Environmental Monitoring and Assessment (2005) 111: 89–112DOI: 10.1007/s10661-005-8040-9 c© Springer 2005

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES

YURI N. SKIBA1,∗, DAVID PARRA-GUEVARA1

and VALENTINA DAVYDOVA BELITSKAYA2

1Laboratorio de Modelacion Matematica de Procesos Atmosfericos, Centro de Ciencias de laAtmosfera, Universidad Nacional Autonoma de Mexico, Circuito Exterior, Ciudad Universitaria,

Mexico, D.F., Mexico; 2Departamento de Meteorologıa, Servicio Meteorologico Nacional,Comision Nacional del Agua, Av. Observatorio 192, Col. Observatorio, Mexico, D.F., Mexico

(∗author for correspondence, e-mail: skiba@servidor.unam.mx)

(Received 19 October 2004; accepted 22 December 2004)

Abstract. Mathematical methods based on the adjoint model approach are given for the air-pollutionestimation and control in an urban region. A simple advection–diffusion-reaction model and itsadjoint are used to illustrate the application of the methods. Dual pollution concentration estimatesin ecologically important zones are derived and used to develop two non-optimal strategies and oneoptimal strategy for controlling the emission rates of enterprises. A linear convex combination ofthese strategies represents a new sufficient strategy. A method for detecting the enterprises, whichviolate the emission rates prescribed by a control, is given. A method for determining an optimalposition for a new enterprise in the region is also described.

Keywords: adjoint model, control and identification of emission rates, pollution estimates

1. Introduction

The release of toxic gases from enterprises and cars constitutes a serious hazardto human health and environment. Therefore, it is of great importance to monitorand control such dangerous emissions. There are two approaches for protecting theambient air against the violation of air quality sanitary norms in large industrialregions. The first one, so-called “technological way”, deals with “green” technolo-gies allowing to maintain the lowest level of a harmful pollutant emission. Thesecond approach provides control of emissions from main pollution sources and isof considerable mathematical interest. Since the pollution transport model is usedhere just for illustrating the methods, we will consider a simple 2D (vertically in-tegrated) model and quasi-passive pollutants. Thus, chemical reactions are ignoredor described by a linear law.

All the environmental optimization problems can be divided into two groups.The first group problems deal with control of emissions from the known sourcessuch as industrial plants (Marchuk, 1982; Penenko and Raputa, 1983; Skiba, 1993,1997, 2003; Parra-Guevara and Skiba, 2000, 2003; Skiba and Parra-Guevara, 2000;Skiba and Davydova Belitskaya, 2002), roadways (Skiba and Davydova Belit-skaya, 2003), oil spill (Skiba, 1996, 1999), etc. The unique solvability of the trans-port pollution problems of this type in a limited area was proved in Skiba and

90 Y.N. SKIBA ET AL.

Parra-Guevara (2000). We will not consider here problems of the second group.The main object of these problems is to identify the positions and intensity of un-known sources such as nuclear or chemical plant accidents, nuclear bomb testing,terrorist acts, etc. (Kovgar, 1996; Pudykiewicz, 1998; Penenko and Baklanov, 2001;Seibert, 2001; Issartel and Baverel, 2003) and data assimilation (Lewis and Derber,1985; Robertson and Persson, 1993; Vautard et al., 2000). The unique solvabilityof the source identification problem was proved by Kovgar (1996), and the methodused consists in adjusting the sources so that the model output fits the observationsbest minimizing a cost function.

The main method of this study is the adjoint technique widely used in the modelsensitivity study and control theory (Marchuk and Orlov, 1961). Note that solutionsof the adjoint equations serve as information functions (see also Lewins, 1965). Inthe meteorological, oceanological and pollution problems, this method was first ap-plied by Marchuk (1974, 1986). The sensitivity theory for general nonlinear equa-tions was developed by Cacuci et al. (1980), Cacuci (1981), Marchuk (1995), andMarchuk et al. (1996). The adjoint method has proved to be fruitful in the analysisof atmospheric temperature anomalies (Marchuk and Skiba, 1976, 1990; Marchuket al., 1983; Robertson, 1992), model sensitivity study (Penenko, 1975, 1979; Halland Cacuci, 1983; Hall, 1986), calculation of the moments of normally distributedrandom fields (Marchuk et al., 1985a,b), data assimilation (Lewis and Derber, 1985;Robertson and Persson, 1993; Shutyaev, 1997; Talagrand and Courtier, 1987), iden-tification of unknown pollution sources (Pudykiewicz, 1998) and others.

In the present work, the adjoint method is applied to solve a few importantair quality problems produced by industrial (point) and automobile (linearly dis-tributed) sources. A simple 2D (vertically integrated) advection–diffusion-reactionmodel and its adjoint are described in Sections 2 and 3 and are used in this study toillustrate the application of the methods suggested. Of course, the methods can beapplied to any 3D pollution transport model. The adjoint approach is used to obtaindual pollution concentration estimates in ecologically important zones and developtwo non-optimal strategies (Section 5) and one optimal strategy (Section 6) forcontrolling the emission rates of enterprises. We give a simple example of applyingthese strategies and also show that there is an infinite number of sufficient strategies,since a linear convex combination of the aforementioned three strategies representsa new sufficient strategy (Section 7). A method of detecting the enterprises, whichviolate the emission rates prescribed by a control, is given in Section 8. In Section 9,we describe a method for determining an optimal position for a new enterprise inthe region. Conclusions are given in Section 10.

