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Water supply operations during drought: A discrete hedging rule
EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER
European Journal of Operational Research 82 (1995) 163-175
Theory and Methodology
Water supply operations during drought: A discrete hedging rule J h i h - S h y a n g S h i h a, C h a r l e s R e V e l l e
b,.
a Center for Energy and Environmental Studies, Carnegie Mellon University, Pittsburgh, PA 15213, USA b Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, MD 21218-2686, USA Received December 1992; revised July 1993
Abstract
A mixed integer programming model is constructed for the operation of a single water supply reservoir during drought and impending drought. Operating on a critical sequence of low flows, the model determines trigger volumes of storage plus anticipated inflow which signal the need for each of the several phases of rationing. The trigger volumes are examined as the allowed frequency of the various phases of rationing are varied.
Keywords: Drought; Water rationing; Reservoir operating rule; Mathematical programming; Mixed integer linear programming
1. Introduction and literature review
As a water resource system composed of reservoir storage approaches full utilization, the management of d e m a n d / o p e r a t i o n of the system becomes especially important -particularly during drought and incipient drought. When drought strikes, water managers strive to reduce overall damages associated with an inability to meet normal demands to the greatest extent possible. In most situations, however, specific measures of economic damages are not available, and water managers must resort to minimizing surrogates for damage such as the expected maximum shortage or the expected total shortage. The aim of the present research is the creation of demand management rules that can be followed during the drought and its onset which will mitigate the damages due to shortages. Although many definitions of drought are possible, we interpret the term to mean a time period during which ordinary demands cannot be steadily met from stream flow and reservoir storage. Although there has been steady and systematic progress in both reservoir design and operation since the early 1960s, operation during and prior to a drought has received very little attention. As the following brief review suggests, most modern efforts have focussed on the operation and design of multi-reservoir, multi-purpose systems. Techniques for operation of the single reservoir, dedicated only to water supply during drought or impending drought, have not really improved significantly in recent * Corresponding author. 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 2 3 7 - R
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years. Rippl's graphical design procedure (1883) for the single water supply reservoir evolved into the mass diagram procedure (Hazen, 1914). Babbit and Doland (1939) then modified the mass diagram approach to accommodate monthly varying demands rather than constant demands. Thomas (Fair, Geyer and Okun, 1966; Thomas and Burden, 1963) suggested the 'sequent peak' methodology for reservoir design under varying demands. The introduction of synthetic hydrology by Fiering (1967) and others provided a means both to extend the historical record and to generate equally likely stream flow sequences, thereby allowing reservoir designers to approach the calculation of the reliability of a design. The first suggestion of the application of linear programming to the operation and design of a water supply reservoir was offered by Dorfman (1962); his model was principally illustrative and conceptual but established the potential for optimization in design and operation. Hirsch (1979) developed two methodologies for the operation of a water supply reservoir. The generalized risk analysis model coupled simulation with reconstructed flows for the calculation of water emergency probabilities. The position analysis model used reconstructed flows and current reservoir status to calculate the likelihood of water emergencies. Hashimoto et al. (1982) and Moy et al. (1986) have investigated the interplay of reliability, vulnerability and resilience in reservoir design and management. More recently, Palmer and Holmes (1988) applied expert systems to create a set of rules to operate a water supply system during drought. Multi-objective linear programming has also recently been utilized (Randall et al. 1986, 1990) in the management of a water supply system during drought. That model uses recursive programming with streamflow and demand forecasts to modify operating policy as inflows occur. Simulation has also been utilized (Wright et al., 1986) in a multi-participant setting to develop policies which can serve cooperating jurisdictions during drought. Wilhite and Easterling (1989) suggest a tenstep planning process to promote drought preparedness. The process includes creation of operating/ management rules for application during drought. Despite the importance of planning for drought, apparently only these few papers have focussed solely on operation/management during drought of the water supply reservoir. The field has considered instead systems of reservoirs, sometimes operated only for water supply and sometimes operated for multiple purposes. The single reservoir operating during drought, however, has apparently received only the scant attention referred to above. Roefs (1968) and Yeh (1985) have both provided reviews of mathematical programming applications to reservoir modelling as has Croley (1979). Neither these authors nor the text of Major and Lenton (1979), however, focus on the important issue of the single water-supply-only reservoir operating during drought. Missing from most of the procedures for the operation of a single water supply only reservoir is the notion of an allowable failure that might occur or the consequences of such a failure. With the exceptions of Hashimoto et al. (1982) and Moy et al. (1986) who do allow failures in their models, none of the mathematical programming studies referred to consider either the possibility or the impact of failures to deliver anticipated demand. Nor do they therefore seek rules to reduce these shortfalls and the damages that accompany them. The two exceptions cited do allow failures but seek no general rules of management to reduce the extent of failure. That is, release/demand reduction choices are determined for each time period and are not generalizable to the conditions present at another moment in time. In the general situation of actual or impending drought, water supply managers have been observed to prefer a number of smaller shortages to a few very large ones, suggesting that the damages are convex in the amount of a shortfall. Water managers utilize restrictions or rationing to reduce demand levels temporarily and to preserve storage for future use (Water Science and Technology Board, 1986). If the water supply reservoir is entering a period of drought, a water manager needs specific quantitative values to signal the onset and extent of the restrictions that should be utilized.
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It should be noted that the operating rule used in single reservoir water supply simulations, often called the standard operating policy (SOP) (Mass et al., 1962), is only optimal if the objective is to reduce total shortfalls (Hashimoto et al., 1982). The SOP is a rule suggested for use in simulation models and not for actual and real-time operation of a reservoir in a practical situation. This is because it does not provide a mechanism for reducing demand in small increments in the early stages of, or during indications of, impending drought. The minimization of total shortfall, it will be seen, is not the objective of the water manager. Realistic demand m a n a g e m e n t / o p e r a t i n g rules would suggest that, during periods of incipient drought, reductions be made in demand even when it can be fully delivered from storage and current inflow; the reductions prevent larger shortages and much larger damages in later periods of operation. The goal of the present work is the creation of demand m a n a g e m e n t / o p e r a t i n g rules that can sustain the operation of the reservoir through a drought while avoiding severely damaging shortages. Such rules are needed because the damage function associated with shortfalls is apparently convex in the extent of the shortage. The evidence is seen in the preference of water managers for more small shortages to a few larger shortages. Rationing rules can utilize as their trigger information either a value of storage or a value of storage plus projected inflow. Although storage has been utilized as the trigger in studies on operating rules for multi-purpose reservoirs (ReVelle et al., 1969; Houck et al., 1980), storage plus anticipated inflow is probably a better candidate for a rationing signal. This research utilizes triggers consisting of values of current storage plus the conditional expected value of inflow, but the information used for the rationing signals remains a subject for further research. A common sequence of drought management steps as they might be taken by a water agency is 1) forecasting of inflows and demands, 2) a consideration of drought management options, 3) the establishment of levels of storage a n d / o r inflow that trigger the various options of a demand reduction program, and 4) the adoption of a management plan at the levels indicated by storage/inflow signals. It is common for water agencies faced with impending or existing drought to use a phased demand reduction program in their m a n a g e m e n t of a drought situation. A hierarchically-phased demand-reduction program is a water conservation program in which successively more stringent policies are brought into play as the signals of the drought's severity worsen. Voluntary measures are often chosen during the initial period of concern. Mandatory measures may be taken next should the drought situation become worse. Each set of measures may have gradations that can be drawn on successively as the drought appears to become more severe, and each set of measures has associated with it an anticipated savings in water demand. The likely water savings of each of the phases is probably obtained from past drought experiences. With this information in hand, the water agency must still decide when the public should be informed of impending problems. It must also decide at what point voluntary restrictions should be initiated and when voluntary restrictions are insufficient and mandatory restrictions are required. In addition, it must determine the nature of the composition of the signals that trigger the various phases of rationing. Further, the agency must decide when operations can be returned to normal. Water supplies are managed by a combination of professional judgement, experience, and politics. For example, if reservoir storage plus projected streamflow (or other information) fall below some critical level, the first emergency phase may be announced to the public. If the reservoir storage plus projected streamflow should fall even further, to some second benchmark level, a second phase of rationing may be declared. Further phases of rationing may follow. The signals of water availability information that are used in practice to declare different emergency phases are not, however, determined in any uniform or even prescribed way. Water engineering texts offer no information on the determination of these signals. Such signals are generally peculiar to the water agency and may even be variable, i.e., not fully set in advance. In this paper, we suggest a
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J.S. Shih et al./ European Journal of Operational Research 82 (1995) 163-175
gt
Draft D ~
i.......i
infeasible
/
to Water I~ID
Supply
~2D
1~ Full Demand Released
" 1
First Pha~e Rationing |
I
I
I Secbnd Phas~ Rationing i
,
~
I
I
I
V3p V2p
Vlp
C+D
/
st-l+ ]'t
Storage Plus Projected Inflow Fig. 1. An example of the discrete hedging rule.
