An application of semi-infinite programming to air pollution control

Transcription

1 An application of semi-infinite programming to air pollution control A. Ismael F. Vaz 1 Eugénio C. Ferreira 2 1 Departamento de Produção e Sistemas Escola de Engenharia Universidade do Minho aivaz@dps.uminho.pt 2 IBB-Institute for Biotechnology and Bioengineering, Centre of Biological Engineering, University of Minho, Campus of Gualtar, Braga, Portugal ecferreira@deb.uminho.pt EURO XXII - July 8-11, 2007 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

2 Contents 1 Introduction and notation 2 Dispersion model 3 Problem formulations 4 Numerical results 5 Conclusions A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

3 Introduction and notation Contents 1 Introduction and notation 2 Dispersion model 3 Problem formulations 4 Numerical results 5 Conclusions A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

4 Introduction and notation Semi-infinite programming (SIP) Consider the following semi-infinite programming problem min f(u) u R n s.t. g i (u, v) 0, i = 1,..., m u lb u u ub v R R p, where f(u) is the objective function, g i (u, v), i = 1,..., m are the infinite constraint functions and u lb, u ub are the lower and upper bounds on u. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

5 Introduction and notation Coordinate system Z H X b h H θ Y a d (a, b) d h H H = h + H θ stack position stack internal diameter stack height plume rise effective stack height mean wind direction A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

6 Dispersion model Contents 1 Introduction and notation 2 Dispersion model 3 Problem formulations 4 Numerical results 5 Conclusions A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

7 Dispersion model Gaussian model Assuming that the plume has a Gaussian distribution, the concentration, of gas or aerosol (particles with diameter less than 20 microns) at position x, y and z of a continuous source with effective stack height H, is given by Q C(x, y, z, H) = 2πσ y σ z U e 1 2 Y σy 2 ( e z H σz + e ) z+h σz where Q (gs 1 ) is the pollution uniform emission rate, U (ms 1 ) is the mean wind speed affecting the plume, σ y (m) and σ z (m) are the standard deviations in the horizontal and vertical planes, respectively. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

8 Dispersion model Change of coordinates The source change of coordinates to position (a, b), in the wind direction. Y is given by Y = (x a) sin(θ) + (y b) cos(θ), where θ (rad) is the wind direction (0 θ 2π). σ y and σ z depend on X given by X = (x a) cos(θ) (y b) sin(θ). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

9 Dispersion model Plume rise The effective emission height is the sum of the stack height, h (m), with the plume rise, H (m). The considered elevation is given by the Holland equation H = V od U ( T o T e d T o ), where d (m) is the internal stack diameter, V o (ms 1 ) is the gas out velocity, T o (K) is the gas temperature and T e (K) is the environment temperature. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

10 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

11 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

12 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

13 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

14 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

15 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

16 Dispersion model Stability classes The σ y and σ z are computed accordingly to the weather stability class. Stability classes: Highly unstable A. Moderate unstable B. Lightly unstable C. Neutral D. Lightly stable E. Moderate stable F. For example the Pasquill-Gifford equations for stability class A is σ y = 1000 tg(t )/2.15 with T = ln(x). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

17 Problem formulations Contents 1 Introduction and notation 2 Dispersion model 3 Problem formulations 4 Numerical results 5 Conclusions A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

18 Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

19 Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

20 Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

21 Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

22 Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

23 Problem formulations Formulations Assuming n pollution sources distributed in a region; C i is the source i contribution for the total concentration; Gas chemical inert. We can derive three formulations: Minimize the stack height; Maximum pollution computation and sampling stations planning; Air pollution abatement. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

24 Problem formulations Minimum stack height Minimizing the stack height u = (h 1,..., h n ), while the pollution ground pollution level is kept below a given threshold C 0, in a given region R, can be formulated as a SIP problem min u R n n c i h i i=1 s.t. g(u, v (x, y)) v R R 2, n C i (x, y, 0, H i ) C 0 i=1 where c i, i = 1,..., n, are the construction costs. Note: higher complex objective function can be considered. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

25 Problem formulations Maximum pollution and sampling stations planning The maximum pollution concentration (l ) in a given region can be obtained by solving the following SIP problem min l R l s.t. g(z, v (x, y)) v R R 2. n C i (x, y, 0, H i ) l i=1 The active points v R where g(z, v ) = l are the global optima and indicate where the sampling (control) stations should be placed. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

