This paper presents a mathematical analysis of water supply reservoir operation. The analysis illustrates one-stage, two-stage, and three-stage formulations of multiple-period reservoir operation depending on the effects of operational constraints. There is a one-stage model when storage capacity constraints are nonbinding. Release decisions depend on total water availability and exhibit equal marginal utilities. Binding upper (lower) storage capacity constraint blocks the effect of decreased (increased) water availability in the subsequent stages on release decisions in the preceding stages. When one storage capacity constraint is binding, multiple periods become two stages and a gap occurs between marginal utilities of water. When there are one upper and one lower binding storage capacity constraints, reservoir operation is characterized as a three-stage model. Effects of forecast uncertainty and ending storage on reservoir operation are affected by reservoir storage capacity. When the storage capacity constraints are nonbinding, the reservoir can regulate streamflow in an extended timeframe, and current release decision is affected by forecast uncertainty of total streamflow and ending storage. When the storage capacity constraints are binding, the reservoir can regulate streamflow only in a short timeframe, and current release decision is primarily affected by forecast uncertainty of streamflow in the current stage.
Optimization models have been widely used in reservoir operation to determine optimal decisions [
Reservoir operation is a sequential process [
Mathematical analysis provides insights on the optimization of reservoir operations, particularly those related to water supply problems. Trade-offs exist between current and future water usages because water is a scarce resource. Hedging rules, which reduce water delivery to cope with water shortage risks caused by uncertainty of future streamflow, have been widely discussed in the literature and used in practice [
Mathematical analysis of reservoir operation is usually limited to conceptual two-stage models [
The rest of this paper is organized as follows. Section
Reservoir operation of water supply balances the release of multiple periods to maximize total utility. Release in one period yields economic utility and affects WA and utilities in other periods. This section formulates the optimization model for water supply reservoir operation and illustrates the optimality conditions of operation decisions.
Consider a reservoir problem with a study horizon of : reservoir storage at the beginning of period
The parameters are as follows:
Based on the aforementioned variables and parameters, we select
Reservoir storage, which is refilled by streamflow and drawn down by release, carries over water between periods and functions as a link in multiple-period reservoir operation (Figure
Schematic of formulation of multiple-period water supply reservoir operation (Solid line indicates the water balance relationship; dashed line represents economic utility).
Based on (
The optimization model of water supply in ((
Dual values are applied to characterize the effects of operational constraints on optimal decisions [ Equation ( Equation ( Equation (
Following the KKT conditions, we denote dual values for the total WA constraint (
Dual values of constraints in reservoir operation optimization.
Dual value | Meaning of dual value |
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Marginal value of WA, for example, marginal utility from one unit increases in WA. |
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Marginal value of |
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Marginal value of |
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Marginal value of |
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Marginal value of |
Among the KKT conditions, (
Release decisions have a balanced relationship, the sum of which is equal to the total WA. As a result, the dual value
Consider a baseline Case 0 in which both release and storage capacity constraints are nonbinding; that is, the reservoir has enough capacity to regulate streamflow variability. Dual values of release and storage capacity constraints must be zero according to the complementary slackness condition (see (
Release among periods exhibits equal marginal utilities when release and storage capacity constraints are nonbinding (Case 0).
A positive monotonic relationship exhibited by WA and
There are in total
Despite the monotonic relationship between
In Case 1, we examine the binding constraint of upper release capacity, which occurs with ample WA and large storage capacity. In addition to the diminishing marginal utility of
The binding constraint of the upper release capacity hinders increase in release because of increased WA and causes spill (Case 1).
Based on Case 1, we analyze Case 2, where
The binding constraint of the low release capacity in period
A comparison of Cases 1 and 2 with baseline Case 0 reveals that total WA remains the major determinant of
Reservoir storage carries over WA between periods. During high-flow periods, storage is refilled to reserve water for future use; during low-flow periods, storage is drawn down to supply water. The storage refill and drawing down are bounded by the upper and lower storage capacity constraints, respectively. Incorporating dual values of the storage capacity constraints yields the following:
In Case 3, we suppose that an upper storage capacity constraint bounds in period
The marginal utilities of WA in the two stages generate a gap that indicates the economic incentive of storing water in stage 1; that is, the opportunity cost of storing water in stage 1 is lower than the potential utility of supplying water in stage 2. Reservoir storage is refilled until it is bounded by the upper storage capacity
The binding constraint of the upper storage capacity in period
Based on Case 3, we analyze Case 4 where a binding constraint of lower storage capacity occurs in period
The binding constraint of the lower storage capacity in period
The binding constraint of storage capacity leads to multistage reservoir operation. For Cases 3 and 4, water carried over between the two stages is driven by the marginal value of water but constrained by storage capacity [
The binding constraints of the lower storage capacity in period
A comparison of Cases 3, 4, and 5 with Cases 0, 1, and 2 reveals that binding storage capacity constraints result in multistage reservoir operation. WA in one stage remains an important determinant of that stage’s release decisions, which exhibit equal marginal utilities (marginal value principle). However, the marginal utilities of WA in two consecutive stages generate a gap because of the binding storage capacity constraint and is thus indicated by the corresponding dual value. The upper storage capacity sets the limit of storage refill that carries over WA from one stage to its subsequent stages, and the binding of this storage capacity blocks the effect of decreased WA in subsequent stages on release decisions in preceding stages. By comparison, the lower storage capacity sets the limit of storage drawn down, and its binding blocks the effect of increased WA in subsequent stages. Dual value of the binding storage capacity constraint indicates utility gain from one unit of additional storage carried over between stages.
