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Low-Flow and Flood Control: Distributed Versus Lumped Reservoir Model
IFAC, Wat~r Water and Related Copyright © IFAC. R~latoo Land Resource Resourc~ Systems Syst~ms Cleveland, Cl~v~land. Ohio 1980
LOW-FLOW AND FLOOD CONTROL: DISTRIBUTED VERSUS LUMPED RESERVOIR MODEL R. Harboe*, G. A. Schultz* and L. Duckstein** *Lehrstuhl fur Wasserw';rtschaft Wasserwirtschaft und Umwelttechn';k Umwelttechnik I, *Lehrstuhlfur Ruhr·University Bochum, Federal Republic of Germany Ruhr-Un';vers';ty Engineering Department, University of Anzona, Arizona, **Systems and Industrial EngineenOng rizona, USA Tucson, A nzona, Abstract. Two possible approaches to the problem of optimal operation of a reservoir system are applied, namely a distributed model in which each reservoir is individually considered and a lumped model with only one equivalent reservoir to be op~ rated. These models yield near optimal solutions to the reservoir operation problem. The solutions are then analysed under various criteria: operating rules, marginal value of storage, computational aspects, stochastic considerations, environmental factors and multicriteria analysis. As a practical application the Wupper River Reservoir System, consisting of six major multipurpose mUltipurpose reservoirs, is operated to meet three objectives, namely low-flow augmentation, flood control and recreation. recreation . Keywords. Water resources; water pollution; operations research; optimization; dynamic programming; reservoir operation; multicriterion decision-making.
2 ], is located at the city of [km2], [km Wuppertal (Fig. 1) and has presently a mean flow of 7,52 [m 3 /s]. Flood control is provided by leaving a flood control reservoir storage space free in each reservoir according to a seasonal time schedule as indicated in Table 1. No flood control is provided between April 1. and October 31. Recreation is provided mainly at reservoir 5, but no fixed constraints exist. Kerspe reservoir is operated only for drinking water supply and almost never spills. Neye reservoir (4) is operated to satisfy a water supply and as intermediate storage for water from reservoir 3. It can release downstream or overflow into reservoir 5.
INTRODUCTION The Wupper river system has six main reservoirs which are under control of a single river basin authority except for drinking water supplies. Up to now reservoirs 1 through 5 (Fig. 1), which are in parallel, have been operated using dispatcher dia(Wupperverband) grams and experience (Wupperverband). Now that reservoir 6 is to be constructed, an important reservoir in series with all others, the operation of the whole system has to be reviewed. The reservoirs are of very different capacities, as shown in Table 1, but all of them will be considered of equal importance in the solution of the operational problem. The reservoirs serve three main purposes, low-flow augmentation, flood control and recreation. Important social, political and economic constraints, sometimes antagonistic, have also to be considered in real r e al world operation by the river basin authority. The low-flow augmentation target is to have a flow as high as Possible during the dry periods (especially during summer) at a given control gage. The low-flow augmentation thus provided will help dilute residual wastewater improving water quality and the environment. The control gage with a catchment of 338
DISTRIBUTED RESERVOIR MODEL The solution of a complex system like the Wupper river system is not possible through a single mathematical model given todays algorithms and computer capabilities (Hall and Dracup, 1970). The distributed model consists in applying a sequence of deterministic single reservoir optimization models to the given reservoirs. These monthly operation models
393
R. Harboe, G. A. Schultz and L. Duckstein
394
have as objective function the maximization of minimum flow at the control gage subject to the flood control constraints. Since they are applied in sequence, for the optimum operation of the second reservoir the results of the optimum operation of the first reservoir are used, and so on.. on Deterministic dynamic programming optimization (Harboe, (Harboe , 1976 a) was applied to each reservoir using 32 years of historic inflow records. records . The histohisto ric record of flow from the intermeinterme diate catchment between reservoirs and the control gage was utilized as a basic flow to be augmented. augmented . The follofollo wing recursive equation was used for a single reservoir (B6hle and others, others , 1979) :
f
n
(S
n
) =max [min { (REL +Q ), f 1 (S 1) }] REL n n nnn
(1) S
:amount of water at the beginning of each period n :release from reservoir during month n :periods, numbered backwards, n = 1,2, l,2 , ... ••• N N ::flow flow to be augmented at the control gage (natural flow plus optimal releases of previously optimized reservoirs) :optimum return (maximum of minimum flow at control gage) from periods n through 1.
n
REL
n
n
f
n
(S
n
)
The state transformation equation can be written for one reservoir as: S
n-l n- l
=S +1 +I -REL -E (S , n n n n n
with I E
n n
(2 )
S n-l)
:inflow to the reservoir during period n :net evaporation losses from water surface of the reservoir during period n
The following constraints are satissatis izati on: fied during the optim optimization:
-
mandatory release:
REL
- flood control
S
- spilling
S
S
max n
n
n
n- l n-l
~
MANREL
~
S
~
n
max n CAPACITY
maximum storage level due to flood control in period n
The results of this sequence of optimizations are shown in Table 2. 2 . The lowest flow from the intermediate m3 /s] catchment was 0.23 [m /sl and i t was successively augmented by the reserreser voirs up to 3.34 [m m3 /sl /s].. The dry summer periods implied the need for augaug menting low flow through reservoir releases in 7.8 months per year on the average and i t was necessary every
reser year. The order in which the reservoirs were optimized was such that Bever reservoir re s ervoir was kept full most t he time for recreational purposes. purpo s es. of the During the opt optimization, i mization , the flood control constraints were satisfied in each reservoir. Neye reservoir was considered to be, as Kerspe reservoir, reservo i r , for drinking water supply only.
LUMPED RESERVOIR MODEL An alternative approach was to re re-present the complex system by a single equivalent reservoir. reservoir . This rere servoir is located where reservoir 6 is (Fig. (Fig . 1). The capacity is the sum of individual capacities of reserreser 1 , 2 , 3,5 and 6, that is, 55.83 voirs 1,2,3,5 3 ], [10 [1 0 66 m3 l, dead storage and flood control storages are similarly added. added . The inflow to the equivalent reserreser voir is the sum of historic inflows to r e servoirs 1,2,3 1,2 , 3 and 5 plus the flow reservoirs form the intermediate catchment bebe tween the upper six reservoirs and reservoir 6. 6. The dynamic dynam i c programming model was applied to this single reservoir and low - flow augmenaugmen results show that a low-flow tation target of 3,52 [m 3 /s] /sl can be attained. attained . This Th i s level is only 5.4 % higher than the one attained with the distributed reservoir model. model . The result of the lumped system reprerepre sents an upper limit to optimal opeope ration of the distributed system befavo cause its conditions are more favorable, with less restrictions. ThereThere fore, the sum of local optima that constitute the result of the distridistr ibuted model should be close to the system . global optimum for the whole system. The lumped ffiodel was easily solved for several levels of flood control in order to show the transformation function between low-flow low - flow and flood control purposes. purposes . These results are presented in Fig. 2, were 100 % flood control storage represents the actual flood control of reservoirs 1,2,3,5 and 6 together. As flood control is reduced to zero, low-flow low - flow can inin crease from 3,52 to 4,05 [m 3 /s], /sl, an increase of 15 %.
MODEL ANALYSIS The foregoing models and their results can be tested for their appliappli cability to the solution of several problems with different emphasis. emphasis . In the following, six aspects which are considered to be of interest to water resources planners and managers are
395
Low-flow and Flood Control kk(x) (x)
analyzed.