methodology to determine the parameters of a discrete hedging rule which proceeds in phases to reduce water demand. 2. A mathematical programming formulation The central idea of this research is to determine by use of mathematical programming methods optimal values of the volumes that trigger rationing phases. Trigger volumes for various levels of rationing, volumes that consist of flows, storage or some combination, need to be determined for individual months or seasons. The water availability trigger used here will be based on the current reservoir storage and projected inflow. Other measures can be utilized, indeed, may be utilized with some success, but were not pursued in this initial exploration, A consideration of these other measures would, in itself, constitute an ambitious research project. The idea of these triggers and their corresponding rationing phases is illustrated in Fig. 1. For a specific month p, if the previous month's storage plus the projected inflow for this month are greater than Vlp , then the full expected demand can be drafted from the reservoir without recourse to rationing. If the storage plus projected inflow is greater than Vzp but less than Vlp , then the water manager would announce to the public that phase one rationing will be initiated for the coming month. That is, demand will be reduced to only al fraction of usual demand. If the storage plus projected inflows is less then V2p but is greater than V3p, the water manager would announce the initiation of phase two rationing. That is, the demand will be reduced to only a 2 proportion of usual demand. In this model, it is assumed that the minimum trigger volume will always be maintained in the reservoir. Clearly, more phases could be added to the schema. It is the trigger volumes, Vlp, Vzp, V3p, etc., that are being sought for different phases of rationing for each month or season as a means to provide guidance for decision making by the water manager. Trigger volumes depend on both the hydrology of the inflows and the water savings that result from the various demand reduction measures. A more formal statement of the assumptions of this model needs to be provided. First, water use reduction is assumed to be discrete. The fractional reduction in water use corresponding to each phase of rationing should be known beforehand, i.e., through past experience or extrapolation from other similar communities operating in similar seasons. Second, in the model studied here, the fractional reduction for a specific phase is assumed to be the same for all seasons. This assumption, however, can be modified very easily. Third, for simplicity, there are only two phases of rationing in the model, an
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167
assumption that can also be relaxed easily, although with an increase in computational burden. Planned research will consider more phases of rationing. This optimization model seeks the trigger volumes, Vlp, V2p, and V3p, for the different rationing phases in all months/seasons p. The volumes will be chosen such that some performance criterion is maximized or minimized. If the storage at the end of the previous period plus projected inflow are together greater than Vlp, then no water restrictions are announced. If the storage at the close of previous period plus projected inflow falls between Vlp and Vzp, then rationing phase I is assumed to be declared, reducing demand to 0/1 proportion of its expected value. If the storage plus projected inflow fall below Vzp, rationing phase II will be declared and only the proportion 0/2 of usual demand will be expected to occur, and so on. These notions have been formulated into a mathematical programming problem: T max
12
~
Ylt -- tO E ( V l p + V2p -[- V3p) t=l p=l
(1)
s.t. Ylt >-
(st_l
(Vl,-,)
M
Vt, p,
(2)
Vt, p,
(3)
Vt, p,
(4)
Vt, p,
(5)
Vt,
(6)
Vt,
(7)
Vt,
(8)
vl,Ylt ~ 1 --
(s,_, Y2t >-
Y2t ~ 1 --
M
M
M
R t = ( 1 . 0 - al) * D * Yl, q- (°[1St=St_ , +It-R
Ol2) *
D * Y2t + 0/2 * O
t- Wt
St < Co
(9)
S o < S T, Ut < S t / C
Vt,
(10)
W t < M * Ut
Vt,
(11)
Vlp > (1 +/31) * Vzp
Vp,
(12)
Vzp> (1 +/32) * V3p
Vp,
(13)
V3p -> 0/2 * D
Vp,
(14)
Vt, p,
(15)
Vt,
(16)
Vt,
(17)
Vt,
(18)
S t - 1 + I t > V3p + e
Yu-1 +Ylt+x < 1 +Yu Ylt ~ Y2t+l N
~-, Y2t = N - n t=l
where: Ylt =
1 if full demand is available during period t; O, otherwise; unknown.
J.S. Shih et al. / European Journal of OperationalResearch 82 (1995) 163-175
168
1 if we are in phase one rationing or better; 0, otherwise; unknown. Small number, given weight. Storage at the end of period t; unknown. Release in the period t; unknown. The Value of storage and inflow above which no restrictions on water use are placed; unknown. The value of storage and inflow below which phase two is implemented for month p; unknown. The lower bound of storage plus inflow for month p; known. 2 . . . . ,12}, corresponding to different month p = t - [ l ( t - 1)]12. Month in the time horizon. Inflow in period t; known. I,= Projected inflow in period t; known. w , = Spill in period t; unknown. C= Reservoir capacity; known. D= Demand, assumed known and the same throughout the year. 1, if reservoir is full at the end of period t; 0, otherwise; unknown. M= Big number; given. Small number; given. Percentage of demand that obtains during phase one rationing; known. Ol I = Percentage of demand that obtains during phase two rationing; known. O/2= N= Total number of operating periods; known. /,/= Number of periods in which phase two rationing is allowed: a parameter. /31= A fractional separation value, used here as 0.05. /32 = A fractional separation value, used here as 0.05. The primary objective in this model is to maximize the number of months in which no rationing is required. The frequency of phase two rationing is varied over a range by (18) in order to observe the tradeoff between months without rationing and months with phase two rationing. In order to avoid any alternative solutions, the second term of the objective is to minimize the total of the triggering volumes. Smaller trigger volumes mean a lower frequency of rationing in general; hence, it makes sense to have trigger volumes as small as possible. A weight is placed on the volume terms, but the value of the weight is small enough to prevent any effect on the primary objective. Constraint set (2) in combination with the integer requirements says that if the storage at the end of last period plus projected inflow is greater than or equal to the trigger volume Vlp, then Ylt will be one; i.e. that no rationing is needed. Constraint (3) in combination with the integer requirements requires that Ylt be zero if available water is less than Vlp. If there were no e in constraint (2), when
Y2t tO= St= Rt= V~p= V2p= V3p= p~{1, t=
S _l
= vl ,
the variable Ylt could be zero or one. To correct this, e is incorporated into constraint (2) so that when St_l q - ~ = Vlp, the variable Ylt must be equal to 1. Constraint set (4) in combination with the integer requirements indicate that if the storage at the end of previous time period plus projected inflow is greater than or equal to trigger volume Vap, then Y2t will be equal to one. Constraint (5) forces Yet to zero if the end storage plus projected inflow is less than Vap. These constraint sets also assure that Ylt will not equal one unless Y2t equals one, and that if Y2t equals zero, then Ylt must equal zero. This special characteristic of these variables is important to the formulation of the objective. Constraint set (6) sets release equal to full demand if Ylt and Y2t are both unity. It also says that release is alD, if only Yat is one. Constraints (6) further set release to aaD if neither Ylt nor Y2t is one. In this model, the fraction a e of water demand is always released.
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169
Constraint set (7) is the mass balance equation of inflow and outflow in each time period t. T h e r e are many ways to express this mass balance. It is assumed that no significant evaporation or seepage losses occur. The mass balance equation states that the storage at the end of the current period is equal to the storage at the end of the previous time period plus any inflow minus the release and any spill in this period. Constraint set (8) is the capacity restriction on each storage volume. Constraint set (9) requires that through the period of operation water not be borrowed from the initial storage. In order to assure that spill can only h a p p e n when the reservoir is at capacity, constraint sets (10) and (11) are added. Constraints (10) say that if the storage at the end of time period t is less than the reservoir capacity, then Ut will be zero. Constraints (11) force spill to be zero when U t is zero, i.e., force spill to be zero when storage is less than capacity. If the right hand side of constraint (10) equals one (storage equal to capacity), then the right hand side of constraint (11) is a large number, and spill is constrained to that level. Constraints (12) and (13) are designed to separate the trigger volumes of phase one and phase two and the trigger volumes of phase two and phase three by at least some percentages, /31 and/32" Constraints (14) say that the trigger volume V3p should be at least greater than ol2 proportion of demand. As long as this condition is enforced, no more than what the reservoir has in storage can be released. Constraint (15) says that in order to release a 2 percent of water demand, the storage at the end of the previous time period plus projected inflow should be at least greater than V3p. This also implies that a 2 * D (the minimum release) can always be provided. Constraints (16) say that if release in period t 1 and release in period t + 1 are both at full demand, then the current period must have a release of full demand, as well. T h e rationale for this constraint is to prevent a 'flip flopping' or back and forth movement between rationing and non-rationing. The constraints do not prevent this from happening in the real situation, but prevent it in the determination of trigger volumes. Constraints (17) say that if full d e m a n d is released in period t then at least a a fraction of water d e m a n d must be released in period t + 1. This constraint prevents the system from going directly from the release of full demand to the much lower phase two requirement. Constraint (18) is written to assure that no m o r e than n periods of phase two occur during the time horizon. Actually, the relationship of the integer variable Yi, is: Y2t (N - - t_~l t----~lYlt ',) + t=l ~ Ylt = N " = Yzt j'1+ ( t----~1
(6)
The first t e r m is the n u m b e r of periods of phase two rationing, the second t e r m is the n u m b e r of periods of phase one rationing, and the third t e r m is the n u m b e r of periods in which no rationing occurs. The above equation can be generalized to m o r e than two phases of rationing. The m a x i m u m shortage is not an interesting objective in this model of successively more strict rationing. This is because the percentage of water demand to be released in the worst phase is known by the water authority. As a consequence, the maximum shortage (if it is necessary to have phase two rationing) is actually given in advance. The important objectives of this model can be several. Clearly the maximization of the n u m b e r of periods without any rationing is a significant goal. Another significant goal is the minimization of the n u m b e r of periods in which phase two rationing must be declared. The trade-off between these two objectives is of particular interest. The integer variables provide a great deal of f r e e d o m to manipulate these objectives.