26 Problem formulations Air pollution abatement Minimizing the pollution abatement (minimizing clean costs, maximizing the revenue, minimizing the economical impact) while the air pollution concentration is kept below a given threshold can be posed as a SIP problem min u R n n p i r i i=1 s.t. g(u, v (x, y)) v R R 2, n (1 r i )C i (x, y, 0, H i ) C 0 i=1 where u = (r 1,..., r n ) is the pollution reduction and p i, i = 1,..., n, is the source i cost (cleaning or not producing). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

27 Numerical results Contents 1 Introduction and notation 2 Dispersion model 3 Problem formulations 4 Numerical results 5 Conclusions A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

28 Numerical results Modeling environment and solver SIPAMPL (Semi-Infinite Programming with AMPL) was used to code the proposed examples. The NSIPS (Nonlinear Semi-Infinite Programming Solver) was used to solve the proposed examples. NSIPS discretization methods is the only one allowing finite constraints. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

29 Numerical results Modeling environment and solver SIPAMPL (Semi-Infinite Programming with AMPL) was used to code the proposed examples. The NSIPS (Nonlinear Semi-Infinite Programming Solver) was used to solve the proposed examples. NSIPS discretization methods is the only one allowing finite constraints. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

30 Numerical results Modeling environment and solver SIPAMPL (Semi-Infinite Programming with AMPL) was used to code the proposed examples. The NSIPS (Nonlinear Semi-Infinite Programming Solver) was used to solve the proposed examples. NSIPS discretization methods is the only one allowing finite constraints. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

31 Numerical results Example - Minimum stack height (Wang and Luus, 1978) Consider a region with 10 stacks. The environment temperature (T e ) is 283K and the emission gas temperature (T o ) is 413K. The wind velocity (U) is 5.64ms 1 in the 3.996rad direction (θ). The stack height in the table were used as initial guess and a squared region of 40km was considered (R = [ 20000, 20000] [ 20000, 20000]). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

32 Numerical results Example - Minimum stack height (Wang and Luus, 1978) Consider a region with 10 stacks. The environment temperature (T e ) is 283K and the emission gas temperature (T o ) is 413K. The wind velocity (U) is 5.64ms 1 in the 3.996rad direction (θ). The stack height in the table were used as initial guess and a squared region of 40km was considered (R = [ 20000, 20000] [ 20000, 20000]). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

33 Numerical results Example - Minimum stack height (Wang and Luus, 1978) Consider a region with 10 stacks. The environment temperature (T e ) is 283K and the emission gas temperature (T o ) is 413K. The wind velocity (U) is 5.64ms 1 in the 3.996rad direction (θ). The stack height in the table were used as initial guess and a squared region of 40km was considered (R = [ 20000, 20000] [ 20000, 20000]). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

34 Numerical results Example - Minimum stack height (Wang and Luus, 1978) Consider a region with 10 stacks. The environment temperature (T e ) is 283K and the emission gas temperature (T o ) is 413K. The wind velocity (U) is 5.64ms 1 in the 3.996rad direction (θ). The stack height in the table were used as initial guess and a squared region of 40km was considered (R = [ 20000, 20000] [ 20000, 20000]). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

35 Numerical results Data for the 10 stacks The stacks data is Source a i b i h i d i Q i (V o ) i (m) (m) (m) (m) (gs 1 ) (ms 1 ) A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

36 Numerical results Numerical results Two threshold values were tested. C 0 = gm 3 without a lower bound on the stack height, C 0 = gm 3 with a stack lower bound height of 10m 1 and C 0 = gm 3 2. The stack height can only be inferior to 10m if some legal 3 requirements are met. One way to prove that the requirements are met is by simulation, using a proper model, of the air pollution dispersion. 1 Decree law number 352/90 from 9 November Decree law number 111/2002 from 16 April Decree law number 286/93 from 12 March A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

37 Numerical results Numerical results Two threshold values were tested. C 0 = gm 3 without a lower bound on the stack height, C 0 = gm 3 with a stack lower bound height of 10m 1 and C 0 = gm 3 2. The stack height can only be inferior to 10m if some legal 3 requirements are met. One way to prove that the requirements are met is by simulation, using a proper model, of the air pollution dispersion. 1 Decree law number 352/90 from 9 November Decree law number 111/2002 from 16 April Decree law number 286/93 from 12 March A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