= = 0
(3c)
Using two Lagrange multipliers, A and ~, problem (3) is replaced by the unconstrained optimization:
Operating Rules Certainly, if the interest is on obtaining real time operating rules for the reservoirs, a distributed model has advantages. The lumped model yields in this case only one target for low-flow augmentation and no indication as to how this target can be met with the five reservoirs. Additional criteria would have to be established to distribute the target among the reservoirs. For instance, reservoir releases could be such that the resulting free storage spaces in each reservoir are in proportion to the respective mean monthly inflow. The distributed model, on the contrary, yields one target for each reservoir and the following operating rule can be designed. The release of each one of the reservoirs should be the maximum of four calculated possible releases (Harboe and others, 1970) namely: - mandatory release (fish and wildlife, aesthetical reasons, prior rights of water users) - release necessary for meeting the low flow augmentation target found in the optimization (this release is calculated on the basis of natural flow at control gage plus releases of reservoirs already considered) - release necessary to satisfy the monthly flood control reservation - release necessary to avoid spilling. This kind of release policy would be extremely useful for developing simulation models of the degree of complexity of the Wupper system.
Marginal Value of Storage The two approaches are separately analyzed in the following sections.
Lumped model. In this case, the decision variable is the release x(t) at time t. Consider first a single time period model Then let a benefit function f (x) for low- flow augmentation be maximized, subject to a storage constraint (h(x)) (h(x» with full utilization of available storage S, and other constraints, such as recreational (k (x» (x) )
max Z(x,A,~)=f(x)-A[h(x)-S]-~k(x) Z(x,A,~)=f(x)-A[h(x)-sl-~k(x) (4 (4)) * ,A * ,~ * 1 is the solution of this If [x [X*,A*,~*] problem, then the marginal value V of storage is simply A*, also called the "shadow cost" in mathematical programming:
r* (x*) a f*( x*)
V V
Next,
as
A* = A*
(5 )
consider a multiperiod model.
S (t): t = 1,2, .. Let the storage values S(t): T or, equivalently, the replenishment Q(t) in every period t, be given. Then the optimization problem may be written as Z=L:f[t,S(t)] t=l, ... ,T max Z=Ef[t,S(t)] {x(t)}t {xC t)}t
(6a) (Ga)
subject to storage constraints: h(t,x(t), sS (t» (t))
(6b) (Gb)
= 0
and mass balance constraints: k(t,x(t) ,S(t) ,Q(t))=S(t+l) ,Q(t) )=S(t+l) k(t,x(t),S(t)
(Gc) (6c)
Let the solution of the maximization problem (6) (G) be: {x*(t): t
..• 1, . . . ,T},
Z[x*(t)]=
Z*(S(l), S(2), ... , S(T)) S(T»
(7)
The value of storage in any time period t is by definition V (t) V
az* as (t)
(8 )
An analytic evaluation of this expression would involve a multiple application of the chain rule of derivatives, because of the time dependence between storages (Equation 6c): Gc): again, for more than two or three stages, the calculations become unmanageable. Thus, a numerical evaluation of V(t) in Equation (8) is in order and can be accomplished as follows:
<
Max z=f(x)
S
Change storage value S(t) at time t to S(t) + 6S(t), leaving all other storage values as before
* ** and Z ** 2. Calculate x * + 6x+ 6X+
(3a)
3. Repeat with a negative storage increment (8 (t) c r e me n t (S ( t ) - 68 6S (t» ( t) )
(3b)
4. The marginal value of increasing (or decreasing) storage at time t is
subject to h (x)
1.
R. Harboe, G. A. Schultz and L. Duckstein
396
v (t)
~z* L'lz* +
Stochastic Considerations (9 )
M(t) ~S (t)
Note that in the dynamic case, one calculates the value of storage at a given time period. If the present value of storage is desired, then a discount factor a (t) should be used to yield Ea(t)V(t). ~a(t)v(t). a(t) aCt) does not necessarely t have the same value as the discount (6a) factor used (implicitly) in Eq. (6a). The marginal value of storage for the case of both single and multiperiod models can be estimated by the methods presented in Duckstein and Gablinger ( 1978) . Distributed model. For a single-period model, there are six marginal values of storage, one for each reservoir of the Wupper system. Lagrange multipliers can be used provided the functions (3) are not too complex; an alternative approach is to use a simulation algorithm similar to the one presented above. For a multi-period model, calculation of the marginal value V(k,t) of storage for each reservoir k at each time stage t seems to be unwieldy; the total marginal value of storage V(t) vet) at time t is, in any case: V(t) vet)
EV(t,k) ~V(t,k)
k =
1, .. ,6
( 10)
k
In the case of a stochastic model, use of a distributed model appears fraught with unsurmountable numerical difficulties.
The solution of a stochastic model of a system of reservoirs is practically impossible given todays algorithms and computer capabilities. Therefore, a stochastic model could only be developed for a lumped system, i.e. for only one equivalent reservoir. In this case i t is still necessary to add one state variable to the dynamic programming model in order to take serial correlation of inflows, if significant into account (Harboe, 1976 b). The computational effort increases due to the additional values of the state variables and the discretization of the random variable (inflow), which can take a number of discrete values only. The dissaggregation of the results of a stochastic equivalent reservoir model would nevertheless present serious difficulties. Only a single reservoir operating rule would be obtained from a dynamic programming model. This result could be useful for the determination of a marginal value of storage to be used in other models, for example in simulation models.
Environmental and Recreational Factors Depending upon the criteria used to evaluate these factors (Dwyer and others, 1977), a lumped model can or cannot be used. Thus, if the criterion for recreational benefit in lake k is number of visitor-days at stage t, say n(k,t), then n(t) net)
~n (k, t) L:n(k,t)
( 11)
k
Computational Aspects The distributed model implies the solution of one optimization problem through the dynamic programming algorithm for each reservoir. The lumped model has clearly the advantage of requiring only one of such optimizations. In terms of computer time this means approximately 4 minutes in the TR 440 computer for each reservoir when 32 years of monthly historical inflow record and 41 values of the state and decision variables (reservoir storage and release) in each stage (monthly periods) are used. Compu~er Compu~er storage requirement for each program is 31 K-bytes. The computer time for more complex systems would increase in proportion to the product of the following three factors: length of the streamflow record, number of feasible values of ·the the state variable and number'of feasible values of the decision variable.
Conversely, i t is conceivable to disaggregate a lumped number of visitordays to the various lakes if the lake level stays roughly proportional to the lumped equivalent reservoir level. On the other hand, if the criteria for lake k accounts for fluctuations of water level or for local pollutant loading effects, then there seems to be no way to disaggregate a lumped value. If aesthetic considerations come into the picture - for example, the upstream lakes are more scenic than the downstream one cons ideraoness - or if environmental considerations cannot be expressed in the same unit as the hydrology or economics of the problem, then the control becomes multicriterion in nature. The next section deals with such a multicriterion problem, in which we first assume that recreational (or environmental) benefits can be quantified, but in units that are non-com-
Low-flow and Flood Control mensurable with the hydrologic criteria units.
Multicriterion Analysis multi criteria analysis is The use of multicriteria first reviewed for the lumped and then for the distributed reservoir model.
