International Journal of Applied Mathematics

International Journal of Applied Mathematics ————————————————————– Volume 23 No. 3 2010, 549-569

A LINEAR PROGRAMMING MODEL FOR CONTROLLING AIR POLLUTION D. Parra-Guevara1 , Yu.N. Skiba2 § , A. Pérez-Sesma3 Centro de Ciencias de la Atmósfera Universidad Nacional Autónoma de México Circuito Exterior, Ciudad Universitaria México D.F., C. P. 04510, MEXICO 1 e-mail: [email protected] 2 e-mail: [email protected]

Abstract: A three-dimensional dispersion air pollution model with point sources is considered in a limited region. The adjoint model and the duality principle are used to pose a linear programming problem with the aim to determine optimal emission rates of the sources and meet the standards of air quality. The existence of the optimal control problem solution is proved. An interiorpoint method and the simplex method are used to approximate the optimal solution. Numerical results obtained in the case of point sources demonstrate the method’s ability. AMS Subject Classification: 49J20, 90C05, 62P12 Key Words: dispersion model, adjoint model, linear programming, optimal control 1. Introduction Air pollution problems exist at all scales from extremely local to global. On each scale the air pollution events have different features determined by the form, location and intensity of the pollution sources, the type of sustances emitted into the atmosphere and the atmospherical dispersion conditions such as the wind and turbulence [18]. Due to all these factors, the primary and Received:

March 21, 2010

§ Correspondence

author

c 2010 Academic Publications

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D. Parra-Guevara, Yu.N. Skiba, A. Pérez-Sesma

secondary pollutants cause various damages in the receptors which include from the irritation of mocous membranes in eyes and nose to acid rain and global warming [13]. As an example, in the last two decades several epidemiological, toxicological and clinical studies around the world have shown an association between the concentration of PM10 (particles whose diametre is less than 10 μm) and various respiratory and cardiovascular symptoms as well as mortality [6], [21]. It follows that, in order to avoid the negative effects produced by air pollutants on the human beings and the environment, it is necessary to develop quantitative criterions to reduce the emissions and define comprehensive air pollution control programs. The control over emission rates of industrial plants is a kind of inverse problem in the context of air pollution modelling [3], [15]. This control is an important part of the atmospheric pollution control programs since it develops quantitative criterions to restrict the pollutant emissions in order to fulfill some sanitary and ecological goals [4], [5], [23]. Such criteria are designed taking into account the complexity inherent to the processes of dispersion and transformation of pollutants in the atmosphere, the number of point sources to control, their locations in a region under consideration and the corresponding ecological laws [9], [25]. In general, the objective of any atmospheric pollution control program is to establish a set of actions allowing to satisfy existing air quality norms. In practice, it is required to reduce the concentration of each atmospheric pollutant to a level not exceeding the corresponding sanitary norm, or at least, to minimize the number of hours or days when the air quality norms are violated. All the control programs may be classified under two types: long-term and short-term controls [18]. These control programs complement each other and differ by their specific objectives and strategies applied to achieve them. A long-term control (from months to years) is usually implemented for largescale regions (from urban to global) and consists in applying the strategies, which reduce the total mass of the pollutants emitted during the whole period. It should be stressed that the main goal of such a control is to minimize the number of days per annum when the air quality norms are violated, and hence, its application does protect the region under consideration from the emergency days with rather dangerous pollution concentration levels. Such a flexibility is inherent in the very nature of the long-term control that, in addition to air quality norms, takes into account a cost-benefit criterion. On the other side, the short-term control (from hours to days) is implemented for small scales (from local to urban) when the atmosphere stability (inversion, calm) creates favorable conditions for the accumulation of pollu-

A LINEAR PROGRAMMING MODEL...