We detail the effects of water balance, release capacity, and storage capacity constraints on water supply reservoir operation. The one-stage, two-stage, and three-stage formulations of reservoir operation result from the effects of the constraints (Table
Characteristics and properties of operation decisions in the six cases.
Case | Formulation | Characteristics | Properties |
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0 | One-stage (Figures |
No binding release and storage capacity constraints (Figure |
Release decisions depend on total WA and follow the marginal value principle |
1 | One-stage (Figure |
Upper release capacity constraint binding in period |
Increased total WA leads to spill |
2 | One-stage (Figure |
Lower release capacity constraint binding in period |
Minimum requirement is satisfied in period |
3 | Two-stage (Figure |
Upper storage capacity constraint binding in period |
[ |
4 | Two-stage (Figure |
Lower storage capacity constraint binding in period |
[ |
5 | Three-stage (Figure |
Upper and lower storage capacity constraints binding in periods |
[ |
The analysis above highlights the importance of WA in one stage to release decisions in that stage. In decision-making, WA is usually estimated based on streamflow forecast and ending storage [
Forecast uncertainty leads to bias in EWA. Considering the forecast uncertainty (
In Case 0, the reservoir operation has a one-stage formulation. Release decisions depend on total WA and follow the marginal value principle; that is,
In Cases 1 and 2, the reservoir operation also has a one-stage formulation. In Case 1, the WA is sufficient to satisfy maximum demands in all periods.
In Cases 3 and 4, the reservoir operation has two stages. In Case 3, the binding constraint of the upper storage capacity at period
In Case 5, the reservoir operation has three stages. The binding upper (lower) storage capacity constraint blocks the effect of decreased (increased) WA in subsequent stages on release decisions in preceding stages. As a result,
The one-stage, two-stage, and three-stage formulations for the six cases indicate that the reservoir can efficiently regulate streamflow from one stage up to the period when the storage capacity constraint is binding. The effects of forecast uncertainty on
The value of ending storage, which represents water saved for periods beyond
In Case 0, the reservoir operation has a one-stage formulation. Release decisions depend on total WA. We have (Appendix)
In Cases 1 and 2, the one-stage formulation still characterizes reservoir operation. In Case 1, maximum demands in periods 1 to
In Cases 3 and 4, the reservoir operation has two stages. Variation in
In Case 5, the reservoir operation has three stages. The binding lower storage capacity constraint blocks the effects of decreased WA in stage 3 on decisions in stages 1 and 2. The binding upper storage capacity constraint blocks the effects of increased WA in stages 2 and
Water is the scarce resource in reservoir operation. Increasing
The one-stage, two-stage, and three-stage formulations provide information for real-time reservoir operation that confronts a long operation horizon (OH) and a limited forecast horizon (FH). For example, the OH of water supply can be one hydrological year, whereas reliable streamflow forecast has an FH of only several days in the rainy season and several weeks in the dry season [
Based on Monte-Carlo experiments with fixed ending storage, Zhao et al. [
This study presented a mathematical analysis of water supply reservoir operation and discussed the effects of three typical constraints of water balance, release capacity, and storage capacity on decision-making in detail. Water balance determines the total water availability, of which the dual value functions as a link in reservoir operation. Upper release capacity constraint hinders release increase caused by increased water availability and leads to spill, whereas lower release capacity hinders release reduction caused by decreased water availability. The effect of storage capacity constraint is more extensive than that of the release capacity constraint affecting single-period decision. The binding upper (lower) storage capacity constraint blocks the effect of decreased (increased) water availability in subsequent stages on release decisions in preceding stages. Analysis of reservoir operation constraints illustrates that multiple-period reservoir operation can be formulated as a one-stage, two-stage, or three-stage model.
Based on the one-stage, two-stage, and three-stage formulations, we explored the effects of forecast uncertainty and ending storage on water supply decisions. In reservoir operation, forecast uncertainty indicates uncertain streamflow within forecast horizon and ending storage deals with unknown streamflow beyond the forecast horizon. The one-stage formulation illustrates that, in cases without binding constraints of storage capacity, current release decision is subject to the effects of both forecast uncertainty and ending storage. In these cases, the reservoir can regulate streamflow in an extended timeframe and water supply decision is affected by streamflow conditions both within and beyond the forecast horizon. By contrast, binding constraints of storage capacity result in multiple stages in reservoir operation. The two-stage and three-stage formulations illustrate that the current release decision is mainly affected by forecast uncertainty in stage 1 because the binding storage capacity constraints block the effect of forecast uncertainty of streamflow in subsequent stages and the effect of ending storage. In these cases, the reservoir can regulate streamflow only in a short timeframe. This study explicitly addressed the effects of constraints in water supply reservoir operation and highlighted the importance of reservoir storage capacity.
In baseline Case 0, no binding constraint of release and storage capacity exists. WA is the determinant of
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors are grateful to the editor and the anonymous reviewers for their helpful suggestions, which have led to major improvements in this paper. This research was supported by the Ministry of Science and Technology of China (Projects nos. 2011BAC09B07 and 2013BAB05B03) and the National Natural Science Foundation of China (Projects nos. 51179085 and U1202232).