Adaptive Reservoir Operation Model Incorporating Nonstationary Inflow Prediction
Publication: Journal of Water Resources Planning and Management
Volume 141, Issue 8
Abstract
Long-term changes in reservoir inflow due to climate change and human interferences have caused doubts on the assumption of hydrologic stationarity in reservoir design and operation. Incorporating uncertain predictions that consider nonstationarity into an adaptive reservoir operation is a promising strategy for handling the challenges that result from nonstationarity. This study proposes rules for multistage optimal hedging operations that incorporate uncertain inflow predictions for large reservoirs with multiyear flow regulation capacities. Three specific rules for determining the optimal numerical solution are derived. A solution algorithm is then developed based on the optimality conditions and the three rules. The optimal hedging rules and the solution algorithm are applied to the Miyun Reservoir in China, which exhibited a statistically significant decline in reservoir inflow trend from 1957 to 2009, to determine an annual operating schedule from 1996 to 2009. Nonstationary inflows are predicted by using an autoregressive integrated moving average (ARIMA) model on a period-by-period basis. The actual operation (AO) of the reservoir is compared with different operational policy scenarios, including a standard operating policy (SOP; matching the current demand as much as possible), a hedging rule (i.e., leaving a certain amount of water for the future to avoid the risk of a large water deficit) with a prediction from ARIMA (HR-1), and a hedging rule with a perfect prediction (HR-0). With a predefined benefit function, the utility of the reservoir operation under HR-1 is 3.7% lower than that under HR-2, but the utility under HR-1 is 3.1% higher than that of AO and 13.7% higher than that of SOP.
Introduction
Hydrologic stationarity is a fundamental assumption of reservoir design and operation. The validity of this assumption is challenged by alterations due to climate change and human interferences (Milly et al. 2008). Statistically speaking, stationarity implies that reservoir inflows have a time-invariant probability density function whose properties can be estimated from historical records. However, hydrological processes are not always stationary. Agricultural and urban development, deforestation, climatic variability, and changes in regional resource management can all alter hydrologic factors over time (Loucks et al. 2005). For most conventional reservoir operation models, including both simulation and optimization models, the assumption of stationarity is made either by directly taking historical inflows as an input or by using synthesized inflow based on historical streamflow statistics (Labadie 2004). Nonstationarity is challenging these conventional models and the operational policies generated from these models; thus, new methods are being developed to handle this challenge (Haguma et al. 2014; Borgomeo et al. 2014).
In particular, interyear reservoir operation may be influenced significantly by the nonstationarity of annual average inflow. The Miyun Reservoir, about 80 km from the urban area in the northern part of the city of Beijing, exemplifies these problems. It was built from 1957 to 1961 and designed mainly for flood control. However, the use of this reservoir has been modified several times due to changes in both water demand and water availability. Initially, operations switched from flood control to irrigation to boost agricultural production, and then switched again from irrigation to urban water supply to satisfy a growing municipal water demand. Today, the Miyun Reservoir is a backup water source for Beijing, which will receive water from the South-to-North Water Transfer Project beginning in 2015. The total annual inflow volume of the Miyun Reservoir has a clear decreasing trend from 1957 to 2009 as shown in Fig. 1 due to climate impact and human activities (Ma et al. 2010). However, since 2003, the Miyun Reservoir has received extra transferred water from other reservoirs to maintain its inflow, thereby reversing the decreasing inflow trend. The Miyun Reservoir was chosen as a case study to demonstrate the optimization of reservoir operation with nonstationary inflows.
