q-Factor Optimization for 2-WEC Arrays
Lawrence V. Snyder Dept. of Industrial and Systems Engineering Lehigh University, Bethlehem, PA
COR@L Technical Report 14T-001
q-Factor Optimization for 2-WEC Arrays Lawrence V. Snyder∗1 1
Dept. of Industrial and Systems Engineering , Lehigh University, Bethlehem, PA
February 7, 2014
Abstract We derive an analytic solution for the wave energy converter (WEC) layout problem with N = 2 devices, assuming regular (sinusoidal) incident waves, under the point-absorber approximation.
Assume WEC 1 is located at the origin and WEC 2 is located at (d, α). We assume there is a minimum-separation constraint of the form d ≥ d0 λ, (1) where λ is the incident wavelength and d0 > 0 is a constant. Then (d1 , α1 ) = (0, 0); (d2 , α2 ) = (d, α); d12 = d21 = d. Let z ≡ kd cos(β − α). Then 1 1 L∗ = 1 e−iz (2) L = ikd cos(β−α) = iz e e 1 J0 (kd) J= J0 (kd) 1
J
−1
1 1 −J0 (kd) = 1 1 − J0 (kd)2 −J0 (kd)
since J0 (0) = 1. Therefore: 1 q = L∗ J−1 L 2 1 1 −J0 (kd) 1 −iz 1 e = 2 −J0 (kd) 1 eiz 2(1 − J0 (kd) ) 1 1 −iz −iz 1 − J0 (kd)e −J0 (kd) + e = eiz 2(1 − J0 (kd)2 ) 2 − J0 (kd)(e−iz + eiz ) 2(1 − J0 (kd)2 ) 2 − J0 (kd)(2 cos z) = 2(1 − J0 (kd)2 ) =
∗
[email protected]
2
(3)
since e−iz + eiz = (cos z + i sin z) + (cos z − i sin z) = 2 cos z =
1 − J0 (kd) cos(kd cos(β − α)) 1 − J0 (kd)2
Now, we can choose any α, so we can get any β −α ∈ [−π, π] ( mod 2π), so we can get any value of cos(β −α) in [−1, 1], so we can get any value of cos(kd cos(β − α)) in [−1, 1], assuming kd ≥ π (which is almost certainly true). Note that J0 (0) = 1 and for x > 0, J0 (x) resembles a cosine function whose amplitude decreases as x increases. (See Figure 1.) Let jn be the nth local optimizer (min or max) of J0 (·). Maximizers have odd indices n while minimizers have even indices. Approximate values are given in Table 1. (Ignore the last three columns for now.) Figure 1: Bessel function of the first kind of order 0. 1
0.5
0
−0.5
0
5
10
15
20
25
Table 1: Local optima of Bessel function, and resulting optimal q (for any k, β) and optimal locations of WEC 2 (for k = 0.2, β = 0). n 1 2 3 4 5 6 7 8
jn 0.0000 3.8317 7.0156 10.1735 13.3237 16.4706 19.6158 22.7601
J0 (jn ) 1.0000 -0.4028 0.3001 −0.2497 0.2184 −0.1965 0.1801 −0.1672
q∗ ∞ 1.6744 1.4288 1.3328 1.2794 1.2445 1.2196 1.2007
d∗ — 19.1585 35.0780 50.8675 66.6185 82.3530 98.0790 113.8005
α∗ — −1.5708 −1.1065 −1.5708 −1.3328 −1.5708 −1.4099 −1.5708
x∗ — 0.0000 15.7080 0.0000 15.7080 0.0000 15.7080 0.0000
y∗ — −19.1585 −31.3644 −50.8675 −64.7401 −82.3530 −96.8130 −113.8005
(Local optima were found using Matlab fminsearch with default settings.) Consider a fixed value of d. We consider two cases: Case 1: J0 (kd) ≥ 0. Then q is maximized when α is chosen such that cos(kd cos(β − α)) = −1, in which case q=
