CURRENT RESEARCH Economic Air Pollution Control Model for Los Angeles County in 1975 General Least Cost-Air Quality Model John C . Trijonis’ Caltech-EQL,
Pasadena, Calif. 91 109
An economic air pollution control model is formulated for determining the least cost of reaching various air quality levels. The model takes the form of a general, nonlinear, mathematical programming problem. The two basic inputs to this model are the least cost of attaining various emission levels and the relationship between emission levels and air quality. A linear programming submodel is presented for deriving the first input, the control costemission level relationship. Empirical-statistical air quality submodels are outlined for obtaining the second input, the emission level-air quality relationship. The combination of these two inputs to solve the nonlinear programming problem is illustrated graphically. Part II of this paper (page 816) applies the model to photochemical smog in Los Angeles in 1975.
Environmental economics indicates that air pollution control policy should be based on a systematic comparison of all control costs associated with pollution abatement and all damage costs associated with atmospheric pollution levels. The most simplistic economic optimization is cost-benefit analysis. According to cost-benefit analysis, control strategies and resulting air quality levels should be chosen so as to minimize total social cost, the sum of pollution control and damage costs. T o perform this optimization, two key inputs are required: the least control cost associated with reaching various air quality levels and the damage costs associated with various air quality levels. These two inputs would also be of fundamental importance in economic analyses that are more general than the cost-benefit approach. This paper develops a methodology for deriving the first of the fundamental economic inputs described above. Part I formulates a general, mathematical model for determining the least control cost needed to attain various air quality levels. Part I1 applies this model to a specific situation, photochemical smog in Los Angeles County in 1975.
fogs. At the present time, however, these sporadic natural air pollution episodes are not nearly so important as the persistent air quality problems resulting from human activities, particularly in metropolitan areas. Here, only anthropogenic air pollution will be considered. Second, man affects air quality by altering meteorological and topographical conditions, as well as by emitting contaminants into the air. An example of this type of interference is the destruction of vegetation in certain areas leading to high dust levels and low organic gas concentrations. However, this type of effect is usually not of major importance; contaminant emissions are the dominant problem. Third, two general control methods, other than emission control, are available for air pollution problems. Meteorological control can be used to improve the assimilative capacity of the atmosphere. An example is the recently proposed scheme to suck freeway and street air into sewer systems. Receptor control, established on plants, animals, materials, or humans damaged by pollution, is also possible. Examples of this form of control are developing air-pollution-resistant vegetation, staying in air-filtered buildings, or changing place of residence. However, most meteorological control schemes are impractical, and a general policy of receptor control appears ruled out by aesthetic considerations. Only emission control will be considered in this paper. The control problem to be investigated is the determination of the minimum (control) cost of reaching given air
7 EMISSION CONTROL
I
EMISSIONS
Model Development Figure 1 presents the basic air pollution system that will be under consideration. Air pollution, a result of the interaction of emissions from human activities and meteorology, is to be abated through emission control. Certain qualifications are in order with regard to Figure 1; this basic conceptual model is not inclusive of all air pollution systems. For one, air pollution arises from natural processes as well a s from man-made sources. Examples are volcanic eruptions, forest fires, dust storms, and heavy Present address; TRW, Environmental Services, R4/1120,Redondo Beach, Calif. 90278
METEOROLOGY
AIR POLLUTION
Figure 1 .