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Handling nonstationarity, for which adaptive management with a rolling horizon decision procedure is promising, is still a complex topic. Although adaptive management has been used in multiple natural resources management contexts, such as in the works of Holling and Meffe (1996) and Walters (1997), the current study considers adaptive management based on the definition provided in Brekke et al. (2009). This definition states that decisions are made sequentially over time, and adjustments are made according to new forecast or prediction information. Waage and Kaatz (2011) also proposed a similar method called real options for nonstationary water planning and management. The real options method is defined as adjusting water supply strategies over time with risk consideration to maximize total utility, which implies a rolling horizon decision procedure based on rolling hydrological predications. Methods for predicting streamflow include the time series models (Bras and Rodríguez-Iturbe 1985; Mishra and Desai 2005; Mohammadi et al. 2006), neural network models (Baratti et al. 2003; Coulibaly and Baldwin 2005; Jain and Kumar 2007), and physically based models (Beven 1989; Beven et al. 1984). A time series model, the autoregressive integrated moving average (ARIMA) model, is used in this paper to predict nonstationary inflow because it can represent a trend explicitly.
For adaptive management, incorporating prediction uncertainty into adaptive system operation remains a research challenge, and reflects part of the gap between hydrology and water resources system operations (Rogers 1998). One of the key issues in adaptive management is the manner of approaching prediction uncertainty in optimization models. Stochastic dynamic programming (SDP) is a common approach to deal with such uncertainty. For instance, Vicuna et al. (2010) incorporated climate change into sample SDP (SSDP), and applied the algorithm to three cases studies under climate change. Another effective approach is the hedging rule policy (HRP), which was first introduced to reservoir operation by Masse et al. (1962) and Bower et al. (1962), who pointed out that the HRP can be used to integrate economic principles and hydrological uncertainty. Since then, HRP has been widely discussed and applied in the field of reservoir operation. Hashimoto et al. (1982) optimized reservoir operations with the objective functions of minimizing utility loss or water supply deficit in the future. Draper and Lund (2004) derived the principles of hedging rule as “at optimality, the marginal benefit of storage must equal the marginal benefits of release.” You and Cai (2008a) developed a conceptual two-stage model that explicitly includes future inflow and its uncertainty. The principles of the hedging rule are described as “when the two periods have the same marginal utility, the total utility from two periods reaches the optimal value” (You and Cai 2008a). Applying Karush-Kuhn-Tucker (KKT) conditions to a typical two-stage reservoir operation model, Zhao et al. (2011) explicitly merged hydrological forecast/prediction and its uncertainty (i.e., forecast/prediction variance) into a two-stage stochastic model. The concept of risk-adjusted marginal benefit (RAMB) was developed by Zhao et al. (2011) to interpret the economic implications of the HRP in a stochastic model, and to represent the economic value of predictions for reservoir operation. With an additional term that contains the third-order derivative of utility function and prediction uncertainty, RAMB presents the optimal risk defense under uncertain prediction, which is used in developing operation rules in this paper.
Although the two-stage HRP model is theoretically important, in practice, a multistage optimization model that can incorporate prediction uncertainty is required for adaptive reservoir operation because the decisions of the current stage should be made by consideration of certain future stages. This paper extends the study of Zhao et al. (2011) from a two-stage to a multistage reservoir operation problem with uncertain predictions, and develops an algorithm to solve the multistage model based on the optimality conditions derived from the KKT conditions of a nonlinear optimization model. More importantly, a specific prediction updated after each time period is coupled with the reservoir operation model in a rolling time framework, and provides prediction as an input for the reservoir model. The model is applied to the Miyun Reservoir to demonstrate its effectiveness.
The rest of the paper is organized as follows. The “Methodology” section develops a theoretical framework for multistage hedging rule, and a theory-based algorithm is introduced to incorporate uncertain hydrological predictions. Using the algorithm, a rolling operation framework involving nonstationary inflow prediction is described. Then, the case study results of the Miyun Reservoir are presented. Subsequently, the impacts of nonstationary terms, prediction variances, and discount rates are discussed. Finally, conclusions are given.
Methodology
Multi-Stage Framework
Considering a prediction and its uncertainty, the actual inflow of a stage can be expressed as (Vogel 1999)where = prediction value; and = prediction error that follows a normal distribution.
