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,-
A Model for Optimal
Dynamic Control of Emissions
by
Nitin R. Patel
OR 009-72
May, 1972
Work Performed Under
Contract No. DAHC04-70-C-0058
U. S. Army Research Office (Durham)
Dept. of the Army Project No. P-9239-M
M. I. T. DSR 72650
Operations Research Center
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Cambridge, Massachusetts 02139
Reproduction in whole or in part is permitted for any purpose of the United
State s Government.
A MODEL FOR OPTIMAL DYNAMIC CONTROL OF EMISSIONS
by
Nitin R. Patel
ABST RACT
This paper argues that, for large air pollution sources, it may be
better to set emission standards which differ for days with differing
ventilation conditions instead of the common method of setting a single
emission standard.
A highly simplified mathematical model is treated
to illustrate how such dynamic standards may be set and how their
advantages may be quantified.
A more realistic model, utilizing results
from the theory of optimization over Markov chains, is similarly analyzed.
1.
INT RODUCT ION
On±e of the
asic problems in the implementation of air poliution
ontr
.
programs is the translation of air quality standards into emission standards.
Air quality standards specify how often various pollutant concentrations can
be permitted to occur.
Air quality standards specify pollutant levels which
have negligible or no effect on the health of men, animals and plants.
In the
U.S., for example, the Environmental Protection Agency specifies that the
3
daily average concentration of sulfur dioxide should not exceed 365 tg/m
more than once a year, and that the annual average concentration should be
less than 80 1gm/m3 [1].
In order to achieve a given air quality standard it
is necessary to decide by how much emissions from the various sources will
have to be reduced. For example, if a source is emitting 2000 lbs/day of S02
it may be possible to meet the required air quality by reducing emissions to 1000
lbs/day.
The figure of 1000 lbs/day would then be the emission standard for
that source.
Emission standards thus specify how much pollutant an emitter
will be allowed to release into the atmosphere.
The most widely used method of setting emission standards is to fix an
upper limit on the quantity of emissions that may be released on any day.
kind of standard will be referred to as a static emission standard.
I propose to argue the merits f setting dynamic emission standards.
This
In this paper,
These are
emission standards which will meet the required air quality standards but which
set down different emission levels for different meteorological conditions.
this approach, emission standards can be set so that when atmospheric
By
2.
conditions favor rapid dispersal of pollutants they will permit larger amounts of
--. s;::nd
of emission.
- n
'he atrosphere is stagnant
will only alilcv/
,rrz.'l
amo.
ts
The setting of dynamic standards leads to greater efficiency in
the utilization of air resources in that given air quality standards can be met at
lower cost than static standards, or alternatively, better air quality can be
achieved for a given cost.
I do not mean to imply that dynamic standards are
universally superior to static standards.
to set and to administer.
They are considerably more complex
However, I feel that they have much to offer in
controlling large sources of pollution such as power stations, kraft paper mills
and municipal incinerators.
automobiles,
For space heating of homes and offices and for
static standards are, perhaps, the only feasible means of control.
In the next few sections I indicate how operations research techniques may
be used in setting dynamic emission standards and in quantifying the advantages
of dynamic over static emission control policies.
A SIMPLE MODEL
In this section, I will describe and analyze a simple model for air pollution
control which, though it is quite crude, does bring into focus the principal
elements involved in setting dynamic emission standards.
I shall take as my starting point a model, suggested by Marcus [21.
model, based on a 'box' model for diffusion, relates concentrations of a
pollutant on two successive days as follows:
This
3.
C(t) = (C(t-l) + e(t)) v(t)
wh
Cit),
C,
(1)
C(t- 1) are the concentrl:ioas of the pclutar;
. da -
t and t-l respectively
e(t) is the amount of pollutant emitted on day t per
unit volume of the diffusion 'box.'
v(t) is the 'ventilation factor' for day t.
For our simple model we will assume that v(t) can take on values of either 0 or
1 (that is,
days either have perfect ventilation or complete stagnation).
