Comparison of the Deterministic and Stochastic Solutions

--

Sensitivity Analysis

In o r d e r t o gain e x p e r i e n c e with systems d i f f e r e n t from Lake Balaton, w e changed some of t h e p a r a m e t e r s of t h e Balaton problem and continue o u r presentation on t h e application of EMOM with t h i s modified example. W e call t h e hypothetical water body: Lake Alanton.

Lake Alanton d i f f e r s from Balaton in t h e following r e s p e c t s : (i) The volume of Basin IV i s only 6 0 - 1 0 ~ m ~ (see Table 1 , line 5);

(ii) Two d i r e c t sewage loads

are

increased in t h e region of Basin IV ( s e e Figure 5 ) t o 50 and 70 kg/d, resulting in t h e

same

absolute BAP load t h a n t h a t of Basin I (the volumetric load i s however l a r g e r due

to

(i); 2.3 mg/m3/d, see Table 1 , line 7);

(iii) The nominal water quality indicator and t h e slope of t h e response line was modified f o r Basin IV in such a manner, t h a t response lines of Basins I and IV coincide (see Figure 6);

(iv) Cost functions of t h e two treatment plants were modified (they became similar t o t h e one illustrated in Figure 9).

A s a consequence of these changes, t h e water quality of Lake Alanton is approximately equally "bad" a t i t s two ends. Still, however, a n important difference exists between Basins I and IV: t h e load of Basin I is governed by

"stochastic" t r i b u t a r y load, while t h a t of Basin IV, is given by "determinis- tic" sewage discharges. This way t h e low water quality of Basin I is associ- a t e d with l a r g e fluctuations (as seen in previous Sections), but randomness is of secondary importance for Basin IV.

The longitudinal distribution of t h e

water

quality is now quite different from t h a t of Lake Balaton, and a manager may have t h e intention t o estab- lish a uniform quality by control decisions. Accordingly we fixed t h e goals

to

71

=

30 mg/m3 i =I.

. .

^,4, and maintained t h e same parameter values as used in Section 8.3. Results f o r t h e "basic situation" (Section 8.2) a r e given in Figures 1 3 and 14.

From Figure 1 3 t h e same conclusions can b e drawn f o r Basin I than ,

from Figure 9. The only difference i s t h a t

at

a fixed budget t h e

water

qual- ity improvement i s slightly smaller than for Lake Balaton as a p a r t of t h e budget is utilized for Basin IV. A s seen f r o m t h e figure, P precipitation i s a n effective tool for improving t h e

water

quality of Basin IV: t h e concentra- tion (Chla),, i s reduced from 80

to

about 40

mg/m3

already a t a budget of 2 . 5 . 1 0 ' ~ / yr. The i n c r e a s e of @ r e s u l t s in nearly no f u r t h e r change and

Basin I

min { Y 1 )

-

Basin I V

m x

Iy41

-

- - _e

E W 4 I

min

{y4}

I I I

b

10 20 30 fl

>

TAC [ 1

o7

F t l ~ r I

FIGURE

13. Lake

Alanton: water quality indicator

as a

function of t h e to-

tal

annual cost.

Basins I and I V

-

Stochastic model with recourse,

-.-.-

Deterministic model, Eq. (5.12)

XS'l

FIGURE

14. Lake Alanton: major decision variables.

Basins I and I V

Eq. (5.1 1)

FIGURE 15. Sensitivity with r e s p e c t to water quality goals.

t h e fluctuations a l s o remain approximately constant, nevertheless s m a l l in comparison t o Basin I.

In Figure 14,

xss

belongs t o t h e l a r g e s t treatment plant in t h e region of Basin IV. The c h a r a c t e r of xD1(@) and xsN1(@) gained from t h e stochastic m o d e l i s t h e same

as

f o r Lake Balaton (Figure l o ) , while

xsq

i s practically constant above @

=

2.5.107R / yr

.