551

tants. The specific goal of this control is to maintain the concentration of different pollutants below the sanitary norms. The actions normally set as its goal the immediate decrease of the pollutants emissions, up to the total stoppage of some enterprises. We now define a short-term control problem for several pollutants. Let N the number of pollution sources acting in a simply connected bounded domain D ⊂ R3 , and let K be the maximum number of quasi-passive pollutants emitted by each pollution source. Let M be a short-term dispersion model which is used to forecast the concentration φk of the kth pollutant in domain D and finite time interval [0, T ] . → M:− q → φk− → q,

k = 1, ..., K,

→ where − q = (q1 , . . . , qN ) is defined by the nonstationary and nonnegative pollutant emission rates qi , which are located at the points ri ∈ D. We stress that emissions from each source may be composed of various pollutants. The mean concentration of the kth pollutant in the domain Ω ⊂ D and interval [T − τ, T ] of length τ > 0, is defined by the functional T 1 JkΩ,τ = JΩ,τ φk− φk− (1) = → → drdt, q | Ω | τ T −τ Ω q where | Ω | denote the measure (volume) of Ω. → − Let Q = (Q1, . . . , QN ) be a vector that represents actual emission rates, and let JΩ,τ φk− → > Jk for some k, where Jk > 0 is the air quality norm for the Q

kth pollutant. Then the emission rates are excessive, and the control problem → − k ≤ Jk . Note that consists in determining such q = (q1 , . . . , qN ) that JΩ,τ φ− → q each choosen subcomponent qik of qi must be a nonnegative function in interval [0, T ]. In general, this inverse problem is ill-posed, because it can have multiple solutions or no solutions depending on the initial distribution of pollutants concentrations φ0k = φk (0). In order to establish a well-posed problem, we introduce the following regularization method. Let → → − → F (− q ) = − q − Q (2) be a functional defined in the domain − K → q (t), 0 ≤ q (t); q | qi = ik ik k=1 . Q= ≤ Jk , k = 1, .., K JΩ,τ φk− → q

(3)

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D. Parra-Guevara, Yu.N. Skiba, A. Pérez-Sesma → − The optimal control problem consists in determining such q ∗ ∈ F ⊆ Q that − → → q ), (4) F q∗ = − inf F (− → q ∈F

where F is a suitable feasibility space. → − Evidently, q ∗ represents the least restriction imposed by the control on the punctual sources, and it depends on the norm · and the feasibility space F used. Thus, it is possible to find solutions for this inverse problem through different mathematical programming problems [10], [19] or variational problems [3], [4], [17]. It is important to note that the solution of each problem is usually searched with an iterative optimization method, which applies a successive evaluation of the dynamic model M, see [14], [24]. In general, this process is not so efficient, since requires a lot of computations due to the complexity of M. In this work, we describe a technique, based on the adjoint operators and linear programming, which allows to resolve the optimal control problem without using the successive evaluation of M. 2. Dispersion and Adjoint Models 2.1. Dispersion Model Let D =D ×(0, H) be a simply connected bounded domain in R3 with the boundary ∂D = S0 ∪ S ∪ SH which is the union of the cylindric lateral surface S, the base S0 at the bottom, and top cover SH at z = H (see Figure 1). The short-term dispersion model M, considered for K quasi-passive pollutants in the domain D is ∂φk + U·∇φk + σk φk − ∇ · (μ∇φk ) ∂t ∂φk ∂ + ∇ · φsk = fk (r, t) (5) − μz ∂z ∂z (6) φsk = −νks φk e3 in D φk (r, 0) = φ0k (r)

in D

μ∇φk · n − Un φk = 0

μz

on S −

(7) (8)

μ∇φk · n = 0 on

S+

(9)

μ ∇φk · n = 0 on

S0

(10)

− on SH

(11)

∂φk − Un φk = −νks φk ∂z

A LINEAR PROGRAMMING MODEL... ∂φk + = −νks φk on SH ∂z ∂u ∂v ∂w ∇·U= + + = 0 in ∂x ∂y ∂z

553

(12)

μz

D.

(13)

Here φk (r,t) ≥ 0 represents the concentration of kth primary pollutant with a distribution φ0k (r) in D at initial moment t = 0, σk (r,t) ≥ 0 is the chemical transformation coefficient, and μ(r,t) > 0 and μ (r,t) > 0 are the turbulent diffusion tensors, ⎛ ⎞

μx 0 0 μx 0 μy 0 ⎠ , , μ (r, t) = ⎝ 0 (14) μ(r, t) = 0 μy 0 0 μz respectively. The term ∇ · φsk in (5), describes the change of concentration of particles in unit time because of sedimentation with constant velocity νks > 0. The wind velocity U(r,t) = (u, v, w) is assumed to be known and to satisfy the continuity equation (13 ) in D. Assume that the forcing fk (r, t) =