Further,
let us suppose that {v(t)} forms a homogenous Markov process such that
pr (v(t) = 0 Iv(t-l)=O) = po0
pr (v(t) = 1 v(t-l)= 1) =
pr (v(t) = 1 v(t-l) = 0) =
P 1 , pr (v(t) = 0
v(t-l) = 1) =
-P
-pl
Let us suppose that we wish to set two levels of emission, level 1 corresponds
to an emission of 1 unit of pollutant/day and level 2 corresponds to an emission
of 2 units of pollutant/day.
Let us suppose level 2 corresponds to a cost of $ O/day
(no abatement) whereas level 1 corresponds to a cost of $ d/day due to abatement
(fuel switching, curtailed production value,
etc. ).
We have a single air quality standard to meet, namely that a given level
of concentration C
should not be exceeded more often than 100a % of days.
We wish, in other words, to set a dynamic emission standard such that
pr (a randomly chosen day has concentration > C ) < a
(2 )
Clearly if we can meet (2) using level 2 only, then the optimal standard is to
specify this level for all days.
abatement is ever required.
be satisfied in this way.
This would be the uninteresting case where no
Let us suppose, to avoid triviality, that (2) cannot
4.
We shall determine the optimal dynamic policy in the class of policies
of the following type:
Use level 1 on days when concentration reaches or exceeds a level k,
otherwise use level 2 as the emission standard.
Now k < C
otherwise we will not be able to meet requirement (2).
0
wish to find what should be the optimal k from k = 0, 2, 4, ...
C
C
We
(assume
is even).
o
If we select a day at random from a long time period, the probability
that we obtain a day which has concentration >0 is the probability that V=1 on
that day.
From the theory of Markov chains we know that this probability is given
by the steady state probability of being in ventilation state 1.
Let
0'
1
be
the steady state probabilities of being in ventilation states 0 and 1 respectively.
TO
0
and
1
are given by
p0 T0 +(
and
T
0
+
T
i
=
-Pl)
1 1
0
= 1
Solving, we obtain
-P
1
0 = 2-n -n
T
and
l-Po
2-n
=
o
0
a)
U
0
o
U
0
- -
-
2
Figure 1
rrrr
5
5.
In other words pr (a randomly selected day belongs to some stepped
block of the type shown in Fig. 1) = rI
1
l-Po
2-PO-P 1
Now, pr (random day belongs to a 'block' of base - length b it belongs to
some block)
b x pr (random day belongs to a block of base-length b)
00
k pr (random day belongs to a block of base-length k)
=1
b
Pl
b-1
b-1
(1-Pl)
b P1
E (base of blocks)
(l-P 1)
b
b-l (-P
)2
b= 1, 2, 3...
/( 1-P )
(4)
Also,
pr (random day has concentration > C | it belongs to a block of baselength b)
1
k
(b-C
+-)
b
o 2
for b
0
otherwi s e
C
o
--
k
2
(5)
=
Combining (3), (4) and (5) we obtain
pr (random day has concentration > C )
k
1
(b-C +
b=C -o 2
C -k/2
('-Po) P 1
(2-Po
0-
1)
k
) b
b-i
1
(1-p)2
(l-P )
(2 -Po-P1)
6.
Clearly to minimize cost we should arrange to choose k as large as
possible consistent with satisfying requirement (2)i.e., k must be the
largest even integer for which
k
C
( )0 p
Wz
Pk/2
a (2-Po-P1 )
(6)
Since the log function in monotone increasing
log (l-Po) + C
so that
k
2
log P 1 - log a - log ( 2 -Po - P 1)
must be the largest integer (k
kk_
(1-p)
plog
o)
2
(2-Po-Pl)a
+
2 log P
C ) such that
0
C
log P 1
1
or more simply
log (rl/a) +
log P
k
2
(7)
o
We have thus obtained the optimal dynamic emission policy that will meet air
quality standards at lowest cost.