3. Computational results A thirty-six month period was selected out of the 86 years of G u n p o w d e r River (Maryland) streamflow records. This period was months 577 to 612. Flows are shown in Table 1.
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Table 1 36 Months of streamflow data (in billion gallons per month) from the Gunpowder River in Maryland Month
Year One
Year Two
Year Three
1 2 3 4 5 6 7 8 9 10 11 12
4.33 3.41 4.15 4.88 3.96 3.32 1.75 3.87 1.75 1.57 1.66 2.30
4.70 3.50 5.71 5.71 6.08 3.96 2.39 2.03 2.12 5.07 8.75 5.62
7.46 9.12 11.88 20.08 14.65 7.92 6.45 19.25 8.57 6.26 5.53 5.16
T h e s e m o n t h s r e p r e s e n t the most severe low flow p e r i o d in the historical record. I n the m a t h e m a t i c a l p r o g r a m m i n g f o r m u l a t i o n , the d e m a n d level is set large e n o u g h to m a k e r a t i o n i n g necessary f r e q u e n t l y d u r i n g this critical period. T h e critical low flow o c c u r r e d in this 3 6 - m o n t h sequence, l e a d i n g to several p h a s e two cutbacks of u p to 60%. T h a t is, r a t i o n i n g was u s e d to r e d u c e d e m a n d to 40% of its n o r m a l level. It is a s s u m e d that t h e r e were only two phases of r a t i o n i n g , a n d the following i n p u t p a r a m e t e r s were set: d e m a n d equals 7.5 billion g a l l o n s / m o n t h , a 1 = 0.6, 13/2= 0.4, /~1 a n d 8 2 , the s e p a r a t i o n p a r a m e t e r s = 0.05, a n d reservoir capacity = 30 b i l l i o n gallons. T h e m o d e l was r u n o n the C o r n e l l s u p e r c o m p u t e r , I B M 3090-600J, u s i n g the p a c k a g e M P S X , with i n t e g e r capability. R e s u l t s a n d execution times are shown in T a b l e 2. I n T a b l e 2, it is n o t e d that, given the p a r a m e t e r set above, the m i n i m u m n u m b e r of m o n t h s of p h a s e two r a t i o n i n g is 10. T h e r e was n o feasible s o l u t i o n to the mixed i n t e g e r p r o g r a m w h e n the n u m b e r of m o n t h s of p h a s e two r a t i o n i n g was set to less t h a n 10. This is b e c a u s e t h e r e was n o t e n o u g h w a t e r to have o n e m o r e m o n t h of p h a s e o n e r a t i o n i n g or of n o r a t i o n i n g . Also notice that the w a t e r saved from each a d d i t i o n a l m o n t h of p h a s e two r a t i o n i n g gains us n o m o r e t h a n o n e a d d i t i o n a l m o n t h of n o r a t i o n i n g a n d s o m e t i m e s n o a d d i t i o n a l m o n t h s at all. This is b e c a u s e the c u t b a c k of p h a s e o n e r a t i o n i n g
Table 2 Results and execution time of the trigger volume model (demand = 7.5 billion gallons/month, Reservoir capacity = 30 billion gallons, a I = 0.6 and a z = 0.4). Computations were performed on an IBM 3090-600J, a supercomputer at Cornell University, utilizing IBM's mixed integer programming package R u n No.
Number of months without rationing
Number of months with phase one rationing
Number of months with phase two rationing
Execution time (mins)
0 1 2 3 4 5 6 7 8
infeasible 15 16 17 17 18 18 18 18
infeasible 11 9 7 6 4 3 2 1
9 10 11 12 13 14 15 16 17
17.62 18.75 2.64 19.53 1.43 1.47 0.82 0.69
J.S. Shih et aL / European Journal of Operational Research 82 (1995) 163-175
171
Run No.
40
30
Months of the
20
Three Rationing I0 Phases
0 10 11 12 13 14 15 16 17 Number of Months of Phase T w o Rationing [] Months of No Rationing •
Months of Phase One Rationing
•
Months of Phase T w o Rationing
Fig. 2. Bar chart display of the results in Table 2 (demand = 7.5 billion ganons per month, reservoir capacity = 30 billion gallons, a 1 = 0.6 and a 2 = 0.4).
is 40% of demand. Thus, the ratio of additional months of no rationing to months of phase two rationing is at most one. In other examples with a different rate of water saving, this ratio would be different. T h e results of Table 2 can also be shown to the reservoir m a n a g e r using Fig. 2. This figure conveys the tradeoff that exists between rationing phases and thus may assist a decision m a k e r in choices between modest and severe rationing. T h e horizontal axis in this figure shows the n u m b e r of months of phase two rationing, while the constant height of the bars is the total of months under consideration as well as the sum of the months of all different phases. Each bar is divided into three components, which corresponds to the three different rationing phases. The most densely shaded area corresponds to the n u m b e r of months of phase two rationing; the intermediate area corresponds to the n u m b e r of months of phase one rationing; the least densely shaded area reflects the n u m b e r of months of no rationing. The two objectives of concern are the n u m b e r of months with no rationing and the n u m b e r of months with phase two rationing. A careful check of Table 2 reveals that the result of the run with 13 months of phase two rationing is inferior in terms of the two stated objectives to the run with 12 months of phase two rationing. That is, increasing the n u m b e r of months with phase two rationing from 12 did not result in any increase in months without rationing - although it did decrease the n u m b e r of months with phase one rationing by one. H e n c e the solution generated by the run with 13 months of phase two rationing may be regarded as inferior in terms of the two objectives. Trigger volumes associated with this solution would not be selected. Similarly, the solutions generated by the runs with 15, 16, and 17 months of phase two rationing are inferior to the solution generated by the run with 14 months of phase two rationing. The trade-off between phase two rationing and no rationing can be read from the bar diagram of Fig. 2. As the n u m b e r of months of no rationing increases, the number of months of phase two rationing also increases. On the other hand, the n u m b e r of months of phase one rationing decrease. This same information, can also be conveyed by a m o r e conventional tradeoff curve, Fig. 3. The vertical axis of Fig. 3 is the n u m b e r of months of no rationing, the objective being maximized. The horizontal axis is the n u m b e r of months of phase two rationing which is, in effect, being minimized, although it is operated on in the constraint set (constraint 18). As can be seen from Fig. 3, additional months with no rationing are achieved only at the expense of m o r e months with phase two rationing. Additional months of phase two rationing beyond 14 achieve no additional months without rationing. In Table 3, results are displayed for the same level of d e m a n d and a 2, but a different value of al, namely, 0.75. In this case, note first that the n u m b e r of months of phase two rationing can not be less
172
Z S. Shih et al. /European Journal of Operational Research 82 (1995) 163-1 75 19 18 17 Number of Months of No Rationing
16,
15' 14
9
1'0
11
12
13
14
15
Number of Months of Phase Two Rationing Fig. 3. The trade-off between months with no rationing and months with phase two rationing.
than 13; no feasible solution exists at values of less than 13. This value is larger than that in the previous example, because the value of oq is greater than in the previous case; that is, more water is released in phase one rationing. If more water is always released in phase one, it follows that a greater number of months of phase two rationing will occur. Second, over a range each additional m o n t h of phase two rationing gains two months of no rationing. This is because a cutback of phase one rationing, 25%, in the present example, is smaller than in the previous example, 40%. Thus, the water saved by having one more month in phase two is enough to generate two more months of no rationing. Another observation is that when the number of months of phase two rationing is set equal to 13, the fraction of months with no shortage is 0.472 (17 out of 36 months) when o11 0.6. The fraction of months with no shortage is one-third (12 months out of 36 months) when a 1 = 0.75. Such results indicate the sensitivity of the solution to the selection of oq and oz2. An extensive sensitivity analysis in which the o~ values are continuously varied would be an interesting topic for future research. Such an analysis would require much computation, but would provide water managers scope to examine the manner in which they define rationing phases. In Table 3, the solution generated by the run with 17 months of phase two rationing appears to be inferior to that generated by the run with 16 months of phase two rationing since no additional months without rationing are gained by the additional month of phase two rationing. On the other hand, the additional month of phase two rationing does decrease the months with phase one rationing by one. A bar chart display of Table 3 is provided in Fig. 4 wherein the trade-off of months with no rationing against months with phase two rationing given a 1 0.75 can be seen. =
=
Table 3 Results and execution time of the trigger volume model (demand = 7.5 billion gallons/month, a 1 = 0.75 and a 2 = 0.4) Run No.