The most important conclusion of t h i s f i g u r e c a n b e drawn from t h e comparison of t h e stochastic and deterministic solutions: t h e deterministic quadratic model (5.12) excludes (incorrectly) t h e reed-lake p a r t of t h e l a r g e s t r e s e r v o i r p r o j e c t even under t h e l a r g e r budget values, which i s a consequence of neglecting t h e random elements of t h e problem. The (Chl-a),,, concentration i s r e d u c e d on a n a v e r a g e t o about 40 mg/m3; however, extreme values still c a n e x c e e d 1 0 0 mg/m3; as c o n t r a s t e d t o t h e solution of t h e stochastic m o d e l when t h e maximum i s about 60 mg/m3 This example shows clearly if t h e analyst i s not a b l e t o recognize t h e s t o c h a s t i c f e a t u r e s of t h e problem, t h e c o n t r o l s t r a t e g y worked out may lead t o s e r i o u s failures.

The sensitivity of t h e solution with r e s p e c t t o t h e water quality goals i s illustrated in Figure 15 r e f e r r i n g t o

a

typical budget 1 5 . 1 0 ~ ~ / yr. From t h e analysis performed and t h e figure, t h e following conclusions c a n b e drawn:

(i) If t h e goal is set unrealistically f a r from t h e nominal value and from t h e values which c a n b e realized by t h e available c o n t r o l measures (e.g. 0 o r 200 mg/m3 in t h i s c a s e ) , t h e penalty function h a s n e a r l y no influence and t h u s t h e solution i s equivalent t o t h e deterministic one (xD1

=

0);

(ii) The solution is quite sensitive on t h e choice of t h e y i . The vari- able z~~ has a maximum

at

about 40 mg/m3. If t h e goals uniformly o r individually f o r Basins I and IV are close t o 75-80 mg/m t h e 3 corresponding major decision variables. are close to z e r o (no action is taken).

In summary, w e can

state

t h a t t h e water quality goal h a s a major influ- ence on t h e solution, i t f o r c e s indeed t h e solution towards t h e desired lev- els of water quality in t h e different lake basins. This f e a t u r e is t h e primary advantage of t h e stochastic model with r e c o u r s e

as

contrasted

to

t h e expectation-variance model f o r t h e Lake Balaton c a s e and i t s variant Lake Alanton.

A s t o t h e r o l e of t h e o t h e r parameters of t h e penalty function of t h e stochastic model is concerned, t h e systematic analysis performed did not lead t o unambiguous conclusions. For both examples, Lake Balaton and Lake Alanton, t h e influence of ei and qi is of secondary importance

as

compared t o t h e effect of yi

.

In general, i t c a n b e said t h a t t h e influence is minor f o r small and l a r g e budgets. In t h e middle budget r a n g e i t is difficult t o s e p a r a t e t h e impact of ei and q i , especially because it strongly depends also on t h e p r e s e t goals y i , and t h e stochastic f e a t u r e s of the water quality f o r t h e different basins.

The experience gained in t h e frame of t h e p r e s e n t study suggests t h e choice of a piecewise linear-quadratic utility function with q / 2 e

=

1 as a f i r s t step, see (5.9) and Figure 3, and t o perform a thorough sensitivity analysis on t h e p a r a m e t e r s y i , ei and qi in t h e subsequent steps.

W e complete t h i s section with t h e following conclusions:

(i) The s t o c h a s t i c optimization model with r e c o u r s e justified t h e applicability of t h e much simpler expectation-variance model f o r Lake Balaton;

(ii) Deterministic version of t h e s t o c h a s t i c objective function and t h e solution of t h e corresponding deterministic q u a d r a t i c optimization problem leads t o strikingly different and i n c o r r e c t management

s t r a t e g y

as

compared t o t h e s t o c h a s t i c model;

(iii) The major p a r a m e t e r of t h e s t o c h a s t i c optimization model with r e c o u r s e i s t h e water quality goal p r e s c r i b e d f o r d i f f e r e n t basins.

The inclusion of t h e goal in t h e objective function i s t h e primary advantage i n comparison with t h e expectation-variance model.

The advantage of t h e latter model is, of c o u r s e , simplicity and f a s t implementation;

(iv) F u r t h e r experimentation i s needed in t h e selection of p a r a m e t e r s ei and qi (in t h e objective function of t h e s t o c h a s t i c model).

In t h i s p a p e r w e d e a l t with t h e development and application of stochas- t i c optimization models f o r Lake eutrophication management. W e considered primarily shallow l a k e s which are strongly influenced by hydrologic and meteorologic f a c t o r s and t h u s stochasticity should b e a key component of

water

quality control.