(1)
Following the two-stage model presented by Zhao et al. (2011), a multistage model with prediction and its uncertainty for water supply can be formulated as follows:where for the th stage, = delivery for water use; = expected utility function value of the delivery , which is usually assumed to be concave for water supply; = spill; = water storage at the end of the th stage; = initial water storage at the beginning of the first stage; = ending water storage at the end of the last stage; and and = maximum and minimum storage capacities, respectively. To focus on the main modeling issues, this study assumes that evaporation and seepage losses from the reservoir are sufficiently small to be negligible.
(2)
Under similar assumptions and derivation process by Zhao et al. (2011) in the two-stage problem, the KKT conditions for Eq. (2) arewhere stands for the expected solution based on prediction, is the expected water delivery, and is the expected ending storage. are the marginal values of mass balance, nonnegative release, and the minimum and maximum reservoir capacity constraints, respectively. Considering the impact of prediction error, the actual release is set as and the actual ending storage is (except for the special case of setting when ), which means the storage is treated as an operational target (Hsu and Wei 2007; Karamouz et al. 2005; Turgeon 2007). , , , and represent the mass balance constraints, nonnegative release constraints, as well as minimum and maximum reservoir capacity constraints, respectively; (Zhao et al. 2011).
(3)
The KKT conditions can be stated as follows: if there are constants and a feasible solution satisfying Eq. (3), then is the global optimal solution of the uncertain model represented by Eq. (2).
With Eq. (3), one can extend the optimal principle from the two-stage problem to the multistage problem, i.e., make the RAMBs of all stages as close to each other as possible while satisfying all constraints. This principle provides important rules for designing an algorithm to solve a multistage reservoir operation problem. A general relationship between the RAMBs of two adjacent stages is derived according to Eqs. (3) as follows:
(4)
Rule 1: The ideal situation is that the RAMB of all stages are identical. If only mass balance constraints are binding and all nonnegative and storage constraints are nonbinding ( and ), an optimal solution can then be obtained when the RAMBs of all stages are identical.
Rule 2: A binding nonnegative constraint leads to zero release for the corresponding stages. In three adjacent stages, namely the ()th, th, and ()th stages, if a nonnegative constraint is only binding in the th stage but storage constraints are nonbinding, then and . The th stage with a binding nonnegative constraint has zero release (), whereas other stages with nonbinding nonnegative constraints [()th and ()th stages] have the same RAMB, which is larger than the th stage at optimality.
Rule 3: A binding storage constraint results in change points of RAMBs. If the maximum capacity constraint is binding at one stage but nonnegative and minimum storage constraints are nonbinding ( and ), the RAMB of that stage must be lower than that of the next stage, resulting in a change point of RAMBs. By contrast, a binding minimum storage constraint ( and ) leads to higher RAMBs in the earlier stages.
The relationships between the RAMBs of two adjacent stages are summarized in Table 1.
| Conditions | ||||
|---|---|---|---|---|
| Undetermined | Undetermined | |||
| Undetermined | Undetermined | |||
| Undetermined |
Optimization Algorithm Design
Following the three rules derived in the preceding section, a hedging rule-based algorithm (HRBA) for finding an optimal solution for a multistage reservoir operation problem can be designed. The guiding principle is to make the RAMBs of all stages as close to each other as possible while satisfying all constraints. The procedure includes the following steps:
1.
An ideal solution (solution_0) with identical MBs or RAMBs is sought. This step aims to find a solution following Rule 1. Considering only mass balance constraints, a solution that makes the RAMBs of all stages (from the 1 st stage to the th stage) identical can be obtained using a simple Newton iteration method (Zhao et al. 2011).
2.
The nonnegative constraints of solution_0 are checked. This step follows Rule 2 and one of the following two actions is followed:
a.
If solution_0 satisfies all nonnegative release constraints, solution_0 is renamed as solution_1.
b.
If solution_0 violates nonnegative release constraint(s), then the stage(s) with negative release(s) is (are) reset to zero release(s) and the model is resolved according to the mass balance, i.e., the RAMBs of all the stages are rebalanced except for the stages fixed to zero releases. The new solution is defined as solution_1.
3.
The storage constraints of solution_1, including the maximum and minimum storage constraints, are checked. Rule 3 is applied in this step and one of the following two actions is followed:
a.