Number of months without rationing
Number of months with phase one rationing
Number of months with phase two rationing
Execution time (mins)
0 1 2 3 4 5
infeasible 12 14 16 18 18
infeasible 11 8 5 2 1
12 13 14 15 16 17
76.77 20.13 5.68 0.39 0.30
J.S. Shih et al. / European Journal of Operational Research 82 (1995) 163-175
173
Months of the Three Rationing Phases
10 t l 12 13 14 15 16 17 Number of Months of Phase Two Rationing [] Monthsof No Rationing [] Months of PhaseOne Rationing [] Months of Phase Two Rationing Fig. 4. Bar chart display of the results in Table 3 (demand = 7.5 billion gallons per month, reservoir capacity = 30 billion gallons, a 1 - 0.75 and a 2 = 0.4).
In each of the runs described the results include not only the number of months in various rationing phases, but, most importantly, the trigger volumes that correspond to each phase for each month. The trigger volumes obtained for a I = 0.60, a 2 = 0.40, and for the number of months of rationing at the phase two level set equal to 17, 14 and 10 are given in Table 4. This table shows that trigger volumes are sensitive to the number of months of rationing at the phase two level. In general, as the number of months of phase two rationing is allowed to decrease, the number of months of no rationing also decreases. At the same time, the number of months with phase one rationing increases. It should be expected then that the range of the trigger volumes which lead to phase one rationing will increase. In general, the difference between Vlp and V2p increases as the number of months of phase two rationing decreases from 17 to 10.
Table 4 Trigger volumes for various levels of phase 2 rationing. ( D e m a n d = 7.5 billion g a l l o n s / m o n t h , reservoir capacity = 30 billion gallons, cq = 0.6, c~2 = 0.4) Month
N u m b e r of months of phase 2 rationing 17
1 2 3 4 5 6 7 8 9 10 11 12
14
10
~p
~p
~p
½p
~p
~p
~p
~p
~p
12.22 15.60 15.70 17.18 19.46 22.40 21.49 19.99 14.69 12.12 10.51 9.36
11.64 14.86 14.95 16.36 18.53 21.33 20.47 19.04 13.99 11.54 10.01 8.91
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
7.50 10.36 8.95 9.10 10.08 12.95 12.04 10.54 9.97 7.39 5.79 4.63
7.14 3.15 3.15 3.15 9.60 12.33 11.47 10.04 9.49 7.04 5.51 4.41
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
7.50 10.36 14.97 12.10 12.60 11.51 10.38 7.14 9.97 7.39 8.81 4.63
7.14 3.15 3.15 3.15 8.74 10.96 6.65 6.80 9.49 7.04 5.51 4.41
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
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J.S. Shih et al. / European Journal of Operational Research 82 (1995) 163-175
These results can be displayed for the water manager as guidance using plots that function as rule curves. The values of Vlp can be connected for all seasons, and this graph is defined as rule curve A. For any specific month, if the reservoir storage plus projected inflow is above rule curve A, then full demand is released. If the reservoir storage plus projected inflow is below the curve, then rationing is required. Similarly, rule curve B can be constructed by connecting the values of V2p for all seasons. Then rule curve B is used to distinguish between phase one rationing and phase two rationing. In this example, rule curve C is obtained by connecting the values of V3p for all seasons. Rule curve C represents the minimum level of storage plus projected inflow that is required for feasibility. Ideally, rule curves A and B should display smooth patterns such as increasing to a peak in the summer. However, because operation is over a short period (which is a limitation of the machine capability), not all seasons will display all three phases of rationing when the mathematical program is solved. For example, suppose that over a short flow sequence, January always has a high storage level. Then the program will not be forced to find trigger volumes for January. Therefore, model runs covering a longer operating period are more likely to require different phases. Alternatively, a four-season model rather than a 12-month model could over the same length record be used to generate a more complete curve. The four season model could have 4 values of Vlp, V2p, and V3p and operate with the same number of integer variables. Or, the four season model could use only one-third of the integer variables and triple the record length that could be investigated.
4. Conclusions and further research
A mixed integer programming model has been presented that will operate a water supply reservoir through discrete rationing phases. The model determines the volumes of reservoir storage plus projected inflow that trigger the several phases of rationing under the objective of maximizing months without rationing given a limit on the number of months with phase two rationing. That is, the model is designed to provide guidance to the water manager by selection of the signals that should be used in determining the phase of rationing to be called. Earlier models in this lineage chose release levels and counted periods of shortage as well as total and maximum shortages, but did not determine the volumes which triggered the various rationing phases. No claim is made that the present model is fully ready for use. It is a construct with an initial method of solution for further study. The construct poses the questions that must be answered for a meaningful approach to the problem of temporal shortage allocation. The solution procedure that has been provided is exact in the sense that globally optimal solutions are produced, but the exact methodology is computationally intensive. However, it can be confidently anticipated that computer power will continue to grow as it has in the past, and that what is computationally intensive today will be quite tractable tomorrow. Nonetheless, the construct still needs to be extended to longer droughts and to multiple droughts in order to capture the full variability of inflow sequences. As well, it needs to be extended to stochastic inputs, multiple-purpose reservoirs, and systems of reservoirs. Such extensions are likely to require the creation of heuristic (non-exact) procedures to approach solutions. Heuristic procedures are valuable methodologies for otherwise intractable problems, although they do not necessarily provide optimal solutions. Confidence in the use of heuristics can be established by comparison o f their solutions to the solutions provided by exact procedures. It is in the establishment and verification of heuristic procedures that the exact procedure provided here is likely to find its greatest use. Thus, the problem statement and formulation provided here are meant to pose questions for further study, and the methodology is suggested as a baseline procedure for use in further research.
