Dynamics and control of flexible, free standing solar towers
Michael (a)
Chi(a) ,
Vakhtang Putkaradze(a,b) Francois Gay-Balmaz(c) , Peter Vorobieff
(d)
Mathematical and Statistical Sciences, University of Alberta (b) Chemical Engineering, University of Alberta (c) Ecole Normale Superiore and CNRS, Paris (d) Mechanical Engineering, University of New Mexico putkarad@ualberta.ca
October 27, 2015 Waterloo Institute for Complexity and Innovation 1
M. Chi, F. Gay-Balmaz, V. Putkaradze and P. Vorobieff, Dynamics and control of flexible solar updraft towers, Proc. Roy. Soc A, 471, 20140539 Vakhtang Putkaradze Dynamics and control of flexible solar towers (2014).
Solar updraft towers: design considerations
The general idea
VP, P. Vorobieff, A. Mammoli and N. Fahti, Inflatable free-standing flexible solar towers, Solar Energy 98 (2013).
Vakhtang Putkaradze
Dynamics and control of flexibleSolar solar towers towers – p. 3/35
Advantages of solar tower technology
1
Huge thermal mass means very low variability of the output (unlike any other solar power facility)
2
Continuous production of dispatchable energy possible
3
Low maintenance (only for turbines)
4
Does not need continued supply of water
5
Can be built from cheap (and locally sourced) materials
Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Solar updraft tower in Manzanares, Spain
Figure : Tower in Manzanares seen through polyester roof Vakhtang Putkaradze
Dynamics and control of flexible solar towers
History of solar towers Invented by colonel Isidoro Cabanyes, La energia electrica (1903) Built in Manzanares, Spain for âˆź $1 mill; 195 m tall tower, 10 m in diameter, 122 m collector radius. Produced 50kW (peak), 36 kW (average) (Funded by Germany). Manzanares plant built in 1982; tower collapsed in 1989 due to rusting of guy wires; decommissioned. Jinshawan Updraft Tower (Inner Mongolia, China); cost $200 mill. Current power 200kW; target power 27.5 MW. Planned: 40MW Ciudad Real Torre Solar (Spain, EU funded); 750 m tower. No information is available on current state; cost unknown. Planned: peak 200MW tower in Arizona; height âˆź 800m, cost $750 mill; all power produced contracted with California for 30 years. Construction goal: 2015. However, funding only partially secured (35 mill out of 800). Namibian government approved proposal for 400MW Greentower, 1.5km high, 280 m in diameter, collector of 37 km2 . Cost unknown. Never built. Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Tower is the obstacle!
Rigid towers are too expensive and difficult to built at these scales and holding back the technology. Technology is not competitive on $-to-$ basis with either renewable or non-renewable sources. Projects have to be extremely large and expensive due to tower Is there any way to make the tower cheaper? YES!
Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Shape of the Eiffel tower 10/27/2015
1
Equation for shape of Eiffel tower [Weidman and Pinelis, C. R. Acad Sci. Paris, 2004]
2
Eiffel’s idea: At every level, torque produced by the wind is balanced by torques due to tangent forces
3
Resulting nonlinear integro-differential equations; can’t solve it analytically but w = Ae −λz fits for all A, λ
4
Trying to fit exponential to the tower does not work
5
Answer Eiffel started with one value of λ and then changed to another (larger) value Vakhtang Putkaradze
8c0a1ea7_11895163134_de71183349_o.xxxlarge_2x.jpg (204
Dynamics and control of flexible solar towers
General idea
Figure : A sketch of proposed design Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Tower design: general considerations
Make the tower consisting of adjacent toroidal elements Deformation of the tower under the wind is distributed proportionally to the tori size along the tower Gives tower shapes given the ratio inner/outer radii for tori and max allowable wind. √ Restoring force is proportional to relative tilt growing faster than any linear function ⇒ super-stability.
Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Tower design: explicit tower Fixed inner radius, varying outer tower – explicit tower.
Produces very little energy due to constriction on top (âˆź 1MW ). Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Tower design: implicit tower Fixed inner radius, varying outer tower – implicit tower.
Efficient design: power production âˆź 50MW. Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Dynamical theory M. Chi, F. Gay-Balmaz, V. Putkaradze and P. Vorobieff, , Dynamics and optimal control of flexible solar updraft towers, Proc. Roy. Soc. A 2014.0539 (2014). Setup of the problem: evolution of n-th torus is described in terms of orientation Λn and position rn .