Lumped model. This important case isfirst is first described, then applicable theory and an example are presented. a) Problem description. Let three objectives be defined for the system as a function of release x and time t: f 1 (x,t) (x, t) f1
low-flow Iow-flow target
f 2 (x,t) 2
flood reduction
f 3 (x,t) 3
recreational benefit
The multicriterion problem is thus: Max z xe:D x£D
Ef(x,t) l:f(x,t) t-
L
397
D,~
lip
J
a,P D,rl
~L
P
~
L
( 14 14))
in which weights a ii are either assessed subjectively by the decision maker (DM), derived from the preference structure P = {p}, or obtained by ffuzzy u z z y set aanalysis n a I y s i s ((Zeleny, Ze le n y, 11973~ 9 7 3 ), and D Dii is a deviation from the ideal value. The parameter p is a reflection of the decision maker's preference: p = 1 corresponds to group viewpoint (all deviations are counted linearly); p = 2 gives larger weight to greater deviations and is thus closer to individual utility; p = ~ minimizes the maximum deviation and generally results in the solutions where all deviations are equal, thus corresponding to individual utility. In general, I.p the ~ increases with p: balancing of deviations is made at the expense of the overall utility measure Lp. Lp'
( 12 )
with !.(x,t)={f i(x,t)={f1 (x,t) ,f 2 (x,t) ,f 3 (x,t)} 1 3 2 ( 13) and D: set of feasible releases. To solve problem (12), several multiobjective programming methods are available, as reviewed for example in Cohon (1978) and Duckstein (1978). An appropriate approach seems to be the Compromise Programming (CP) algorithm, initially developed by Zeleny (1973) and applied to a water resource problem in Duckstein and Opricovic (1979). It should be noted that particular cases of Compromise Proprogramming include Multi-objective simplex (Benayoun and others, 1971; Loucks, 1978) and Goal Programming. A brief description of this methodology follows. b) Compromise programming. Compromise solutions are those which are the closest, by some distance measure, to the ideal one (Zeleny, 1973, 1977; Starr and Zeleny, 1977). Among all achievable scores for the ith criterion there is at least one value that is preferred to all remaining ones; for example, ffi* i * = max f i , i = l, ... n. The vector f* whose elements are all such maxima is called the ideal vector: f* = (fl*, ... ,f n *); it is generally not a feasible solution of the multiobjective problem. One of the most frequently used measures of "closeness" is a family of Lp-metrics, defined as follows:
Since objective functions are usually not expressed in commensurable units, D Dii is a normalized distance along the ith criterion, obtained by subtracting~rom fi the ideal value sUbtracting~rom fi* and dividing by the range. The objective of compromise programming is thus to minimize the following norm:
L
P
P a.l
II II M.-* - m. I Pj P f.
f.
l
l
l
l
I.
p
(15)
in which M Mii = max fi and mi = min fi. fi' For example, in a maximizing problem, ffi* Mi' The CP approach leads to a i * = Mi. Pareto Pare to optimum point pOint located on a small portion of the Pareto boundary. This small region is defined by varya i . In ing p in (1,00) and the weights ai. this sense, CP may be viewed as a decision-making aid, since it reduces the choice set from a whole Pareto boundary (also called the set of nondominated solutions) to a small portion thereof. The application to the lumped Wupper Valley model would be straightforward analogously to the regional minewater problem presented in Duckstein and others (1979). However, the problem of sequential decision-making involves use of a more complex algorithm. c) Example. For example, let f1 = flood control storage and f2 f 2 = lowflow, as shown in Figure 2. Let us calculate L1 and L L22 for points A, B, C, D, E, given that Q is the ideal point 4.05). Let po in t (lOO, (1 00, 4. 05). Le t aal1 = a 22 = 1. The range of f1 is 100, lOO, that of ff22 is 0.53; hence the results shown in
R. Harboe, G. A. Schultz and L. Duckstein
398 Table 3.
CONCLUSIONS
It is thus seen that the Lp-norm is minimum at point B for p = = 1 and at point C for p = 2 as well as p = ~; = ~ is since the minimax solution p = obtained when D1 = D22 , which is almost true at point C. To sum up. Compromise programming reduces the choice set from line ABCDE to (approximately) the segment BC. When curves of f1 versus f3 f 3 (recreational objective) and f2 versus f3 will be available, an exact determination of the compromise region can be made.
Distributed model. For the kth reservoir of the Wupper System, let let us define three objectives at a given time period t, namely: ff11 (k ,~) (k,~)
low flow target
ff2 (k,!.) (k,~)
flood reduction
f3 (k,~) f3(k,~)
recreational benefit
2
in which x is the decision vector
x
(release)
{x(k): k == 1, ... ,6}
D be the feasible domain of ~; Let ~ this includes satisfying water supply multi criteria constraints. Then the multicriteria problem for one reservoir is written as : Max z xe:D x£D
Two main approaches to the solution of the optimal operation problem for a system of reservoirs were described The first one consists of a sequential optimization of individual reservoirs (distributed system model). The second implies the solution of one equivalent reservoir (lumped system model) . Results for a case study, the Wupper River System, including six multipurpose reservoirs, were presented and the merits of the two approaches analyzed according to a set of six different criteria. It is concluded that for some purposes, for example developing real time operating rules, the distributed system approach was to be prefered, and for others, like multicriteria analysis, the lumped system model should be developed. Future research along these lines would include the establishment of real world parameters for the objective function in multicriteria analysis, the analysis of stochastic equivalent reservoir models, the consideration of environmental chance constraints and the evaluation of the marginal value of storage through simulation models.
{.f(k)}
For the complete system, is written as:
the problem
Max z = = {!.O) ,!.(2) ,!.(6)} Cf ( 1 ) ,i. (2 ) , ... ,! (6 )} x£D ~e:.Q.
(17) (1 7)
which represents a maximum of 18 objectives at every time stage. A summation over time is then necessary. However, the low flow target fl(k,~) f1(k,~ may be a factor at one or two locations only, and the other objectives may be combined according to the situation, which would reduce the dimension of the objective function vector. In any case, the multicriterion analysis would not be simple, and serious consideration must be given to the necessity of using a distributed model to justify the additional complexity. In the case when recreational benefits cannot be quantified-for example, they are rated on a qualitative scalemulticriterion methods such as ELECTRE I (Roy, 1971, David and Duckstein, 1976) or Concordance Analysis (Nijkamp and Vos, 1977) can be used.