2. Pollution Transport Model

The dispersion of the pollutants emitted in the atmosphere is a non-stationary 3Dphenomenon. However, during short-term extreme meteorological events such as

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 91

thermic inversions, the atmospheric stability is rather strong and the pollutantsare accumulated within a thin surface layer. Then as a first approximation, thevertical structure of the process can be ignored, and a simple 2D vertically averaging(integrated over a lower atmosphere layer) model can be considered (Pudykiewicz,1998; Parra-Guevara and Skiba, 2000, 2003; Skiba and Parra-Guevara, 2000; Skibaand Davydova Belitskaya, 2002, 2003; Skiba, 2003). Such simple model, althoughit represents a rather poor approximation of the real air pollution transport, is quitesuitable in this study for the illustrative objectives.

Thus, we consider in an urban area D and time interval (0, T ) a 2D advection–diffusion-reaction equation

∂φ

∂t+ U · ∇φ + σφ − ∇ · µ∇φ = Q(r, t) (1)

with initial condition

φ(r, 0) = φ0(r ) at t = 0 (2)

and boundary conditions

µ∂

∂nφ − Unφ = 0 at S− (3)

µ∂

∂nφ = 0 at S+ (4)

Here φ(r, t) is the concentration of a pollutant at point r = (x, y) and moment t,Q(r, t) is the pollution source, µ(r, t) > 0 is the turbulent diffusion coefficient, and∇ is the gradient. The parameter σ (r, t) > 0 describes the exponential decay ofφ(r, t) because of various physical (or/and chemical) processes (Marchuk, 1986).

The wind velocity vector U(r, t) = {u(r, t), v(r, t)} is assumed to be known inD and to satisfy the continuity equation

∇ · U = 0 (5)

We denote by Un = U · n the projection of U on the unit outward normal n to theboundary S of the domain D, and by S− and S+ the inflow (Un < 0) and outflow(Un ≥ 0) parts of the boundary S, respectively (Marchuk, 1982). For example, pointsA and B in Figure 1 belong to parts S+ and S−, respectively. By (3), the combined(diffusive plus advective) pollution flow is assumed to be zero at the inflow part S−.In other words, we suppose that there are no pollution sources outside D. Condition(4) means that at boundary S+, the diffusive pollution flow µ∂φ/∂n is assumedto be negligible as compared to the advective pollution outflow Unφ from D. In acase when Un = 0, condition (4) can be considered as satisfactory if there are no

92 Y.N. SKIBA ET AL.

Figure 1. A limited area D of the pollution transport problem, and the inflow part (S−) and the outflowpart (S+) of the boundary S of the domain D.

pollution sources in immediate proximity to the boundary. It can always be attainedby extending the domain D. In the non-diffusion limit (µ = 0), condition (3) isreduced to φ = 0 (there is no pollution on the inflow boundary), while (4) vanishes,as it must be. Indeed, for the pure advection problem (µ = σ = 0), no conditionis required at the outflow part of the boundary where solution is predetermined bythe method of the characteristic lines (Godunov, 1971). Thus, in the non-diffusionlimit (µ → 0), Equations (3) and (4) approximate the correct boundary conditionsfor the hyperbolic pure advection problem. It should be noted that the boundaryconditions considered by Kovgar (1996) and Pudykiewicz (1998) do not work inthe non-diffusion limit. So, conditions (3) and (4) are quite good from the physicalpoint of view and allow avoiding the problem related with the absence of data onthe pollution flux at the boundary. They are also mathematically good, becausethe problem (1)–(5) is well posed in that it has the unique solution continuouslydepending on the initial distribution φ0(r ) and forcing Q(r, t) (Skiba and Parra-Guevara, 2000).

Integrating Equation (1) over D and taking into account (3)–(5) we obtain thepollution balance equation

∂

∂t

∫D

φ dr =∫

DQ(r, t) dr −

∫D

σφ dr −∫

S+Unφ d S (6)

Similarly, multiplying Equation (1) by φ(r, t) and integrating the result over D leadsto

∂

∂t‖φ‖2 = 2

∫D

Qφ dr − 2∫

D(σφ2 + µ|∇φ|2) dr −

∫S+

Unφ2 d S

+∫

S−Unφ

2 d S (7)

where ‖φ‖ = (∫

D φ2 dr )1/2 is the norm of solution φ(r, t). By (6), the total pollutionconcentration

∫D φ dr in domain D increases because of the emissions Q(r, t), and

decreases by reason of both dissipation (σ > 0 ) and advective pollution outflowacross S+ (where Un ≥ 0). In the particular case that there are no sources in D

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 93

(Q(r, t) = 0), both∫

D φ dr and∫

D φ2 dr decrease with time since Un ≥ 0 at S+

and Un < 0 at S−. Moreover, if dissipation is absent either (σ = 0 , µ = 0) andUn = 0 everywhere at the boundary S, both integrals are conserved with time:

∂

∂t

∫D

φ dr = 0,∂

∂t

∫D

φ2 dr = 0 (8)

Although the conservation laws (8) are obtained under artificial boundary conditionsUn = 0, they (as well as the balance equations (6) and (7)) are very useful in testingthe numerical algorithms and computing programs (Skiba, 1993).

