The aerodynamic solver (Radially Independent Actuator Disc model – RIAD) used though out this paper is further described in Part 1, and therefore only a brief overview is given here. It makes an explicit relationship between the local-thrust coefficient (CLT – normalized ∂T/∂r) and the local-power coefficient (CLP – normalized ∂P/∂r) with given operational conditions such as the global tip speed ratio (λ) and the local glide ratio (Cl/Cd) and may include tip loss as well. A diagram showing the relationship graphically can be seen in Fig. 

In this section, the optimization methodology that allows for the fast and very efficient solution to the optimization is derived. It finds the optimal power for a fixed rotor increase. In principle, the rotor radius could also be an optimization parameter, but as is shown later, the optimal global power turns out to be unbounded, and having the solution for the fixed rotor radius increase allows for optimization with a simple cost function, which is further explained later. The main outcome of this section is a function that through solving an optimization problem gives the optimal power for a given set of constraints with a fixed radius increase and fixed wind speed (Popt(R, V)).

The purpose of this section is to extend the optimization methodology to include optimization for maximum annual energy production (AEP) with load constraints across all wind speeds as well as fixed rated power and a fixed radius increase.

AEP is computed as the average power over a year multiplied by the time of a year. The average power can be computed from the wind distribution (fwei, i.e., the frequency at which a wind turbine is operating at a given wind speed) and the power curve. Mathematically it can be computed as 17AEP=Tyear∫VCIVCOPV,∂T∂r(V),RfweidV, where Tyear is the time of a year, P is the power curve function, and VCI and VCO are the cut-in and cut-out wind speed respectively. The optimization problem for AEP optimization can be stated as 18max∂T∂r(V)Tyear∫VCIVCOPV,∂T∂r(V)⋅fweidVsubj.TV,∂T∂r(V)≤T0MfV,∂T∂r(V)≤Mf,0PV,∂T∂r(V)≤PratedforallV, where it should be noted that the loading is allowed to change freely with changing wind speed, which is indicated by the ∂T∂r(V). We apply the same normalization as in Sect. , where the wind speed is normalized with the rated wind speed for a rotor operating at max CP and a unit rotor radius (R=1). The normalized optimization problem is given as 19maxCLT(Ṽ)∫ṼCIṼCOCPCLT(Ṽ)⋅R̃2Ṽ3fweidṼsubj.CTCLT(Ṽ)⋅R̃2Ṽ2≤T̃0CFMCLT(Ṽ)⋅R̃3Ṽ2≤M̃f,0CPCLT(Ṽ)⋅R̃2Ṽ3≤P̃0forallṼ. Using the assumption that CLT can change independently with wind speed the maximization can be taken within the wind speed integration. Since the constraint is for all wind speeds, the optimization problem is now a power optimization for each wind speed. 20∫ṼCIṼCOmax⁡CLTCPCLT⋅R̃2Ṽ3fweidṼsubj.CTCLT(Ṽ)⋅R̃2Ṽ2≤T̃0CFMCLT(Ṽ)⋅R̃3Ṽ2≤M̃f,0CPCLT(Ṽ)⋅R̃2Ṽ3≤P̃0forallṼ,⇓(poweroptimization)21∫ṼCIṼCOmaxW0,W1CPW0,W1⋅R̃2Ṽ3fweidṼsubj.CTW0,W1⋅R̃2Ṽ2≤T̃0CFMW0,W1⋅R̃3Ṽ2≤M̃f,0CPW0,W1⋅R̃2Ṽ3≤P̃0forallṼ, where the boldface W0 and W1 signify that it is changing with wind speed.

It can be further simplified as 22AEP̃opt(R̃)=∫ṼCIṼCOP̃opt(Ṽ,R̃)fweidṼ, where the function for the output from the power optimization (P̃opt) is used. It shows that the AEP optimization can be reduced to a power optimization for each wind speed in the integration. A flowchart for the AEP optimization can be seen in Fig. . The output from the optimization is denoted as AEP̃opt.

The optimizations presented so far have been for a fixed radius increase, but in this section the optimization for rotor radius will be presented. The power optimization and AEP optimization could in principle easily be extended for radius optimization as well by simply adding the rotor radius as a design variable, but as discussed in Sect.  the optimization problem is unbounded with the global optimum at R̃→∞, which is clearly not feasible for turbine design. To get a feasible rotor design, the optimization for rotor size will also include a cost function.

This section shows the result of applying the optimization methodology described in Sect.  for increasing the rotor radius.

The input for the aerodynamic solver (Part 1, ) is as simple as possible with no viscous loss (Cd/Cl=0) and without tip loss for two different tip speed ratios (λ→∞, λ=5).

In Fig.  the optimization problem is solved for increasing values of R̃, and the power is relative to the baseline power (P̃0), which is the power at R̃=1.