ACKNOWLEDGEMENTS The authors wish to acknowledge the Wupperverband for the provision of all the basic data and the helpfull comments during the preparation of this paper. REFERENCES Benayoun, R., J. de Montgolfier, J. Tergny and o. O. Laritchev (1971). Linear programming with multiple objective functions: Step Method (STEM), Math. Progr., 1(3), pp. 366-375.---- ----B6hle, W., R. Harboe and G.A. Schultz (1979). Low-flow augmentation for water quality improvement in a river-system in Germany. Proceedings, XVIII Congress of the International Association for Hydraulic Research, Cagliari, Italy. Cohon, J.L. (1978). Multiobjective Programming and Planning, Academic Press, New York. David, L. and L. Duckstein (1976). Multi-criterion ranking of alternative long-range water resources systems, water Water Resources Bulletin, 12(4), pp. 731-754. Duckstein, L. (1978). Irnbedding Imbedding uncertanties into multiobjective mUltiobjective decision models in water resources, Proceedings, International Symposium on Risk and Reliability in Water Resources, June,Waterloo,
Low-flow and Flood Control Ontario, Canada Canada,, pp. 3-30. Duckstein, L., I. Bogardi and F F.. Trade-off Szidarovszky (1979) (1979).. between regional mining development and environmental impact, -~ Proceedings, 16th International Com Symposium on Application of Computer and Operations Research in SME / AIME . the Mineral Industry, SME/AIME. Fall Meeting, Oct. 17-19, Tucson, 355 - 364 . pp. 355-364. ( 1978) Duckstein, L. and M. Gablinger (1978). The dynamic value of conjunctive storage of water, Proceedings, International Symposium on ConCon junctive Use of Water, Erice-Trapani, Italy, October. (1979) Duckstein, L. and S. Opricovic (1979). Multiobjective optimization in river basin development, to appear, Water Resources Research. Dwyer, J., J. Kelly and M. Bowes (1977) evalu Improved procedures for evaluation of the control of recreation to national economic developRes . Rep. Rep . 128, Water Rement, Res. sources Center, University of Illinois, September. (1970) . Hall, W.A. and J.A. Dracup (1970). Water Resources Systems Engineering. McGraw-Hill, New York. York . Harb~R. (1976 a). Harb~R. Deterministische Optimierungsmodelle fur Speichersysteme , Wasserwirtschaft, Wasserwirtschaft , Vol. Vol . systeme, 66, No. 7/8, pp. 221-226. 221 - 226. MObasheri , F. and W.W.-G. Harboe, R., Mobasheri, Yeh (1970). Optimal Policy for Reservoir Operation, Journal of the Hydraulics Division, Division , A.S.C.E., A.S.C.E . , Vol . 96, No. HY11, pp. pp . 2297-2308. 2297-2308 . Vol. b) . Harboe, R. (1976 b). A Stochastic Optimization and Simulation Model for the Operation of the Lech River System, Mitteilungen, Hydraulik und Gewasserkunde, TU Munchen, Heft Nr. 21. 2l. Loucks, D.P. Interactive mulD. P. (1978). (1978) . tiobjecti~ tiobj ecti ~ water resources planning: An application and some extensions, tensi ons , Proceedings, International Symposium on Risk and Reliability in Water Resources, Waterloo, Ontario, Canada, pp. pp . 30 -40. Nijkamp, P. and J.B. Vos (1977) (1977).. A multicriteria analysis for water resource and land use development, Water Resources Research, Vol. 13, No. 3, 3 , pp. 515-518 515 - 518 Roy, (1971).. Problems and Methods ROY, B. (1971) with multiple objective functions, Mathematical Programming I, pp. pp . 239-266. M. K. and M. Zeleny (1977). (1977) . Starr, M.K. State and future of the arts, in Multiple Criteria Decision Making, edited by M.K. Starr and M. Zeleny, North-Holland Publishing Co Co., . , Amsterdam-New sterdam - New York, pp. 5-29. Wupperverband Wupperverband, Unpublished operation opera tion records. Wuppertal, Wuppertal , Federal Repu6
399
blic of Germany. Germany . Zeleny, M. (1973). Compromise programming, in Multiple Criteria Decision Making, JJ.L. . L . Cochrane and M. Zeleny (editors), (editors) , University of South Carolina Press, Press , Columbia, pp. 262-301. Zeleny Zeleny,, M. (1977). Adaptive displacement of preferences in decision making, in Multiple Criteria Decision Making, M.K. Starr and M. Zeleny (editors), North Holland Publishing Company, Amsterdam-New Amsterdam - New York York,, pp. 147-157. 147-157 .
R. Harboe, Ha rb oe , G. A. Schultz Schul t z and L. Duckstein
400
CITY OF WUPPERTAL WUPPER-RIVER
-----_ "\\ ------,,-\11 \ \ ..
CONTROL GAGE
Suas~~~-~ _--I
\
\
--- ------------
,--
1...-L----
\
\
\ \ \
\
\ \
\ \ \ \ 1£.\'I\ 11.J Suas~S_--
_t---\---\
\ \ \
----------- ----, ------ , ------\ \
\
\
\ .-\---
1
BRUCHER
2
LINGESE LINGE SE
-R£.S£.R'J~J
-\\
RESERVOIRS: RESERVOIRS :
¥-~~---_-i
----- -\.----
---- ---
\ 1\
SCHEVELINGER NEYE
Suas~':"I~~:J
SEVER BEVER WUPPER
Fig. F ig . 1. 1.
\
L------
-- -- --
Wupper-River-System. Wupper - River - System .
f2
4.1 4. 1
LOW
A
FLOW
(m 3 Is) /s)
4.05 ~ 05_________________
+0
4. 0 4.0 NON FEASIBLE REGION
3.9 3. 9
p=l
3.8 3. 8
0,53 0,,53
p~2
3. 7
FEASIBLE REGION
3. 6
fl 3. 52
FLOOD CONTROL STORAGE ( ·/0 )
3. 5 0 Fig.
2. 2.
\
\ \
KERSPE
3 4 5 6
\
25
50
75
1 00
Low-flow re~ervoir for different flood Low - flow augmentation by equivalent reservoir control levels.
Low-flow and Flood Control TABLE 1
Reservoir
401
Characteristics of the Wupper Reservoirs
6 [ 10 6 m33 ] Flood Control Storage [10
Dead Storage 6 3 [10 [ 10 6m3 ]
Capacity 6 3 [ 10 6m3 ]
1.Nov .-31 .Jan. 1.Nov.-31.Jan.
1 .Feb .-28 .Feb. 1.Feb.-28.Feb.
1.Mar.-31.Mar. 1 .Mar .-31 .Mar.
11.. Brucher
3.340
0
0.400
0
0
2. Lingese
2.600
0
1.100 1 .100
0
0
3. Schevelin-
0.290
0
0
0
0
6.000
0
0.750
0
0
0
0
ger 4. Neye
5. Bever
23.700
1.500
5.000
6. Wupper
25.900
2.500
9.900
TABLE 2
Clecreasing ~ecreasing
from decreasing from 9.9 to 5.0 5.0 to 0
Augmented Flow at Control Gage
Augmented nu.nl.mum minimum flow at con- Length of historical sample: trol gage 32 years
6 3 [10 m/month]
3 Is] [m 3 /s]
Augmented year, years
Augmented months/year
Flow from catchment between Wupper reservoir and control gage
0.588
0.23
-
F
Bever
3.958
1.53
31
130/31=4.2
Bever Schevelinger
3.998
1.54
30
120/30=4.0
Bever Schevelinger Lingese
4.339
1.67
30
125/30=4.2
Bever Schevelinger Lingese Brucher
4.744
1.83
30
124/30=4.1 124/3 0=4.1
8.666
3.34(X) 3.34(x)
32
250/32=7 .8 250/32=7.8
1
b
0
Y
w a u g m e n t e d
R e s e r v 0
i
r s
Flow 9ug~ented.by ~u~ented.by W~pper W~pper.. reserVOl.r servo~r l.n ~n conJunctl.on conJunct~on Wl.~ W~Ln all other reservoirs
-
(x) Higher levels (4 or 5 m33/s) / s) can be attained with historical record but with probabilities less than 100 %.