Basic conceptual model for
air pollution
control
Volume 8 , Number 9, September 1974
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quality levels. This problem will be formulated mathematically, in terms of equations, in such a way that the solution of these equations provides the answer to the control problem. Already, by examining Figure 1, one can see the basic form that this mathematical model will take. In the least cost-air quality control problem, there are three basic sets of parameters, emission control cost, emission levels, and air quality. Among these parameters there are two basic relationships, the control cost-emission level relationship and the emission level-air quality relationship. Once these two relationships have been put in mathematical form, the model is completed by their synthesis. Before taking up the mathematical formulation, it should be noted that some type of time specification must be made for the problem. One alternative is to examine a fixed time period. Such a static model could be either a short-term or a long-term model-e.g., it could find the cost of achieving given air quality levels during either a two-day smog episode or some entire year in the future. A second alternative is to formulate a more complex, dynamic model that examines the cost of various “air quality paths” for “n” successive time periods. Seinfeld and Kyan ( I ) have examined the dynamic problem and have indicated how it might be solved with dynamic programming techniques. However, the complexity of the dynamic problem has precluded actual solutions for large-scale real-life air polluttion systems. Here, a simpler, longterm, and static model is constructed to calculate the cost of reaching various air quality levels for some given year in the future. The first step in formulating the least cost-air quality control problem mathematically is to put the basic parameters of the system into symbolic notation. Total control cost can be represented by a scalar, C, measured in dollars. To allow systematic comparison of initial and recurring expenditures, the costs can be put in an “annualized” form. Emission levels for N primary contaminants can be represented by N source functions, & L ( l , i~=) ,1, . . . , N,giving the rate of the ith contaminant emitted per unit volume a t all points, 5 and times, T , in the air basin and year under consideration. The final pollution resulting from these emissions is also a vector-valued function of space and time. It can be specified by functions, (P ,(x,t),j = 1, , K, giving the levels of K final pollutants a t positions, x, and times, t.
CONTROL COST RELATIONSHIP
CONTROL COST -
EMISSION
Ps = the expected number of days per year that a certain pollutant exceeds some standard for 1 hr a t a given monitoring station
Ps = the probability that the maximum 1-hr concentration of a certain pollutant exceeds some standard a t any point in the basin and a t any time during the year Figure 2 shows the least cost-air quality control problem with the mathematical notation introduced so far. This figure emphasizes the two functional relationships that must be found before the model can be completed. First, there is the control cost-emission level relationship. This is a techno-economic relationship which gives the minimum cost of achieving any given level and pattern of emissions. It is found by taking each emission level, 8 ,(E,T), 1 = 1, . .V, technically determining the subset of controls that exactly achieves that emission level, and choosing the specific control plan with minimum cost, C. This relationship, the minimum cost of reaching various emission levels, will be denoted by G.
Second, there is the emission level-air quality relationship. This is a physical relationship which gives expected air quality levels, P,, as functions of emission levels, & < ( E , r ) .Finding these functions is a problem in atmospheric physics and chemistry. Here, these relationships will be denoted by F,:
-
- EMISSION LEVEL
CONTROL
Actually, the rigorous specification of air quality (final pollutant levels), is even more complicated than indicated above. Since we are dealing with some future year, and since meteorology is uncertain, final pollution is more appropriately specified by probability distributions of the functions, 6 , ( x , t ) .For this investigation, in order to simplify the statement of air quality, it is assumed that an integration over these probability distributions and over space and time have been made so that air quality is specified by a set of air quality indices, P,, j = 1, . . . , M . Such indices are the type of air quality measures actually used by control agencies. Example would be: PI = the expected yearly average concentration of a certain pollutant a t a given monitoring station
EMISSION LEVEL AIR QUALIlY RELATIONSHIP
+
AIR QUALIN
EMISSIONS
1 METEOROLOGY
Figure 2. Formulation of least cost-air quality model in mathematical notation
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Pj = F j [ & & .
7)1
= Fj[&1(57 . ).
...
9
8,445,7)l
j = 1, . . . ,2bI
(2)
When we assume the techno-economic relationship, G, and the physicochemical relationship, F,, have been determined, the problem of finding the minimum cost of a t least reaching air quality goals, P,O, reduces to the following mathematical problem:
.
Choose that
Ei(5, 7). i
which minimizes
c
s u b j e c t to
PJn2 FjICi(5. 7)1 j = I , . . . ,.\!I (3)
= 1, . , , LL’
= G / & i ( < 7)1 ,
Stated in words, one chooses that emission level which has the minimum control cost subject to the constraint that the emission level will a t least reach the air quality goals.