38 Numerical results Numerical results Two threshold values were tested. C 0 = gm 3 without a lower bound on the stack height, C 0 = gm 3 with a stack lower bound height of 10m 1 and C 0 = gm 3 2. The stack height can only be inferior to 10m if some legal 3 requirements are met. One way to prove that the requirements are met is by simulation, using a proper model, of the air pollution dispersion. 1 Decree law number 352/90 from 9 November Decree law number 111/2002 from 16 April Decree law number 286/93 from 12 March A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

39 Numerical results Numerical results Two threshold values were tested. C 0 = gm 3 without a lower bound on the stack height, C 0 = gm 3 with a stack lower bound height of 10m 1 and C 0 = gm 3 2. The stack height can only be inferior to 10m if some legal 3 requirements are met. One way to prove that the requirements are met is by simulation, using a proper model, of the air pollution dispersion. 1 Decree law number 352/90 from 9 November Decree law number 111/2002 from 16 April Decree law number 286/93 from 12 March A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

40 Numerical results Numerical results Instance 1 Instance 2 Instance 3 h h h h h h h h h h Total A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

41 Numerical results Constraint contour 2 x x 10 4 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

42 Numerical results Example - Maximum pollution level and sampling stations planning (Gustafson et al., 1977) Computing the maximum pollution level (l ) by fixing the stack height h i. Consider a region with 25 stacks. The region considered was R = [0, 24140] [0, 24140] (square of about 15 miles). Environment temperature of 284K, and wind velocity of 5ms 1 in direction 3.927rad (225 o ). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

43 Numerical results Example - Maximum pollution level and sampling stations planning (Gustafson et al., 1977) Computing the maximum pollution level (l ) by fixing the stack height h i. Consider a region with 25 stacks. The region considered was R = [0, 24140] [0, 24140] (square of about 15 miles). Environment temperature of 284K, and wind velocity of 5ms 1 in direction 3.927rad (225 o ). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

44 Numerical results Example - Maximum pollution level and sampling stations planning (Gustafson et al., 1977) Computing the maximum pollution level (l ) by fixing the stack height h i. Consider a region with 25 stacks. The region considered was R = [0, 24140] [0, 24140] (square of about 15 miles). Environment temperature of 284K, and wind velocity of 5ms 1 in direction 3.927rad (225 o ). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

45 Numerical results Example - Maximum pollution level and sampling stations planning (Gustafson et al., 1977) Computing the maximum pollution level (l ) by fixing the stack height h i. Consider a region with 25 stacks. The region considered was R = [0, 24140] [0, 24140] (square of about 15 miles). Environment temperature of 284K, and wind velocity of 5ms 1 in direction 3.927rad (225 o ). A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

46 Numerical results Data for the 25 stacks Source a i b i h i d i Q i (V o) i (T o) i (m) (m) (m) (m) (gs 1 ) (ms 1 ) (K) A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

47 Numerical results Numerical results - contour The maximum pollution level of l = gm 3 in position (x, y) = (8500, 7000). x x 10 4 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

48 Numerical results Example - Air pollution abatement (Gustafson and Kortanek, 1972) Consider: three plants (P 1, P 2 and P 3 ), with emissions of e 1, e 2 and e 3, where 0 e i 2, (i = 1, 2, 3) of a certain pollutant. Q = 1gs 1. By legal imposition the pollution level must not exceed a given threshold (C 0 = 1 2 ) under mean weather conditions, i.e., θ = 0 and U = ( ) 1 2 2π ms 1. The remaining stacks data is Source a i b i h i A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

49 Numerical results Example - Air pollution abatement (Gustafson and Kortanek, 1972) Consider: three plants (P 1, P 2 and P 3 ), with emissions of e 1, e 2 and e 3, where 0 e i 2, (i = 1, 2, 3) of a certain pollutant. Q = 1gs 1. By legal imposition the pollution level must not exceed a given threshold (C 0 = 1 2 ) under mean weather conditions, i.e., θ = 0 and U = ( ) 1 2 2π ms 1. The remaining stacks data is Source a i b i h i A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

50 Numerical results Example - Air pollution abatement (Gustafson and Kortanek, 1972) Consider: three plants (P 1, P 2 and P 3 ), with emissions of e 1, e 2 and e 3, where 0 e i 2, (i = 1, 2, 3) of a certain pollutant. Q = 1gs 1. By legal imposition the pollution level must not exceed a given threshold (C 0 = 1 2 ) under mean weather conditions, i.e., θ = 0 and U = ( ) 1 2 2π ms 1. The remaining stacks data is Source a i b i h i A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