For the case of λ→∞ (which means that there are no aerodynamic losses) it is seen that P̃opt is increasing to a flat plateau (a saddle point) at R̃=1.34 and ΔP=12 %. This is a similar result to that found by p. 810, Sect. 3 (they parameterized axial induction and included tip loss) where the optimal solution is said to be at R̃=1.34 and ΔP=12 %. From this analysis (without aerodynamic loss) it is found that this point is a saddle point, but including any aerodynamic loss (or non-optimal loading, like approximate optimal induction) it is found that a local optimum is formed, as can be seen for the case with λ=5 (wake rotation loss) where a local optimum is found at ΔR=23 % and ΔP=11 %.

Common to both cases is that the curve is seen to increase again beyond the saddle point/local optimum, and the curves are seen to still increase at ΔR=200 %. The global optimum is found to have an asymptotic limit as R̃→∞, with the optimal power going towards the thrust constraint limit for the case without aerodynamic losses (P̃opt→T̃0, ΔP→50 %). Similar behavior is observed for the case with aerodynamic losses. This author observed a similar behavior using 1D momentum theory but only with a thrust constraint p. 163, Fig. 6, which was also observed by p. 809. To understand why this is also the case when the loading is allowed to vary along the span with thrust and flap moment constraints, it should be noted that the loading at the tip is negative for large radius increases. The negative loading makes it possible to find a set of load distributions where the flap moment is zero (CFM=0), but crucially it can still have a positive power (CP>0). This is all possible while making the thrust loading arbitrary small (CT←0), which in turn means it is always possible to satisfy the constraints for any rotor radius increase while having a positive power (P̃opt→T̃0). Applying a similar argument for the case without a thrust constraint, it can be found that the power will grow unbounded (P̃opt→∞) since the power coefficient remains finite for increasing rotor radius (CFM=0→CP>0).

The unbounded behavior of Popt clearly leads to unfeasible designs, and for the coming rotor design example a cost function is included to make a realistic rotor design.

This section will show the result of applying WOwRI for power-per-cost (PpC) optimization at a single wind speed (assumed to be Ṽ=1).

For the rotor design, the aerodynamic losses will be included (i.e., wake rotation loss, viscous loss, tip loss). To include viscous loss, the glide ratio (Cl/Cd) needs to be given as input, and to get a realistic input for the glide ratio the DTU 10 MW reference turbine is used as a basis, in particular the aerodynamic polars as well as the relative airfoil profile thickness distribution along the span. The glide ratio used for this design can be seen in Fig. d, where the polars with relative airfoil thickness th=[24 %, 30 %, 36 %, 48 %] have been used and the design point for the polar is found as described in Sect. 3.5; some smoothing is then applied to ensure the design will be continued. The Cl and α in Fig.  are used for creating the chord and twist distributions later.

With the glide ratio from Fig. d the optimal tip speed ratio (λ) can be found as described (Part 1, Sect. ). The optimal λ and the one used in this section is λ=8.23.

A plot of the relative power per cost P̃optP̃0fcost for increasing rotor radius can be seen in Fig. . The optimum is found at a radius increase of ΔR=7.9 %, leading to an increase in power per cost of ΔPPC=4.2 % and a power increase of ΔP=9.0 %. From the plot it can also be seen that around R̃≈1.21 the impact of increasing the rotor radius leads to a lower power per cost relative to the baseline.

A comparison of the loading distribution is shown in Fig. . The plot shows that the loading distribution tapers towards the tip for the optimal design relative to the baseline design. Figure a shows the thrust loading density (∂T/∂r) per blade (assuming three blades) and Fig. b the power density (∂P/∂r) per blade as a function of the rotor radius (r). The solid black line is the value for the PpC-optimized rotor, and the difference to the baseline is highlighted with shaded regions, where green indicates a positive impact and red indicates a negative impact. The striking thing to see here is how large the decrease is (the shaded green region) in Fig. a and how little impact this lower loading has on the loss of power in Fig. b (shaded red region). This has all to do with the fact that operating at maximum CLP a change in CLT will not lead to a proportional change in CLP, much like the observation made by this author in p. 157, Fig. 1 using only 1D momentum theory. Another interesting thing is that it is only the Mf constraint that is active, which means that this PpC-optimized rotor also comes with a lower thrust of ΔT=-3.8 %.

The rotor planform (blade chord and twist) can be found from the loading distribution (CLT, Fig. ), the lift coefficient and angle of attack (Cl, α, Fig. a and c), through Eqs. (36) and (37) in Part 1 (, Sect. 4.1). A plot of the rotor planform can be seen in Fig. . The figure shows chord and twist for the PpC-optimized rotor, the baseline rotor (R̃=1) and the DTU 10 MW reference turbine. A clear thing to see from these plots is that the optimization did not include a max chord constraint, with the max chord being ≈27 m, which is much larger than the DTU 10 MW reference turbine where the max chord is 6.2 m. Looking at Fig. , it is seen that the region from r<35 m has a similar loading as the baseline. Thus the optimization is not exploiting the maximum chord for significant gains, and one can safely correct these aberrations after. For r>35 m the chord is seen to be smaller than the DTU 10 MW reference for both the baseline and the PpC-optimized rotor, with the exception of the longer blade for the PpC-optimized rotor. Comparing the baseline and the PpC-optimized rotor, the chord is seen to be the same around r≈60 m, with the cost-optimized chord being slightly smaller from this point until the tip loss starts to become significant (which is the reason that the chord is going to zero at the tip). The smaller chord is an effect of the tapering CLT for the PpC-optimized rotor. Thus, the lower loading distribution leads to a reduction in the chord. This may have structural implications (i.e., reduced strength and stiffness) that are not accounted for in this optimization.