In order to solve the model numerically, a balanced and absolutely stable second-order finite-difference scheme based on the application of the splitting method andCrank–Nicolson scheme is used, which possesses two conservation laws in theabsence of sources and sinks (discrete analogs of (8)) (Skiba, 1997).

The form of function Q(r, t) depends on the type of pollution sources. Forexample, in the case when there are N industrial plants in the domain D (Figure 2)we take

Q(r, t) =N∑

i=1

Qi (t)δ(r − ri ) (9)

Figure 2. The main roads of Guadalajara City and positions of 52 main utilities burning fossil fuelsand producing the sulfur dioxide (SO2).

94 Y.N. SKIBA ET AL.

where ri = (xi , yi ) and Qi (t) are the position and emission rate of the ith plant,respectively, and δ(r − ri ) is the Dirac function of the ith plant position (i =1, 2, . . . , N ). If, instead of industrial sources, automobile emissions along N mainroadways Ri are considered (Figure 2) then

Q(r, t) ={

Qi (r, t), if r ∈ Ri

0, if r /∈ Ri(i = 1, . . . , N ) (10)

where Qi (r, t) is the time-dependent emission rate along the ith road Ri (i =1, 2, . . . , N ). However, it is significant that linearly and superficially distributedsources can be incorporated into this model by means of partitioning the corre-sponding lines or areas occupied by the sources and discretizing the forcing in theform (9) (Parra-Guevara and Skiba, 2003). Thus, (9) can be considered as a generalform of the problem forcing.

3. Adjoint Model and Dual Pollution Estimates

Using the Hilbert space of square integrable functions defined in the domain D andtime interval (0, T ), we can introduce the operator adjoint to problem (1)–(5) andconsider in D × (0, T ) the adjoint problem

− ∂

∂tg − U · ∇g + σg − ∇ · µ∇g = P(r, t) (11)

µ∂

∂ng = 0 at S− (12)

µ∂

∂ng + Ung = 0 at S+ (13)

g(r, T ) = 0 at t = 0 (14)

(Marchuk, 1986; Skiba, 1993), which is also well posed if solved backward intime (from t = T to t = 0) (Skiba and Parra-Guevara, 2000). Let � ⊂ D be anecologically sensitive zone in the domain D. The direct problem (1)–(5), the adjointproblem (11)–(14) with the forcing

P(r, t) =

1

τ |�| , if (r, t) ∈ � × (T − τ, T )

0, otherwise(15)

and the Lagrange identity (Marchuk and Orlov, 1961) allow deriving two equivalentestimates of the average pollution concentration in zone � and time interval (T −τ ,

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 95

T ) of length τ (Marchuk and Skiba, 1976), namely, the direct estimate

J (φ) = 1

τ |�|∫ T

T −τ

∫�

φ(r, t) dr dt, (16)

and adjoint estimate

J (φ) =N∑

i=1

∫ T

0g(ri , t)Qi (t) dt +

∫D

g(r, 0)φ0(r ) dr (17)

The duality principle, i.e., the equivalence of (16) and (17), allows us to chooseone of these two estimates depending on a concrete situation. Usually in practice,number K of ecological zones �k ⊂ D (k = 1, . . . , K ) is much less than numberN of industrial sources. In this case, the adjoint estimates of the type of (17) arevery useful and efficient in the sensitivity study of concentrations Jk(φ) to varia-tions in such parameters as emission rates Qi (t), positions ri and number N of theenterprises, and initial pollutant distribution φ0(r ).

4. Control of Industrial Emissions

The control of industrial emission rates is one of the most important inverse prob-lems in the air pollution modeling. This control provides quantitative criteria torestrict the industrial emissions and to reduce the air pollution concentrations. Suchquantitative criteria are designed taking into account the complexity inherent inthe processes of dispersion and transformation of pollutants in the atmosphere,a number of point sources to be controlled, their locations in the region underconsideration and corresponding ecological laws.

In general, the objective of any air pollution control program consists in in-stituting emergency measures to insure the fulfillment of existing sanitary normsfor each atmospheric pollutant, or at least, in reducing to a minimum the numberof hours or days when these norms are violated. All the control strategies can beclassified under two types: long-term and short-term (Flagan and Seinfeld, 1988).Both types complement each other and differ in their objectives and the methodsused to achieve them.