en+1 = 3 n+1 n n
hn(µn, en3 =
1
en3
1
=
n+1 E3 n)
contact
qn n E3
dn
n 1 E3
qn θn
hn (µn , θn )
Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Symmetry invariance (no gravity) Fundamental assumption: If there is no gravity, interaction of n and n + 1-th tori is invariant with respect to rotation and translation of the whole system Mathematically: define the element of SE (3) group (Λ, r) with group multiplication Λ, r · Λ0 , r0 = ΛΛ0 , Λr0 + r Then, potential energy of deformation Ud between n and n + 1 torus is given by the combination −1 −1 Λn , rn · Λn+1 , rn+1 = Λ−1 n Λn+1 , Λn (rn+1 − rn ) We can define SO(3) (rotationally)-invariant variables ωk ρk µk
˙ = Λ−1 k Λk ∈ so(3), −1 = Λk rk ∈ R3 , = Λ−1 k Λk+1 ∈ SO(3),
3 γ k = Λ−1 k r˙ k , ∈ R , k = 1, ..., N −1 3 κk = Λk g ∈ R , k = 1, ..., N k = 1, ..., N − 1
Euler-Poincar´e theory of symmetry-reduced Lagrangian Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Euler-Poincar´e dynamics of a rigid body I Configuration of a rigid body is described by its rotation with respect to a fixed frame Λ, with ΛT Λ = Id. Ë™ is rotationally invariant, and depends Lagrangian L = L(Λ, Λ) b = Λ−1 Λ˙ from physics on the left-invariant quantity â„Ś (antisymmetric matrix): Antisymmetric matrices are tangent vectors of SO(3) at unity, or belong to the Lie Algebra so(3): take Λ( ) ∈ SO(3), Λ(0) = Id and differentiate wrt at = 0:
d T Λ( ) Λ( )
= Ab + AbT = 0 d =0 Antisymmetric matrices are equivalent to R3 vectors through b = A Ă— x, for all x ∈ R3 . the hat map Abij = ijk Ak : Ax Lagrangian is ` = 21 â„Ś ¡ Iâ„Ś where I is a constant matrix (inertia tensor). Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Euler-Poincar´e dynamics of a rigid body II One wayR to write equations: Critical action principle ˙ δS = δ L(Λ, Λ)dt with the constraint ΛT Λ = Id (possible, but tends to be awkward). b = Λ−1 δΛ ∈ so(3) and Another way: define free variations Σ compute b = δ Λ−1 Λ˙ =−Λ−1 δΛΛ−1 Λ˙ + Λ−1 δ Λ˙ = −Σ bΩ b + Λ−1 δ Λ˙ δΩ b = ∂t Λ−1 Λ˙ =−Λ−1 ΛΛ bΣ b + Λ−1 δ Λ˙ ˙ −1 δΛ + Λ−1 δ Λ˙ =−Ω ∂t Σ h i b =Σ b˙ − Ω, b Σ b ⇔ δΩ = Σ ˙ −Ω×Σ Subtract to obtain δ Ω Take variations for arbitrary Σ Z Z Z ˙ − Ω × Σ dt 0 =δ `(Ω)dt = IΩδΩ = IΩ · Σ Z ˙ + Ω × IΩ · Σ = 0 =− IΩ ˙ + Ω × IΩ = 0. Equations of motion for the rigid body IΩ Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Tower dynamics: variations Introduce gravity using advected quantities κn = Λ−1 n g Explicitly separate over-pressure dependence p in all quantities (index 0) N
1X (pn +p0 ) I0 ω n ·ω n +M0,n |γ n |2 `= 2 n=1
−pn U0,n (µn , ρn , ρn+1 )−(pn +p0 )M0,n ρn · κn .
−1 Introduce free variations Σk = Λ−1 k δΛk and Ψk = Λk δrk and compute the variations, for example: −1 ˙ −1 ˙ c ck + Λ−1 δ Λ˙ k ˙ δb ωk = δ Λk Λk =−Λ−1 δΛk Λ−1 k Λk + Λk δ Λk = −Σk ω k k −1 ˙ −1 ˙ −1 −1 ˙ b k + Λ−1 δ Λ˙ k ∂t Σk = ∂t Λk Λk =−Λk Λk Λk δΛk + Λk δ Λk =−b ωk Σ k
Subtract equations to get rid of δ Λ˙ k to obtain the variations: i bk h dΣ dΣk bk, ω δc ωk = + Σ bk ⇔ δω k = + Σk × ω k dt dt Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Tower dynamics: Variations and Critical action Variations are computed as ˙ k − Σk × ω k δω k = Σ ˙ k + ω k × Ψk − Σ k × γ k δγ k = Ψ δρk = Ψk − Σk × ρk δµk = µk Σk+1 − Σk µk δκk = −Σk × κk , Critical action principle for symmetry-reduced Lagrangian: N X δ` · δω k δω k 0 0 k=1 ! N−1 X δ` δ` δ` δ` + ·δγ k + ·δρk + · δκk + ,δµk dt δγ k δρk δκk δµk
Z
0 =δ
T
Z
T
`(ω, γ, ρ, µ, κ)dt =
k=1
Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Calculation of reduced variations
Equations of motion are computed by collecting terms proportional to Σk and Ψk together with kinematic equation for µk : δ` δ` δ` δ` d + ωk × + γk × + ρk × + κk × = Zk +τ k dt δω k δγ k δρk δκk d δ` δ` + ωk × − = 0, dt δγ k δρk µ˙ k = µk ωk+1 − ωk µk where A∨ is inverse of hat map (antisymmetric matrices→ vectors in R3 ), τ k is wind torque, and Zk is torque due to deformations !∨ !∨ δ` T δ` T δ` δ` T T Zk := µk−1 − µk−1 + µk − µ δµk−1 δµk−1 δµk δµk k Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Dynamics in two dimensions The Lagrangian is L = L(ϕ, ϕ) ˙ =
N N−1 X 1 X Skl ϕ˙ k ϕ˙ l − pn U0 (∆ϕn ) , 2 k,l=1
Skl = Slk
n=1
Equations of motion for fixed pressures pn : (Sϕ) ¨ n + pn−1 U00 (∆ϕn−1 ) − pn U00 (∆ϕn ) = 0, n = 1, ..., N,
Figure : Snapshots of dynamics with 20 tori. All sizes are in m. Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Experimental prototype 3m prototype build on the roof of UNM ME building, easily withstood 100km/r wind.