Major elements of t h e study performed are

as

follows:

(i) Identification of important s t e p s of eutrophication management in p r a c t i c e ,

(ii) Based on t h e principle of decomposition and aggregation, an a p p r o a c h is presented how t o develop a eutrophication manage- ment optimization model, EMOM, which p r e s e r v e s t h e scientific details of d i v e r s e in-lake and watershed p r o c e s s e s needed

at

t h e decision-making level. The p r o c e d u r e combines simulation and optimization in t h e framework of EMOM.

(iii) W e d e s c r i b e t h e proposed stochastic, planning t y p e lake eutrophi- cation and nutrient load models, LEMP and NLMP2, respectively, which are major components of EMOM.

(iv) W e discuss control variables, c o s t functions and various con- s t r a i n t s t o b e used in EMOM.

(v) Alternative management optimization models are formulated which use t h e same LEMP, NLMP2, control variables etc., and d i f f e r pri- marily in t h e objective function and solution technique t o b e adopted. T h r e e of them were selected f o r implementation: a full stochastic model, a n expectation-variance model, and t h e d e t e r - ministic (quadratic) version of t h e full stochastic method. For t h e f i r s t one a new stochastic programming p r o c e d u r e had t o b e developed, while f o r t h e o t h e r two s t a n d a r d packages could b e employed. The objective function of t h e full stochastic model h a s one more p a r a m e t e r

as

compared t o t h e expectation-variance model: including t h e possibility of selecting t h e water quality goal t o b e achieved by t h e management, which makes t h i s model espe-

cially a t t r a c t i v e .

(vi) Application t o Lake Balaton. This p a r t of t h e study had a d i r e c t impact on t h e policy making p r o c e d u r e performed in Hungary in 1982 which ended up with a government decision in 1983.

(vii) Comparison of t h e two stochastic models, f u r t h e r m o r e determinis- t i c and stochastic a p p r o a c h e s on t h e example of Lake Balaton and on a modified, hypothetical system (the comparison

w a s

associated with a detailed sensitivity analysis).

ACKNOWLEDGEMENT

In addition to t h e development of t h e computer code f o r t h e method d e s c r i b e d in Section 7, Alan King (University of Washington, S e a t t l e ) h a s contributed much to t h i s p r o j e c t by his p e r t i n e n t comments concerning algorithmic implementation and d a t a analysis. W e hope h e finds a small sign of o u r a p p r e c i a t i o n in seeing t h e v a r i a n t of Lake Balaton named in his honor.

REFERENCES

Beck, M.B. 1982. Time-Series Analysis of Zala River Nutrient Loadings.

WP-82-116. Laxenburg, Austria: International Institute f o r Applied Systems Analysis.

Biswas, A.K. (ed.) 1981. Models f o r Water Quality Management, pp.329. N e w York: McGraw-Hill International Book Company.

Bogdrdi, J., L. Ddvid, and L. Duckstein. 1983. Trade-off Between Cost and Effectiveness of Control of Nutrient Loading into a Water Body. RR-83- 19. Laxenburg, Austria: International Institute f o r Applied Systems Analysis.

Canale, R. 1976. Modeling Biochemical Processes in Aquatic Ecosystems.

Ann Arbor Science. Ann Arbor, Michigan.

Cochran, W.G. 1962. Sampling n c h n i q u e s . N e w York: John Wiley.

Dubois, D.M. (ed.) 1981. P r o g r e s s in Ecological Engineering and Manage- ment by Mathematical Modelling. Liege, Belgium: Cebedoc.

Haith, D.A. 1982. Models f o r Analyzing Agricultural Nonpoint-Source Pollu- tion. RR-82-017. Laxenburg, Austria: International Institute f o r Applied Systems Analysis.

Herodek, S. 1983. Biochemical P r o c e s s e s in Lake Balaton, in E u t r o p h i c a - t i o n of ShaLlow Lakes: Modeling a n d Management. The Lake BaLaton Case S t u d y , pp. 101-146. L. Somlyddy, S. Herodek, and J. Fischer (eds.). CP-83-53. Laxenburg, Austria: International Institute f o r Applied Systems Analysis.