achieving given emission levels for several pollutants in an air basin. This model, in a modified form, can be used to determine the function, C = G(E1, . . . , E.,-), the minimal cost of attaining G(E1, . . . , E s ) . The modification relates to the fact that Kohn’s model expresses emission level constraints as inequalities. That is, Kohn’s model finds the set of controls which minimizes the cost of a t least achieving specific emission levels, G(E1, , E , v ) . In this study, the function, G, should represent the cost of exactly achieving G(E1, . . . , E s ) . Thus, some of the constraints in Kohn’s model become equalities. The linear programming model to be used here employs the following definitions:
s,,i = 1, . . ., s
L,, m
= 1, . . ., 1
Need for Simplifications Equation 3 is the complete mathematical control model. To find the minimum cost of reaching given air quality levels, one determines the control cost-emission level relationship, G, finds the air quality-emission level functions, F , , and solves Equation 3. This problem though very simply stated, is extremely complex to actually solve. The main factor that leads to complexity is that in finding G, finding the F,’s, and solving Equation 3, one must consider all possible space and time patterns of emissions as well as total emission levels. In applying the model to a real-life air pollution problem, some simplifications would have to be made in describing the space and time distributions of emissions. For instance, freeways might be considered line sources, power plants as point sources, and nonfreeway traffic as a uniform surface source. In the example that will be taken up here, the ultimate simplification is made; the space and time pattern of emissions is not considered a t all. It is assumed (although this is certainly not exactly true), that emission control programs do not alter the pattern, so that emission level changes occur in the same proportion a t all points in space and time. With the emission pattern fixed, emissions can simply be measured by total emission levels in the air basin for each primary contaminant and represented by a vector, E, independent of space and time. With this simplification, the model becomes the following: Choose
E i . i = 1...
. .s
t h a t m i n i m i z e s c‘ = G ( E i ) = G ( E i . . . . ,E,\,)
subject t o
x , , j = 1, . . ., r
c,, j = 1, . . ., r
Eke, h = 1, . . ., N
,N
Dml, m = 1, . . ., 1 ,r
.
Pj“ 2 Fj(E1, . . . E Y ) j = 1, . . .
,,\I (4)
where (;(El, . . . , E , ) is the minimum control cost of , E\), and F,(E1,. . , , E Y ) ,j = 1, . . . , M , give expected air quality levels as functions of emission levels. The solution to the complete model proceeds in three steps: (1) find G, (2) find F , , and (3) solve Equation 4. The next three sections will discuss the general methodology that will be used in Part I1 to accomplish each of these steps.
.,N ., r
The ith component of a vector specifying the magnitude of all emission sources (e.g., the number of large power plant boilers, or the number of refinery heaters). The ith component of a vector specifying the supply limits of fixed supply inputs into control activities (e.g., the total available natural gas. Natural gas is a clean-burning fuel that can be used to reduce emissions from automobiles as well as power plant boilers). The j t h component of a vector specifying the levels of control activities, X,(e.g., the number of a certain control device added to 1970 vehicles). The j t h component of a vector specifying the costs of one unit of each control activity (e.g., the cost of adding a certain control device to one 1970 vehicle). The hth component of a vector specifying the yearly emission levels of the N contaminants under consideration with “no control” (x, = 0) (e.g., with no controls installed, the average NO, emission level, say in tons/day). The hth component of a vector specifying the yearly emission levels of the N contaminants to be obtained by the control activities X,. The number of units of source i controlled by one unit of activity j (e.g., one 1970 vehicle is controlled by adding one unit of a certain control device to 1970 vehicles). The amount of the mth limited supply input used by one unit of the j t h control activity (e.g., the amount of natural gas used by a unit of a given control activity). The emission reduction of the hth contaminant resulting from one unit of the j t h control activity (e.g., the reduction in NO, emissions resulting from addition of one control device to a 1970 vehicle).
The mathematical statement of the cost minimization problem, once the constants S,, L,,,, c Et 0 . E / , A , ,, D,, ,, and B , are given is: Find I,that minimizes ),
,
Control Cost-Emission Lecel Relationship: G(E,) Kohn ( 2 ) has formulated a mathematical linear programming model for determining the minimal cost of
r
c,
=
c
cjsj
j=i
Volume 8. Number 9 , September 1974 813
subject to the constraints: r
~ B , , S ,= E,'
-
E , k = 1, . . . , -V, (b)
151
( e m i s s i o n s a r e reduced to E b )
2AijXj5
si i
= 1,. . . , s
(c)
Then, by Equations 1 and 2 and the linearity of the equation of advective diffusion describing the dispersion of an inert pollutant, pollution levels on any day are strictly proportional to the yearly emission level, E = aEO. Thus, a t any emission level, E, the corresponding distribution function, N ( P ) ,is
N(P)dP= NO(P/cu)d(P/a) = the expected number of days per year that the
j =1
-
maximum n-hour pollutant concentration is in the range P P + dP a t emission level E =
(no m o r e of s o u r c e i t h a n S i c a n b e c o n t r o l l e d )
CYEO
7
But, when we know N ( P ) , the air quality a t any emission level is simply calculated by (supply l i m i t s of fixed i n p u t s . L,, a r e not exceeded)
s,
P
0 j = 1. . . .