The long-term control (from months to years) is usually implemented for large-scale regions (from urban to global) and consists in applying the strategies, whichreduce the total mass of the pollutants emitted during the whole period. It is im-portant to emphasize that the central goal of the long-term control is to reduce thenumber of days per annum when due to weak and slow dispersion, the air-qualitynorms are violated and concentration of pollutants reaches dangerous levels. In otherwords, its application does not protect from the emergency days. Such flexibility is

96 Y.N. SKIBA ET AL.

inherent in the very nature of the long-term control that takes into account not onlythe air quality norms, but also a cost-benefit criterion.

On the other side, the short-term control (from hours to days) is implemented forsmall scales (from urban to local) during atmospheric stability events (inversions,calm) which favor the accumulation of pollutants. The actions characteristic forthis type of control are those which immediately decrease pollutants emissionsincluding a full stop of some enterprises. The specific goal of the control during theatmospheric stability event is to maintain the concentration of pollutants below thecorresponding sanitary norms. The control strategies can be optimal in some senseor non-optimal.

5. Non-Optimal Short-Term Control Strategies

In order to develop some elemental short-term control strategies, we shall considerthe dispersion model (1)–(5) and its adjoint model (11)–(15). As it has been men-tioned earlier, the adjoint models are power tools for inverse modeling. Besides,they have increasingly been used in meteorology and oceanology for the sensitivitystudy, data assimilation and identification of parameters of various models, cover-ing the range from simple box models to sophisticated General Circulation models(Giering, 2000). Such wide heterogeneity in the application of adjoint equations ismainly due to the duality principle like (17), which will be used below in developingvarious control strategies.

The adjoint models and estimates help to monitor the air quality in ecologicallyimportant zones inside D. The direct estimate (16) requires solving the pollutiontransport problem (1)–(5) again whenever emission rates or/and initial pollutiondistribution is changed. Unlike this, solutions to the adjoint model (11)–(15) areindependent of these characteristics and serve in (17) as the weight functions,providing valuable information on the contribution of each source in polluting azone. The adjoint estimate (17) can be immediately rewritten as

δ J (φ) =N∑

i=1

∫ T

0g(ri , t)δQi (t) dt +

∫D

g(r, 0)δφ0(r ) dr (18)

It makes the adjoint estimates (17) rather economical and convenient for the sen-sitivity study of the model solution to variations in the emission rates Qi (t) andinitial pollution distribution φ0(r ), and also in the control of emission rates. Indeed,the adjoint solution g(r, t), once calculated, can repeatedly be used to evaluate thecontribution of variations in the emission rates δQi (t) and initial distribution δφ0(r )to variation δ J (φ) in the pollutant concentration in the zone �. Formulae (17) and(18) also allow estimating the role of the number N and positions ri of enterprisesin polluting this zone.

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 97

In order to describe various control strategies, assume that the model forecastis unfavorable in the sense that pollution concentration J (φ) exceeds the sanitarynorm J0 in the zone �: J (φ) > J0. Then one of the control strategies shouldbe applied to reduce emission rates and prevent excessive polluting of �. Threestrategies differing in the ways of restricting the emission rates are suggested inParra-Guevara and Skiba (2000, 2003). Each strategy uses solutions to the adjointproblem and assures the fulfillment of the air quality norm in the zone:

J (φ) ≤ J0 (19)

The first strategy. This non-optimal strategy imposes restrictions on the totalamount of the pollutant emitted by each enterprise throughout time interval (0, T ).Let γi be a fraction of the total pollution emitted in a unit time by the ith enterprisewhen all the plants run at full capacity (i = 1, 2, . . . , N ):

γ1 + γ2 + · · · + γN = 1 (20)

For example,

γi = δi

/N∑

k=1

δk (21)

where δi denotes the maximal emission rate of the ith enterprise when it operatesat full power. Then (19) is fulfilled if

‖Qi‖ ≤ γiα

‖g(ri )‖ , i = 1, 2, . . . , N (22)

where

‖Qi‖ =(∫ T

0|Qi (t)|2 dt

)1/2

, ‖g(ri )‖ =(∫ T

0|g(ri , t)|2 dt

)1/2

(23)

and

α = J0 −∫

Dg(r, 0)φ0(r ) dr (24)

is the parameter characterizing the contribution of initial pollution (Parra-Guevaraand Skiba, 2000, 2003).

The second strategy. This non-optimal strategy prescribes a temporal behaviorof emission rates in interval [0, T ]:

Qi (t) ≤ γi min{δi , Gi (t)}, i = 1, 2, . . . , N (25)

98 Y.N. SKIBA ET AL.

where

Gi (t) =

α

|Ai |g(ri , t)if t ∈ Ai

δi if t ∈ [0, T ] − Ai

(26)

where Ai is the subset of [0, T ], g(ri , t) > 0, and |Ai | is the length of Ai , |Ai | ≤ T .