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175 h ¸
References Babbit, H.E., and Doland, J.J. (1939), Water Supply Engineering, McGraw-Hill, New York. Croley, T.E.; (1979), Risk in reservoir design and operation: A state-of-the-art review" in: E. McBean et al. (eds.), Reliability in Water Resources Management, Water Resources Publications, Fort Collins, CO. Dantzig, G.B., (1965), Linear Programming and Extensions, Princeton University Press, Princeton, NJ. Dorfman, R. (1962), "Mathematical models: The multi-structure approach", in: A. Mass (ed.), Design of Water Resources Systems, Harvard University Press, Cambridge, MA. Fair, G.M., Geyer, H.C., and Okun, D.A. (1966), Water and Wastewater Engineering: Volume 1. Water Supply and Waste water Removal, Wiley, New York. Fiering, M.B. (1967), Stream Flow Synthesis, Harvard University Press, Cambridge, MA. Gill, P.E., Murray, W., and Wright, M. (1981), Practical Optimization, Academic Press, New York. Hashimoto, T., Stedinger, J.R., and Loucks, D.P. (1982), "Reliability, resilience and vulnerability criteria for water resources system performance evaluation", Water Resources Research 18/1, 14-20. Hazen, A. (1994), "Storage to be provided in impounding reservoirs for municipal water supply", Transactions of the ASCE 77/1539. Hirsch, R., (1978), "Risk analysis for a water supply system: Occaquon Reservoir, Fairfax and Prince William Counties, Virgina, USA", Hydrologic Science Bulletin 23/4, (1978) 475-505. Houck, M.H., Cohort, J.L., and ReVelle, C.S. (1980), "Linear decision rule in reservoir design and management, 6. Incorporation of economic efficiency benefits and hydroelectric energy generation", Water Resources Research 16/1, 196-200. IBM (1979), "IBM Mathematical Programming System Extended/370 Program Reference Manual", IBM, New York. IMSL (1987), Inc., "User's Manual, Math/Library. FORTRAN subroutines for mathematical applications", Houston, TX. Linsley, Jr., R.K., Kohler, M.A., and Paulus, J.L.H. (1982), Hydrology for Engineers, McGraw-Hill, New York. Maass, A., Hufschmidt, M.M., Dorfman, R., Thomas, Jr., H.A., Marglin, S.A., and Fair, G.M. (1962), Design of Water-Resource Systems, Harvard University Press, Cambridge, MA. Major, D.C., and Lenton, R.L. (1979), Applied Water Resources Systems Planning, Prentice-Hall, Englewood Cliffs, NJ. Matalas, N.C. (1963), "Probability distribution of low flows", USGS Professional Paper 434-A, US Government Printing Office, Washington DC. Moy, W.-S. (1983), "Reliability, resilience, and vulnerability in reservoir operation", Ph.D. Dissertation, The Johns Hopkins University. Palmer, R., and Holmes, K.J. (1988), "Operational guidance during droughts: Expert System approach", Journal of Water Resources Planning and Management 114/6, 647-666. Randall, D., Houck, M.H., and Wright, J.R. (1990), "Simulating water system operation during periods of drought", Journal of the American Water Works Association, 53-60. Randall, D., Houek, M.H., and Wright, J.R. (1990), "Drought management of existing water supplysystem", Journal of Water Resources Planning and Management 116/1, 1-20. ReVelle, C.S., Joeres, E., and Kirby, W. (1969), The linear decision rule in reservoir management and design, 1. Development of the stochastic model", Water Resources Research 5/4, 767-777. Rippl, W. (1983), "The capacity of storage reservoirs for water supply", Proceedings of the Institution of Civil Engineers, 71, 270-278. Roefs, T G. (1968), "Reservoir management: The state of the art", IBM Washington Scientific Center, Wheaton, MD. Stedinger, J.R. (1984), "The performance of LDR models for preliminary design and reservoir operation", Water Resources Research, 20/2, 215-224. Thomas, H.A., and Burden, R.P. (1963), "Operations research in water quality management", Harvard University, Cambridge, MA. Water Science and Technology Board (1986), "Drought management and its impact on public water systems", Colloquium Report, National Research Council, National Academy Press, Washington, D.C. Whipple, Jr., W. (1966), "Regional drought frequency analysis", Journal of Irrigation Drainage 9/2, 11-31. Wilhite, D A., and Easterling, W.E. (1989), "Coping with drought: toward a plan of action", Eos, February. Wright, J.R., Houck, M.H., Diamond, J.T., and Randall, D. (1986), "Drought contingency planning", Civil Engineering Systems 3, 210-221. Yeh, W.W.-G. (1985), "Reservoir management and operation models: A state of the art review", Water Resources Research 21/12, 1797-1818. Yevjevieh, V.M. (1967), "An objective approach to definitions and investigations of continental hydrologic droughts", Hydrology Paper No. 23, Colorado State University, Fort Collins, CO.
European Journal of Operational Research 82 (1995) 163-175
Theory and Methodology
Water supply operations during drought: A discrete hedging rule J h i h - S h y a n g S h i h a, C h a r l e s R e V e l l e
b,.
a Center for Energy and Environmental Studies, Carnegie Mellon University, Pittsburgh, PA 15213, USA b Department of Geography and Environmental Engineering, The Johns Hopkins University, Baltimore, MD 21218-2686, USA Received December 1992; revised July 1993
Abstract
A mixed integer programming model is constructed for the operation of a single water supply reservoir during drought and impending drought. Operating on a critical sequence of low flows, the model determines trigger volumes of storage plus anticipated inflow which signal the need for each of the several phases of rationing. The trigger volumes are examined as the allowed frequency of the various phases of rationing are varied.
Keywords: Drought; Water rationing; Reservoir operating rule; Mathematical programming; Mixed integer linear programming
1. Introduction and literature review
As a water resource system composed of reservoir storage approaches full utilization, the management of d e m a n d / o p e r a t i o n of the system becomes especially important -particularly during drought and incipient drought. When drought strikes, water managers strive to reduce overall damages associated with an inability to meet normal demands to the greatest extent possible. In most situations, however, specific measures of economic damages are not available, and water managers must resort to minimizing surrogates for damage such as the expected maximum shortage or the expected total shortage. The aim of the present research is the creation of demand management rules that can be followed during the drought and its onset which will mitigate the damages due to shortages. Although many definitions of drought are possible, we interpret the term to mean a time period during which ordinary demands cannot be steadily met from stream flow and reservoir storage. Although there has been steady and systematic progress in both reservoir design and operation since the early 1960s, operation during and prior to a drought has received very little attention. As the following brief review suggests, most modern efforts have focussed on the operation and design of multi-reservoir, multi-purpose systems. Techniques for operation of the single reservoir, dedicated only to water supply during drought or impending drought, have not really improved significantly in recent * Corresponding author. 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 3 ) E 0 2 3 7 - R
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years. Rippl's graphical design procedure (1883) for the single water supply reservoir evolved into the mass diagram procedure (Hazen, 1914). Babbit and Doland (1939) then modified the mass diagram approach to accommodate monthly varying demands rather than constant demands. Thomas (Fair, Geyer and Okun, 1966; Thomas and Burden, 1963) suggested the 'sequent peak' methodology for reservoir design under varying demands. The introduction of synthetic hydrology by Fiering (1967) and others provided a means both to extend the historical record and to generate equally likely stream flow sequences, thereby allowing reservoir designers to approach the calculation of the reliability of a design. The first suggestion of the application of linear programming to the operation and design of a water supply reservoir was offered by Dorfman (1962); his model was principally illustrative and conceptual but established the potential for optimization in design and operation. Hirsch (1979) developed two methodologies for the operation of a water supply reservoir. The generalized risk analysis model coupled simulation with reconstructed flows for the calculation of water emergency probabilities. The position analysis model used reconstructed flows and current reservoir status to calculate the likelihood of water emergencies. Hashimoto et al. (1982) and Moy et al. (1986) have investigated the interplay of reliability, vulnerability and resilience in reservoir design and management. More recently, Palmer and Holmes (1988) applied expert systems to create a set of rules to operate a water supply system during drought. Multi-objective linear programming has also recently been utilized (Randall et al. 1986, 1990) in the management of a water supply system during drought. That model uses recursive programming with streamflow and demand forecasts to modify operating policy as inflows occur. Simulation has also been utilized (Wright et al., 1986) in a multi-participant setting to develop policies which can serve cooperating jurisdictions during drought. Wilhite and Easterling (1989) suggest a tenstep planning process to promote drought preparedness. The process includes creation of operating/ management rules for application during drought. Despite the importance of planning for drought, apparently only these few papers have focussed solely on operation/management during drought of the water supply reservoir. The field has considered instead systems of reservoirs, sometimes operated only for water supply and sometimes operated for multiple purposes. The single reservoir operating during drought, however, has apparently received only the scant attention referred to above. Roefs (1968) and Yeh (1985) have both provided reviews of mathematical programming applications to reservoir modelling as has Croley (1979). Neither these authors nor the text of Major and Lenton (1979), however, focus on the important issue of the single water-supply-only reservoir operating during drought. Missing from most of the procedures for the operation of a single water supply only reservoir is the notion of an allowable failure that might occur or the consequences of such a failure. With the exceptions of Hashimoto et al. (1982) and Moy et al. (1986) who do allow failures in their models, none of the mathematical programming studies referred to consider either the possibility or the impact of failures to deliver anticipated demand. Nor do they therefore seek rules to reduce these shortfalls and the damages that accompany them. The two exceptions cited do allow failures but seek no general rules of management to reduce the extent of failure. That is, release/demand reduction choices are determined for each time period and are not generalizable to the conditions present at another moment in time. In the general situation of actual or impending drought, water supply managers have been observed to prefer a number of smaller shortages to a few very large ones, suggesting that the damages are convex in the amount of a shortfall. Water managers utilize restrictions or rationing to reduce demand levels temporarily and to preserve storage for future use (Water Science and Technology Board, 1986). If the water supply reservoir is entering a period of drought, a water manager needs specific quantitative values to signal the onset and extent of the restrictions that should be utilized.
