Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban...
Transcript of Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban...
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Urban heat islands:An optimal control approach
L.J. Alvarez-Vazquez, F.J. Fernandez,
N. Garcıa-Chan, A. Martınez,
M.E. Vazquez-Mendez.
Departamento de Matematica Aplicada II
Universidad de Vigo
Spain
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The Urban Heat Island: a metheorological phenomenon
(published on website. 30 June, 2010)http://blog.rtve.es/eltiempo/2010/06/isla-de-calor.html
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Connected problems: “Beret polution”
http://elpais.com/diario/2011/10/07/catalunya/1317949650_850215.html (published on website. 7 October, 2011)
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Improvement Strategy: Build green zones
http://www.construyeargentina.com/wp-content/uploads/2013/06/green_roof3b.jpg
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Objective: Control UHI by building green zones and parks
http://www.fincasfiol.com/wp-content/uploads/2013/09/VERDEVERTICAL_0721.jpg
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Numerical Simulation Optimal Control Numerical Results
Table of contents
1 Numerical Simulation: air temperature and velocityMathematical modelNumerical Resolution
2 Optimal Control: location of green zonesMathematical FormulationNumerical Resolution
3 Numerical ResultsNumerical simulation without green zonesOptimal location of two green zones
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Numerical Simulation Optimal Control Numerical Results
Table of contents
1 Numerical Simulation: air temperature and velocityMathematical modelNumerical Resolution
2 Optimal Control: location of green zonesMathematical FormulationNumerical Resolution
3 Numerical ResultsNumerical simulation without green zonesOptimal location of two green zones
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Numerical Simulation Optimal Control Numerical Results
Table of contents
1 Numerical Simulation: air temperature and velocityMathematical modelNumerical Resolution
2 Optimal Control: location of green zonesMathematical FormulationNumerical Resolution
3 Numerical ResultsNumerical simulation without green zonesOptimal location of two green zones
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Numerical Simulation Optimal Control Numerical Results
Domain Ω (3D)
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Numerical Simulation Optimal Control Numerical Results
Domain Ω (2D)
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Numerical Simulation Optimal Control Numerical Results
State System
Air velocity u(x, t) (m/s) and pressure p(x, t)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂u
∂t+ u.∇u−∇.(Km∇u) +∇p =
θ
θrefg
−cdf
NGZ∑
k=1
LADk1Ωk∥u∥u in Ω× (0, T ),
∇.u = 0 in Ω× (0, T ),u.n = 0 on (Γr ∪ Γw ∪ Γs)× (0, T ),u.n = −u∗ on Γ1 × (0, T ),u.n = 0 on Γ2 × (0, T ),u.n = u∗ on Γ3 × (0, T ),u(0) = u0 in Ω,
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Numerical Simulation Optimal Control Numerical Results
State System
Air temperature θ(x, t) (K)
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
∂θ
∂t+ u.∇θ −∇.(Kh∇θ) =
NGZ∑
k=1
LADk1Ωk
θkf − θ
rin Ω× (0, T ),
θ = θin on Γ1 × (0, T ),∇θ.n = 0 on (Γ2 ∪ Γ3)× (0, T ),Kh∇θ.n = γ1(T
4rw − θ4) on Γw × (0, T ),
Kh∇θ.n = γ1(T4rr − θ4) on Γr × (0, T ),
Kh∇θ.n =NGZ∑
k=1
(σk1γ1(T
4rp − θ4) + σk
2γ2(θkf
4− θ4)
)1Γgk
+γ1(T4r − θ4)
(1−
NGZ∑
k=1
Γgk
)on Γs × (0, T ),
θ(0) = θ0 in Ω,
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Numerical Simulation Optimal Control Numerical Results
State System
Foliage temperature θkf (x, t) (K) at parkland Ωk, k = 1, . . . , NGZ
θkf − θ
r= σk
1γ1(T4rf − θkf
4) + σk
2γ2(θk4− θkf
4) in Ωk × (0, T )
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Time discretization: the method of characteristics
We choose a natural number N ∈ N, define the time step ∆t = TN , and
consider the discrete times tnNn=0 ⊂ [0, T ] given by tn = n∆t, forn = 0, . . . , N. The characteristic method is based on
Dc
Dt=
∂c
∂t+ u.∇c ≃ 1
∆t(cn+1 − cn Xn),
where Xn(x) = X(x, tn+1, tn) is given by
⎧⎨
⎩
dX
dτ= u(X(x, t, τ), τ),
X(x, t, t) = x.