1 + J0 (kd) 1 . = 1 − J0 (kd)2 1 − J0 (kd)
(4)
We want J0 (kd) to be as small as possible. Since local mins become progressively larger as d increases, we want the smallest local argmin possible, which means choosing the smallest d ≥ d0 λ so that kd is a local argmin of 3
J0 (·), unless J0 (kd0 λ) is already smaller than J0 of this argmin, in which case we should set d = d0 λ. Therefore, ( ∗ jn , if J0 (jn∗ ) ≤ J0 (kd0 λ) ∗ d = k (5) d0 λ, otherwise, where n∗ = 2m∗ and m∗ is the smallest integer m such that j2m ≥ kd0 λ. (In other words, n∗ is the smallest argmin greater than or equal to kd0 λ.) We want cos(kd cos(β − α∗ )) = −1, i.e., π arccos(−1) ∗ α = β − arccos = β − arccos . (6) kd kd In the special case of β = 0, this solution translates to the following rectangular coordinates (assuming the first case in (5) holds): x∗ = d∗ cos(α∗ ) j∗ π π = n· ∗ = k jn k ∗ ∗ ∗ y = d sin(α ) π jn∗ sin − arccos = ∗ k kd jn∗ π =− sin arccos k jn∗ s 2 p jn∗ π =− 1− (sin(arccos x) = 1 − x2 ) k jn∗ s π2 jn∗ 1− 2 =− k jn∗ Case 2: J0 (kd) < 0. Then q is maximized when α is chosen such that cos(kd cos(β − α)) = 1, in which case q=
1 − J0 (kd) 1 = . 2 1 − J0 (kd) 1 + J0 (kd)
(7)
We want J0 (kd) to be as large as possible. Since local maxes become progressively smaller as d increases, we want the smallest local argmax possible, which means choosing the smallest d ≥ d0 λ so that kd is a local argmax of J0 (·), unless J0 (kd0 λ) is already larger than J0 of this argmax, in which case we should set d = d0 λ. Therefore, ( jn∗ if J0 (jn∗ ) ≥ J0 (kd0 λ) ∗ k , d = (8) d0 λ, otherwise, where n∗ = 2m∗ + 1 and m∗ is the smallest integer m such that j2m+1 ≥ kd0 λ. (In other words, n∗ is the smallest argmax greater than or equal to kd0 λ.) We want cos(kd cos(β − α∗ )) = 1, i.e., arccos(1) π ∗ α = β − arccos =β− . (9) kd 2
4
In the special case of β = 0, this solution translates to the following rectangular coordinates (assuming the first case in (8) holds): x∗ = d∗ cos(α∗ ) π jn∗ = cos − =0 k 2 y ∗ = d∗ sin(α∗ ) π jn∗ = sin − k 2 jn∗ =− k Theorem 1 Let n∗ be the smallest integer n such that jn ≥ kd0 λ. Then the optimal solution (d∗ , α∗ ) and the corresponding q ∗ , are as follows: ( j n∗ if |J0 (jn∗ )| ≥ |J0 (kd0 λ)| ∗ k , d = d0 λ, otherwise ( β − arccos kdπ∗ , if J0 (jn∗ ) ≥ 0, ∗ α = β − π2 , if J0 (jn∗ ) < 0 1 q∗ = 1 − |J0 (kd∗ )| In the special case of β = 0, and assuming |J0 (jn∗ )| ≥ |J0 (kd0 λ)|, the translation of the optimal solution to rectangular coordinates (x∗ , y ∗ ) is: (
x∗ = y∗ =
π k,
if J0 (jn∗ ) ≥ 0, 0, if J0 (jn∗ ) < 0 ( j ∗q 2 − nk 1 − jπ2 , if J0 (jn∗ ) ≥ 0, − jnk∗ ,
n∗
if J0 (jn∗ ) < 0
Numerical values for the solutions under different values of n∗ , assuming β = 0 and k = 0.2, are given in the last set of columns in Table 1.
5
