Iterative learning control applied to a non-linear vortex panel model for improved aerodynamic load performance of wind turbines with smart rotors

Figures & data

Figure 1. NREL S825 airfoil.

Figure 2. Flow at the trailing edge. The×marks the vortex creation point.

Figure 3. Controller (11(11) $$u^k=\mu \Delta t{E}^{k-1}$$(11) ) with μ = 10. Error E^k with no control, u^k = 0 (red). Error E^k with control (green). Control input u^k (blue).

Figure 4. Error E^k for controller (11(11) $$u^k=\mu \Delta t{E}^{k-1}$$(11) ) with μ = 10 (red), μ = 20 (green) and μ = 30 (blue).

Figure 5. Controller (13(13) $$u_j^{k_c}=u_{j-1}^{k_c}+\mu \Delta t{E}_{j-1}^{k_c+\Delta }$$(13) ) with μ = 1 and Δ = 0. Error E^k (red) and control input u^k (green).

Figure 6. The controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 1, μ₁ = 30 and Δ = 0. Error E^k (red) and control input 30×u^k (green).

Figure 7. Error E^k for oscillatory flow past the airfoil with two vortices with no control (case 1).

Figure 8. No control error E^k for case 1 (red). Error E^k for controller (11(11) $$u^k=\mu \Delta t{E}^{k-1}$$(11) ) with μ = 30 (green).

Figure 9. No control error E^k for case 1 (red). Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) -16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₁ = 30 (green), μ₀ = 1 and Δ = 0 (green).

Figure 10. No control error E^k for case 1 (red). Control input u^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ applied to case 1 (green).

Figure 11. No control error E^k for case 4 (red). Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 (green) (error scaled by 100, magenta), and control input u^k (blue).

Figure 12. No control error E^k for case 6 (red). Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 applied to case 6 (green).

Figure 13. No control error E^k for case 6 (red). Control input u^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 applied to case 6 (green).

Table 1. Parameters for selected cases using the two-term ILC.

Case	M	AoA	A	T	Lr	T0	T1
1	2	0°	0.1	2.5	0.379	10	50
2	2	7°	0.1	2.5	0.802	10	50
3	2	7°	0.1	2.5	0.802	10	50
4	2	0°	0.1	10	0.379	50	250
5	2	7°	0.1	10	0.802	50	250
6	12	0°	0.05	20	0.379	100	250

Table 2. Error norms for selected cases using the two-term ILC.

Case	Norm	No control	Control	Ratio×100
1	${\mathcal{L}}_2$	0.0411	0.0013	3.1
	${\mathcal{L}}_{\infty }$	0.0812	0.0100	12.3
2	${\mathcal{L}}_2$	0.0843	7.5×10^−4	0.9
	${\mathcal{L}}_{\infty }$	0.1469	0.0045	3.1
3	${\mathcal{L}}_2$	0.0923	0.0020	2.2
	${\mathcal{L}}_{\infty }$	0.2324	0.0135	5.8
4	${\mathcal{L}}_2$	0.0444	5.6×10^−5	0.1
	${\mathcal{L}}_{\infty }$	0.0916	4.5×10^−4	0.5
5	${\mathcal{L}}_2$	0.0900	3.4×10^−5	0.04
	${\mathcal{L}}_{\infty }$	0.1608	1.9×10^−4	0.1
6	${\mathcal{L}}_2$	0.0235	1.4×10^−4	0.6
	${\mathcal{L}}_{\infty }$	0.0406	0.0013	3.2

Figure 14. Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = applied to case 1 with no delay (red), λ = 100 and λ = 10 (blue). The errors for no delay and with λ = 100 are almost the same when plotted at this level.

Figure 15. Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 applied to case 1 with no delay (red), λ = 1 (green), λ = 0.1 (blue) and λ = 0.01 (magenta).

Figure 16. Control inputs u^k for the actuator delays λ in Figures 14 and 15. No delay (red), λ = 100 (green), λ = 10 (blue), λ = 1 (magenta), λ = 0.1 (light blue), λ = 0.01 (yellow).

Figure 17. Error norms for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 for different λ; left, ${\mathcal{L}}_2$ and right, ${\mathcal{L}}_{\infty }$. Case 1 (red), case 2 (green) and case 3 (blue). R is the ratio of the norms against the values for no delay.

Figure 18. Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 applied to case 6 with no delay (red) and λ = 10 (green). The control with λ = 100 produces identical results to that with no delay at this scale.

Figure 19. Error E^k for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) )–(16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 15, Δ = 0 and μ₁ = 100 applied to case 6 with no delay (red), λ = 1 (green), λ = 0.1 (blue) and λ = 0.01 (magenta).

Figure 20. Error norms for controller (14(14) $${\widehat{u}}_j^{k_c}=u_{j-1}^{k_c}+{\mu }_0\Delta t{E}_{j-1}^{k_c+\Delta }$$(14) –16(16) $$u_j^{k_c}={\widehat{u}}_j^{k_c}+{\bar{u}}^k$$(16) ) with μ₀ = 30, μ₁ = 1 and Δ = 0 for different λ; left, ${\mathcal{L}}_2$ and right, ${\mathcal{L}}_{\infty }$. Case 4 (red), case 5 (green) and case 6 (blue). R is the ratio of the norms against the values for no delay.

Figure 21. Free stream velocity oscillation (red) and control input (green) for case 5 (left) and case 6 (right).