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Semi-discrete problem
Then, given initial fields u0 and θ0, we are interested in finding, for eachn = 0, . . . , N − 1, the fields un+1, θn+1, pn+1, and θkf,n+1, k = 1, 2,solving the following system of partial differential equations:
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
αun+1 −∇.(Km∇un+1) +∇pn+1 =θn+1
θrefg
−cdf
2∑
k=1
LADk1Ωk∥un+1∥un+1 + α(un Xn) in Ω,
∇.un+1 = 0 in Ω,un+1.n = 0 on Γr ∪ Γw ∪ Γs,un+1.n = −u∗ on Γ1,un+1.n = 0 on Γ2,un+1.n = u∗ on Γ3.
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
αθn+1 −∇.(Kh∇θn+1) =2∑
k=1
LADk1Ωk
θkf,n+1 − θn+1
rn+1+ α(θn Xn)
θn+1 = θin on Γ1
∇θn+1.n = 0 on Γ2 ∪ Γ3,Kh∇θn+1.n = γ1(T
4rw,n+1 − θ4n+1) on Γw,
Kh∇θn+1.n = γ1(T4rr,n+1 − θ4n+1) on Γr,
Kh∇θn+1.n =2∑
k=1
(σk1γ1(T
4rp,n+1 − θ4n+1) + σk
2γ2(θkf,n+1
4− θ4n+1))1Γgk
+γ1(T4ra,n+1 − θ4n+1)(1− 1Γg1
− 1Γg2) on Γs
θkf,n+1 − θn+1
rn+1= σk
1γ1(T4rf ,n+1 − θkf,n+1
4) + σk
2γ2(θkn+1
4− θkf,n+1
4)
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩
αθn+1 −∇.(Kh∇θn+1) =2∑
k=1
LADk1Ωk
θkf,n+1 − θn+1
rn+1+ α(θn Xn)
θn+1 = θin on Γ1
∇θn+1.n = 0 on Γ2 ∪ Γ3,Kh∇θn+1.n = γ1(T
4rw,n+1 − θ4n+1) on Γw,
Kh∇θn+1.n = γ1(T4rr,n+1 − θ4n+1) on Γr,
Kh∇θn+1.n =2∑
k=1
(σk1γ1(T
4rp,n+1 − θ4n+1) + σk
2γ2(θkf,n+1
4− θ4n+1))1Γgk
+γ1(T4ra,n+1 − θ4n+1)(1− 1Γg1
− 1Γg2) on Γs
θkf,n+1 − θn+1
rn+1= σk
1γ1(T4rf ,n+1 − θkf,n+1
4) + σk
2γ2(θkn+1
4− θkf,n+1
4)
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Space discretization: the finite element method
We consider a mesh τh of Ω and, associated to this mesh, we define thefollowing finite element spaces:
Wh = v ∈ [C(Ω)]n : v| !T ∈ [P1b(T )]n, ∀T ∈ τh, v · n = 0 on ∂Ω,Mh = q ∈ C(Ω) : q| !T ∈ P1(T ), ∀T ∈ τh,Vh = z ∈ C(Ω) : z| !T ∈ P1(T ), ∀T ∈ τh, z|Γ1
= 0,Xh = z ∈ L2(Ω) : z| !T ∈ P0(T ), ∀T ∈ τh.
Fully discrete problem
For each n = 0, . . . , N − 1, we look for(uh,n+1, ph,n+1, θh,n+1, θ1f,h,n+1, θ
2f,h,n+1) ∈ Wh ×Mh × Vh ×Xh ×Xh,
solution of the variational formulations of the previous semi-discreteproblems −→ Freefem++.
[1] F.J. Fernandez, et al. Optimal location of green zones in metropoli-tan areas to control the urban heat island. J. Comput. Appl.Math., in press (2015)
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Space discretization: the finite element method
We consider a mesh τh of Ω and, associated to this mesh, we define thefollowing finite element spaces:
Wh = v ∈ [C(Ω)]n : v| !T ∈ [P1b(T )]n, ∀T ∈ τh, v · n = 0 on ∂Ω,Mh = q ∈ C(Ω) : q| !T ∈ P1(T ), ∀T ∈ τh,Vh = z ∈ C(Ω) : z| !T ∈ P1(T ), ∀T ∈ τh, z|Γ1
= 0,Xh = z ∈ L2(Ω) : z| !T ∈ P0(T ), ∀T ∈ τh.