If solution_1 satisfies all storage constraints, then it is the best solution in this multistage problem.
b.
If solution_1 violates storage constraint(s), then one chooses the stage with the largest violation and set its storage value to the feasible boundary (i.e., minimum or maximum storage). In other words, if the storage value of the th stage is the maximum value in solution_1 [] but larger than the maximum capacity (), then the storage value is set to capacity (). On the other hand, if it is the minimum in solution_1 [] but lower than the minimum capacity (), the minimum capacity () can be set. In this case, the whole multistage problem is divided into two subproblems with fewer stages. One is the multistage problem from the 1 st to th stages, and the other is the problem from the ()th to th stage. In both subproblems, the initial and ending storage values are known. Hence, for both subproblems, one can loop from Steps (1) to (3) until a final solution is found. The dividing process is shown in Fig. 2.
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Eventually, a solution that satisfies all constraints is obtained. This algorithm results in an optimal solution that satisfies the KKT conditions with the assumption of a convex objective function and a convex solution space. One can conduct a simple test on the efficiency of HRBA by comparing it with traditional dynamic programming (DP). The comparison is made with a deterministic case because the two approaches handle prediction errors differently under uncertain conditions. In this test, the Thomas–Fiering model is used (Thomas and Fiering 1962) to generate synthetic streamflow for 100 stages. The mean of inflow is set as unity, as 0.3, and correlation as 0.4. A cubic function () is set as the hypothetical utility function of reservoir water supply, which satisfies the characteristics of the utility function of water supply (You and Cai 2008a, b). The reservoir capacity is varied from 1 to 5 to illustrate its impact on the HRBA, as the HRBA invokes its subprocesses when is binding. To compare the efficiency in different hydrological scenarios, 100 synthetic streamflow series are employed for each algorithm. Table 2 shows the total computational time for each algorithm to obtain optimal solutions for the 100 scenarios.
| Algorithm | Discretization (n) | Total computational time (s) | ||||
|---|---|---|---|---|---|---|
| HRBA | 100 | 4.02 | 2.15 | 1.93 | 1.68 | 1.47 |
| 1,000 | 18.62 | 14.14 | 10.82 | 8.99 | 7.16 | |
| DP | 100 | 20.02 | 19.67 | 19.57 | 19.44 | 19.25 |
| 1,000 | 1,968.69 | 1,944.74 | 1,923.76 | 1,910.66 | 1,902.61 | |
The efficiency of the HRBA is higher than that of DP in terms of computational time. With an increased discretization number , the computational time of HRBA exhibits a linear increase [], whereas DP shows a quadratic increase []. When considering the impact of the storage capacity , DP has the same efficiency regardless of the value of . On the other hand, the HRBA spends more computational time if is small because capacity constraints are frequently binding. Every time the capacity constraints are binding, the HRBA divides the problem into subproblems and recalculates according to Step (3) of the algorithm.
Under uncertain conditions, when the forecast/prediction variance is included, one only needs to substitute the MB term with the RAMB term in the HRBA in order to have the same level of computational efficiency. Compared with the increasing computational cost from DP to stochastic DP, the HRBA is theoretically more efficient under uncertainty conditions.
Dynamic Operation Framework Incorporating Nonstationary Inflow Prediction
Based on the multistage HRBA, a dynamic reservoir operation framework is designed with hydrologic predictions updated periodically with a rolling horizon decision procedure (RHDP). In the RHDP, a hydrologic prediction model is coupled with the reservoir operation model and the operating decisions are updated periodically with the updated prediction. The details of the framework are illustrated with the Miyun Reservoir case in this section.
To consider nonstationary annual inflows, the authors chose an ARIMA model to predict the inflow for future years. The ARIMA model was first introduced by Box et al. (1976). Based on autocorrelation function (ACF), partial ACF, and manual modification, a practical ARIMA (4, 1, 0) model is specified as a prediction model of the RHDP framework, where 1 is chosen to reflect a linear change. The (4, 1, 0) structure has the lowest Akaike Information Criterion (AIC) compared to other ARIMA (, 1, ) model in which p and q range from 0 to 5.