TABLE 3
Point A
D D1 1
1
B B
0.75
C
0.50
D D
0.25
E
0
Analysis by Compromise Programming programming
D D2 2
0 0.09 =0.170 =0.17C 0.53 0.27 =0.510 =0.51C 0.53 0.46 =0.856 0.53 1
(for p = 1 ) L 1 =D 1 +D 2 1 1 2
1
(for p = 2 ) D 2 L 2 =D 11 2+ 2+Di 2 2 1
0.92*
0.589
1 .01
0.510*
1 .106
0.813
1
1
LOW-FLOW AND FLOOD CONTROL: DISTRIBUTED VERSUS LUMPED RESERVOIR MODEL R. Harboe*, G. A. Schultz* and L. Duckstein** *Lehrstuhl fur Wasserw';rtschaft Wasserwirtschaft und Umwelttechn';k Umwelttechnik I, *Lehrstuhlfur Ruhr·University Bochum, Federal Republic of Germany Ruhr-Un';vers';ty Engineering Department, University of Anzona, Arizona, **Systems and Industrial EngineenOng rizona, USA Tucson, A nzona, Abstract. Two possible approaches to the problem of optimal operation of a reservoir system are applied, namely a distributed model in which each reservoir is individually considered and a lumped model with only one equivalent reservoir to be op~ rated. These models yield near optimal solutions to the reservoir operation problem. The solutions are then analysed under various criteria: operating rules, marginal value of storage, computational aspects, stochastic considerations, environmental factors and multicriteria analysis. As a practical application the Wupper River Reservoir System, consisting of six major multipurpose mUltipurpose reservoirs, is operated to meet three objectives, namely low-flow augmentation, flood control and recreation. recreation . Keywords. Water resources; water pollution; operations research; optimization; dynamic programming; reservoir operation; multicriterion decision-making.
2 ], is located at the city of [km2], [km Wuppertal (Fig. 1) and has presently a mean flow of 7,52 [m 3 /s]. Flood control is provided by leaving a flood control reservoir storage space free in each reservoir according to a seasonal time schedule as indicated in Table 1. No flood control is provided between April 1. and October 31. Recreation is provided mainly at reservoir 5, but no fixed constraints exist. Kerspe reservoir is operated only for drinking water supply and almost never spills. Neye reservoir (4) is operated to satisfy a water supply and as intermediate storage for water from reservoir 3. It can release downstream or overflow into reservoir 5.
INTRODUCTION The Wupper river system has six main reservoirs which are under control of a single river basin authority except for drinking water supplies. Up to now reservoirs 1 through 5 (Fig. 1), which are in parallel, have been operated using dispatcher dia(Wupperverband) grams and experience (Wupperverband). Now that reservoir 6 is to be constructed, an important reservoir in series with all others, the operation of the whole system has to be reviewed. The reservoirs are of very different capacities, as shown in Table 1, but all of them will be considered of equal importance in the solution of the operational problem. The reservoirs serve three main purposes, low-flow augmentation, flood control and recreation. Important social, political and economic constraints, sometimes antagonistic, have also to be considered in real r e al world operation by the river basin authority. The low-flow augmentation target is to have a flow as high as Possible during the dry periods (especially during summer) at a given control gage. The low-flow augmentation thus provided will help dilute residual wastewater improving water quality and the environment. The control gage with a catchment of 338
DISTRIBUTED RESERVOIR MODEL The solution of a complex system like the Wupper river system is not possible through a single mathematical model given todays algorithms and computer capabilities (Hall and Dracup, 1970). The distributed model consists in applying a sequence of deterministic single reservoir optimization models to the given reservoirs. These monthly operation models
393
R. Harboe, G. A. Schultz and L. Duckstein
394
have as objective function the maximization of minimum flow at the control gage subject to the flood control constraints. Since they are applied in sequence, for the optimum operation of the second reservoir the results of the optimum operation of the first reservoir are used, and so on.. on Deterministic dynamic programming optimization (Harboe, (Harboe , 1976 a) was applied to each reservoir using 32 years of historic inflow records. records . The histohisto ric record of flow from the intermeinterme diate catchment between reservoirs and the control gage was utilized as a basic flow to be augmented. augmented . The follofollo wing recursive equation was used for a single reservoir (B6hle and others, others , 1979) :
f
n
(S
n
) =max [min { (REL +Q ), f 1 (S 1) }] REL n n nnn
(1) S
:amount of water at the beginning of each period n :release from reservoir during month n :periods, numbered backwards, n = 1,2, l,2 , ... ••• N N ::flow flow to be augmented at the control gage (natural flow plus optimal releases of previously optimized reservoirs) :optimum return (maximum of minimum flow at control gage) from periods n through 1.
n
REL
n
n
f
n
(S
n
)
The state transformation equation can be written for one reservoir as: S
n-l n- l
=S +1 +I -REL -E (S , n n n n n
with I E
n n
(2 )
S n-l)
:inflow to the reservoir during period n :net evaporation losses from water surface of the reservoir during period n
The following constraints are satissatis izati on: fied during the optim optimization:
-
mandatory release:
REL
- flood control
S
- spilling
S
S
max n
n
n
n- l n-l
~
MANREL
~
S
~
n
max n CAPACITY
maximum storage level due to flood control in period n
The results of this sequence of optimizations are shown in Table 2. 2 . The lowest flow from the intermediate m3 /s] catchment was 0.23 [m /sl and i t was successively augmented by the reserreser voirs up to 3.34 [m m3 /sl /s].. The dry summer periods implied the need for augaug menting low flow through reservoir releases in 7.8 months per year on the average and i t was necessary every
reser year. The order in which the reservoirs were optimized was such that Bever reservoir re s ervoir was kept full most t he time for recreational purposes. purpo s es. of the During the opt optimization, i mization , the flood control constraints were satisfied in each reservoir. Neye reservoir was considered to be, as Kerspe reservoir, reservo i r , for drinking water supply only.
LUMPED RESERVOIR MODEL An alternative approach was to re re-present the complex system by a single equivalent reservoir. reservoir . This rere servoir is located where reservoir 6 is (Fig. (Fig . 1). The capacity is the sum of individual capacities of reserreser 1 , 2 , 3,5 and 6, that is, 55.83 voirs 1,2,3,5 3 ], [10 [1 0 66 m3 l, dead storage and flood control storages are similarly added. added . The inflow to the equivalent reserreser voir is the sum of historic inflows to r e servoirs 1,2,3 1,2 , 3 and 5 plus the flow reservoirs form the intermediate catchment bebe tween the upper six reservoirs and reservoir 6. 6. The dynamic dynam i c programming model was applied to this single reservoir and low - flow augmenaugmen results show that a low-flow tation target of 3,52 [m 3 /s] /sl can be attained. attained . This Th i s level is only 5.4 % higher than the one attained with the distributed reservoir model. model . The result of the lumped system reprerepre sents an upper limit to optimal opeope ration of the distributed system befavo cause its conditions are more favorable, with less restrictions. ThereThere fore, the sum of local optima that constitute the result of the distridistr ibuted model should be close to the system . global optimum for the whole system. The lumped ffiodel was easily solved for several levels of flood control in order to show the transformation function between low-flow low - flow and flood control purposes. purposes . These results are presented in Fig. 2, were 100 % flood control storage represents the actual flood control of reservoirs 1,2,3,5 and 6 together. As flood control is reduced to zero, low-flow low - flow can inin crease from 3,52 to 4,05 [m 3 /s], /sl, an increase of 15 %.
MODEL ANALYSIS The foregoing models and their results can be tested for their appliappli cability to the solution of several problems with different emphasis. emphasis . In the following, six aspects which are considered to be of interest to water resources planners and managers are
395
Low-flow and Flood Control kk(x) (x)
analyzed.
