Consideration of Sensitivity and Parameter Uncertainty in Optimal Process Design Michael S. K. Chenl, Larry E. Erickson, and Liang-tseng Fan Department of Chemical Engineering, Kansas State University, Manhattan, Kan. 66502
Sensitivity analysis i s applied t o reveal important parameters in the optimal design of a two-stage continuous-flow stirred-tank reactor system with recycle. The specific example of a two-tank biological waste treatment system is examined to detect parameters to which the optimal total holding time i s most sensitive. The total holding time is most sensitive with respect t o the efficiency of the thickener, next the recycle ratio, and thirdly the maximum specific growth rate (in order of decreasing sensitivity). Several optimal design strategies which take into account parameter uncertainty are proposed. One of these, the expected value criterion, has been used to find new optimal results a n d quantitative safety factors for the two-tank biological waste treatment system. In general, the optimal total holding time should be increased as the range of uncertainty increases. Results are presented for normal a n d uniform parameter distributions.
P a r a m e t e r variations occur in a number of processes because of changes in feed flow rate, feed composition, ambient conditions, and other factors. One process which i s repeatedly subjected to changes in these parameters is the biological waste treatment process. In previous papers (Erickson and Fan, 1968; Erickson et al., 1968), the optimization of the hydraulic regime of the activated sludge process and the step aeration process were considered. In each of these studies, the optimal design variables which minimize a preassigned objective function were determined based on a set of fixed numerical values of the parameters, such as the yield factor, flow rate and concentration in the influent, temperature, pH, and separation efficiency in the thickener. Unfortunately, the values of the parameters in practice are subject to change owing either to the uncertainties in the experimental evaluation or to variations of the operating and surrounding conditions. Much of this uncertainty usually is partially resolved after the system is built and its operation observed. In this study we describe a strategy for hedging against design errors caused by this persistent uncertainty. I Present address, School of Chemical Engineering, University of Pennsylvania, Philadelphia, Pa. 19104
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Specifically, an attempt is made to answer the following two questions: What is the effect of a change in parameter values on optimal values of the objective function and the optimal design variables? How can the uncertainty of parameters be built into the process model and the selection of optimal design variables, if it is found that the optimal performance of the system is strongly dependent on the selected parameter values which are subject to variation? These questions are closely connected to each other, in the sense that the answer to the first is the first step in answering the second. I n the past, these problems were treated qualitatively only through the designer’s experience. Such an experience often takes the form of “safety factors” in the final design. I n this study we discuss how we may incorporate such empirical factors into design procedures based on a more quantitative foundation by using the statistical decision theory. We first apply sensitivity analysis t o the optimal design of the activated sludge process to identify the parameters that deserve particular attention, Then the optimal system design with parameter uncertainty is considered.
Sensitivity Analysis of Activated Sludge Process
The activated sludge process considered here is the same as the one in the previous paper (Erickson and Fan, 1968) (Figure 1). The model is composed of a sequence of N conipletely mixed tanks connected in series, followed by a secondary clarifier. A portion of the sludge from the bottom of the clarifier is removed and sent to the sludge disposal system, and the remainder is recycled. The system equations are
' - x; -
not be used; however, for fixed values of r and w the value of fl must satisfy the constraint
The system equations, Equations 1 through 4, are transformed into dimensionless form based on the nominal or expected values of flow rate q, inlet organic concentrations, ?/, yield factor, P, and recycle ratio, F, as follows:
] = 0, n=l,2,
...,N
(1)
n = 1, 2, . . . , N
n = l , 2 , ...,N
(2)
n = 1, 2, rxi'
(la)
(3)
y:'
volumetric flow rate, ft '/ hr recycle flow rate ratio, dimensionless flow rate ratio to sludge digestor, dimensionless concentration of organic waste in nth tank, lbift' concentration of active microorganisms in nth tank, lb/ ft volume of nth tank, f t ' separator concentration efficiency, dimensionless nutrient conversion yield factor, dimensionless maximum growth rate when organic concentration is not limiting rate of growth, hr Michaelis-Menten constant, lh/ft ' specific endogenous microbial attrition rate, hr influent organic waste concentration, lh/ft3 d is defined as the ratio of organism concentration in
the sedirnented phase of the clarifier to the organism concentration in the inlet stream. This definition has the advantage that, in the following analysis, Equation 5 need
- -
34
R(l
+ F) - [l
-
R(l
.. .,N
1 y;
+ r)
=
o
(2a)
(3a)
where
y; = yz =
?$IF{
0" =
v",q(l+ r)
x;/P
z
K1 = K / E !