N

qik (t)δ(r − ri )

(15)

i=1

is formed by the point sources (industries) located at the points ri ∈ D, i = 1, ..., N ; besides, qi (t) is the emission rate of the ith industrial plant formed by the emission rates of K different pollutants, qi (t) =

K

qik (t) ,

(16)

k=1

and δ(r − ri ) is the Dirac delta centered at the plant position ri (i = 1, ..., N ). The conditions on the open boundary ∂D of the limited domain D lead to the well-posed problem in the sense of Hadamard, see [12]. We denote by Un = U · n the projection of the velocity U on the outward unit normal n to the boundary S, which is divided into the outflow part S + where Un ≥ 0 (advective pollution flow is directed out of D) and the inflow part S − where Un < 0 (advective pollution flow is directed into D). The region D is assumed to be large enough to include all important pollution sources. Thus, we suppose that there is no sources outside D, and by condition (8), the combined (diffusive plus advective) pollution flow is zero on the inflow part S − . The pollution flow is non-zero only on S + , besides, according to (9), the diffusive pollution flow on

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Figure 1: Cross section of region D S + is assumed to be negligible as compared with the corresponding advective pollution flow. The conditions (11) and (12) have similar meanings on SH , where the sedimentation of the particles has been taken into account. Equation (10) indicates no flow of the substances through S0 , since U · n and νks are both zero on the irregular terrain (see Figure 1). In general, (11) and (12) are necessary because w = 0 on S0 and (13) lead to a non-zero vertical velocity component at SH : w(x, y, z, t) = −

0

z

∂u ∂v + ∂x ∂y

dz.

(17)

The boundary conditions are mathematically good, because problem (5)(13) is a well posed one (its solution exists, is unique and continuously depends on the initial condition and forcing [27]). This follows from the fact that the


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problem operator A: Aφk = U·∇φk + σk φk − ∇ · (μ∇φk ) −

∂ ∂φk μz + ∇ · φsk , ∂z ∂z

(18)

is nonnegative: 2 1 12 2 ∇φk dr + σk φk dr + νks φ2k |e3 · n| dS (Aφk , φk ) = μ 2 2 D D S0 1 s 2 2 2 νk φk dS + Un φk dS − Un φk dS ≥ 0. (19) + + − 2 SH S + ∪SH S − ∪SH Here (φ, η) = D φ η dr is the inner product. It can then be shown (see [27]) that (20) φk 2 ≤ T max fk (r, t) 2 + φ0k 2 , 0≤t≤T

1/2 . where ϕ 2 = D ϕ2 dr The boundary conditions are also physically appropriate, since the integration of (5) over domain D leads to a mass balance equation ∂ ∂t

D

φk dr =

qik (t) −

i=1

−

N

D

σk φk dr −

S0

+ S + ∪SH

Un φk dS

νks φk |e3 · n| dS.

(21)

Thus, the total mass of the pollutants increases due to the nonzero emission + , sources qik (t), and decreases because of advective outflow across S + ∪ SH chemical transformations and settlement on the ground. Finally, in order to get the numerical solution of the dispersion model (23)-(30), and its adjoint, a balanced and absolutely stable second-order finitedifference scheme based on the application of the splitting method and CrankNicolson scheme can be used, see [11], [29]. 2.2. Adjoint Model As it was mentioned above, the main objective of this study is to develop an optimal short-term control allowing to maintain the mean concentration (1) of each pollutant below a respective sanitary air quality norm Jk : JkΩ,τ ≤ Jk ,

k = 1, . . . , K.

(22)

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The functional (1) does not provide explicit dependence of JkΩ,τ on the pollution emission rates, which is required to develop efficient emissions control and satisfy (22). Such explicit relation can be obtained with the adjoint dispersion model [8], [9] and duality principle (Lagrange identity) [8], [22]. The adjoint approach not only provides an effective and economical technique for the sensitivity study of the model solution with respect to the model parameters [28], but also permits to solve such important problems as optimal allocation of new industries [9], control of pollution emissions [3], [4], [23], [25], detection of the industrial plants that violate prescribed emission rates [26], remediation of aquatic systems [2], etc. To this end, we now consider in the domain D × (0, T ) the adjoint model associated with the original dispersion model by means of the Lagrange identity (Aφ, g) = (φ, A∗ g): −