Kallberg, J., and M. Kusy. 1976. Code Instruction f o r S.L.P.R., A Stochastic Linear Program with Simple Recourse. Vancouver: University of Brit- ish Columbia.

King, A. 1985. An Implementation of t h e Lagranpian Finite-Generation Method, in Numericat Methods f o r Stochastic P r o g r a m m i n g , Y . Ermo- liev and R. Wets (eds.). IIASA Collaborative Volume.

Kovdcs, L.B., E. Boros, and F. Inotay. 1983. A Two Stage Approach f o r Large Scale Sewer Systems Design with Application t o t h e Lake Balaton Resort Area, in E u t r o p i c a t i o n of Shallow Lakes: Modeling a n d Management. 7he Lake B a l a t o n Case S t u d y , pp.315-333. L. Somlyddy, S. Herodek, and J. Fischer (eds. ). CP-83-53. Laxenburg, Austria:

International Institute f o r Applied Systems Analysis.

Kusy, M., and W. Ziemba. 1981. A Bank

Asset

and Liability Management Model. Technical Report, University of British Columbia, Vancouver, Operations R e s e a r c h (forthcoming).

Lam, D.C.L., and L. Somlybdy. 1983. A Comparison Between Lake E r i e and Lake Balaton Water Quality Studies, in E u t r o p h i c a t i o n of Shallow Lakes: Modeling a n d Management. The Lake B a l a t o n Case S t u d y , pp.287-314. L. Somlybdy, S. Herodek, and J. Fischer (eds.). CP-83-53.

Laxenburg, Austria: International Institute f o r Applied Systems Analysis.

Monteith, T. J., R.C. Sullivan, and W.C. Sonzogni. 1980. Phosphorus Control S t r a t e g i e s

at

Municipal

Waste-Water

Treatment Plants in t h e U.S. G r e a t Lakes Basin. GLEPS, Contribution No. 1 4 , pp.43.

Murtagh, B., and M. Saunders, 1983. MINOS 5.0 User's Guide, Technical Report 83-20. Systems Optimization Laboratory, Stanford University.

Nazareth, L. 1984. Design and Implementation of Stochastic Optimizer with Recourse and Tenders, in Numerical Methods f i r S t o c h a s t i c Program- m i n g , Y . Ermoliev and R. Wets (eds.). IIASA Collaborative Volume.

OECD. 1982. Eutrophication of

Waters.

Monitoring, Assessment and Control, pp.154. Paris.

Orlob, G.T. (ed.) 1983. Mathematical Modeling of Water Quality: Streams,

Rockafellar, R.T., and R. Wets, 1985. A Lagrangian finite Generation Tech- nique f o r Solving Linear-Quadratic Problem in Stochastic Program- ming, in S t o c h a s t i c Programming: 1984, A. P r e k o p a and R.

Wets

(eds.)

.

Mathematical Programming S t u d y .

Scavia, D., and A. Robertson. 1979. Perspectives on Lake Ecosystem Model- ing. Ann Arbor Science. Ann Arbor, Michigan.

Schussler, H. 1981. Kosten d e r Phosphatelimination in kommunalen KLiiranlagen. GWF-Wasser/Abwasser 122. H.6, pp.251-257.

SomlyCMy, L. 1982. Modeling a Complex Environmental System: The Lake Balaton Case Study. Mathematical Modeling, Vol. 3.

SomlyCMy, L. 1983a. A Systems Approach

to

Eutrophication Management with Application

to

Lake Balaton. Water Q u a l i t y B u l l e t i n Vol. 9 Shallow Lakes: Modeling and Management, pp.367. The Lake BaLaton Case S t u d y . CP-83-53. Laxenburg, Austria: International Institute

Wets. R. 1983a. Stochastic Programming: Solution Techniques and

Approximation Schemes, in Mathematical Programming: The State-of- the-Art, pp.566-603. A. Bachem, M. Grotschel and B. Korte (eds.) Springer Verlag.

Wets, R. 1983b. Solving Stochastic Programs with Simple Recourse, Sto-

c h a s t i c ~ 10:219-242.

Im Dokument Stochastic Optimization Models for Lake Eutrophication Management (Seite 85-98)