For the twist (Fig. b), the difference between the baseline and PpC-optimized rotor is relatively small, with an almost constant offset of 1.5∘ as can be seen from the Δθtwist plot. The change is fairly small since the flow angle is approximately ϕ=tan⁡-11/λr̃, and the change in CLT only has a small impact.

In this section the result of solving for the optimal annual energy production (AEP) is shown, as explained in Sect. , which resulted in AEP̃opt (Eq. ). The aerodynamic input is the same as for power optimization with no viscous loss (Cd/Cl=0), no tip loss and no wake rotation loss (λ→∞), but also including a case with large wake rotation loss (λ=1.5) to show that a local optimum is formed when aerodynamic loss is added.

When solving the optimization problem in Eq. () the wind speed integration was discretized in 200 steps, which was found to make the discretization error insignificant. The integration is then performed using the trapezoidal rule.

The solution for solving the AEP optimization problem can be seen in Fig. . The AEP optimization is seen to have similar behavior as the power optimization in Fig.  with an initial large slope, followed by a flatter region, and then the AEP begins to improve again. The AEP optimization does not reach a saddle point or local maximum for the case of λ→∞, as was the case for the power optimization. The slope is always positive. For the case of λ=1.5 a local optimum is found, but the formation of this local maximum required a significant amount of aerodynamic loss (λ=1.5, which leads to a large wake rotation loss) compared to the power optimization where any aerodynamic loss would lead to the formation of a local maximum.

As was the case for the power optimization the global optimum for AEP optimization is found to be a similar asymptotic limit with the optimum as R̃→∞ (ΔAEP→70 %). This is the case both for λ→∞ and λ=1.5. The global optimum for the AEP optimization tends to a power curve which almost runs at rated power for all wind speeds, but the maximum power for a given wind speed is P̃=T̃0Ṽ, and for a small region of the power curve the power will follow this limit before it reaches rated power. This limit is mostly of academic interest since it is not feasible for practical turbine design, and it is not investigated further here.

We turn to the power and load curves for the four highlighted points in Fig. . As expected, the baseline is simply operating at maxCP until rated power, creating the familiar Ṽ3 behavior for the power and Ṽ2 for the loads, where all the peak loads occur at rated conditions. However, comparing the different optimal solutions along this curve reveals different load profiles than typical modern turbines. Small increases in rotor radius increase the AEP by reaching rated power earlier. In all the extended rotor cases, the root flap-wise bending moment constraint becomes active before rated conditions are reached. Initially, this relaxes the thrust constraint. This bending moment constraint seems to limit the maximum achievable power over a greater range of the power curve. This seems to impose a minimum wind speed that rated power can be achieved; furthermore, increases in AEP must be achieved at lower wind speeds. Finally, for very large rotors, it seems that the moment constraint is active at all wind speeds, and the optimization starts to become further constrained by the thrust constraint. In general, the power curve is found to fall into three regimes in terms of wind speed (Ṽ), which are

max CP (no active constraints),

In this section, the result of solving for the optimal AEP per cost (AEPpC) is presented. At first, the optimal power curve is presented, and at the end common wind turbine technology is used to see how close it can get to the optimal power curve.

The optimization will use the same aerodynamic input as in Sect.  (rotor design with cost function), with λ=8.23, glide ratio as in Fig. d) and including tip loss.

AEPpC for increasing values of R̃ can be seen in Fig. , where the AEPpC optimal power curve is highlighted as well as the baseline. The optimal AEPpC is found to increase by ΔAEPpC=2.9 %, with a fairly large radius increase of ΔR=17 % as well as a fairly large AEP increase of ΔAEP=13 %. In Fig.  it is also possible to see the power curve as well as the difference to the baseline. The increase in the power is seen to also increase for increasing Ṽ until rated power.

Figure  shows this power curve along with the loads and loading distribution in greater detail. Three operational regimes can be seen in Fig. , separated by vertical dashed lines. The optimal power curve is seen to only have an active Mf constraint starting at Ṽ≈0.79 up until rated power at Ṽ≈0.94. In this region, the thrust curve is seen to change the slope and become linear, but it does not reach the constraint limit (vertical dashed line). The loading distribution (CLT) for four selected points can also be seen. Starting from the point just before the Mf constraint becomes active, the loading is the one that maximizes CP as it has been all the way up until this point. CLT is then seen to progressively taper towards the tip as the wind speed increases.

The presented optimal power curve can not be made into a blade design as was done in Sect. , since the loading distribution was varied independently at each wind speed. The presented AEPpC optimization can therefore be seen as the idealized power curve much like the Betz limit is the idealized maximum power a turbine can achieve. It is therefore not possible, within the design constraints and aerodynamic modeling, to do any better than this optimal power curve. In the next section, it is investigated how close it is possible to get to the optimal power curve using common wind turbine technology.