6. Optimal Short-Term Control Strategy

We now consider the third strategy, which is optimal. This strategy sets the greatestpossible emission rates in (0, T ), and hence, is less restrictive and more attractivefor the enterprise. It requires reducing the original rates Qi (t) to

Q∗i (t) = βi Qi (t) (27)

where positive amortization constants βi (0 ≤ βi ≤ 1, i = 1, 2, . . . , N ) are foundas a solution to the following control problem (Parra-Guevara, 2001; Parra-Guevaraand Skiba, 2003):

minβi

H ≡N∑

i=1

||Qi ||2(1 − βi )2

provided thatN∑

i=1

aiβi = α

(28)

If

ai =∫ T

0g(ri , t)Qi (t) dt > 0 (29)

then the ith point source has a nonzero impact on the zone �. The control is optimalin the sense that all the constants βi are searched to be as close to the unit as possiblein order to minimize material losses of the enterprises and at the same time satisfy(19).

The optimization problem can be written in the equivalent form as

minxi

H ≡N∑

i=1

(xi − x0

i

)2,

subject to

N∑i=1

bi xi = α

x1 ≥ 0, . . . , xN ≥ 0

(F)

(30)

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 99

where xi = ||Q∗i || (see (23)), x0

i = ||Qi || > 0, and bi = ai/||Qi || > 0, besides,∑Ni=1 bi x0

i > α, and the feasibility set F is defined by condition (F). This problemhas a unique solution. The existence immediately follows from the fact that the goalfunction H is continuous and the feasibility set F is compact (Luenberger, 1989;Weierstrass theorem). The uniqueness is the consequence of the facts that H is astrictly convex function, and set F is convex, too (Luenberger, 1989). Moreover,due to convexity, the optimal values β∗

i of the constants βi satisfy the conditions

β∗i ≤ 1, i = 1, . . . , N (31)

(Parra-Guevara, 2001) and hence, industrial sources must diminish their activity:

Q∗i (t) ≤ Qi (t), i = 1, . . . , N ; 0 ≤ t ≤ T (32)

The optimization problem solution x∗ = (x∗i )N

i=1 is found by using its differentialcharacterization theorem (Luenberger, 1989; Parra-Guevara, 2001), which statesthat x∗ is the solution if

−∇H(x∗) · (x − x∗) ≤ 0 (33)

for all x ∈ F, x = x∗. The first approximation of that x∗ is obtained as the or-thogonal projection in the minimization process using the Lagrange multiplicators(Luenberger, 1989):

x∗i = x0

i − λ

2bi , i = 1, . . . , N (34)

where

λ

2=

∑Ni=1 bi x0

i − α∑Ni=1 b2

i

(35)

This approximation satisfies the condition∑N

i=1 bi x∗i = α. To improve the approx-

imation, we use an algorithm of successive orthogonal projections, which consistsof the two steps:

Step I. If x∗i ≥ 0 for i = 1, . . . , N , then x∗ ∈ F and

−∇H(x∗) · (x − x∗) = λ

N∑i=1

bi xi − λ

N∑i=1

bi x∗i = λα − λα = 0 (36)

100 Y.N. SKIBA ET AL.

for all x ∈ F, x = x∗. By the differential characterization theorem (Luenberger,1989; Parra-Guevara, 2001), x∗ is the minimum.

Step II. If at least one component of x∗ is negative, then it is required to redefinethe projection. Without loss of generality, assume that x∗

i ≤ 0 for i = l + 1, . . . , N(since α > 0, at least one of the components is always positive). Let us definex∗∗

i = 0 for i = l + 1, . . . , N , and the rest variables are taken from the newprojection with the functional H and set F restricted to the first l variables:

x∗∗i = x0

i − η

2bi , i = 1, . . . , l (37)

where

η

2=

∑li=1 bi x0

i − α∑li=1 b2

i

(38)

This approximation also leads to the condition∑N

i=1 bi x∗∗i = α. If x∗∗

i ≥ 0 fori = 1, . . . , l, then x∗∗ ∈ F and

−∇H(x∗∗) · (x − x∗∗) = 2N∑

i=l+1

xi

(x0

i − η

2bi

), x ∈ F, x = x∗∗ (39)

Since∑l

i=1 bi x∗i > α, (34) leads to η > λ that, together with (34), gives

−∇H(x∗∗) · (x − x∗∗) ≤ 2N∑

i=l+1

xi x∗i ≤ 0 for all x ∈ F, x = x∗∗ (40)

Due to the differential characterization theorem (Luenberger, 1989; Parra-Guevara,2001), x∗∗ is the minimum.

This algorithm converges to the exact solution at most in N iterations (pro-jections), because α > 0. Moreover, it is not computer-time consuming, since thenumber of arithmetic operations does not exceed 3(N 2+ N ) (Parra-Guevara, 2001).Thus, every time a short-term forecast is unfavorable, the control of emission ratescan rapidly be carried out. This property is especially important when the number ofemission sources is large (it is the case when linearly and/or superficially distributedsources are incorporated in the form of point sources as well (Parra-Guevara andSkiba, 2003).