. i'
(e1
(nonnegativity of c o n t r o l a c t i v i t i e s ) Letting C denote the minimum of C,, the solution to this problem (for fixed S,, L,, c,, A,,, D,,, and B, !) for various values of E k gives the emission control cost function, c = G(Ek). There are several assumptions inherent in the linear programming emission cost model of Equation 5 . The interested reader is referred to Kohn ( 2 ) or Trijonis (3) for a discussion of these assumptions and the applicabilitv of the model. Eniission Leuel-Air Quality Relationship F(EJ Determination of the relationship between air quality and emissions, P, = F,(E,), is a problem in atmospheric physics and chemistry. There are two basic approaches to finding such a relationship. One, which can be called the physicochemical simulation approach. is to model the atmosphere mathematically and solve the appropriate chemical and physical equations on a computer. Diffusion modeling is a prime example of this approach. The second is a statistical-empirical approach; past atmospheric monitoring data and simple physical assumptions are used to derive the relationship. In the Los Angeles photochemical smog application (Part 11), this study will use "management-type'' air quality indices; air quality will be measured by the expected number of days per year violating state standards. Physicochemical models are not suited for providing this type of air quality function because solutions are needed for the whole distribution of meteorology. Thus, the statistical approach is used. Two statistical models have been developed for this application, one for primary contaminants and one for secondary contaminants. The mathematical forms of these models are summarized below. The first statistical model uses monitoring data taken a t a given yearly emission level, Eo, to determine the expected number of days per year that an n-hour standard for a primary pollutant, P, is exceeded as a function of emission level. Two physical assumptions underlie the model: (1) that the pollutant is an inert, primary contaminant and ( 2 ) that emission level changes are homogeneous (that they occur proportionately in space and time to the emission pattern existing a t Eo-i.e., E ( x , t ) = cuEO (x, t ) . The calculations proceed as follows: A t some base yearly emission level, EO, one determines from atmospheric monitoring data the distribution function, N O P ) ,where
the expected number of d a y. s .p e r y e a r t h a t t h e -hour s t a n d a r d , P,, is e x c e e d e d at a n y e m i s s i o n level. E =
)\
*
=
SO(P/a) d ( P / a) - s
aE,
( = function of
(
i
P,,t h e
standard
CY* t h e e m i s s i o n l e v e l
So,t h e d i s t r i b u t i o n function
m e a s u r e d at o n e e m i s s i o n level
In Part I1 this model is applied to the 1-hr California standard for nifrogen dioxide. Of course, SO2 is certainly not an inert pollutant. Equivalently, the assumption is made that maximal NO2 levels are strictly proportional to KOr emissions. The second statistical model is applicable to mid-day air quality levels of a secondary, photochemical pollutant, 2, produced by reactions stemming from two primary contaminants, X and Y. The basic idea for this model was put forth in a paper by Schuck et al. ( 4 ) . The expected number of days per year that a given standard for the pollutant, 2,is exceeded as a function of emission levels of the two primary contaminants is determined from air monitoring data taken a t given yearly emission levels, (El"(x, t ) . EzO(x, t ) ) , Four physical assumptions underlie the model: (1) Emission reductions of X and Y occur homogeneously. ( 2 ) In the air mass that will lead to the pollution of any day, emissions of X and Y accumulate without reacting to produce final (morning) concentrations x and y, respectively. (3) Accumulation stops and then certain weather variables act on the primary contaminant concentrations, x and y,to produce a (mid-day) level z of the secondary contaminant, 2. (4) The weather factors that determine the level of 2 produced from given x and 3 are statistically independent of the distribution of x and y.