= = 0
(3c)
Using two Lagrange multipliers, A and ~, problem (3) is replaced by the unconstrained optimization:
Operating Rules Certainly, if the interest is on obtaining real time operating rules for the reservoirs, a distributed model has advantages. The lumped model yields in this case only one target for low-flow augmentation and no indication as to how this target can be met with the five reservoirs. Additional criteria would have to be established to distribute the target among the reservoirs. For instance, reservoir releases could be such that the resulting free storage spaces in each reservoir are in proportion to the respective mean monthly inflow. The distributed model, on the contrary, yields one target for each reservoir and the following operating rule can be designed. The release of each one of the reservoirs should be the maximum of four calculated possible releases (Harboe and others, 1970) namely: - mandatory release (fish and wildlife, aesthetical reasons, prior rights of water users) - release necessary for meeting the low flow augmentation target found in the optimization (this release is calculated on the basis of natural flow at control gage plus releases of reservoirs already considered) - release necessary to satisfy the monthly flood control reservation - release necessary to avoid spilling. This kind of release policy would be extremely useful for developing simulation models of the degree of complexity of the Wupper system.
Marginal Value of Storage The two approaches are separately analyzed in the following sections.
Lumped model. In this case, the decision variable is the release x(t) at time t. Consider first a single time period model Then let a benefit function f (x) for low- flow augmentation be maximized, subject to a storage constraint (h(x)) (h(x» with full utilization of available storage S, and other constraints, such as recreational (k (x» (x) )
max Z(x,A,~)=f(x)-A[h(x)-S]-~k(x) Z(x,A,~)=f(x)-A[h(x)-sl-~k(x) (4 (4)) * ,A * ,~ * 1 is the solution of this If [x [X*,A*,~*] problem, then the marginal value V of storage is simply A*, also called the "shadow cost" in mathematical programming:
r* (x*) a f*( x*)
V V
Next,
as
A* = A*
(5 )
consider a multiperiod model.
S (t): t = 1,2, .. Let the storage values S(t): T or, equivalently, the replenishment Q(t) in every period t, be given. Then the optimization problem may be written as Z=L:f[t,S(t)] t=l, ... ,T max Z=Ef[t,S(t)] {x(t)}t {xC t)}t
(6a) (Ga)
subject to storage constraints: h(t,x(t), sS (t» (t))
(6b) (Gb)
= 0
and mass balance constraints: k(t,x(t) ,S(t) ,Q(t))=S(t+l) ,Q(t) )=S(t+l) k(t,x(t),S(t)
(Gc) (6c)
Let the solution of the maximization problem (6) (G) be: {x*(t): t
..• 1, . . . ,T},
Z[x*(t)]=
Z*(S(l), S(2), ... , S(T)) S(T»
(7)
The value of storage in any time period t is by definition V (t) V
az* as (t)
(8 )
An analytic evaluation of this expression would involve a multiple application of the chain rule of derivatives, because of the time dependence between storages (Equation 6c): Gc): again, for more than two or three stages, the calculations become unmanageable. Thus, a numerical evaluation of V(t) in Equation (8) is in order and can be accomplished as follows:
<
Max z=f(x)
S
Change storage value S(t) at time t to S(t) + 6S(t), leaving all other storage values as before
* ** and Z ** 2. Calculate x * + 6x+ 6X+
(3a)
3. Repeat with a negative storage increment (8 (t) c r e me n t (S ( t ) - 68 6S (t» ( t) )
(3b)
4. The marginal value of increasing (or decreasing) storage at time t is
subject to h (x)
1.
R. Harboe, G. A. Schultz and L. Duckstein
396
v (t)
~z* L'lz* +
Stochastic Considerations (9 )
M(t) ~S (t)
Note that in the dynamic case, one calculates the value of storage at a given time period. If the present value of storage is desired, then a discount factor a (t) should be used to yield Ea(t)V(t). ~a(t)v(t). a(t) aCt) does not necessarely t have the same value as the discount (6a) factor used (implicitly) in Eq. (6a). The marginal value of storage for the case of both single and multiperiod models can be estimated by the methods presented in Duckstein and Gablinger ( 1978) . Distributed model. For a single-period model, there are six marginal values of storage, one for each reservoir of the Wupper system. Lagrange multipliers can be used provided the functions (3) are not too complex; an alternative approach is to use a simulation algorithm similar to the one presented above. For a multi-period model, calculation of the marginal value V(k,t) of storage for each reservoir k at each time stage t seems to be unwieldy; the total marginal value of storage V(t) vet) at time t is, in any case: V(t) vet)
EV(t,k) ~V(t,k)
k =
1, .. ,6
( 10)
k
In the case of a stochastic model, use of a distributed model appears fraught with unsurmountable numerical difficulties.
The solution of a stochastic model of a system of reservoirs is practically impossible given todays algorithms and computer capabilities. Therefore, a stochastic model could only be developed for a lumped system, i.e. for only one equivalent reservoir. In this case i t is still necessary to add one state variable to the dynamic programming model in order to take serial correlation of inflows, if significant into account (Harboe, 1976 b). The computational effort increases due to the additional values of the state variables and the discretization of the random variable (inflow), which can take a number of discrete values only. The dissaggregation of the results of a stochastic equivalent reservoir model would nevertheless present serious difficulties. Only a single reservoir operating rule would be obtained from a dynamic programming model. This result could be useful for the determination of a marginal value of storage to be used in other models, for example in simulation models.
Environmental and Recreational Factors Depending upon the criteria used to evaluate these factors (Dwyer and others, 1977), a lumped model can or cannot be used. Thus, if the criterion for recreational benefit in lake k is number of visitor-days at stage t, say n(k,t), then n(t) net)
~n (k, t) L:n(k,t)
( 11)
k
Computational Aspects The distributed model implies the solution of one optimization problem through the dynamic programming algorithm for each reservoir. The lumped model has clearly the advantage of requiring only one of such optimizations. In terms of computer time this means approximately 4 minutes in the TR 440 computer for each reservoir when 32 years of monthly historical inflow record and 41 values of the state and decision variables (reservoir storage and release) in each stage (monthly periods) are used. Compu~er Compu~er storage requirement for each program is 31 K-bytes. The computer time for more complex systems would increase in proportion to the product of the following three factors: length of the streamflow record, number of feasible values of ·the the state variable and number'of feasible values of the decision variable.
Conversely, i t is conceivable to disaggregate a lumped number of visitordays to the various lakes if the lake level stays roughly proportional to the lumped equivalent reservoir level. On the other hand, if the criteria for lake k accounts for fluctuations of water level or for local pollutant loading effects, then there seems to be no way to disaggregate a lumped value. If aesthetic considerations come into the picture - for example, the upstream lakes are more scenic than the downstream one cons ideraoness - or if environmental considerations cannot be expressed in the same unit as the hydrology or economics of the problem, then the control becomes multicriterion in nature. The next section deals with such a multicriterion problem, in which we first assume that recreational (or environmental) benefits can be quantified, but in units that are non-com-
Low-flow and Flood Control mensurable with the hydrologic criteria units.
Multicriterion Analysis multi criteria analysis is The use of multicriteria first reviewed for the lumped and then for the distributed reservoir model.