7. Example: The Application of Control Strategies

Let us consider in the square domain D = (0, 3)× (0, 3) of 9 km2 the two industrial(point) sources with coordinates r1 = (1.5, 0.2) and r2 = (0.6, 2.0), which emit

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 101

sulfur dioxide with the following non-stationary rates (in kg h−1):

Q1(t) =

10000t if t ∈ [0, 0.5]

5000 if t ∈ [0.5, 1]

5000(8 − t) if t ∈ (1, 8]

,

Q2(t) =

5000t if t ∈ [0, 0.5]

2500 if t ∈ [0.5, 1]

2500 + 2000(t − 1) if t ∈ (1, 1.5)

3500 if t ∈ [1.5, 7.5]

3500 − 7000(t − 7.5) if t ∈ (7.5, 8)

Assume that the initial SO2 distribution in D is Gaussian:

φ0(r) = 175 exp{−|r − r1|2} + 125 exp{−|r − r2|2}

The emission rates Qi (t) as well as the parameters σ = 0.36 h−1 and µ = 1.98km2 h−1 are taken from Shir and Shich (1974). The wind velocity U = {u, v} isgenerated with the stream function ψ(r) = (5/3)x2 −5y. We will analyze the meanSO2 concentrations in the zone � = [1, 2] × [1, 2] during the time interval (0, T )where T = 8 h. Calculating the problem within (0, T ) and using (16) or (17), weget that J (φ) = 242.43 µg m−3 that represents a violation of the correspondingsanitary norm J0 = 210 µg m−3 (or 0.08 ppm). Figure 3 shows the isolines ofthe SO2 concentration in D at t = 0, t = 2 h, t = 8 h, and isolines of the adjointsolution at t = 0. The SO2 concentration in the zone � first increases with timedue to the wind directed from southwest to northeast and then at the end of the dayit decreases together with the intensity of sources. Figure 3 also demonstrates thatwhereas the time decreases from t = T to t = 0, the step-like form of the adjointsolution generated by the forcing (15) varies due to advection in the direction of−U and diffusion.

The emission rates for the first and second sources calculated using five differentcontrol strategies are given in Figures 4 and 5, respectively. The first and secondnon-optimal strategies are described in Section 5, and the third optimal strategy isgiven in Section 6. The fourth and the fifth convex strategies representing linearconvex combinations of the previous ones are defined in Table I, which also gives themean SO2 concentrations obtained in the zone � for each of the control strategiesconsidered. Any of the strategies assures the fulfillment of the air quality norm inthe zone. However, only the third (optimal) and the fifth (convex) strategies generatethe emission rates {Q∗

31(t), Q∗32(t)} and {Q∗

51(t), Q∗52(t)} whose profiles are similar

to the corresponding profiles of the original emission rates {Q1(t), Q2(t)}. As amatter of fact, the optimal emission rates represent in this example the best solutionof the control problem, since they are reduced to zero at the beginning and at the

102 Y.N. SKIBA ET AL.

Figure 3. Isolines of the SO2 concentration in D at t = 0, t = 2 h, t = 8 h, and isolines of the adjointsolution at t = 0.

end of a working day. Moreover, the implementation of this strategy is simple andnot too restrictive, since the rates obtained constitute 88% and 84% of the originalrates Q1(t) and Q2(t), respectively. Note that although rates Q∗

21(t) and Q∗22(t) of

the second strategy are stationary (and hence, their implementation is simple inpractice), they are far lower than the maximal values of the corresponding originalrates Q1(t) and Q2(t). Finally, note that the rates Q∗

11(t) and Q∗12(t) of the first

strategy can represent an alternative to the third and the fifth strategies only in casewhen each enterprise is able to start working with its maximal capacity.

As we can see, the control problem has infinite number of solutions. Using (24)and taking into account that

∫D g(r, 0)φ0(r ) dr = 2.98, we obtain α = 207.02, that

is, the impact of the initial distribution of SO2 on the concentration J (φ) in zone� is rather small even if initial field φ0(r ) is not small. It is due to the dispersionprovoked by the wind values between 5 and 10 km h−1 during the 8-h operation ofthe enterprises. Thus, daily operation of the sources is responsible for violating thesanitary norm in the zones adjacent to the enterprises.

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 103

Figure 4. Temporal behavior of the original emission rate Q1(t) of the first source and emission ratesQi1(t) corresponding to the ith control strategy (i = 1, . . . , 5).

8. Detection of the Enterprises Violating Prescribed Emission Rates

Assume that a forecast model runs in some interval (0, T ) and gives an unsatis-factory air quality result in zone �: J (φ) > J0. Also assume that a control wasthen applied in (0, T ) to reduce the actual emission rates Qi (t) to the values Qi (t)guaranteeing the fulfillment of the sanitary norm in �: J (φ) ≤ J0. We now con-sider the situation when some enterprises have ignored this requirement and workedwith the emission rates exceeding prescribed values Qi (t). In other words, in spiteof the control application, the measurements show J (φ) > J0 again. This raisesthe question of detecting and sanctioning the enterprises responsible for excessivepolluting zone �. Though the detection problem is nontrivial for time-dependentemission rates Qi (t), it can easily be solved for invariable emission rates (Skiba,2003). The last requirement is not a strong restriction if interval (0, T ) is smallenough.