Figure : Photographs of UNM solar chimney prototype: a) wind speed 0.7 m/s from the south (left) and overall view, b) wind speed 2.2 m/s, c) wind speed 20.1 m/s. The overpressure in the tori is uniform at 0.17 atm. Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Comparison between theory and experiment 0.25
0.2
φ
0.15
0.1
0.05
0
5
10
15
k
20
25
30
35
Figure : Left: steady deflection angles. Right: comparison of the shape given by a convergent steady state of theory with experiments for 2.2 m/s wind speed.
Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Control of dynamics using pressure in the tori Use of air gives a possibility of inflating and deflating each individual torus to prevent ’breaks’ in shape. R Hamilton-Pontryagin Principle: optimize C (p, φ)dt subject to equations of motion by taking Z X 0 = δ C (p1 , . . . pN , φ1 , . . . , φn )+ vk ·(k-th equation of motion) k
Variations with respect to vk give equations of motion Variations with respect to pk and vk give new set of optimal control equations Example: consider the ’straightening’ cost function C (ϕ1 , ..., ϕN−1 , p1 , ..., pN ) = − Vakhtang Putkaradze
N−1 N 1X 1X βk cos ∆ϕk + αk F (pk ) 2 2 k=1
k=1
Dynamics and control of flexible solar towers
Optimal control equations Equations of motion obtained by variations with respect to δvk ! N X d k (S0 + pk S1 )ϕ ˙ +pn−1 U00 (∆ϕn−1 )−pn U00 (∆ϕn )−τnw = 0 dt k=1
n
Variations with respect to δpk give ˙ + ∆vn U00 (∆ϕn ) = 0 , αn F 0 (pn ) − v˙ T Sn1 ϕ
∆vn := vn+1 − vn
Variations with respect to ϕk give d X k 1 pk S1 v˙ − (βn sin ∆ϕn − βn−1 sin ∆ϕn−1 ) dt 2 n k
− ∆vn pn U000 (∆ϕn ) + ∆vn−1 pn−1 U000 (∆ϕn−1 ) = 0 Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Simplification: single pressure and shape function Assume that all pressures in the tori are the same pn = p (maintained by a single pump) Assume that there is a single function predicting the shape ϕk = ak ϕ, ak =const (confirmed by experiments; optimal design is ak = αk). Assume a general shape control C (φ, p) = αF (p) + βG (φ) Control equations: 0 00 00 ∂t ((A0 + A1 p)v˙ ) + βG (φ) + v (pU1 (φ) + U0 (φ)) = 0 ∂t (A0 + A1 p)φ˙ + pU10 (φ) + U00 (φ) = τ w αF 0 (p) − A1 v˙ φ˙ + vU10 (φ) = 0.
These equations possess a first integral of mysterious origin ˙ (A0 +A1 p)v˙ φ+(αF (p) + βG (φ))+v pU10 (φ) + U00 (φ) = D = const. Vakhtang Putkaradze
Dynamics and control of flexible solar towers
Control dynamics Assume that the force of wind is is known for a finite interval (e.g. using anemometers at a distance from the tower). Φ
U
0.20 0.18
5000
-0.2
0.16
-0.1
0.1
0.2
0.14 Φ 0.12 0.10
-5000
5
10
15
20
t
P 0.80
v 6. ´ 10-7
0.78
4. ´ 10-7
0.76
2. ´ 10-7
0.74 0.72
-2. ´ 10-7 5
10
15
5
10
15
20
t
t -7 20 -4. ´ 10
Figure : Bi-stable configuration with gravity; - no control; - control Vakhtang Putkaradze Dynamics and control of flexibleblack solar towers
Conclusions and future directions
We have derived the equations of motion and optimal control for the towers Will need to extend the control theory to 3D Incorporate individual pressure changes in tori for control Looking for funding to build a 90 ft prototype in AZ Interesting applications for energy generation and CO2 capture from the air
Vakhtang Putkaradze
Dynamics and control of flexible solar towers