Lumped model. This important case isfirst is first described, then applicable theory and an example are presented. a) Problem description. Let three objectives be defined for the system as a function of release x and time t: f 1 (x,t) (x, t) f1
low-flow Iow-flow target
f 2 (x,t) 2
flood reduction
f 3 (x,t) 3
recreational benefit
The multicriterion problem is thus: Max z xe:D x£D
Ef(x,t) l:f(x,t) t-
L
397
D,~
lip
J
a,P D,rl
~L
P
~
L
( 14 14))
in which weights a ii are either assessed subjectively by the decision maker (DM), derived from the preference structure P = {p}, or obtained by ffuzzy u z z y set aanalysis n a I y s i s ((Zeleny, Ze le n y, 11973~ 9 7 3 ), and D Dii is a deviation from the ideal value. The parameter p is a reflection of the decision maker's preference: p = 1 corresponds to group viewpoint (all deviations are counted linearly); p = 2 gives larger weight to greater deviations and is thus closer to individual utility; p = ~ minimizes the maximum deviation and generally results in the solutions where all deviations are equal, thus corresponding to individual utility. In general, I.p the ~ increases with p: balancing of deviations is made at the expense of the overall utility measure Lp. Lp'
( 12 )
with !.(x,t)={f i(x,t)={f1 (x,t) ,f 2 (x,t) ,f 3 (x,t)} 1 3 2 ( 13) and D: set of feasible releases. To solve problem (12), several multiobjective programming methods are available, as reviewed for example in Cohon (1978) and Duckstein (1978). An appropriate approach seems to be the Compromise Programming (CP) algorithm, initially developed by Zeleny (1973) and applied to a water resource problem in Duckstein and Opricovic (1979). It should be noted that particular cases of Compromise Proprogramming include Multi-objective simplex (Benayoun and others, 1971; Loucks, 1978) and Goal Programming. A brief description of this methodology follows. b) Compromise programming. Compromise solutions are those which are the closest, by some distance measure, to the ideal one (Zeleny, 1973, 1977; Starr and Zeleny, 1977). Among all achievable scores for the ith criterion there is at least one value that is preferred to all remaining ones; for example, ffi* i * = max f i , i = l, ... n. The vector f* whose elements are all such maxima is called the ideal vector: f* = (fl*, ... ,f n *); it is generally not a feasible solution of the multiobjective problem. One of the most frequently used measures of "closeness" is a family of Lp-metrics, defined as follows:
Since objective functions are usually not expressed in commensurable units, D Dii is a normalized distance along the ith criterion, obtained by subtracting~rom fi the ideal value sUbtracting~rom fi* and dividing by the range. The objective of compromise programming is thus to minimize the following norm:
L
P
P a.l
II II M.-* - m. I Pj P f.
f.
l
l
l
l
I.
p
(15)
in which M Mii = max fi and mi = min fi. fi' For example, in a maximizing problem, ffi* Mi' The CP approach leads to a i * = Mi. Pareto Pare to optimum point pOint located on a small portion of the Pareto boundary. This small region is defined by varya i . In ing p in (1,00) and the weights ai. this sense, CP may be viewed as a decision-making aid, since it reduces the choice set from a whole Pareto boundary (also called the set of nondominated solutions) to a small portion thereof. The application to the lumped Wupper Valley model would be straightforward analogously to the regional minewater problem presented in Duckstein and others (1979). However, the problem of sequential decision-making involves use of a more complex algorithm. c) Example. For example, let f1 = flood control storage and f2 f 2 = lowflow, as shown in Figure 2. Let us calculate L1 and L L22 for points A, B, C, D, E, given that Q is the ideal point 4.05). Let po in t (lOO, (1 00, 4. 05). Le t aal1 = a 22 = 1. The range of f1 is 100, lOO, that of ff22 is 0.53; hence the results shown in
R. Harboe, G. A. Schultz and L. Duckstein
398 Table 3.
CONCLUSIONS
It is thus seen that the Lp-norm is minimum at point B for p = = 1 and at point C for p = 2 as well as p = ~; = ~ is since the minimax solution p = obtained when D1 = D22 , which is almost true at point C. To sum up. Compromise programming reduces the choice set from line ABCDE to (approximately) the segment BC. When curves of f1 versus f3 f 3 (recreational objective) and f2 versus f3 will be available, an exact determination of the compromise region can be made.
Distributed model. For the kth reservoir of the Wupper System, let let us define three objectives at a given time period t, namely: ff11 (k ,~) (k,~)
low flow target
ff2 (k,!.) (k,~)
flood reduction
f3 (k,~) f3(k,~)
recreational benefit
2
in which x is the decision vector
x
(release)
{x(k): k == 1, ... ,6}
D be the feasible domain of ~; Let ~ this includes satisfying water supply multi criteria constraints. Then the multicriteria problem for one reservoir is written as : Max z xe:D x£D
Two main approaches to the solution of the optimal operation problem for a system of reservoirs were described The first one consists of a sequential optimization of individual reservoirs (distributed system model). The second implies the solution of one equivalent reservoir (lumped system model) . Results for a case study, the Wupper River System, including six multipurpose reservoirs, were presented and the merits of the two approaches analyzed according to a set of six different criteria. It is concluded that for some purposes, for example developing real time operating rules, the distributed system approach was to be prefered, and for others, like multicriteria analysis, the lumped system model should be developed. Future research along these lines would include the establishment of real world parameters for the objective function in multicriteria analysis, the analysis of stochastic equivalent reservoir models, the consideration of environmental chance constraints and the evaluation of the marginal value of storage through simulation models.
{.f(k)}
For the complete system, is written as:
the problem
Max z = = {!.O) ,!.(2) ,!.(6)} Cf ( 1 ) ,i. (2 ) , ... ,! (6 )} x£D ~e:.Q.
(17) (1 7)
which represents a maximum of 18 objectives at every time stage. A summation over time is then necessary. However, the low flow target fl(k,~) f1(k,~ may be a factor at one or two locations only, and the other objectives may be combined according to the situation, which would reduce the dimension of the objective function vector. In any case, the multicriterion analysis would not be simple, and serious consideration must be given to the necessity of using a distributed model to justify the additional complexity. In the case when recreational benefits cannot be quantified-for example, they are rated on a qualitative scalemulticriterion methods such as ELECTRE I (Roy, 1971, David and Duckstein, 1976) or Concordance Analysis (Nijkamp and Vos, 1977) can be used.