Assume that within (0, T ) the ith enterprise has worked at some unknown con-stant rate Qi (Qi = Qi ), and denote δQi = Qi − Qi (i = 1, 2, . . . , N ). Let J (φ)

104 Y.N. SKIBA ET AL.

TABLE IThe SO2 concentrations J (φ) in the zone � for different control strategies

Control strategy Emission rates J (φ) (µg m−3)

Without control Q1(t) and Q2(t) 242.43

The first (non-optimal) control strategy Q∗11(t) and Q∗

12(t) 210.00

The second (non-optimal) control strategy Q∗21(t) and Q∗

22(t) 142.27

The third (optimal) control strategy Q∗31(t) and Q∗

32(t) 210.00

The fourth (convex) control strategy Q∗41(t) = 0.8Q∗

11(t) + 0.2Q∗21(t) 196.45

Q∗42(t) = 0.8Q∗

12(t) + 0.2Q∗22(t)

The fifth (convex) control strategy Q∗51(t) = 0.3Q∗

11(t) + 0.7Q∗31(t) 210.00

Q∗52(t) = 0.3Q∗

12(t) + 0.7Q∗32(t)

Figure 5. Temporal behavior of the original emission rate Q2(t) of the second source and emissionrates Qi2(t) corresponding to the ith control strategy (i = 1, . . . , 5).

and J (φ) be the mean concentrations of a pollutant φ in � forecasted by the modeland measured after applying the control, respectively. Thus,

δ J (φ) = J (φ) − J (φ) (41)

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 105

is the known value. Let us choose K zones in the domain D(K ≥ N ), and denote asδ Jk(φ) the value (41) obtained for the kth zone �k (k = 1, 2, . . . , K ). Assuming thatJk(φ) > J0 at least for one k, we now detect the enterprises, which are responsible forexcessive polluting the zone(s). Denote by gk(r, t) the solution of adjoint problem(11)–(15) for the zone �k and by gk(ri , t) its value at position ri of the ith enterprise(i = 1, . . . , N ). Figure 6 shows isolines of the adjoint problem solution gk(r, t)calculated for zone �k and τ = 60 min at two moments: t = T − 60 min andt = T − 90 min (T = 360 min), and Figure 7 shows a typical temporal behaviorof functions gk(ri , t) in (0, T). Since δφ0(r ) ≡ 0 and δQi is constant for each i, theadjoint estimates (18) for K zones lead to the linear system

A�x = �b, (42)

or

N∑i=1

akiδQi = bk, (k = 1, 2, . . . , K ) (43)

with non-negative elements

aik =∫ T

0gk(ri , t) dt (44)

of the N × K matrix A (i = 1, . . . , N ; k = 1, . . . , K ), and

bk = δ Jk(φ) (45)

The system (43) gives values δQi , and hence, the values Qi = Qi − δQi . Thus,if Qi > Qi for some i then the ith enterprise has violated the prescribed emissionrate.

9. Optimal Allocation of a New Industrial Plant

Assume that in addition to N enterprises located at points ri ∈ D and operating withemission rates Qi (t) (i = 1, 2, . . . , N ), it is required to construct in the domain Done more enterprise whose emission rate Q0(t) does not exceed value

Q0 = max0≤t≤T

Q0(t) (46)

How a suitable position r0 for such new enterprise in D can be found? The solutiondepends on the criteria used and is not unique (Marchuk, 1986). Let us select K

106 Y.N. SKIBA ET AL.

Figure 6. Isolines of the six adjoint problem solutions calculated for Guadalajara City MetropolitanArea for the three zones and two typical winds at two moments: t = T − 60 min (dotted lines), andt = T − 90 min (solid lines); T = 360 min.

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 107

Figure 7. A typical temporal behavior of the adjoint solution (weight function) g(ri , t) within interval(0, T ) at the positions ri corresponding to various enterprises denoted by the letters A, B, C, etc.

zones �k in D (k = 1, . . . , K ) and find position r0 of the new enterprise so that

Jk ≤ J0, (k = 1, . . . , K ) (47)

where Jk is the mean concentration level of φ(r, t) in �k , and J0 is the sanitary norm.For simplicity, we neglect here the initial pollution distribution in D: φ0(r ) ≡ 0.Then

Jk =N∑

i=0

∫ T

0gk(ri , t)Qi (t) dt (48)

is the air quality estimate for zone �k when N + 1 enterprises are in operation.Denoting by

J k =N∑

i=1

∫ T

0gk(ri , t)Qi (t) dt (49)

the same estimate, but for the case of only N already operating enterprises, we get

∫ T

0gk(r0, t)Q0(t) dt ≤ J0 − J k = J k (50)

108 Y.N. SKIBA ET AL.

Taking into account (46) we find that (47) is fulfilled if

∫ T

0gk(r0, t) dt ≤ J k/Q0 (k = 1, . . . , K ) (51)