∂gk − U·∇gk + σk gk − ∇ · (μ∇gk ) ∂t −

∂gk ∂ μz − ∇ · gks = P (r, t) ∂z ∂z gks = −νks gk e3 gk (r, T ) = 0

in in

μ∇gk · n + Un gk = 0 μ∇gk · n = 0

(23)

D

(24)

D

(25)

on S +

(26)

on S −

μ ∇gk · n + gks · n = 0 on

(27) S0

(28)

∂gk + + Un gk = 0 on SH ∂z ∂gk − = 0 on SH . μz ∂z The forcing P (r, t) in (23) is defined as:

(29)

μz

P (r, t) =

1 |Ω| τ

0

(r, t) ∈ Ω × (T − τ, T ) (r, t) ∈ / Ω × (T − τ, T )

(30)

.

The adjoint model (23)-(30) being solved backward in time (from t = T to t = 0) also has a unique solution, which continuously depends on the forcing P (r, t). This result can be immediately shown by the transformation of variable t = T − t, see [27].


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Combining the solutions of the dispersion model and its adjoint [8], [9], one can obtain an alternative (dual) formula for the mean concentration JkΩ,τ in the zone Ω: N T k gk (ri , t)qik (t)dt = JΩ,τ φ− → q + D

i=1

0

gk (r, 0)φ0k (r)dr, k = 1, . . . , K.

(31)

Estimate (31) is the required formula that explicitly relates JkΩ,τ with φ0k (r) and qik (t). Although the adjoint model solution gk depends on the meteorological conditions, characteristics of the pollutant (coefficients σk and νks ), sources positions and parameters τ and |Ω|, it is independent of the emission rates qik (t) and initial pollution distribution φ0k (r). This solution is nonnegative and serves in (31) as the weight function for φ0k (r) and qik (t). The last integral in (31) determines the contribution of φ0k in JkΩ,τ . In the case that this term is large enough, the air quality norm is violated (JkΩ,τ > Jk ) even if all the emission rates are zero. This integral is a key parameter, which determines if a control problem solution exists.

3. Linear Optimal Control Let the dispersion model M be coupled with a weather forecast model predict both meteorological and air quality conditions within (0, T ). Suppose that the → − air quality forecast obtained with emission rates Q = (Q1 , . . . , QN ) is unfa→ > Jk , k = 1, . . . , K. Then in order to prevent excessive vorable: JkΩ,τ φ− Q concentration of a pollutant in Ω, a short-term control problem can be applied by establishing a more appropriate behavior of the industries within (0, T ). In → other words, we determine reduced emission rates − q , optimal to some sense, k → such that JΩ,τ φ− q ≤ Jk , for all k. Remarks: 1. It is assumed that the concentrations of pollutants have been obtained by means of a short-term air quality forecast, and then the method realizes the control of only those passive pollutants whose concentrations violate established sanitary norms. Here we supose that the number of such pollutants is K. 2. The control criterion over the emission rates is defined by the introduction of nonnegative amortization constants λi ≤ 1, i = 1, . . . , N . Such constants, which will be got through an adecuated optimization process, will

558


determine optimal emission rates qi∗ as qi∗ = λi Qi ,

i = 1, . . . , N ,

(32)

where (1 − λi ) × 100 represents the percentage short-term decrease of emissions of the ith pollution source (in other words, the decrease of industrial activity). 3. The control parameters vk are defined from (31) by the equations vk = Jk − gk (r, 0)φ0k (r)dr, k = 1, . . . , K. (33) D

Such parameters allow to separate the following control cases. Control Conditions: 1. If vj < 0, for fixed j, then it must be taken λi = 0 ∀ i, namely, all the industrial activities must be interrupted in order to prevent dangerous concentrations of the pollutants. 2. For each j, such that νj = 0, it must be taken λm = 0 for all sources where T gj (rm, t)Q mj (t)dt > 0, b jm = 0

namely, at this stage the industrial sources that emit the jth pollutant and impact the zone Ω must interrupt their activities. This criterion might completly stop the industrial activity in the region D, however, when just a few sources have interrupted their activities, then the control must be applied over the rest of the amission rates according to the next criterion. 3. Without loss of generality, here we assume that there are N pollution sources and K pollutants in D. Let vk > 0 for all k, due to JΩ,τ φk− → > Jk , Q

(31) and (33) we have N i=1

T 0

gk (ri, t)Qi k (t)dt > vk , k = 1, .., K.

