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
The Urban Heat Island: a metheorological phenomenon
(published on website. 30 June, 2010)http://blog.rtve.es/eltiempo/2010/06/isla-de-calor.html
Connected problems: “Beret polution”
http://elpais.com/diario/2011/10/07/catalunya/1317949650_850215.html (published on website. 7 October, 2011)
Improvement Strategy: Build green zones
http://www.construyeargentina.com/wp-content/uploads/2013/06/green_roof3b.jpg
Objective: Control UHI by building green zones and parks
http://www.fincasfiol.com/wp-content/uploads/2013/09/VERDEVERTICAL_0721.jpg
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
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
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
Numerical Simulation Optimal Control Numerical Results
Domain Ω (3D)
Numerical Simulation Optimal Control Numerical Results
Domain Ω (2D)
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,
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,
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 )
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.
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.
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)
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)
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)
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)
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)
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
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
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
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))
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.
Numerical Simulation Optimal Control Numerical Results
Optimal Control
Optimal Control Problem
minb ∈ Uad
J(b)
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
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
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)
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)
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
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
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)
Numerical Simulation Optimal Control Numerical Results
Results
Optimal location of two green zones
Without green zones
Numerical Simulation Optimal Control Numerical Results
Results
Optimal location of two green zones
Without green zones
Numerical Simulation Optimal Control Numerical Results
Work in progress
3D approach
Preliminary numerical results
Numerical Simulation Optimal Control Numerical Results
Work in progress
3D approach
Preliminary numerical results