As mentioned previously, extra water has been artificially transferred to the Miyun Reservoir to maintain inflow since 2003 (Fig. 1). The constant term () in the ARIMA model, which represents an average periodical change, is used to represent the differences caused by this event. This study included this constant term in the ARIMA model before 2003 to represent the decreasing trend of inflow, and removed it from the ARIMA model after 2003 to represent an artificial stationary process.
In the RHDP framework, decisions are made based on the latest prediction, and both prediction and decision are updated periodically. This paper attempts to optimize the operations of the Miyun Reservoir from 1996 to 2009 on an annual basis assuming that the prediction is available. The actual initial storage at the beginning of 1996 () and the actual ending storage at the end of 2009 () are employed as the boundary conditions to guarantee the same total available water for all policies; thus, other policies can be compared with AO, which is actually implemented in practice. The rolling horizon decision procedure is demonstrated in Fig. 3.
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At the beginning of 1996 [Fig. 3(a)], the first year, only the historical inflow data from 1957 to 1995 are known, which are used as the input data in the prediction (ARIMA) model, to generate the inflow from 1997 to 2009. In this case, the authors assume that the prediction for the year 1996 is perfect (or the error is negligibly small) because the actual inflow from the current year can be observed before the decision for the year 1996 is completed. Thus, considering the prediction result and its uncertainty as inputs can enable decision making using the HRBA by determining how much water should be released in 1996, and the ending storage of that year. Moving to the next year, 1997, the actual inflow of 1996 is added into the input data of the ARIMA model and the determined ending storage of 1996 is set as the initial storage of 1997. Similarly, at the beginning of 1997 [Fig. 3(b)], the known actual inflow data from 1957 to 1996 are inputted into the prediction model to predict the inflows from 1998 to 2009, and the release and ending storages of 1997 can then be determined using the HR algorithm.
The framework can deal with the nonstationarity of inflow if a proper nonstationary prediction model is chosen. This approach has an advantage over most traditional methods that assume hydrologic stationarity, especially for reservoirs with significant inflow changes. Thus, the RHDP framework reflects the process of adaptive management based on new observed information, which is important for system operation under a changing environment.
Results of the Miyun Reservoir Case
Operation Policy Specification
Water from the Miyun Reservoir is delivered to the north part of Beijing for household, commerce, industry, and environmental uses. Such objectives have a decreasing order of priority because their marginal utilities are generally recognized to differ. For example, industrial and environmental water uses are reduced during drought years, especially the water supply for the artificial lakes and river channels in the city, but household water use is usually guaranteed. To evaluate the effects of different policies, this study fits a cubic utility function for the Miyun Reservoir to the actual water supply data of Beijing, whose maximum value at a delivery volume of matches the real total water demand of the target areawhere = water delivery (ranging from 0 to a maximum demand of ); and = discount rate. The minimum and maximum capacities are and , respectively.
(5)
To illustrate the results of different operation rules under nonstationarity, the following policies are defined for comparison: (1) standard operation policy (SOP; matching the current demand as much as possible); (2) RHDP-based hedging algorithm (i.e., leaving a certain amount of water for future use to avoid the risk of a large water deficit) with prediction from ARIMA (HR-1); (3) hedging with perfect prediction (HR-0); and (4) AO. The focus of this case study is water supply operation; thus, flood operation is not discussed in this study. Flood operation can be ignored because flooding has not happened since 1995, which is attributed to the decrease in runoff induced by both climate change and human activities (Ma et al. 2010). In practice, considering the large storage capacity () and the small annual inflow in recent years (less than ), inflow during the flood season has been stored in the reservoir for water supply since then.
Result Analysis
To compare different policies, this study sets the discount rate to zero, which implies that all utility functions in different stages are identical (the impacts of are discussed later). The total benefits of HR-1, HR-0, AO, and SOP are 87.6, 90.9, 85.0, and 77.0, respectively.