The calculations proceeds as follows: At some base yearly emission level of the primary contaminants, (E1O, EzO), one determines from atmospheric monitoring data the following two distribution functions:
No(P)dP= the expected number of days per year that the
No(x,y)dxdy = the expected number of days per year that
maximum n-hour pollutant concentration is in the range P P dP a t emission level EO
the morning concentrations of the primary contaminants are in the ranges x x + dx
-
814
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Environmental Science 8, Technology
-
-
+
and y y d y , respectively, a t emission level ( E l 0 ,EzO) = the probability that t exceeds some standard z s , on a day with morning concentrations x and y of the primary contaminants a t emission level ( E l 0 ,EzO)
P,O(x,y)
€2
Then, [as discussed in Trijonis ( 3 ) ] ,Equations 1-4 imply that a t any emission level, ( E l , E21 = ( a E 1 ~ , 3 E 2 ~ ) , the corresponding distribution functions are
= NO(x/cy,y / p ) d ( r / 4 d ? ' / P )
S ( S . y)dxdy
P&XJ)= PsO(s,y ) But, knowing N ( x , y ) and P,O(x,y), the air quality for any emission level is simply calculated by t h e e x p e c t e d n u m b e r of d a y s p e r y e a r that 2 exceeds t h e s t a n d a r d , z s r at e m i s s i o n level. (El.E,) = n m
( a E i O /3Ezo) .
=
k-I-
s(x, y ) x E,
P,(s, I' )dsdy
n m
Figure 3. air quality
Two-dimensional illustration of solution to least costmodel
(7)
= function of
1
cy and 9, e m i s s i o n l e v e l s Soand P,", d i s t r i b u t i o n functions m e a s u r e d at a base e m i s s i o n l e v e l for a given s t a n d a r d . 2 ,
In Part I1 this model is applied to the problem of midday ozone in central Los Angeles (Downtown, Pasadena, Burbank area). The relevance of this application in regard to assumptions of Equations 1-4 is discussed a t length in Trijonis ( 3 ) .
Solution of Complete Model Once the control cost-emission level relationship. G, and the air quality-emission level relationship, F, have been determined, the least cost of achieving various air quality levels is found by solving the system of Equations 4. It consists of a nonlinear mathematical programming problem. The development of techniques for solving nonlinear programming problems has been a rapidly advancing field of applied mathematics during the past 10 years [Abadie ( 5 ) ] .Many different numerical solution methods have been developed. For a specific problem, the choice of solution method depends on N , M , and the form of the functions G and F,. If G is determined by the linear programming model of Equation 5 , it is necessarily a convex
function; this should aid in the solution of Equation 4 [Baumol ( 6 ) ] .However, it is not the purpose here to become involved in an extended discussion of nonlinear programming; suffice it to note that techniques for the solution of Equation 4 have been developed. For a two-dimensional emission vector, ( N = 21, the solution of the model can be illustrated graphically. Figure 3 presents a schematic diagram for the case of two air quality constraints, (M = 2 ) . The axes of the graph measure total emission levels, E1 and Ez. The uncontrolled or base emission level for the basin is the point, (E10, EzO). The curves labeled C1, C2, etc., are iso-cost curves-i.e., G(E1, Ez) = C1, G(E1, Ez)= CZ, etc. Along any curve labeled C h , the minimum cost of reaching any point on that curve is Ct . As emission levels fall (downward and to the left in the graph), control costs rise. Thus, CI < C2 < , < C g . The air quality constraints are represented by the two curves Fl(E1, E z ) = P1O and Fz(E1, Ez) = P2O. The constraint of a t least reaching air quality level P10 for the first pollutant requires that emissions be reduced below the P1O curve. The constraint that air quality be a t least as good as PzO for the second pollutant requires that emissions be reduced to the left of the PzO curve. The emission levels which satisfy both air quality constraints lie in the cross-hatched region. This is the admissible air quality region. The minimum cost of meeting both air quality constraints is Cg, and the solution is to reduce emissions to point A .
..
(Part 11follous on next p a g e )
Volume 8 , Number 9, September 1974
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