F!
34 =
x:/
R
(1 + r ) / ( l + 7 )
=
Q = ql? Yd = Y I P The minimization of the two objective functions given below was considered in the previous study (Erickson and Fan, 1968). Case I. The total volume of the biological growth chamber is to be minimized-Le.,
Figure 1. Schematic diagram of biological treatment system
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where (ag:aX)-' is the inverse of matrix iiglax. All the partial derivatives of f and g are evaluated a t the optimal conditions and a t the specific values of the parameters. Substituting Equation 12 into Equation 11 gives in vector matrix form
or
Case 11. The sum of the cost of the organic waste being discharged and the cost of the total volume is to be minimized-i.e.,
J? = alq(l - U)X?
+~2Vr
(8)
where a, and a2 are constant. For Case I, the degree of freedom of the system is (N - 1) for the set of fixed parameters xi, x;", q, r , B, Y , h, k D , and K ; for Case 11, the degree of freedom becomes N, since x;" is not fixed. I t would be desirable to see how variations of these parameters affect the optimal design variables and objective functions. This is called the sensitivity analysis of the optimal system (McBeath and Eliassen, 1966; Radanovic, 1965). Let J = f(x, e, 5) where x is the s-dimensional state vector, 0 is the r-dimensional decision (design) vector, and [ is the p-dimensional parameter vector. Then the sensitivity of J to the variation of & which is a component of [ at [a= & is expressed as
TYPEB. EFFECTO F VARIATIONI N O N J , X, A N D 8 OPTIMALDECISION7 UNIFIXED.The sensitivity analysis of this type is to find aJla[,, a 8 l a [ , , and a x l a [ , . WITH
Since 0 and x are constrained and they must change optimally, this problem should be handled carefully. According to the Lagrange multiplier method, s number (equal to the number of equality constraints) of multipliers are introduced to release the s constraints. Then the necessary conditions for the optimal solution to the problem are
h = 1 , 2 , . . . )5
gb(x,o,$)=o,
j = 1,2,.. .,s
(14)
(15)
and where J = f ( Y , B, $) where X is the state variable under the optimal condition corresponding to $. [ a J / d [ , ]is~ called the sensitivity coefficient of J with respect to parameter (,. The sensitivities and sensitivity coefficients of state variables can be defined in the same manner as in Equation 9. Since the state vector, x, the decision vector, 6, and the parameter vector, [, of a system are often related through equality constraints or system equations-for example, Equations 1 through 5
1 = 1, 2 ,
., ., r
(16)
Differentiation of Equations 14, 15, and 16 with respect to ti gives, respectively,
g,(X,8,()=0, k = 1 , 2, . . . , s (10) the sensitivity analysis can become complicated. Two types of sensitivity analysis may be considered (Chang and Wen, 1967, 1968; Radanovic, 1965), with the type of analysis depending upon whether or not the decision vector is fixed. TYPEA. EFFECTOF VARIATION IN tL O N J AND x WITH FIXED OPTIMALDECISIONS.This is to find aJia[, and dx,/a&, j = 1, 2, . . . , s, with the values of the decision variables fixed a t their optimal values. Differentiation of the objective function with respect to yields
k = 1, 2,
[2
The derivatives dx,/a[,-i.e., the sensitivity coefficients of the state variables with respect to parameter [,-are obtained from the equality constraints as follows: Differentiating Equation 10 with respect to .$ gives
The solution may be expressed in vector matrix form as 5 16
The above equations consist of a set of linear algebraic equations with unknowns a x / & dola[,, and axla[,. Sensitivity coefficients ae/a[, and a x l a [ , are obtained from the solution of the above simultaneous equations based on a set of parameter values, and the optimal decision and state variables. All the partial derivatives-of g, h, and P are evaluated a t the optimal values, x and 8, for the nominal parameter values. To find the sensitivity coefficient aJ,a[, in this case, we note that
J =f
+
Mi, t - 1
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
(20)
A=4 y:=o.1
Q = I y; = I
I
s;:
I
Substituting Equations 14, 15, and 16 into Equation 2 1 gives __ i,J = i A * - + dg, -
at)
/
rlE,
I
af
atl
Solution of the set of equations given by Equation 15 for x i , k = 1, 2 . . . . , s and substitution of the result into the above equation yields
02
, 03
05
04
-
'K,
b -04 -0.6 L
