Robustly Optimal Fixed Pitch HAWT

(with Tip Correction & Drag )

•Exact trig solution of BEM optimal rotor

•New criterion of robust optimality

•Max annual power pitch& chord

s= 2psin r/B advance/B. H=I/F, F a function of (R-r)/s.221s. ½ W CL r x W = -dL /dA= 2 I (Wz) the (BEM) eqn (1)

dP= xT (dL+dD)=2 xT (Wz) I (sin - cos tane) dA I=FH H=Nsin(-), Wz =Wsin, W=Ncos(-)

dP/dA= xT N2 F sin2(-)(sin2 -½sin2 tane)

2cote= cot +cot (3-2) =2/3+e/6 +

Robustly Optimal Blade Elements

CL =4Fsin tan(-) CL / 2= e sin, -=½¼e sin ( -r)=Fsin tan(½¼e) =½e =¼eBc/r

cos( -r)=d { Fsin tan(½ ¼e)} / d robust

c o t r = cot d + (1-½de/d) csc(d ½e)+ dF/ Fd

cos (pF/2)=e-f , f=p(R-r)/s=B(R-r)/ 2rsino

cotr cot ½d+½ecscd cotd -½de/dcsc(d) - 2 f cot (½pF) cotd/pF

r½d/h,h=1+¼/d-¼d/d-fcot(½F)/ F

r ½d –/8 Tip r2d /3 –/8

robust r sinr =Fsind tan(½d¼e)

perturb =r cot r effect on d/d vs robust d/d

Fsin tan(-)=sin(-)

Fsinsec2(-)= u= vd/d v=Fsin sec2(-)-....+cos

d/d =u/v (d/d) -(u/v)2 v/ u

u/v=d/d=2/3, v =(cos)=/ sin

d(-)/d-4/9tan(-)cos2(-)/sin2r 4(d-r) h2/ 9tan(½d)

Blade Pitch for highest net mean Annual power dP/dA=-½ xTN2F sind

{6(-o)2 + ½d2/d2sin2d (ad-ao)2} mean 6{4(ad-ar) cot½h2/9}2 + ½d2/d2sin2d(ad-ao)2

Relate , x to T: synchronous generator X as T -1.

p.d. rotary pump fixed torque has T2CT = T2Cp/X so X T2. Say xTg, g2 >0 Then with τ=T/Td , gτ sin 2 . .

peak p(τ)=3{1-4(τ -1)2}/2 between τ =.5 and1.5 (τ)2= .05 (ad-ar)8h4cot2½d+(ad-ao) d2/d2270sin2d/g2sin23d 0

WIND SHEAR as T0(1+ msin β) where m= k x /X

p(β)= (1+ msinβ)3/a averaging over β, a=1+3m2/2. t 0= T/T0 t0= msin β

in τ0= T/T0 aτ0=3 m2/2+3m4/8 and a(τ0)2 = ½ m2+9m4/8.

τ 0 = 1+τ0 1+3m2/2-15m4/8 and the true minimum variance about it is

(τ0) 2 - { τ0}2 ½ m2-15m4/8 , so {τ}2 ½ m2-27m4/8 ;

r at k=1/3 1+k=2(1-k) at x/X=.7, about .017 versus .05 above;

T i p , {τ}2 is exactly .029 .

Such independent variances ADD.

YAW the variance is proportional to the mean 4th power of yaw angle. Need wind

data and yawcontrol algorithm.