The total benefit of HR-1 (87.6) is 3.7% lower than that of HR-0 (90.9). HR-0 is based on the perfect prediction of future inflow and can thus provide the best solution with the maximum total benefit compared with all other policies. Although perfect prediction does not exist in the real world, HR-0 provides an upper bound for all operation results and shows the potential maximum benefit of improving the prediction accuracy of HR-1.
The total benefit of HR-1 (87.6) is 3.1% higher than that of AO (85.0) due to two potential reasons. First, the ARIMA model with a constant term in HR-1 can capture the decreasing inflow trend. Thus, more water is reserved for future use according to the principle of balancing RAMBs, resulting in higher overall utility. Second, the RHDP framework optimizes the annual release solution based on the imperfect prediction, which is better than the empirical judgment of reservoir operators in AO.
The total benefit of SOP (77.0) is the worst among the four policies. Under SOP, water is always released to satisfy the current demand as much as possible without any concern about future inflow. The utility is 10.3% higher under AO than SOP, which implies that a certain level of hedging with empirical knowledge of future inflows is actually being adopted in the real-world operation of the system.
The results of the RHDP contain some interesting features with the releases and storage time series under the various operation policies (Fig. 4). First, in the case of HR-0 with perfect prediction and zero discount rate, the optimal solution is to evenly allocate the total water volume to each year, which makes the marginal benefits of all stages identical and yields the maximum total benefit. This statement is inferred from the straight line in Fig. 4(a). HR-0 is the optimized policy based on perfect prediction; thus, it is the best among all possible policies. In reality, however, perfect prediction is hardly achieved. A release policy closer to HR-0 can produce an improved result.
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Second, an SOP releases as much water as possible from 1996 to 2002. However, after the dead storage volume is reached in 2003, the water releases are the same as inflows until 2007. Part of the inflow in 2008 and all inflows in 2009 are then stored to achieve the prescribed final storage in 2009. The SOP policy employed in this study is to simply deliver as much water as possible to satisfy the demand of the current stage, which is likely to empty the reservoir when inflow is low. Meanwhile, an improved SOP in practice should prevent emptying the reservoir under normal conditions.
Third, a comparison between the releases of AO and SOP verifies that AO does not completely follow an SOP. Rather, it contains an implicit hedging component that stores some water for future use, demonstrating that the operators make an informal decision to limit releases in anticipation of decreased future inflow. With this anticipation, some industrial and environmental water uses are continuously reduced in practice. Although the empirical anticipation may not be sufficiently viable as a prediction model, it raises the utility of AO 10.3% higher than SOP. However, in the early stages, the degree of hedging is not as great as it is under HR-0.
The release of HR-1 fluctuates around HR-0, showing its capacity to change releases dynamically and adaptively. In 1996, the predicted average future annual inflow exceeds the actual average future annual inflow, but the release in 1996 is even smaller than the optimal release under HR-0. This finding shows that the consideration of prediction error can lead to a conservative policy under hedging. Before 2000, the releases under HR-1 are larger than the ideal values under HR-0 due to optimistic predictions of inflow. On the other hand, after 2000, pessimistic predictions and risk aversion lead to excessive storage. Under HR-1, water releases are smaller than the ideal values under HR-0, and the reservoir storage increases. After 2005, water releases continue to increase and exceed the ideal value under HR-0 because the reservoir storage needs to fall to the 2009 boundary condition after experiencing small water releases during 2000–2005.
Discussion
Importance of Nonstationarity and Prediction Variance
In the ARIMA (4, 1, 0) model, the constant term represents the changing inflow trend. Alho and Spencer (2005) suggested to “examine the effect of including a constant, to see how it changes the forecast function” because “the decision to include or not to include the constant can be the single most important aspect of the eventual forecast.” In this paper, the ARIMA (4, 1, 0) model with a constant term provides a prediction with a continuing decreasing trend, which can be considered as a nonstationary prediction under changing hydrological conditions. On the other hand, the ARIMA (4, 1, 0) model without a constant term provides a prediction that maintains current inflow characteristics, which can be regarded as a stationary prediction based on current hydrological conditions.