ACKNOWLEDGEMENTS The authors wish to acknowledge the Wupperverband for the provision of all the basic data and the helpfull comments during the preparation of this paper. REFERENCES Benayoun, R., J. de Montgolfier, J. Tergny and o. O. Laritchev (1971). Linear programming with multiple objective functions: Step Method (STEM), Math. Progr., 1(3), pp. 366-375.---- ----B6hle, W., R. Harboe and G.A. Schultz (1979). Low-flow augmentation for water quality improvement in a river-system in Germany. Proceedings, XVIII Congress of the International Association for Hydraulic Research, Cagliari, Italy. Cohon, J.L. (1978). Multiobjective Programming and Planning, Academic Press, New York. David, L. and L. Duckstein (1976). Multi-criterion ranking of alternative long-range water resources systems, water Water Resources Bulletin, 12(4), pp. 731-754. Duckstein, L. (1978). Irnbedding Imbedding uncertanties into multiobjective mUltiobjective decision models in water resources, Proceedings, International Symposium on Risk and Reliability in Water Resources, June,Waterloo,
Low-flow and Flood Control Ontario, Canada Canada,, pp. 3-30. Duckstein, L., I. Bogardi and F F.. Trade-off Szidarovszky (1979) (1979).. between regional mining development and environmental impact, -~ Proceedings, 16th International Com Symposium on Application of Computer and Operations Research in SME / AIME . the Mineral Industry, SME/AIME. Fall Meeting, Oct. 17-19, Tucson, 355 - 364 . pp. 355-364. ( 1978) Duckstein, L. and M. Gablinger (1978). The dynamic value of conjunctive storage of water, Proceedings, International Symposium on ConCon junctive Use of Water, Erice-Trapani, Italy, October. (1979) Duckstein, L. and S. Opricovic (1979). Multiobjective optimization in river basin development, to appear, Water Resources Research. Dwyer, J., J. Kelly and M. Bowes (1977) evalu Improved procedures for evaluation of the control of recreation to national economic developRes . Rep. Rep . 128, Water Rement, Res. sources Center, University of Illinois, September. (1970) . Hall, W.A. and J.A. Dracup (1970). Water Resources Systems Engineering. McGraw-Hill, New York. York . Harb~R. (1976 a). Harb~R. Deterministische Optimierungsmodelle fur Speichersysteme , Wasserwirtschaft, Wasserwirtschaft , Vol. Vol . systeme, 66, No. 7/8, pp. 221-226. 221 - 226. MObasheri , F. and W.W.-G. Harboe, R., Mobasheri, Yeh (1970). Optimal Policy for Reservoir Operation, Journal of the Hydraulics Division, Division , A.S.C.E., A.S.C.E . , Vol . 96, No. HY11, pp. pp . 2297-2308. 2297-2308 . Vol. b) . Harboe, R. (1976 b). A Stochastic Optimization and Simulation Model for the Operation of the Lech River System, Mitteilungen, Hydraulik und Gewasserkunde, TU Munchen, Heft Nr. 21. 2l. Loucks, D.P. Interactive mulD. P. (1978). (1978) . tiobjecti~ tiobj ecti ~ water resources planning: An application and some extensions, tensi ons , Proceedings, International Symposium on Risk and Reliability in Water Resources, Waterloo, Ontario, Canada, pp. pp . 30 -40. Nijkamp, P. and J.B. Vos (1977) (1977).. A multicriteria analysis for water resource and land use development, Water Resources Research, Vol. 13, No. 3, 3 , pp. 515-518 515 - 518 Roy, (1971).. Problems and Methods ROY, B. (1971) with multiple objective functions, Mathematical Programming I, pp. pp . 239-266. M. K. and M. Zeleny (1977). (1977) . Starr, M.K. State and future of the arts, in Multiple Criteria Decision Making, edited by M.K. Starr and M. Zeleny, North-Holland Publishing Co Co., . , Amsterdam-New sterdam - New York, pp. 5-29. Wupperverband Wupperverband, Unpublished operation opera tion records. Wuppertal, Wuppertal , Federal Repu6
399
blic of Germany. Germany . Zeleny, M. (1973). Compromise programming, in Multiple Criteria Decision Making, JJ.L. . L . Cochrane and M. Zeleny (editors), (editors) , University of South Carolina Press, Press , Columbia, pp. 262-301. Zeleny Zeleny,, M. (1977). Adaptive displacement of preferences in decision making, in Multiple Criteria Decision Making, M.K. Starr and M. Zeleny (editors), North Holland Publishing Company, Amsterdam-New Amsterdam - New York York,, pp. 147-157. 147-157 .
R. Harboe, Ha rb oe , G. A. Schultz Schul t z and L. Duckstein
400
CITY OF WUPPERTAL WUPPER-RIVER
-----_ "\\ ------,,-\11 \ \ ..
CONTROL GAGE
Suas~~~-~ _--I
\
\
--- ------------
,--
1...-L----
\
\
\ \ \
\
\ \
\ \ \ \ 1£.\'I\ 11.J Suas~S_--
_t---\---\
\ \ \
----------- ----, ------ , ------\ \
\
\
\ .-\---
1
BRUCHER
2
LINGESE LINGE SE
-R£.S£.R'J~J
-\\
RESERVOIRS: RESERVOIRS :
¥-~~---_-i
----- -\.----
---- ---
\ 1\
SCHEVELINGER NEYE
Suas~':"I~~:J
SEVER BEVER WUPPER
Fig. F ig . 1. 1.
\
L------
-- -- --
Wupper-River-System. Wupper - River - System .
f2
4.1 4. 1
LOW
A
FLOW
(m 3 Is) /s)
4.05 ~ 05_________________
+0
4. 0 4.0 NON FEASIBLE REGION
3.9 3. 9
p=l
3.8 3. 8
0,53 0,,53
p~2
3. 7
FEASIBLE REGION
3. 6
fl 3. 52
FLOOD CONTROL STORAGE ( ·/0 )
3. 5 0 Fig.
2. 2.
\
\ \
KERSPE
3 4 5 6
\
25
50
75
1 00
Low-flow re~ervoir for different flood Low - flow augmentation by equivalent reservoir control levels.
Low-flow and Flood Control TABLE 1
Reservoir
401
Characteristics of the Wupper Reservoirs
6 [ 10 6 m33 ] Flood Control Storage [10
Dead Storage 6 3 [10 [ 10 6m3 ]
Capacity 6 3 [ 10 6m3 ]
1.Nov .-31 .Jan. 1.Nov.-31.Jan.
1 .Feb .-28 .Feb. 1.Feb.-28.Feb.
1.Mar.-31.Mar. 1 .Mar .-31 .Mar.
11.. Brucher
3.340
0
0.400
0
0
2. Lingese
2.600
0
1.100 1 .100
0
0
3. Schevelin-
0.290
0
0
0
0
6.000
0
0.750
0
0
0
0
ger 4. Neye
5. Bever
23.700
1.500
5.000
6. Wupper
25.900
2.500
9.900
TABLE 2
Clecreasing ~ecreasing
from decreasing from 9.9 to 5.0 5.0 to 0
Augmented Flow at Control Gage
Augmented nu.nl.mum minimum flow at con- Length of historical sample: trol gage 32 years
6 3 [10 m/month]
3 Is] [m 3 /s]
Augmented year, years
Augmented months/year
Flow from catchment between Wupper reservoir and control gage
0.588
0.23
-
F
Bever
3.958
1.53
31
130/31=4.2
Bever Schevelinger
3.998
1.54
30
120/30=4.0
Bever Schevelinger Lingese
4.339
1.67
30
125/30=4.2
Bever Schevelinger Lingese Brucher
4.744
1.83
30
124/30=4.1 124/3 0=4.1
8.666
3.34(X) 3.34(x)
32
250/32=7 .8 250/32=7.8
1
b
0
Y
w a u g m e n t e d
R e s e r v 0
i
r s
Flow 9ug~ented.by ~u~ented.by W~pper W~pper.. reserVOl.r servo~r l.n ~n conJunctl.on conJunct~on Wl.~ W~Ln all other reservoirs
-
(x) Higher levels (4 or 5 m33/s) / s) can be attained with historical record but with probabilities less than 100 %.
TABLE 3
Point A
D D1 1
1
B B
0.75
C
0.50
D D
0.25
E
0
Analysis by Compromise Programming programming
D D2 2
0 0.09 =0.170 =0.17C 0.53 0.27 =0.510 =0.51C 0.53 0.46 =0.856 0.53 1
(for p = 1 ) L 1 =D 1 +D 2 1 1 2
1
(for p = 2 ) D 2 L 2 =D 11 2+ 2+Di 2 2 1
0.92*
0.589
1 .01
0.510*
1 .106
0.813
1
1