(34)

Now, with the use of L1 -norm, and formulas (16) and (32) in (2), the optimal control problem (4) is reduced to determining the amortization constants λi being the solution of the following mathematical programming problem: F (λ1 , . . . , λN ) =

minimize

N i=1

subject to:

N

T

ci

K k=1

T 0

(Qi k − λi Qi k ) dt,

gk (ri, t)Qi k (t)dt ≤ vk , k = 1, . . . , K, and 0 ≤ λi ≤ 1, i = 1, . . . , N. i=1 λi 0

(35)

,

(36)


559

where the coefficient ci represents the cost per unit mass due to the decrease of the activity in the ith source, and the constraints (36) come from industrial → − φ ≤ J and (31). JK k q Ω,τ The linear programming problem (35)-(36) can be written as: minimize

F (λ1 , . . . , λN ) =

N

ai λi ,

(37)

i=1

≤ vk , k = 1, . . . , K, , (38) subject to: and 0 ≤ λi ≤ 1, i = 1, . . . , N . T T where ai = −ci 0 Qi dt and bk i = 0 gk (ri, t)Qi k (t)dt. Note that the term - N i=1 ai has been omitted in the objective function (37) as it does not affect the required minimum. If new vectors a = (a1 , .., aN ), λ = (λ1 , .., λN )t and v =(v1 , .., vK )t are introduced then a matricial form of the linear programming problem is: N

i=1 bk i λi

minimize subject to:

F = aλ,

(39)

Bλ ≤ v, and 0 ≤ λ ≤ 1.

,

(40)

where B is the K × N matrix of nonnegative transition coefficients: B = (bk i )K×N .

(41)

Observe that due to (32) the optimal control problem (4) has been reduced to an optimization problem in N whose feasibility space F is defined by (40), (42) F = λ ∈N | 0 ≤ λ ≤ 1, Bλ ≤ v . Existence of the solutions. The feasibility space is nonempty since 0 ∈ F, moreover, it is a bounded set since 0 ≤ λ ≤ 1. In order to show that F is a closed set, observe that if x ∈ F then there is a sequence {λn }∞ n=1 in F such that λn → x, namely, λin → xi (1 ≤ i ≤ N ). For fixed i, we can estimate that xi < ε + λin ≤ ε + 1

and 0 ≤ λin < xi + ε,

∀ε > 0 and n > n0 ,

hence 0 ≤ xi ≤ 1, i = 1, . . . , N. On the other hand, note that Bx − Bλn 2 ≤ B 2 x − λn 2 → 0 when n → ∞,

560


namely,

Bk λn → Bk x when n → ∞, 1 ≤ k ≤ K.

In this way, Bk x = Bk x − Bk λn + Bk λn < ε + vk

∀ε > 0 and n > n0 ,

hence Bk x ≤ vk , k = 1, . . . , K. It is concluded that x ∈ F, and hence F ⊂ F. Since the feasibility space F is a nonempy, bounded and closed set in N then we can concluded that F is a compact set. On the other hand, to show that the feasibility space F is a convex set, we observe that if t ∈ (0, 1) and λ1 , λ2 ∈ F then 0 ≤ tλ1 ≤ t and

0 ≤ (1 − t) λ2 ≤ 1 − t,

summing the last two estimations, we have that 0 ≤ tλ1 + (1 − t) λ2 ≤ 1. Now, we observe that B (tλ1 + (1 − t) λ2 ) = tBλ1 + (1 − t) Bλ2 ≤ tv + (1 − t) v ≤ v, and hence, tλ1 + (1 − t) λ2 ∈ F. It is important to note that the solution of the linear programming problem (39)-(40) always exists because of the feasibility space F is a nonempty compact set in N and the objective function F is continuous (Weierstrass Theorem), see [1]. Such a solution belongs to the boundary ∂F of the feasibility space [16], and due to (34), at least one optimal amortization constant λj is strictly less than the unit, that is the emissions of at least one pollution source must be decreased in order to fulfill the air-quality norms. Note that in this control strategy of air quality, some optimal amortization constants may be equal to unit, meaning that the emission rates of respective sources do not undergo changes. On the other hand, observe that the linear programming problem (39)-(40) could have many solutions with the same value of the objective function (39) [16] , and, due to the convexity of F and the linearity of F , all these solutions form a convex set C in F. This characteristic is an advantage since allows to choose an optimal solution, at the same cost, among all the solutions: tλ1 + (1 − t) λ2 , t ∈ (0, 1) and λ1 , λ2 ∈ C. The criterion to choose such a solution could be its technical feasibility in a real situation.