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165
It should be noted that the operating rule used in single reservoir water supply simulations, often called the standard operating policy (SOP) (Mass et al., 1962), is only optimal if the objective is to reduce total shortfalls (Hashimoto et al., 1982). The SOP is a rule suggested for use in simulation models and not for actual and real-time operation of a reservoir in a practical situation. This is because it does not provide a mechanism for reducing demand in small increments in the early stages of, or during indications of, impending drought. The minimization of total shortfall, it will be seen, is not the objective of the water manager. Realistic demand m a n a g e m e n t / o p e r a t i n g rules would suggest that, during periods of incipient drought, reductions be made in demand even when it can be fully delivered from storage and current inflow; the reductions prevent larger shortages and much larger damages in later periods of operation. The goal of the present work is the creation of demand m a n a g e m e n t / o p e r a t i n g rules that can sustain the operation of the reservoir through a drought while avoiding severely damaging shortages. Such rules are needed because the damage function associated with shortfalls is apparently convex in the extent of the shortage. The evidence is seen in the preference of water managers for more small shortages to a few larger shortages. Rationing rules can utilize as their trigger information either a value of storage or a value of storage plus projected inflow. Although storage has been utilized as the trigger in studies on operating rules for multi-purpose reservoirs (ReVelle et al., 1969; Houck et al., 1980), storage plus anticipated inflow is probably a better candidate for a rationing signal. This research utilizes triggers consisting of values of current storage plus the conditional expected value of inflow, but the information used for the rationing signals remains a subject for further research. A common sequence of drought management steps as they might be taken by a water agency is 1) forecasting of inflows and demands, 2) a consideration of drought management options, 3) the establishment of levels of storage a n d / o r inflow that trigger the various options of a demand reduction program, and 4) the adoption of a management plan at the levels indicated by storage/inflow signals. It is common for water agencies faced with impending or existing drought to use a phased demand reduction program in their m a n a g e m e n t of a drought situation. A hierarchically-phased demand-reduction program is a water conservation program in which successively more stringent policies are brought into play as the signals of the drought's severity worsen. Voluntary measures are often chosen during the initial period of concern. Mandatory measures may be taken next should the drought situation become worse. Each set of measures may have gradations that can be drawn on successively as the drought appears to become more severe, and each set of measures has associated with it an anticipated savings in water demand. The likely water savings of each of the phases is probably obtained from past drought experiences. With this information in hand, the water agency must still decide when the public should be informed of impending problems. It must also decide at what point voluntary restrictions should be initiated and when voluntary restrictions are insufficient and mandatory restrictions are required. In addition, it must determine the nature of the composition of the signals that trigger the various phases of rationing. Further, the agency must decide when operations can be returned to normal. Water supplies are managed by a combination of professional judgement, experience, and politics. For example, if reservoir storage plus projected streamflow (or other information) fall below some critical level, the first emergency phase may be announced to the public. If the reservoir storage plus projected streamflow should fall even further, to some second benchmark level, a second phase of rationing may be declared. Further phases of rationing may follow. The signals of water availability information that are used in practice to declare different emergency phases are not, however, determined in any uniform or even prescribed way. Water engineering texts offer no information on the determination of these signals. Such signals are generally peculiar to the water agency and may even be variable, i.e., not fully set in advance. In this paper, we suggest a
166
J.S. Shih et al./ European Journal of Operational Research 82 (1995) 163-175
gt
Draft D ~
i.......i
infeasible
/
to Water I~ID
Supply
~2D
1~ Full Demand Released
" 1
First Pha~e Rationing |
I
I
I Secbnd Phas~ Rationing i
,
~
I
I
I
V3p V2p
Vlp
C+D
/
st-l+ ]'t
Storage Plus Projected Inflow Fig. 1. An example of the discrete hedging rule.
methodology to determine the parameters of a discrete hedging rule which proceeds in phases to reduce water demand. 2. A mathematical programming formulation The central idea of this research is to determine by use of mathematical programming methods optimal values of the volumes that trigger rationing phases. Trigger volumes for various levels of rationing, volumes that consist of flows, storage or some combination, need to be determined for individual months or seasons. The water availability trigger used here will be based on the current reservoir storage and projected inflow. Other measures can be utilized, indeed, may be utilized with some success, but were not pursued in this initial exploration, A consideration of these other measures would, in itself, constitute an ambitious research project. The idea of these triggers and their corresponding rationing phases is illustrated in Fig. 1. For a specific month p, if the previous month's storage plus the projected inflow for this month are greater than Vlp , then the full expected demand can be drafted from the reservoir without recourse to rationing. If the storage plus projected inflow is greater than Vzp but less than Vlp , then the water manager would announce to the public that phase one rationing will be initiated for the coming month. That is, demand will be reduced to only al fraction of usual demand. If the storage plus projected inflows is less then V2p but is greater than V3p, the water manager would announce the initiation of phase two rationing. That is, the demand will be reduced to only a 2 proportion of usual demand. In this model, it is assumed that the minimum trigger volume will always be maintained in the reservoir. Clearly, more phases could be added to the schema. It is the trigger volumes, Vlp, Vzp, V3p, etc., that are being sought for different phases of rationing for each month or season as a means to provide guidance for decision making by the water manager. Trigger volumes depend on both the hydrology of the inflows and the water savings that result from the various demand reduction measures. A more formal statement of the assumptions of this model needs to be provided. First, water use reduction is assumed to be discrete. The fractional reduction in water use corresponding to each phase of rationing should be known beforehand, i.e., through past experience or extrapolation from other similar communities operating in similar seasons. Second, in the model studied here, the fractional reduction for a specific phase is assumed to be the same for all seasons. This assumption, however, can be modified very easily. Third, for simplicity, there are only two phases of rationing in the model, an
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167
assumption that can also be relaxed easily, although with an increase in computational burden. Planned research will consider more phases of rationing. This optimization model seeks the trigger volumes, Vlp, V2p, and V3p, for the different rationing phases in all months/seasons p. The volumes will be chosen such that some performance criterion is maximized or minimized. If the storage at the end of the previous period plus projected inflow are together greater than Vlp, then no water restrictions are announced. If the storage at the close of previous period plus projected inflow falls between Vlp and Vzp, then rationing phase I is assumed to be declared, reducing demand to 0/1 proportion of its expected value. If the storage plus projected inflow fall below Vzp, rationing phase II will be declared and only the proportion 0/2 of usual demand will be expected to occur, and so on. These notions have been formulated into a mathematical programming problem: T max
12
~
Ylt -- tO E ( V l p + V2p -[- V3p) t=l p=l
(1)
s.t. Ylt >-
(st_l
(Vl,-,)
M
Vt, p,
(2)
Vt, p,
(3)
Vt, p,
(4)
Vt, p,
(5)
Vt,
(6)
Vt,
(7)
Vt,
(8)
vl,Ylt ~ 1 --
(s,_, Y2t >-
Y2t ~ 1 --
M
M
M
R t = ( 1 . 0 - al) * D * Yl, q- (°[1St=St_ , +It-R
Ol2) *
D * Y2t + 0/2 * O
t- Wt
St < Co
(9)
S o < S T, Ut < S t / C
Vt,
(10)
W t < M * Ut
Vt,
(11)
Vlp > (1 +/31) * Vzp
Vp,
(12)
Vzp> (1 +/32) * V3p
Vp,
(13)
V3p -> 0/2 * D
Vp,
(14)
Vt, p,
(15)
Vt,
(16)
Vt,
(17)
Vt,
(18)
S t - 1 + I t > V3p + e
Yu-1 +Ylt+x < 1 +Yu Ylt ~ Y2t+l N
~-, Y2t = N - n t=l
where: Ylt =
1 if full demand is available during period t; O, otherwise; unknown.
J.S. Shih et al. / European Journal of OperationalResearch 82 (1995) 163-175
168
1 if we are in phase one rationing or better; 0, otherwise; unknown. Small number, given weight. Storage at the end of period t; unknown. Release in the period t; unknown. The Value of storage and inflow above which no restrictions on water use are placed; unknown. The value of storage and inflow below which phase two is implemented for month p; unknown. The lower bound of storage plus inflow for month p; known. 2 . . . . ,12}, corresponding to different month p = t - [ l ( t - 1)]12. Month in the time horizon. Inflow in period t; known. I,= Projected inflow in period t; known. w , = Spill in period t; unknown. C= Reservoir capacity; known. D= Demand, assumed known and the same throughout the year. 1, if reservoir is full at the end of period t; 0, otherwise; unknown. M= Big number; given. Small number; given. Percentage of demand that obtains during phase one rationing; known. Ol I = Percentage of demand that obtains during phase two rationing; known. O/2= N= Total number of operating periods; known. /,/= Number of periods in which phase two rationing is allowed: a parameter. /31= A fractional separation value, used here as 0.05. /32 = A fractional separation value, used here as 0.05. The primary objective in this model is to maximize the number of months in which no rationing is required. The frequency of phase two rationing is varied over a range by (18) in order to observe the tradeoff between months without rationing and months with phase two rationing. In order to avoid any alternative solutions, the second term of the objective is to minimize the total of the triggering volumes. Smaller trigger volumes mean a lower frequency of rationing in general; hence, it makes sense to have trigger volumes as small as possible. A weight is placed on the volume terms, but the value of the weight is small enough to prevent any effect on the primary objective. Constraint set (2) in combination with the integer requirements says that if the storage at the end of last period plus projected inflow is greater than or equal to the trigger volume Vlp, then Ylt will be one; i.e. that no rationing is needed. Constraint (3) in combination with the integer requirements requires that Ylt be zero if available water is less than Vlp. If there were no e in constraint (2), when
Y2t tO= St= Rt= V~p= V2p= V3p= p~{1, t=
S _l
= vl ,
the variable Ylt could be zero or one. To correct this, e is incorporated into constraint (2) so that when St_l q - ~ = Vlp, the variable Ylt must be equal to 1. Constraint set (4) in combination with the integer requirements indicate that if the storage at the end of previous time period plus projected inflow is greater than or equal to trigger volume Vap, then Y2t will be equal to one. Constraint (5) forces Yet to zero if the end storage plus projected inflow is less than Vap. These constraint sets also assure that Ylt will not equal one unless Y2t equals one, and that if Y2t equals zero, then Ylt must equal zero. This special characteristic of these variables is important to the formulation of the objective. Constraint set (6) sets release equal to full demand if Ylt and Y2t are both unity. It also says that release is alD, if only Yat is one. Constraints (6) further set release to aaD if neither Ylt nor Y2t is one. In this model, the fraction a e of water demand is always released.