(Note that the actual value of d plays no role
in the analysis thus far. )
We shall next obtain the average cost per day of the optimal dynamic policy
for comparison with the static policy.
pr (Random day has concentration
b--
k
b2
=
+
for b
k/2
otherwise
k it belongs to a block of
base-length b)
7.
pr (Random day has concentration > k)
co
2b - (k-2)
2b
-
2 (1- p
b-l
(
I
)
1-P (2-pO-Pl)
k
b--
=
b
(8)
Tr 1 pk/2
-1
1k given
by (6),
For optimal k given by (6),
C
E (cost) =
d
1
p1
*
r1 p 1
Pi
2
=
a
C'
o
-1
(9)
a
Let us now develop for purposes of comparison the cost for a static emissions
policy.
Let 1
f
<2 be the static emission standard per day.
Using analysis similar to that used above we obtain
[co
pr (Random day has concentration > C )=
T1 l
p
where the notation [x] indicates the largest integer
x.
To minimize cost we will choose the largest value of f such that
L0-1
Cr
o
f
log
>
a/r
log
<
'
lP 1
- a.
1
(10)
1
Let + (f) be the cost ($/day) of restricting the emission per day to a value
of f.
(Note
(2) = 0 and + (1) = d. )
Then the expected cost for the static policy will be + (f ) where f
is the
solution to (10).
Computation of the average costs for dynamic and static emission control
strategies for a hypothetical example are shown below.
8.
256
Let d = 1, p 0 = pi = 1/2, a
Let
=8
+ (f) be linear
Then we find
T=
r
= 1/2
8-1
1 .
E (cost dynamic) =
1/256
= 1/2
1/256
Also,
C
Cf
log
/Z
7
6
f=8
Ao1/2
7
log 1/2
E (cost static) = 1 (2-8) = 6
Ratio dynamic to static cost
7
12
A MORE COMPLEX MODEL
The model discussed in the previous section needs to be expanded to make
it more realistic.
in order to:
In this section we will consider a model which is expanded
(a) incorporate ventilation states other than total and zero
ventilation, (b) have air quality standards specified by both a constraint of
type (2) and an average annual concentration constraint, (c) more than two
levels of emission standards.
Again we use as our starting point relation (1)
there are n levels of ventilation vl, v,
...
,v
n
(0
.
.v
; 4
3
We shall assume that
1, j = 1, 2, n).
Let there be q levels of emission to be incorporated in the dynamic standard
el, e 2 ,...
e
C
.C..C
1
1'
C,
2'
(with el<e,
.
.
.<e
).
Let us also restrict concentrations to
m
*In fact, any relation of the type C(t) = f(C(t-l), u(t), v(t) ) where f is an
arbitrary function can be treated with the analysis outlined here.
9.
We shall assume that the sequence {v(t)} forms a homogenous Markov
v2 , .
chain on the n states v,
.. vn
. Let q(j L)= pr {v(t) = j Iv(t-l) = L}.
From
relation (1) using discretization we can compute
p(i, j
k, L, d) - pr (C(t) = i,
v(t)=j
C(t-l) = k,
v(t-l) = Land e(t) = ed).
Now we have an induced Markov chain on the states (C(t), v(t) ).
denote the states of this Markov chain by s.., i=l, 2,...m,
Thus s..
Let us
j = 1, 2, ...
n.
state corresponding to concentration level C. and ventilation level v..
1j
1
3
Suppose we have as our air quality standard both requirement (2) and the
requirement that the annual average concentration should be less than P units.
Let k d be the cost of operating one day at the emission level e d .
Then the problem of setting dynamic emission standards so as to minimize
costs while meeting air quality standards can be shown to be the following
linear programming problem [31:
m
Min
q
n
kdwij d
E
i=l
j=l d=l
subject to
wij
-
d=l
q
n
m
q
\
+
p (i j k, L, d)wkLd =0
for i=,
2,..,
m
k=l I,=1 d=l
m
q
n
kis
=l
n
W=kLd=
q
>1f
k>C
L=1 d=l
0
WkLd
<L
(11)
10.
n
Ck WkLd ' P
(12)
L=I
wa
kLd
0
k= 1, 2, ... ,
m
L=i, 2, ... ,
d=l, 2, ... ,
n
q
where w..
are the decision variables which are solved for in the linear
program.