Fully discrete problem
For each n = 0, . . . , N − 1, we look for(uh,n+1, ph,n+1, θh,n+1, θ1f,h,n+1, θ
2f,h,n+1) ∈ Wh ×Mh × Vh ×Xh ×Xh,
solution of the variational formulations of the previous semi-discreteproblems −→ Freefem++.
[1] F.J. Fernandez, et al. Optimal location of green zones in metropoli-tan areas to control the urban heat island. J. Comput. Appl.Math., in press (2015)
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Space discretization: the finite element method
We consider a mesh τh of Ω and, associated to this mesh, we define thefollowing finite element spaces:
Wh = v ∈ [C(Ω)]n : v| !T ∈ [P1b(T )]n, ∀T ∈ τh, v · n = 0 on ∂Ω,Mh = q ∈ C(Ω) : q| !T ∈ P1(T ), ∀T ∈ τh,Vh = z ∈ C(Ω) : z| !T ∈ P1(T ), ∀T ∈ τh, z|Γ1
= 0,Xh = z ∈ L2(Ω) : z| !T ∈ P0(T ), ∀T ∈ τh.
Fully discrete problem
For each n = 0, . . . , N − 1, we look for(uh,n+1, ph,n+1, θh,n+1, θ1f,h,n+1, θ
2f,h,n+1) ∈ Wh ×Mh × Vh ×Xh ×Xh,
solution of the variational formulations of the previous semi-discreteproblems −→ Freefem++.
[1] F.J. Fernandez, et al. Optimal location of green zones in metropoli-tan areas to control the urban heat island. J. Comput. Appl.Math., in press (2015)
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Numerical Simulation Optimal Control Numerical Results
Control 2D
Control variable
We asumme that Ωk = Γgk × [0, Zk], where Zk is known. If Ω ⊂ R2,then
Γgk = [p1, p1 + lk]
We take NGZ = 2 and suppose that l1 + l2 = L is given. Thenl2 = L− l1 and the control variable is
b = (p1, p2, l1) ∈ R3
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Numerical Simulation Optimal Control Numerical Results
Control 2D
Control variable
We asumme that Ωk = Γgk × [0, Zk], where Zk is known. If Ω ⊂ R2,then
Γgk = [p1, p1 + lk]
We take NGZ = 2 and suppose that l1 + l2 = L is given. Thenl2 = L− l1 and the control variable is
b = (p1, p2, l1) ∈ R3
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Numerical Simulation Optimal Control Numerical Results
Control 3D
Control variable
We asumme that Ωk = Γgk × [0, Zk], where Zk is known. If Ω ⊂ R3 andΓgk is a rectangle, then Γgk = [pk1 , p
k1 + lk1 ]× [pk2 , p
k2 + lk2 ]
We take NGZ = 2 and suppose thatNGZ∑
k=1
lk1 lk2 = L is given. Then
l22 = (L− l11l12)/l
21 and the control variable is
b = (p11, p12, p
21, p
22, l
11, l
12, l
21) ∈ R7
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Numerical Simulation Optimal Control Numerical Results
Optimal Control
Objective function
J(b) =
∫ T
0
∫
(Γs\(Γg1∪Γg2 ))×[a,b]θ(x, t) dx dt
T (b− a)µ (Γs \ (Γg1 ∪ Γg2))
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Numerical Simulation Optimal Control Numerical Results
Optimal Control
Admissible set
Uad = b ∈ R4(n−1)−1 : ∀k = 1, 2, Γgk ⊂ Γsjk, for any jk ∈ 1, . . . ,M,
with Γsj1∩ Γsj2
= ∅ and µmin ≤ µ(Γgk) ≤ µmax.
