Power curve in region 3 seems wrong

[rschmehl]

Feedback from Jesse Hummel:

I think that the cycle power in region 3 drops faster than what I expect when looking at the power output and reeling factors. The cycle power is just the time-averaged power

P_cycle = (P_out t_out + P_in t_in) / (t_out + t_in)

The time can be calculated by dividing a distance 's' by the speed:

= (P_out s_out / v_out + P_in s_in / v_in) / (s_out / v_out + s_in / v_in)

If s_out = s_in, they disappear from the equations:

= (P_out / v_out + P_in / v_in) / (1/v_out + 1/v_in).

Multiply by (v_out v_in)/(v_out v_in) to get

= (P_out v_in + P_in v_out) / (v_in + v_out).

Multiply by vw/vw to get the reel-out factor in the equation

= (P_out f_in + P_in f_out) / (f_in + f_out)

For vw = 15 m/s, P_out = 20, P_in = -5, f_out = 0.3, and f_in = 0.6. Plugging that in yields 11.7 kW, however, the figure indicates something around 2.5 kW. So that’s why I think that the calculation of mean cycle power in region 3 is incorrect.

[rschmehl]

Yes, something is not right here. P_cycle = (P_out f_in + P_in f_out) / (f_in + f_out) is the correct formula, if we assume both f_in and f_out are positive (which I don't in the lecture notes).

I will have to dig into the code. Thanks for pointing this out.

[0northlake0]

If the convention $f_{in} < 0$ is adopted, $Pc = (f{in} \cdot P{out} - f{out} \cdot P{in}) / (f{in} - f{out})$. However when analysing the code for the power limited regime, this equality does not hold. $f{in}$ and $f_{out}$ do not seem to be the problem though.

The issue seems to be that in the derivation of the objective function $-pc$, $F{t,n} = 0.5 \cdot \rho \cdot v_{n,P}^2 \cdot S \cdot \gamma_0 \cdot (\cos{\beta0} - f{n,P})^2$ is used as the expression for $F_{t, n}$. However, because of the flight strategy of changing the aerodynamic properties in the power limited regime, $\gamma_0$ is not constant for different wind speeds here, in contrast to the other regimes.

If $F{t,n} = 0.5 \cdot \rho \cdot v{w}^2 \cdot S \cdot \gamma_0 \cdot (\cos{\beta0} - f{0})^2$ is used instead, the power computations become consistent. Attached is a sample plot of the new power curve.

[rschmehl]

Thank you, @0northlake0. Please copy-paste here the corrected part of the code (old vs new) so that I can retrace what exactly you changed.

[rschmehl]

Is this what you did?

[0northlake0]

Sorry for my late reply. The old objective_function _3:

def objective_function_3(x, mu_P, f_nP):
    f_in     = x[0]
    return -(((cosine_beta_out - f_nP) / mu_P)**2  \
             - (force_factor_in / force_factor_out) \
               * (np.sqrt(1 + E2in*(1 - f_in**2)) - f_in)**2/(1 + E2in))  \
           * f_in*f_nP/(mu_P*f_in-f_nP)

The new objective_function_3:

def objective_function_3(x, mu_P, f_nP):
    f_in     = x[0]
    return -((cosine_beta_out - f_nP/mu_P)**2  \
             - (force_factor_in / force_factor_out) \
               * (np.sqrt(1 + E2in*(1 - f_in**2)) - f_in)**2/(1 + E2in))  \
            * f_in*f_nP/(mu_P*f_in-f_nP)

[0northlake0]

Is this what you did?

Yes, the new code now implements the form of the second to last line. I believe the last line should contain $\mu_P \cos{\beta_0}$ instead of $\mu_P^2 \cos{\beta_0}$.

[rschmehl]

Correct, thanks!