Physically wi
is the probability of setting emission level
ijd
Note that in general the solution will give
at ed and being in state sij.
us a randomized strategy as optimal.
That is to say, our solution will
say that when in state s.. the emission level for the optimal dynamic
strategy should be ed for a certain percentage of the time (not necessarily
0 or 100%).
An important advantage of this approach is that the shadow prices
on rows (11) and (12) explicitly give
the air quality restrictions.
us the marginal costs of altering
Thus, it is possible to investigate the
economic impact of raising or lowering the air quality standards.
To illustrate this, a small hypothetical model was solved on a
computer.
For this example:
q = 2, m =
1
= 0,
6, n = 4
v 2 = 0.1, v 3 =
0.2, v 4 = 0.4
C1 = 0, C2 = 2, C3 = 4, C4 = 6,
el
1, e 2 = 4, k
C 5 = 8, C
10
= 1, k = 0
and the transition matrix q(j i) is given by the element in the i throw and
.th
j
column of the following matrix:
11.
0.5
0. 3
0.2
0. 0
0.2
0. 5
0.2
0. 1
0.0
0. 3
0. 5
0.2
0.0
0.2
0. 3
0. 5
I
= 6, a = 0. 02,
The air quality requirements were given by C
= 3. 0.
Table I below shows the results of parametric runs on the model and
illustrates the information these give on the cost effects of tightening or
relaxing air quality standards.
Table I
Shadow Prices
row
on
on a row
Minimum Cost
Case No.
a
P
1
0.020
3.0
.1634
1.61
0
2
0.011
3.0
.1779
1.61
0
3
0. 005
3.0
.1875
1.61
0
4
0.02
1.2
.1634
1.61
0
5
0.02
0.6
. 3095
0
.625
,
,
.
,
,. ,
To obtain the cost for the static emission policy for comparison with the
dynamic, we can use the following procedure.
1.
Set d=q
2.
Solve
V V
2,
k
I1 kLp(i, j
k, L, d) =
L
.
ij
I
k
L
IT
=1
kL
for II ij = steady state probability of being in Sij under static
emission level e d .
12.
C3.II
If
k> C
L
a
and
kLCk <,
then quit.
The
kL
value of d obtained is the optimal for a static policy.
Other-
wise reduce d by 1 unit and repeat step 2.
For our example problem the cost for the static policy is 1.
CONCLUSION
In this paper I have suggested that dynamic standards are more efficient
than static standards for control of large pollutant sources.
I have analyzed
a simple model which provides some insight into the nature of dynamic
standards and a more realistic model which would help in optimally setting
dynamic standards.
These models will also aid quantitative estimation of
the advantages to be gained from dynamic over static control policies.
One aspect of dynamic strategies which has implications for research
and development of anti-pollution devices is worth noting.
The dynamic
strategy may render feasible techniques and apparatus for pollution control
which have high operating costs. In such a case, if one computed the cost
for a static emission standard itmay be too exhorbitant to impose on the
polluter.
However, the dynamic strategy may require only intermittent
use of the expensive pollution-control equipment and hence, may reduce
costs to an acceptable magnitude.
Thus, research efforts could be justifiably
directed towards developing pollution control technology even if it has high
operating costs.
13.
REF ERENCES
1.
National Primary and Secondary Ambient Air Standards, Federal
Register 36: 8186-8201, April 30, 1971.
2.
Marcus, Steven J., "Mathematical Decision Models for Air
Pollution Control Policies, " unpublished PhD thesis, Harvard
University, January 1971.
3.
Wagner, Harvey M., "Principles of Operations Research,"
Prentice-Hall, 1969.