51 Numerical results Example - Air pollution abatement (Gustafson and Kortanek, 1972) Consider: three plants (P 1, P 2 and P 3 ), with emissions of e 1, e 2 and e 3, where 0 e i 2, (i = 1, 2, 3) of a certain pollutant. Q = 1gs 1. By legal imposition the pollution level must not exceed a given threshold (C 0 = 1 2 ) under mean weather conditions, i.e., θ = 0 and U = ( ) 1 2 2π ms 1. The remaining stacks data is Source a i b i h i A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

52 Numerical results Example - Air pollution abatement (Gustafson and Kortanek, 1972) Consider: three plants (P 1, P 2 and P 3 ), with emissions of e 1, e 2 and e 3, where 0 e i 2, (i = 1, 2, 3) of a certain pollutant. Q = 1gs 1. By legal imposition the pollution level must not exceed a given threshold (C 0 = 1 2 ) under mean weather conditions, i.e., θ = 0 and U = ( ) 1 2 2π ms 1. The remaining stacks data is Source a i b i h i A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

53 Numerical results Problem The emission rate reduction is to be minimized. min 2r 1 + 4r 2 + r 3 r 1,r 2,r 3 R 3 s.t. (2 r i )C(x, y, 0, H i ) C 0 i=1 0 r i 2, i = 1, 2, 3 (x, y) [ 1, 4] [ 1, 4]. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

54 Numerical results Numerical results Solution found r = (0.987, 0.951, 0.943) The maximum pollution is attained at (x, y) 1 = (1.100, 0.125), (x, y) 2 = (1.100, 0.100) and (x, y) 3 = (3.675, 0.625), where the sampling stations should be placed. A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

55 Numerical results Constraint contour A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

56 Numerical results Example - Air pollution abatement (Wang and Luus, 1978) The data proposed by (Gustafson and Kortanek, 1972), in spite of illustrating the air pollution abatement problem, is not a real scenario. We have ( used the data from (Wang and Luus, 1978) with the Portuguese 10 limit i=1 (1 r i)c i (x, y, 0, H i ) gm ). 3 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

57 Numerical results Numerical results The initial guess is r i = 0, i = 1,..., 10, corresponding to no reduction in all sources. r 1 r 2 r 3 r 4 r 5 r 6 r 7 r 8 r 9 r 10 Total A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

58 Numerical results Constraint contour 2 x x 10 4 A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

59 Conclusions Contents 1 Introduction and notation 2 Dispersion model 3 Problem formulations 4 Numerical results 5 Conclusions A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

60 Conclusions Conclusions Air pollution control problems formulated as SIP problems; Problems coded in (SIP)AMPL modeling language. vaz1.mod Minimum stack height vaz2.mod Maximum attained pollution and sampling stations planning vaz3.mod Air pollution abatement vaz4.mod Air pollution abatement Publicly available together with the SIPAMPL at Numerical results obtained with the NSIPS solver; A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

61 Conclusions Conclusions Air pollution control problems formulated as SIP problems; Problems coded in (SIP)AMPL modeling language. vaz1.mod Minimum stack height vaz2.mod Maximum attained pollution and sampling stations planning vaz3.mod Air pollution abatement vaz4.mod Air pollution abatement Publicly available together with the SIPAMPL at Numerical results obtained with the NSIPS solver; A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

62 Conclusions Conclusions Air pollution control problems formulated as SIP problems; Problems coded in (SIP)AMPL modeling language. vaz1.mod Minimum stack height vaz2.mod Maximum attained pollution and sampling stations planning vaz3.mod Air pollution abatement vaz4.mod Air pollution abatement Publicly available together with the SIPAMPL at Numerical results obtained with the NSIPS solver; A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

63 THE END The end Web A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36

64 THE END References S.-Å. Gustafson and K.O. Kortanek. Analytical properties of some multiple-source urban diffusion models. Environment and Planning, 4:31 41, S-Å. Gustafson, K.O. Kortanek, and J.R. Sweigart. Numerical optimization techniques in air quality modeling: Objective interpolation formulas for the spatial distribution of pollutant concentration. Applied Meteorology, 16(12): , December B.-C. Wang and R. Luus. Reliability of optimization procedures for obtaining global optimum. AIChE Journal, 24(4): , A.I.F. Vaz and E.C. Ferreira (UMinho) Air pollution control SIP EURO XXII July / 36