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Constraint set (7) is the mass balance equation of inflow and outflow in each time period t. T h e r e are many ways to express this mass balance. It is assumed that no significant evaporation or seepage losses occur. The mass balance equation states that the storage at the end of the current period is equal to the storage at the end of the previous time period plus any inflow minus the release and any spill in this period. Constraint set (8) is the capacity restriction on each storage volume. Constraint set (9) requires that through the period of operation water not be borrowed from the initial storage. In order to assure that spill can only h a p p e n when the reservoir is at capacity, constraint sets (10) and (11) are added. Constraints (10) say that if the storage at the end of time period t is less than the reservoir capacity, then Ut will be zero. Constraints (11) force spill to be zero when U t is zero, i.e., force spill to be zero when storage is less than capacity. If the right hand side of constraint (10) equals one (storage equal to capacity), then the right hand side of constraint (11) is a large number, and spill is constrained to that level. Constraints (12) and (13) are designed to separate the trigger volumes of phase one and phase two and the trigger volumes of phase two and phase three by at least some percentages, /31 and/32" Constraints (14) say that the trigger volume V3p should be at least greater than ol2 proportion of demand. As long as this condition is enforced, no more than what the reservoir has in storage can be released. Constraint (15) says that in order to release a 2 percent of water demand, the storage at the end of the previous time period plus projected inflow should be at least greater than V3p. This also implies that a 2 * D (the minimum release) can always be provided. Constraints (16) say that if release in period t 1 and release in period t + 1 are both at full demand, then the current period must have a release of full demand, as well. T h e rationale for this constraint is to prevent a 'flip flopping' or back and forth movement between rationing and non-rationing. The constraints do not prevent this from happening in the real situation, but prevent it in the determination of trigger volumes. Constraints (17) say that if full d e m a n d is released in period t then at least a a fraction of water d e m a n d must be released in period t + 1. This constraint prevents the system from going directly from the release of full demand to the much lower phase two requirement. Constraint (18) is written to assure that no m o r e than n periods of phase two occur during the time horizon. Actually, the relationship of the integer variable Yi, is: Y2t (N - - t_~l t----~lYlt ',) + t=l ~ Ylt = N " = Yzt j'1+ ( t----~1
(6)
The first t e r m is the n u m b e r of periods of phase two rationing, the second t e r m is the n u m b e r of periods of phase one rationing, and the third t e r m is the n u m b e r of periods in which no rationing occurs. The above equation can be generalized to m o r e than two phases of rationing. The m a x i m u m shortage is not an interesting objective in this model of successively more strict rationing. This is because the percentage of water demand to be released in the worst phase is known by the water authority. As a consequence, the maximum shortage (if it is necessary to have phase two rationing) is actually given in advance. The important objectives of this model can be several. Clearly the maximization of the n u m b e r of periods without any rationing is a significant goal. Another significant goal is the minimization of the n u m b e r of periods in which phase two rationing must be declared. The trade-off between these two objectives is of particular interest. The integer variables provide a great deal of f r e e d o m to manipulate these objectives.
3. Computational results A thirty-six month period was selected out of the 86 years of G u n p o w d e r River (Maryland) streamflow records. This period was months 577 to 612. Flows are shown in Table 1.
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Table 1 36 Months of streamflow data (in billion gallons per month) from the Gunpowder River in Maryland Month
Year One
Year Two
Year Three
1 2 3 4 5 6 7 8 9 10 11 12
4.33 3.41 4.15 4.88 3.96 3.32 1.75 3.87 1.75 1.57 1.66 2.30
4.70 3.50 5.71 5.71 6.08 3.96 2.39 2.03 2.12 5.07 8.75 5.62
7.46 9.12 11.88 20.08 14.65 7.92 6.45 19.25 8.57 6.26 5.53 5.16
T h e s e m o n t h s r e p r e s e n t the most severe low flow p e r i o d in the historical record. I n the m a t h e m a t i c a l p r o g r a m m i n g f o r m u l a t i o n , the d e m a n d level is set large e n o u g h to m a k e r a t i o n i n g necessary f r e q u e n t l y d u r i n g this critical period. T h e critical low flow o c c u r r e d in this 3 6 - m o n t h sequence, l e a d i n g to several p h a s e two cutbacks of u p to 60%. T h a t is, r a t i o n i n g was u s e d to r e d u c e d e m a n d to 40% of its n o r m a l level. It is a s s u m e d that t h e r e were only two phases of r a t i o n i n g , a n d the following i n p u t p a r a m e t e r s were set: d e m a n d equals 7.5 billion g a l l o n s / m o n t h , a 1 = 0.6, 13/2= 0.4, /~1 a n d 8 2 , the s e p a r a t i o n p a r a m e t e r s = 0.05, a n d reservoir capacity = 30 b i l l i o n gallons. T h e m o d e l was r u n o n the C o r n e l l s u p e r c o m p u t e r , I B M 3090-600J, u s i n g the p a c k a g e M P S X , with i n t e g e r capability. R e s u l t s a n d execution times are shown in T a b l e 2. I n T a b l e 2, it is n o t e d that, given the p a r a m e t e r set above, the m i n i m u m n u m b e r of m o n t h s of p h a s e two r a t i o n i n g is 10. T h e r e was n o feasible s o l u t i o n to the mixed i n t e g e r p r o g r a m w h e n the n u m b e r of m o n t h s of p h a s e two r a t i o n i n g was set to less t h a n 10. This is b e c a u s e t h e r e was n o t e n o u g h w a t e r to have o n e m o r e m o n t h of p h a s e o n e r a t i o n i n g or of n o r a t i o n i n g . Also notice that the w a t e r saved from each a d d i t i o n a l m o n t h of p h a s e two r a t i o n i n g gains us n o m o r e t h a n o n e a d d i t i o n a l m o n t h of n o r a t i o n i n g a n d s o m e t i m e s n o a d d i t i o n a l m o n t h s at all. This is b e c a u s e the c u t b a c k of p h a s e o n e r a t i o n i n g
Table 2 Results and execution time of the trigger volume model (demand = 7.5 billion gallons/month, Reservoir capacity = 30 billion gallons, a I = 0.6 and a z = 0.4). Computations were performed on an IBM 3090-600J, a supercomputer at Cornell University, utilizing IBM's mixed integer programming package R u n No.
Number of months without rationing
Number of months with phase one rationing
Number of months with phase two rationing
Execution time (mins)
0 1 2 3 4 5 6 7 8
infeasible 15 16 17 17 18 18 18 18
infeasible 11 9 7 6 4 3 2 1
9 10 11 12 13 14 15 16 17
17.62 18.75 2.64 19.53 1.43 1.47 0.82 0.69
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171
Run No.
40
30
Months of the
20
Three Rationing I0 Phases
0 10 11 12 13 14 15 16 17 Number of Months of Phase T w o Rationing [] Months of No Rationing •
Months of Phase One Rationing
•
Months of Phase T w o Rationing
Fig. 2. Bar chart display of the results in Table 2 (demand = 7.5 billion ganons per month, reservoir capacity = 30 billion gallons, a 1 = 0.6 and a 2 = 0.4).