561

In order to determine a solution of the linear programming problem (39)(40), it can be used a large-scale optimization method based on LIPSOL (Linear Interior Point Solver, [30]), which is a variant of Mehrotra’s predictor-corrector algorithm [20], a primal-dual interior-point method. Also, for medium-scale optimization can be applied a projection method which is a variation of the well-known simplex method for linear programming [7]. Since the methods of interior point reduce the execution times in large size problems [30], these represent a good alternative in air pollution problems with many point sources. In this work we use the LINPROG routine of MATLAB which has implemented both methods above mentioned.

4. A Two-dimensional Example: Optimal Control for Point Sources The dispersion of the pollutants emitted to the atmosphere by fixed or mobile sources is a non-stationary three-dimensional phenomenon. However, during short-term extreme pollution events (T is a few hours), when the vertical motion of the air mass is weak, the pollution concentration increases near the earth surface and emission sources. In the case of a great atmospheric stability (thermic inversion), when the pollutants are accumulated within a thin surface layer of height H, the vertical structure of the process can be ignored to the first approximation (w = 0), and the three-dimensional dispersion equation can be integrated over the surface layer [0, H]. Dividing the resulting equation by the height H, we obtain a simplified two-dimensional dispersion model in the limited area D. Its solution ϕk (r, t) represents the mean concentration (in the air column of height H) of the kth pollutant at the point r = (x, y) and moment t. In the framework of this approximation, the mean concentration (1), during time interval (0, T ), for the kth quasi-passive pollutant satisfy: k = J ϕ , JΩ,τ φk− → → 1 − Ω2 ,τ q q H

→ ∀− q,

(43)

where Ω = Ω2 × [0, H] is the control zone. Thus, according to (43), when the control technique described in Section 3 is applied to the bidimensional model we can find the optimal emission rates for the three-dimensional problem. Let us consider now a surface layer of height H = 200 m, and a domain D = (0, 8000 m) × (0, 6000 m) with four point sources located at r1 = (1000, 3000), r2 = (3000, 2000), r3 = (3000, 4000) and r4 = (5000, 3000) which emitsulfur dioxide (SO2 ) with equal non-stationary emission rates Qi (t) = Q(t) gs−1

562 i 1 2 3 4

D. Parra-Guevara, Yu.N. Skiba, A. Pérez-Sesma ci 1 1 1 1

λi 1 1 1 0.6131

ci 1 1 1 3

λi 0.0072 1 1 1

ci 3 1 1 3

λi 1 0 0.9785 1

ci 3 1 1 3

λi 1 0.9785 0 1

ci 3 1 1 3

λi 1 0.4825 0.4825 1

Table 1 (i = 1, . . . , 4), where ⎧ 0 ≤ t < 1800 ⎨ (5/324)t, Q(t) = 250/9, 1800 ≤ t ≤ 25200 ⎩ 250/9 − (10/1296) (t − 25200) ,