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Numerical Simulation Optimal Control Numerical Results
Optimal Control
Optimal Control Problem
minb ∈ Uad
J(b)
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the Optimal Control Problem
The discrete control problem
minb∈Uad
J∆th (b) =
N∑
n=1
∫
Ω(1− 1Γg1
− 1Γg2)1[a,b]θh,n(x)dx
N
(∫
Ω(1− 1Γg1
− 1Γg2)1[a,b] dx
)−1
Equivalent Formulation: The MINLP Problem
(MINLP)
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
minb,y
f(b,y)
s.t. h(b,y) ≤ 0Ay ≤ cy ∈ 0, 1p
where p = 2M , y ∈ R2M , b ∈ R4(n−1)−1, f : R4(n−1)−1+2M → R,
h : R4(n−1)−1+2M → Rm1 , A ∈ Mm2×2M and c ∈ Rm2
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Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the Optimal Control Problem
The discrete control problem
minb∈Uad
J∆th (b) =
N∑
n=1
∫
Ω(1− 1Γg1
− 1Γg2)1[a,b]θh,n(x)dx
N
(∫
Ω(1− 1Γg1
− 1Γg2)1[a,b] dx
)−1
Equivalent Formulation: The MINLP Problem
(MINLP)
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
minb,y
f(b,y)
s.t. h(b,y) ≤ 0Ay ≤ cy ∈ 0, 1p
where p = 2M , y ∈ R2M , b ∈ R4(n−1)−1, f : R4(n−1)−1+2M → R,
h : R4(n−1)−1+2M → Rm1 , A ∈ Mm2×2M and c ∈ Rm2
![Page 29: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/29.jpg)
Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Numerical procedure
(MINLP)
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
minb,y
f(b,y)
s.t. h(b,y) ≤ 0Ay ≤ cy ∈ 0, 1p
We define Y =y ∈ 0, 1p : Ay ≤ c
. For each y∗ ∈ Y we
denote by b∗ ∈ R4(n−1)−1 the numerical solution of
(NLP)
minb
f(b,y∗)
s.t. h(b,y∗) ≤ 0−→ IPOPT code
(MINLP) is solved by solving the following problem
(IP) miny∗∈Y
f(b∗,y∗) −→ Exhaustive search
[1] F.J. Fernandez, et al. Optimal location of green zones in metropoli-tan areas to control the urban heat island. J. Comput. Appl.Math., in press (2015)
![Page 30: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/30.jpg)
Numerical Simulation Optimal Control Numerical Results
Numerical Resolution of the State System
Numerical procedure
(MINLP)
⎧⎪⎪⎪⎨
⎪⎪⎪⎩
minb,y
f(b,y)
s.t. h(b,y) ≤ 0Ay ≤ cy ∈ 0, 1p
We define Y =y ∈ 0, 1p : Ay ≤ c
. For each y∗ ∈ Y we
denote by b∗ ∈ R4(n−1)−1 the numerical solution of
(NLP)
minb
f(b,y∗)
s.t. h(b,y∗) ≤ 0−→ IPOPT code
(MINLP) is solved by solving the following problem
(IP) miny∗∈Y
f(b∗,y∗) −→ Exhaustive search
[1] F.J. Fernandez, et al. Optimal location of green zones in metropoli-tan areas to control the urban heat island. J. Comput. Appl.Math., in press (2015)
![Page 31: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/31.jpg)
Numerical Simulation Optimal Control Numerical Results
Numerical simulation without green zones
Data
Trw = 352.60K, Trr = Tra = 368.28K, Trp = 358.90K,Trf = 309.71K
Rsw,net(1− am) +Rlw,dow − ϵm σB T 4r = 0
Results
![Page 32: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/32.jpg)
Numerical Simulation Optimal Control Numerical Results
Numerical simulation without green zones
Data
Trw = 352.60K, Trr = Tra = 368.28K, Trp = 358.90K,Trf = 309.71K
Rsw,net(1− am) +Rlw,dow − ϵm σB T 4r = 0
Results
![Page 33: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/33.jpg)
Numerical Simulation Optimal Control Numerical Results
Optimal location of two green zones
Data
Z1 = Z2 = 6m
L = 8m, µmin = 1m, and µmax = 5m
LAD1(z) = LAD2(z)
![Page 34: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/34.jpg)
Numerical Simulation Optimal Control Numerical Results
Results
Optimal location of two green zones
Without green zones
![Page 35: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/35.jpg)
Numerical Simulation Optimal Control Numerical Results
Results
Optimal location of two green zones
Without green zones
![Page 36: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/36.jpg)
Numerical Simulation Optimal Control Numerical Results
Work in progress
3D approach
Preliminary numerical results
![Page 37: Urban heat islands: An optimal control approach › 2015 › 09 › alvarezvazquez-icia… · Urban heat islands: An optimal control approach L.J. Alvarez-V´azquez,F.J.Fern´andez,](https://reader030.fdocuments.in/reader030/viewer/2022041100/5ed7fffdf90a4a344d62eac3/html5/thumbnails/37.jpg)
Numerical Simulation Optimal Control Numerical Results
Work in progress
3D approach
Preliminary numerical results