is 40% of demand. Thus, the ratio of additional months of no rationing to months of phase two rationing is at most one. In other examples with a different rate of water saving, this ratio would be different. T h e results of Table 2 can also be shown to the reservoir m a n a g e r using Fig. 2. This figure conveys the tradeoff that exists between rationing phases and thus may assist a decision m a k e r in choices between modest and severe rationing. T h e horizontal axis in this figure shows the n u m b e r of months of phase two rationing, while the constant height of the bars is the total of months under consideration as well as the sum of the months of all different phases. Each bar is divided into three components, which corresponds to the three different rationing phases. The most densely shaded area corresponds to the n u m b e r of months of phase two rationing; the intermediate area corresponds to the n u m b e r of months of phase one rationing; the least densely shaded area reflects the n u m b e r of months of no rationing. The two objectives of concern are the n u m b e r of months with no rationing and the n u m b e r of months with phase two rationing. A careful check of Table 2 reveals that the result of the run with 13 months of phase two rationing is inferior in terms of the two stated objectives to the run with 12 months of phase two rationing. That is, increasing the n u m b e r of months with phase two rationing from 12 did not result in any increase in months without rationing - although it did decrease the n u m b e r of months with phase one rationing by one. H e n c e the solution generated by the run with 13 months of phase two rationing may be regarded as inferior in terms of the two objectives. Trigger volumes associated with this solution would not be selected. Similarly, the solutions generated by the runs with 15, 16, and 17 months of phase two rationing are inferior to the solution generated by the run with 14 months of phase two rationing. The trade-off between phase two rationing and no rationing can be read from the bar diagram of Fig. 2. As the n u m b e r of months of no rationing increases, the number of months of phase two rationing also increases. On the other hand, the n u m b e r of months of phase one rationing decrease. This same information, can also be conveyed by a m o r e conventional tradeoff curve, Fig. 3. The vertical axis of Fig. 3 is the n u m b e r of months of no rationing, the objective being maximized. The horizontal axis is the n u m b e r of months of phase two rationing which is, in effect, being minimized, although it is operated on in the constraint set (constraint 18). As can be seen from Fig. 3, additional months with no rationing are achieved only at the expense of m o r e months with phase two rationing. Additional months of phase two rationing beyond 14 achieve no additional months without rationing. In Table 3, results are displayed for the same level of d e m a n d and a 2, but a different value of al, namely, 0.75. In this case, note first that the n u m b e r of months of phase two rationing can not be less
172
Z S. Shih et al. /European Journal of Operational Research 82 (1995) 163-1 75 19 18 17 Number of Months of No Rationing
16,
15' 14
9
1'0
11
12
13
14
15
Number of Months of Phase Two Rationing Fig. 3. The trade-off between months with no rationing and months with phase two rationing.
than 13; no feasible solution exists at values of less than 13. This value is larger than that in the previous example, because the value of oq is greater than in the previous case; that is, more water is released in phase one rationing. If more water is always released in phase one, it follows that a greater number of months of phase two rationing will occur. Second, over a range each additional m o n t h of phase two rationing gains two months of no rationing. This is because a cutback of phase one rationing, 25%, in the present example, is smaller than in the previous example, 40%. Thus, the water saved by having one more month in phase two is enough to generate two more months of no rationing. Another observation is that when the number of months of phase two rationing is set equal to 13, the fraction of months with no shortage is 0.472 (17 out of 36 months) when o11 0.6. The fraction of months with no shortage is one-third (12 months out of 36 months) when a 1 = 0.75. Such results indicate the sensitivity of the solution to the selection of oq and oz2. An extensive sensitivity analysis in which the o~ values are continuously varied would be an interesting topic for future research. Such an analysis would require much computation, but would provide water managers scope to examine the manner in which they define rationing phases. In Table 3, the solution generated by the run with 17 months of phase two rationing appears to be inferior to that generated by the run with 16 months of phase two rationing since no additional months without rationing are gained by the additional month of phase two rationing. On the other hand, the additional month of phase two rationing does decrease the months with phase one rationing by one. A bar chart display of Table 3 is provided in Fig. 4 wherein the trade-off of months with no rationing against months with phase two rationing given a 1 0.75 can be seen. =
=
Table 3 Results and execution time of the trigger volume model (demand = 7.5 billion gallons/month, a 1 = 0.75 and a 2 = 0.4) Run No.
Number of months without rationing
Number of months with phase one rationing
Number of months with phase two rationing
Execution time (mins)
0 1 2 3 4 5
infeasible 12 14 16 18 18
infeasible 11 8 5 2 1
12 13 14 15 16 17
76.77 20.13 5.68 0.39 0.30
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173
Months of the Three Rationing Phases
10 t l 12 13 14 15 16 17 Number of Months of Phase Two Rationing [] Monthsof No Rationing [] Months of PhaseOne Rationing [] Months of Phase Two Rationing Fig. 4. Bar chart display of the results in Table 3 (demand = 7.5 billion gallons per month, reservoir capacity = 30 billion gallons, a 1 - 0.75 and a 2 = 0.4).
In each of the runs described the results include not only the number of months in various rationing phases, but, most importantly, the trigger volumes that correspond to each phase for each month. The trigger volumes obtained for a I = 0.60, a 2 = 0.40, and for the number of months of rationing at the phase two level set equal to 17, 14 and 10 are given in Table 4. This table shows that trigger volumes are sensitive to the number of months of rationing at the phase two level. In general, as the number of months of phase two rationing is allowed to decrease, the number of months of no rationing also decreases. At the same time, the number of months with phase one rationing increases. It should be expected then that the range of the trigger volumes which lead to phase one rationing will increase. In general, the difference between Vlp and V2p increases as the number of months of phase two rationing decreases from 17 to 10.
Table 4 Trigger volumes for various levels of phase 2 rationing. ( D e m a n d = 7.5 billion g a l l o n s / m o n t h , reservoir capacity = 30 billion gallons, cq = 0.6, c~2 = 0.4) Month
N u m b e r of months of phase 2 rationing 17
1 2 3 4 5 6 7 8 9 10 11 12
14
10
~p
~p
~p
½p
~p
~p
~p
~p
~p
12.22 15.60 15.70 17.18 19.46 22.40 21.49 19.99 14.69 12.12 10.51 9.36
11.64 14.86 14.95 16.36 18.53 21.33 20.47 19.04 13.99 11.54 10.01 8.91
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
7.50 10.36 8.95 9.10 10.08 12.95 12.04 10.54 9.97 7.39 5.79 4.63
7.14 3.15 3.15 3.15 9.60 12.33 11.47 10.04 9.49 7.04 5.51 4.41
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
7.50 10.36 14.97 12.10 12.60 11.51 10.38 7.14 9.97 7.39 8.81 4.63
7.14 3.15 3.15 3.15 8.74 10.96 6.65 6.80 9.49 7.04 5.51 4.41
3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00 3.00
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These results can be displayed for the water manager as guidance using plots that function as rule curves. The values of Vlp can be connected for all seasons, and this graph is defined as rule curve A. For any specific month, if the reservoir storage plus projected inflow is above rule curve A, then full demand is released. If the reservoir storage plus projected inflow is below the curve, then rationing is required. Similarly, rule curve B can be constructed by connecting the values of V2p for all seasons. Then rule curve B is used to distinguish between phase one rationing and phase two rationing. In this example, rule curve C is obtained by connecting the values of V3p for all seasons. Rule curve C represents the minimum level of storage plus projected inflow that is required for feasibility. Ideally, rule curves A and B should display smooth patterns such as increasing to a peak in the summer. However, because operation is over a short period (which is a limitation of the machine capability), not all seasons will display all three phases of rationing when the mathematical program is solved. For example, suppose that over a short flow sequence, January always has a high storage level. Then the program will not be forced to find trigger volumes for January. Therefore, model runs covering a longer operating period are more likely to require different phases. Alternatively, a four-season model rather than a 12-month model could over the same length record be used to generate a more complete curve. The four season model could have 4 values of Vlp, V2p, and V3p and operate with the same number of integer variables. Or, the four season model could use only one-third of the integer variables and triple the record length that could be investigated.
4. Conclusions and further research
A mixed integer programming model has been presented that will operate a water supply reservoir through discrete rationing phases. The model determines the volumes of reservoir storage plus projected inflow that trigger the several phases of rationing under the objective of maximizing months without rationing given a limit on the number of months with phase two rationing. That is, the model is designed to provide guidance to the water manager by selection of the signals that should be used in determining the phase of rationing to be called. Earlier models in this lineage chose release levels and counted periods of shortage as well as total and maximum shortages, but did not determine the volumes which triggered the various rationing phases. No claim is made that the present model is fully ready for use. It is a construct with an initial method of solution for further study. The construct poses the questions that must be answered for a meaningful approach to the problem of temporal shortage allocation. The solution procedure that has been provided is exact in the sense that globally optimal solutions are produced, but the exact methodology is computationally intensive. However, it can be confidently anticipated that computer power will continue to grow as it has in the past, and that what is computationally intensive today will be quite tractable tomorrow. Nonetheless, the construct still needs to be extended to longer droughts and to multiple droughts in order to capture the full variability of inflow sequences. As well, it needs to be extended to stochastic inputs, multiple-purpose reservoirs, and systems of reservoirs. Such extensions are likely to require the creation of heuristic (non-exact) procedures to approach solutions. Heuristic procedures are valuable methodologies for otherwise intractable problems, although they do not necessarily provide optimal solutions. Confidence in the use of heuristics can be established by comparison o f their solutions to the solutions provided by exact procedures. It is in the establishment and verification of heuristic procedures that the exact procedure provided here is likely to find its greatest use. Thus, the problem statement and formulation provided here are meant to pose questions for further study, and the methodology is suggested as a baseline procedure for use in further research.
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175 h ¸
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