tilt

[luccasportiolli]

Hi!

I would like to better understand the differences in the way FLORIS computes the power and the thrust coefficient, when using the "cosine-loss" turbine operation model.

By seeing the code, the power is computed firstly applying the air density correction, as well as the yaw and tilt ones, and then it goes to the look-up table and interpolates it to give the power.

    # Compute the power-effective wind speed across the rotor
    rotor_average_velocities = average_velocity(
        velocities=velocities,
        method=average_method,
        cubature_weights=cubature_weights,
    )

    rotor_effective_velocities = rotor_velocity_air_density_correction(
        velocities=rotor_average_velocities,
        air_density=air_density,
        ref_air_density=power_thrust_table["ref_air_density"]
    )

    rotor_effective_velocities = rotor_velocity_yaw_cosine_correction(
        cosine_loss_exponent_yaw=power_thrust_table["cosine_loss_exponent_yaw"],
        yaw_angles=yaw_angles,
        rotor_effective_velocities=rotor_effective_velocities,
    )

    rotor_effective_velocities = rotor_velocity_tilt_cosine_correction(
        tilt_angles=tilt_angles,
        ref_tilt=power_thrust_table["ref_tilt"],
        cosine_loss_exponent_tilt=power_thrust_table["cosine_loss_exponent_tilt"],
        tilt_interp=tilt_interp,
        correct_cp_ct_for_tilt=correct_cp_ct_for_tilt,
        rotor_effective_velocities=rotor_effective_velocities,
    )

    # Compute power
    power = power_interpolator(rotor_effective_velocities) * 1e3 # Convert to W
    print("rotor_effective_velocities")
    return power

For the thrust coefficient, instead, it doesn't account for the air density correction, it computes the average velocity at the rotor plane and goes directly to the look-up table to take a thrust coefficient value. With it, it applies the yaw and tilt correction just by multiplying the interpolated value by the cosine of the respect angles, without any cosine loss exponent.

   # Construct thrust coefficient interpolant
    thrust_coefficient_interpolator = interp1d(
        power_thrust_table["wind_speed"],
        power_thrust_table["thrust_coefficient"],
        fill_value=0.0001,
        bounds_error=False,
    )

    # Compute the effective wind speed across the rotor
    rotor_average_velocities = average_velocity(
        velocities=velocities,
        method=average_method,
        cubature_weights=cubature_weights,
    )

    # TODO: Do we need an air density correction here?
    thrust_coefficient = thrust_coefficient_interpolator(rotor_average_velocities)
    thrust_coefficient = np.clip(thrust_coefficient, 0.0001, 0.9999)

    # Apply tilt and yaw corrections
    # Compute the tilt, if using floating turbines
    old_tilt_angles = copy.deepcopy(tilt_angles)
    tilt_angles = compute_tilt_angles_for_floating_turbines(
        tilt_angles=tilt_angles,
        tilt_interp=tilt_interp,
        rotor_effective_velocities=rotor_average_velocities,
    )
    # Only update tilt angle if requested (if the tilt isn't accounted for in the Ct curve)
    tilt_angles = np.where(correct_cp_ct_for_tilt, tilt_angles, old_tilt_angles)

    thrust_coefficient = (
        thrust_coefficient
        * cosd(yaw_angles)
        * cosd(tilt_angles - power_thrust_table["ref_tilt"])
    )

    return thrust_coefficient

Why is that? Shouldn't the yaw and tilt corrections for the thrust coefficient also consider a cosine loss exponent each? Shouldn't it be computed as the power, by first computing the rotor effective velocity accounting for all the corrections and then going to the look-up table to take the final value? Could you help me with that, please?

Thank you in advance for your attention to this matter!

[misi9170]

Hi @luccasportiolli ,

Thank you for the question. The answer is a little bit complicated, but I'll take a first pass at it, and I'll let others chime in if they have any extra thoughts.

Let's first consider the power. For power, as you noticed, we first compute a rotor-averaged velocity; then apply an air density correction; then account for a reduction in the inflow wind speed due to yaw and tilt misalignment. The result is an effective wind speed that the turbine would operate at---we query the power curve at that wind speed to compute the power.

Note that in rotor_velocity_air_density_correction, there is a term **(1/3)---this corrects the velocity for power, because according to actuator disk theory, power is proportional to velocity cubed. The actuator disk theory also suggests that we should reduce power by $(\cos \gamma)^3$, where $\gamma$ is the yaw or tilt misalignment, to account for yaw and tilt; however, field campaigns have reported that this gives an over estimate of the power loss, and instead exponents closer to 2 are found to be better estimates. In FLORIS, the default is 1.88, but this can be changed by setting cosine_loss_exponent_yaw and cosine_loss_exponent_tilt in the turbine input files.

Now, moving to your question on why we don't follow the same process for the thrust coefficient. The thrust coefficient is more difficult to measure than the power in field campaigns, so we mostly still rely on actuator disk theory when it comes to thrust. Actuator disk theory tells us that the turbine's thrust depends on the inflow velocity squared. When operating in yaw/tilt misalignment, the correction to the actuator disk theory is $(\cos \gamma)^2$. I tend to think of this being the product of two cosines: the first accounts for the reduction in wind speed that is incident on the rotor due to the yaw/tilt; and the second accounts for the fact that the force applied by the turbine on the flow is no longer aligned with the wind direction; so, to compute the "thrust" (which should be flow-aligned), we need to take the component of the force in the flow direction.

The computation of the thrust_coefficient that you reported accounts for the first of these two terms. That is, the thrust coefficient here accounts for the reduction in wind speed, but not for the fact that the forcing on the flow is no longer purely in the flow direction. The second part, that accounts for the forcing term not being aligned with the flow direction, is applied in the wake models, for example here in the Gauss velocity deficit model. Combined with the single cosine term computed in the thrust_coefficient calculation, this give us the squared effect of yaw misalignment.

Neither of these is included simply as a change to the flow velocity, after which the thrust coefficient is found from the lookup table using an "effective" velocity, because it's not entirely clear how that should be handled. Turbine controllers are usually based on power, and will either try to maximize power capture for a given wind speed (below rated) or produce rated power (above rated), which is what the method for computing power conveys. However, that does not necessarily mean that their thrust coefficient will follow along nicely, and without clear data for how the thrust coefficient will behave once we move away from aligned conditions, we have kept with the standard "cosine squared reduction" that the actuator disk model implies. It is possible that an air density correction may be made that could be similar to that used in computing power, but now, we would need to use a **(1/2) term rather than the **(1/3) used in the power calculation (because thrust depends on velocity squared, while power depends on velocity cubed). This means that the air density corrected wind speed would not be the same for power and thrust! That feels odd, and likely points to the fundamental assumptions of the actuator disk model not really being upheld. For now, we have left this out (and put a #TODO comment in place).

I'll finally note that in FLORIS the thrust is essentially an intermediate variable used when computing the wake deficits, whereas the power is a final output. All engineering wake models are empirical/based on simplifications, and the thrust computed part-way through the calculations may not perfectly represent the actual aerodynamic thrust imparted on the flow. As such, the thrust_coefficient (and thrust computed using it) should be treated with a bit of caution and skepticism as to the "true" meaning, especially when operating in conditions that don't perfectly match the specified thrust coefficient curve (matched air density and perfectly aligned with the flow).

I hope that this helps!
Misha

[misi9170]

I should also have mentioned that there are some new turbine operation models coming to FLORIS soon that improve upon our standard "modified power and thrust curve" approaches. In particular, see #832 and #924 .

[luccasportiolli]

Hi @misi9170,

Thank you so much for your explanation, it's more clear now!