which is in vector matrix form. I t turns out that the sensitivity coefficient a J / & in this case is exactly the same as dJ at, in the previous case, as can be seen by comparing Equations 13 and 22. This coincidence is by no means casual, if we recall that the sensitivity analysis is carried out a t the optimal condition where aJ,aO = 0. This implies that there is no variation in aJi a(, through i d as, regardless of whether e is fixed or not. However, the sensitivities of the state and decision variables are different in these two cases. The first case will result in the sensitivities of the state variables where the design variables are fixed, while the second case will result in the sensitivities of the state variables as well as those of design variables. The sensitivity coefficients aO/ a& in the second case are particularly useful in estimating and determining the margins of design variables to compensate for the changes in system parameters in order to maintain the optimal system performance. The sensitivity analysis of Type B has been applied to the two-tank activated sludge system a t its optimal design. The optimal results for the Case I objective function have been found for the following set of parameter values (Erickson and Fan, 1968): .I = 0.5
li = 0.1. hr hi, = 0.002, hr r = 0.25
'
d = 4.0, 4.5, a n d 4.75
(23)
K , = 0.01, 0.02, 0.5, 0.1, 0.2, a n d 0.5 .,,I I -l,Q=l. Y,,=l,andR=l These parameter values are used as nominal values in the sensitivity analysis with the sensitivities evaluated a t the nominal values (Chen, 1969). Typical sensitivities of the optimal total holding time %T = 0' + 0' with respect to each parameter are shown as a function of K 1in Figure 2. From the figure several interesting and important observations can be made. The optimal total holding time, % T , is very sensitive to the variations in 3 and r, moderately sensitive to those in Q. k , K , , yf, and yi, and insensitive to those in hi, and Yd. Should there be a great uncertainty in sensitive parameters such as p and r, the resultant design would deviate greatly from the optimal one. This also indicates
SJ, x 10-1
1
Yl
- 1.4 Figure 2. Sensitivities of total holding time, 81, with, respect to each parameter vs. K1 for case d = 4 a n d y i = 0.1
which parameters should be controlled carefully during the operation of the existing plant. If the mathematical model is a good representation of the actual process, the positive and negative signs associated with the sensitivities will clearly indicate that operating and environmental conditions may be improved to obtain a more optimal design than the original one. For example, negative sensitivities such as S:, S < , S:. and S i as shown in the figure indicate that if we can increase y ( , 0,r, and k , we can reduce total holding time BT. It is obvious that improvement may be made increasing the thickener separation efficiency, 0, the recycle ratio, r, and the maximum growth rate, k . However, it is not apparent that by increasing influent organic concentration 34, the total holding time is reduced when, a t the same time, the effluent organic concentration, yi,is maintained a t a fixed value. T o verify this result of the sensitivity analysis, an optimization study has been carried out to with respect to y { . minimize the total holding time The optimal results are shown in Figure 3 for various sets of parameter values. I t is seen that BT decreases monotonely as y: increases from 1 (nominal value) t o 11 and it appears that no minimum value for OT exists. The greater the K 1value, the greater the reduction of BT. This result apparently contradicts the expected result, because in practice B will decrease as y{ increases. In practice, the return sludge concentration from the thickener is often more or less a constant and is usually specified as one of the design variables. One would expect that more holding time would be required t o achieve a fixed effluent organic concentration whenever the influent concentration of the incoming stream is increased. To investigate this point further, additional optimization studies, by assuming that return organism (sludge) concentrations are constant, have been carried out. As expected. Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4,1970
517
example, given that y: = 0.01, K 1 = 0.5, and 0 = 4 (other parameters such as F = 0.25, h = 0.1, k D = 0.002, and Y = 0.5 are given previously), we find that when yi = 1, y: is 3.085 and /IT is 28.3, and when y'i = 10, y: is 9.75 and BT is 36.2. Thus, the organic loading is increased from 0.439 to 1.23 lb BODi/lb MLSSiday, while the optimal total volume per unit waste per hour is reduced from 35.4 to 4.52. The efficiency is greatly improved. Mean and Variance of System Performance