For each k, one can draw isolines of the function

Fk(x, y) ≡ Fk(r ) =∫ T

0gk(r, t) dt (52)

(k = 1, . . . , K ). Then the isoline Fk(x, y) ≡ J k/Q0 will separate a subdomainDk , in which (51) is valid, from its complement D\Dk where Fk(x, y) > J k/Q0.As a result, every point of the set D = ∩K

k=1 Dk being the intersection of all thesubdomains Dk is suitable for the allocation of a new industrial plant. These cal-culations can be repeated for typical (climatic) winds in D for each season, and theintersection of domains D obtained for different winds gives a narrower domainfor the new enterprise. It is evident that the larger the number K of zones �k the

Figure 8. The intersection D of the six subdomains Dk defined by (51) for the three zones and for thetwo types of climatic wind. For simplicity, only 11 industrial plants have been taken into account. Thenon-shaded area D is a suitable place for the allocation of a new industrial plant. The field Fk(x, y)has been premultiplied by 104. The numbers indicate the values J k/Q0 of the isolines that separatethe shaded and non-shaded regions.

AIR QUALITY ASSESSMENT AND CONTROL OF EMISSION RATES 109

narrower is the set D. Of course, other criteria (social, economical, etc.) shouldalso be taken into account to specify an optimal position of the new enterprise inD. Figure 8 shows the zone D (the non-shaded area) as the intersection of the sixsubdomains Dk defined by (51) for the three zones (K = 3) including the zoologi-cal garden (Zoo), Historical Center (CH) and Colomos park (CP) for the two typesof climatic winds in Guadalajara City (Skiba and Davydova Belitskaya, 2002). Forsimplicity, only 11 enterprises were taken into account here (N = 11), besides, allthe enterprises had the same SO2 emission rate: Qi (t) = 60 g s−1 (i = 1, . . . , 11).

10. Conclusions

The main objective of the mathematical modeling in atmospheric environment pro-tection is not only the prediction of concentrations of various pollutants, but alsothe development of the methods helping to avoid the situations when these con-centrations reach dangerous levels. In the present work, we suggest a few methodsof solving important air pollution problems related to industrial (point) and auto-mobile (linearly distributed) sources. Although the methods are illustrated with avery simple 2D advection–diffusion-reaction model and its adjoint (see Sections 2and 3), they can be applied to more complicated 3D pollution transport models.

The methods are based on the adjoint approach utilizing dual (direct and ad-joint) pollution concentration estimates in ecologically important zones. These twoequivalent estimates complement each other well in the assessment and control ofindustrial emissions. The direct estimates require a solution of the pollution trans-port problem and enable a comprehensive analysis of the ecological situation inthe whole area. By contrast, the adjoint estimates just use solutions to the adjointproblem, and are very effective and economical in studying the sensitivity of theestimates to variations in the model parameters. The dependence of these estimateson the number, positions and emission rates of enterprises, as well as on the windand initial distribution of pollutants in the urban region is given in an explicit form.Besides, the adjoint model solutions serve as influence (weight) functions provid-ing valuable information on the role of each enterprise in polluting a zone underconsideration.

Using the adjoint estimates we have developed two non-optimal strategies andone optimal strategy for controlling the emission rates of enterprises (Sections 4–6).The solution of the optimal control problem is found with a method of successiveorthogonal projections. The main advantage of the optimal control is that it offersemission rates that are as close as possible to the original emission rates of theenterprises. Thus, this strategy affects just the rhythm, and not the form of industrialoperations. Obviously, there is an infinite number of sufficient strategies, since anylinear convex combination of the above-mentioned three strategies represents anew sufficient strategy. A simple example, which illustrates application of variousstrategies, is described in Section 7. It should be noted that the existence of control

110 Y.N. SKIBA ET AL.

strategies strongly depends on the sign of the parameter α given by (24). Indeed,there is an infinite number of solutions of the control problem if α > 0. At the sametime, if α = 0 then the only solution is to stop any industrial activity in the urbanzone. Finally, there is no solution if α < 0. By (24), parameter α can efficientlybe calculated only in the case when both the adjoint solution g(r, 0) and the initialpollutant distribution φ0(r ) are known with a rather good accuracy. Therefore, thesign of α can be determined correctly only if the monitoring net is rather dense andwell situated in industrial urban zones.

A method for detecting the enterprises, which violate the emission rates pre-scribed by a control, is given in Section 8. It can be used to detect the infractionsand to apply the corresponding sanctions. In Section 9, a method for determiningan optimal position for a new enterprise in the region is described. The criterionfor optimization is the requirement that a new plant position will not violate thesanitary norm currently in force. As an example, the method have been appliedto the Guadalajara City Metropolitan Area for a quite simple case when only twotypes of climatic wind (the rainy season wind and dry season wind), 11 enterprisesand 3 ecologically most important zones are taken into account.

Acknowledgements

This work was supported by the PAPIIT grant IN122401 (UNAM, Mexico), CONA-CyT project 32247-T (Mexico), SIMORELOS project (CONACyT, Mexico), andSNI grants (Mexican National System of Investigators).

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