. 25200 < t ≤ 28800

The initial condition of SO2 gm−3 is the gaussian distribution

2

2 x − 4000 y − 3000 − ϕ01 (r) = 10−2 exp − 1000 1000 in D (note that K = 1). The chemical decay and diffusion coefficients σ and μ are assumed to be equal to 0.0002 s−1 and 55.55 m2 s−1 , respectively. The nondivergent wind velocity U = (u, v) is generated by the streamfunction ψ = −1 v = − ∂ψ 1.0y : u = ∂ψ ∂y = 1.0 ms , ∂x = 0 , and streamlines have the form of straight lines with the direction from the east to west. The bidimensional dispersion model and its adjoint are considered in the interval (0, T ) with T = 28800 s. We will monitor the mean pollution concentration J1Ω2 ,τ (defined in (1)) in the zone Ω2 = [6000 m, 8000 m] × [2000 m, 4000 m] during the last four hours (τ = 14400 s). The sanitary norm J1 for SO2 is 80 μgm−3 , see [13]. Isolines of the dispersion model solution ϕ1 (r, t) at t = 28800 s are shown in Figure 2. It is seen an increase of the pollutant concentration in Ω2 due to the wind. The mean pollution concentrations J1Ω2 ,τ calculated with (1) is 97.58 μgm−3 , and since the result is unsatisfactory (it exceeds the sanitary norm), we apply the optimal control method described in section 3, namely, we solve the linear programming problem (39)-(40). To do this, the adjoint model solutions gi = g1 (ri , t) are calculated (see Figure 3), and different cost coefficients are considered in Table 1. The table summarizes the optimal amortization constants for each source in five different situations. If all the sources have the same cost coefficient (see the second column of the table), then only the source located at r4 must


Figure 2: Isolines of the dispersion model solution ϕ1 (r, t) in domain D at t = 28800 s. The zone Ω2 is highlighted

563

564


decrease its emmision 39 % (the third column) in order to fulfill the sanitary norm: J1Ω,τ = 80 μgm−3 , which is natural since it is the closer source to the control zone Ω2 . However, if this source has a cost coefficient three times bigger (the forth column), then the source located at r1 must totally stop its activity (the fifth column). Such a result is obtained because the objective function minimizes the global cost. On the other hand, when c1 = c4 = 3 and c2 = c3 = 1 (the sixth column), then the linear programming problem has many solutions (see columns 7 y 9 of the table). In this case one can average them to find a more balanced solution (the eleventh column). Such a solution requires that only the sources located at r2 and r3 decrease its emmision 52 % in order to fulfill the sanitary norm: J1Ω,τ = 80 μgm−3 . We observe that all the above examples use v1 = J1 (see (33)), since the initial distribution of SO2 has no impact on the control zone Ω2 during the time interval (T − τ, T ). Finally, it is interesting to note that the temporal behavior of each optimal emission rate qi (t) = λi Qi is similar to that of the corresponding original rate Qi (see Figure 3). This fact is something useful for the factories, since they must not drastically change their routine of work.

5. Conclusion The progress achieved in numerical short-term weather forecasting and pollution transport modelling have opened up fresh opportunities for the development of the methods capable not only of predicting pollutants concentrations, but also of controlling the industrial emission rates with the aim of preventing undesirable atmospheric conditions with dangerous levels of such concentrations. The development of the various control strategies is based on using different regularization methods for the inverse problem and the adjoint technique. It is important to observe that the adjoint technique allows one to obtain dual (direct and adjoint) pollution concentration estimates in a few ecologically important zones. These two equivalent estimates complement each other nicely in the assessment and control of industrial emissions. The direct estimates require the solution of the pollution transport problem and enable a comprehensive analysis of the ecological situation over the whole area. By contrast, explicit dependence of the adjoint estimates on the number, positions and emission rates of industries, as well as on the initial distribution of pollutants in the urban region, make them very effective and economical in a model sensitivity study and short-term pollution control in the above-mentioned zones. Indeed, the adjoint model solutions serve as influence functions providing valu-


Figure 3: Temporal behavior of adjoint solutions and original and optimal emission rates

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able information on the role of each of the industries in polluting a zone under consideration. Thus, the coefficient bki is the transition coefficient which estimates the contribution of the rate Qik in the mean concentration of the kth pollutant in the Ω zone, during the time interval [0, T ]. Since such coefficients can be calculated one-by-one with any dispersion model M (linear or nonlinear), then the linear programming model with the aim to determine optimal emission rates can be posed and solved as an independient second stage of the inverse problem. Due to the increase of computing time, this process is more suitable for long-term air pollution control studies, when an inmediate decision is not required. In the present work, a tridimensional short-term dispersion air pollution model with point sources is considered in a limited region. The adjoint model and the duality principle are used to pose well a linear programming problem with the aim of determining optimal emission rates of the sources. The existence of the optimal solution is proved. The numerical results obtained in the case of point sources demonstrate the method’s ability.

Acknowledgments This work was supported by the projects: FOSEMARNAT 2004-01-160 (CONACyT, Mexico), PAPIIT project IN105608-3 (UNAM, Mexico), and by the two SNI grants (National System of Investigators, CONACyT, México).

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