The sensitivity analysis of the optimal system design can reveal parameters which may affect the system performance critically. I t is desirable to be able to estimate quantitatively the extent of uncertainty in the over-all system performance due to the uncertainty in such parameters. The system performance index can be generally expressed as
J = f ( x , 8, E )
Figure 3. Optimal holding time v s . influent organic concentration for p = 4.0
the results show that if y{ is increased, the total holding time is generally increased, but in some cases it is decreased as y{ is further increased. The latter decrease in the total holding time appears t o occur only when the specific endogenous microbial attrition rate, k o , is greater than the specific growth rate, ( h y l / K 1 + yJ, in the second tank for p = 4, y: = 0.01, and K 1= 0.5. Since any biological waste treatment process involves essentially an autocatalytic type reaction, a higher organic concentration gives rise t o a higher organism concentration in the process even when a constant return sludge concentration is maintained. As a result, the total holding time required to treat a fixed amount of waste can be reduced by treating it in its most concentrated form. The optimal volume per unit quantity of waste is plotted against influent organic concentration y: in Figure 4. As y: increases, the required volume per unit of waste decreases, the greatest decrease occurring for large values of K 1 . The use of the results obtained in this study may be limited in practical use by many technical difficulties; however, they indicate the feasibility of using a high organic loading to achieve economy and space saving. The use of high BOD in activated sludge processes has been discussed (Kalinske and Shell, 1968; McLellan and Busch, 1967; Pasveer, 1954, 1955; von der Emde, 1960). Increase in the organic loading, often expressed in lb BOD5/lb MLSSiday, as y: increases can be computed from the equation
(25)
where x, e, and 6 are in vector form. Assume that we can represent the uncertainty of variables by probability distributions, regardless of their nature (either fixed but unknown or fluctuating but randomly). I n other words, we shall accept the premise that probability reflects our understanding of a physical condition as well as the physical condition itself. Let 2 and (7) be the optimal state vector and the design vector, respectively, for a set of nominal or expected values $ of the components of parameter vector [. By expanding J in Equation 25 in a multivariable Taylor series up to and including the second-order term about $, i = I,2, . . , , p , we obtain
(r)
e
n
,
T
By taking expectation and variance on both sides of Equation 25a, it can be shown (Hahn and Shapiro, 1967) that the mean and variance of the system performance index (or the objective function of the system) are, respectively,
and
if L, = 1, 2, . . . , p are uncorrelated, and
E [J Organism concentrations in both tanks are approximately the same and for illustration the organism concentration may be assumed to be the same as MLSS. For 5 18
Ind. Eng. Chem. Process Des. Develop., Vol. 9, No. 4, 1970
50 t
y2 I
={O,I
-
This equation is frequently a satisfactory approximation even when the parameters are correlated and not distributed symmetrically. It indicates that the variance of the system performance is proportional to the product of the square of the sensitivity coefficient and the variance of each parameter. As shown in Figure 2, sensitivity S< is the largest. Therefore, if p possesses a great uncertainty, the optimal total holding time will deviate from its expected value due to the parameter. The combined effect of variation in q and /3 may give rise to a considerably larger design error.
0.01 ----
Optimal Design with Parameter Sensitivity and Uncertainty
O & - r - 2 - - 3 4 5
6
e
7
9
IO
yf Figure 4. Optimal total volume per unit waste per hour vs. influent organic concentration for two-tank system with constant return organism concentration for @ = 4.0
If parameter sensitivity and parameter uncertainty are to be taken into consideration in the optimal design of a treatment plant, several optimal design strategies may be used. Four of these are presented and the use of one is illustrated. OPTIMALDESIGNWITH PARAMETER SENSITIVITY CONSTRAINTS. This is a problem of finding the optimal decision vector 0 which minimizes
J = J(x, 8 , E )
(31)
subject to the sensitivity constraints
where ui is the tolerance of the sensitivity with respect to the ith parameter. For example, sensitivity S$ may be limited by imposing a condition of the form of Equation 32. The solution so obtained will result in a system that is less sensitive to parameter variation. OPTIMALDESIGNBY INCLUDING A SENSITIVITY FUNCTION IN THE OBJECTIVE FUNCTION. For this case we minimize
if
