Sensitivity Analysis Models

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* Centro Teoria dei Sistemi. C.N.R.. Via Ponzio. 3415, Milano, Italy.
This work was supported jointly by IIASA and the Centro Teoria dei Sistemi. C.N.R., Via Ponzio. 3415. Milano, Italy. Research Reports provide the formal record of research conducted by the International Institute for Applied Systems Analysis. They are carefully reviewed before publication and represent, in the Institute's best judgment, competent scientific work. Views or opinions expressed herein, however, d o not necessarily reflect those of the National Member Organizations supporting the Institute or of the Institute itself.
International lnstitute for Applied Systems Analysis 2361 Laxenburg, Austria

This report is one of a series describing IIASA research into approaches for comparing alternative models that could be applied t o the establishment of control policies t o meet water-quality standards. In addition to model evaluation, this project has focused on problems of optimization and conflict resolution in large river basins.

Sensitivity theory is applied in this paper t o a class of generalized Streeter-Phelps models in order t o predict the variations induced in BOD by the variations of some parameters characterizing the river system.
The paper shows how simple and elegant this technique is. and at the same time proves that many relatively complex phenomena can be explained by Streeter-Phelps models.

Sensitivity Analysis of Streeter-Phelps Models
The aim of this paper is twofold. First we show how the sensitivity of a given river-quality model can be analyzed by means of the so-called sensitivity theory. For this we first survey the main ideas of sensitivity theory and then as an exercise apply it to simple Streeter-Phelps models. Second, we point out that the result of this study proves that Streeter-Phelps models are flexible and abound with relevant consequences if one knows how to analyze them.
Here, we discuss how a given model is influenced by the variations of some of its main parameters (sensitivity analysis). This can be done in two different ways depending upon the purpose of the sensitivity analysis. One way is to simulate the system several times for different parameter values that cover the expected range of parameter variations and then compare the different solutions. The second way consists in calculating, at a nominal parameter value, the derivatives of the system solution with respect to the parameter. If the purpose of the sensitivity analysis is, for example, to make sure that an oxygen standard is not violated if temperature or flow rate varies over a certain range, one can show by decision-theoretical arguments that the first type of sensitivity analysis should be preferred (Stehfest, 1975a). If the sensitivity is to be discussed in general, without reference to a particular application, calculation of the derivative is most appropriate, because the result can be presented more succinctly than in the other case. Therefore, this approach is used in the following for a sensitivity discussion of the Streeter-Phelps model. Before doing this, however, we

briefly present the elements of t h i s type of sensitivity analysis (see, f o r example, Cruz, 1973).
Assume t h a t a c o n t i n u o u s , lumped parameter system i s des c r i b e d by t h e v e c t o r d i f f e r e n t i a l equation
where x is an n-th order vector and 8 is a constant parameter
with nominal value 8, and l e t t h e i n i t i a l s t a t e xo of t h e sys-
t e m depend upon t h e parameter, i.e.
The s o l u t i o n o f Eq. (1) w i t h t h e i n i t i a l c o n d i t i o n ( 2 ) i s a function
which, under very general conditions, can be expanded i n s e r i e s i n the neighborhood of t h e nominal value of t h e parameter, i.e.
where x ( t ) = x ( t , 8 ) i s t h e nominal s o l u t i o n . The v e c t o r
[ax/aOlg , namely t h e d e r i v a t i v e of t h e s t a t e vector with respect
t o the parameter, is called the sensitivity vector (or sensitivi t y c o e f f i c i e n t ) and f r o m now o n w i l l b e d e n o t e d by s , i . e .

Thus the perturbed solution of Eq. (1) can be easily obtained as
once the sensitivity vector is known. When there are many parameters 81,82,...,8q, the knowledge
of the sensitivity vectors s1 , ~ 2 r . . . r ~alqlows the association of specific characteristics of the system behavior with particular parameters. If, for example, the nominal solution x(t) of a first-order system is the one shown in Figure 1, where sl(t) and s2 (t) are the sensitivity coefficients of x with respect to two parameters €I1 and €I2,one can say that the first parameter
is responsible for the overshoot of x while the second is
responsible for the asymptotic behavior of the system. This characterization of the parameters very often turns out to be of great importance in the validation of the structure of a model; in fact some of the best-known methods of parameter estimation are based on manipulation of the sensitivity vectors.
Figure 1. Nominal solution iiand sensitivity coefficients s l and sg.

By t a k i n g t h e t o t a l d e r i v a t i v e of Eq. (1) it c a n be concluded that the sensitivity vector s ( t ) satisfies the following vector differential equation:
with i n i t i a l conditions

Thus, t h e s e n s i t i v i t y vector is t h e s t a t e vector of the system (31, c a l l e d s e n s i t i v i t y system, which is always a l i n e a r system, even i f system (1) i s n o n l i n e a r . Because o f t h i s p r o p e r t y t h e s e n s i t i v i t y vectors can often be analytically determined. In any case, they can always be computed by means of simulation following t h e scheme shown i n F i g u r e 2 .


x 4)



S~ b

Figure 2. Computation of sensitivity vectors.

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Sensitivity Analysis Models