To demonstrate the impacts of nonstationarity and prediction variance, three comparable HR policies are defined based on HR-1 in Table 3. For HR-1 and HR-1NV, the ARIMA (4, 1, 0) model with a constant term is applied before 2003. For HR-2 and HR-2NV, the ARIMA (4, 1, 0) model without a constant term is employed to represent a stationary prediction. Here NV means that the prediction variance is not used in the decision process.
| Policy | Constant term | Prediction variance | |
|---|---|---|---|
| Before 2003 | After 2003 | ||
| HR-1 | Used | ||
| HR-1NV | Not used | ||
| HR-2 | Used | ||
| HR-2NV | Not used | ||
The release processes are shown in Fig. 5. The total benefits of HR-1, HR-1NV, HR-2, and HR-2NV are 87.6, 87.4, 87.3, and 83.4, respectively. Compared with the other three polices, HR-1 considers both the nonstationarity (i.e., decreasing trend of inflow) and prediction variance; thus, it delivers less water in earlier stages to prepare fully for future water deficits and maintain a high level of reservoir storage. HR-1 is the best in terms of total benefits and takes full advantage of nonstationary prediction and prediction variance. The total benefit of HR-1 (87.6) is 0.25% larger than HR-1NV (87.4) and 0.32% larger than HR-2 (87.3), showing slight improvement when both nonstationarity and prediction variance are included. The comparison between HR-1NV and HR-2 shows that the consideration of variance or nonstationarity can lead to similar release processes and total benefits in the Miyun Reservoir case. This result can be attributed to the consideration of either the continuing decreasing trend or prediction error, leading to conservative decisions to reserve for future water deficits. Hence, either HR-1NV or HR-2 already stores some water for future use and combining these two factors (HR-1) only yields slight improvement. HR-2NV considers neither nonstationarity nor prediction variance, so HR-2NV delivers more water than the other three policies in earlier stages (before 2003) with optimistic and overconfidently used predictions, which results in the worst total benefit (83.4, 4.8% less than HR-1). The comparison shows that simultaneously ignoring the decreasing trend of inflow and prediction variance leads to risky policies due to the potential loss of future water deficits.
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The impacts of nonstationary term and prediction variance are theoretically important but not necessarily very significant for every case. Some factors such as the structure of prediction model and formulation of utility function affect their impacts. Although the improvement in HR-1 compared with HR-1NV and HR-2 are relatively small, they can be large in other cases. The RHDP framework considering prediction variance provides the expected best policy in terms of long-run operation. However, this policy cannot be guaranteed to be the best every time.
Impacts of Discount Rate
The effect of the discount rate on water resource management has been widely discussed for different goals, such as determining reservoir capacity (Vicuna et al. 2010), water release policy (Vicuna 2007; You and Cai 2008b), forecast horizon (Zhao et al. 2012), and floodplain protection decisions (Zhu et al. 2007). In reservoir operation, a higher discount rate reduces the value of the carryover storage and represents a more myopic reservoir operation (Vicuna 2007). Apart from indicating the time value of water releases at different time points, the discount rate can also reflect storage loss (e.g., evaporation loss) (You and Cai 2008b).
To analyze the impacts of the discount rate, this study modified HR-1 with two different discount rates: (HR-R5) and (HR-R10). According to Eq. (5), the utility functions differ for each stage under HR-R5 and HR-R10. The water releases and storages of HR-1, HR-R5, HR-R10, and AO are compared. As shown in Fig. 6, the release patterns of different scenarios remain similar. With a higher discount rate, more water is released in early stages (1996 to 2004) and less in later stages (2005 to 2009), because a higher discount rate causes a higher value of water use in early years. In practice, the discount rate is an important factor affecting reservoir operation, as it directly influences releases and storages (Fig. 6). This point increases rationality in the AO policy employed by the operators of the Miyun Reservoir. The maximum storage is also very close to the maximum capacity in HR-1. When the discount rate is relatively low, the reservoir needs to be larger to store more water for future use. If the discount rate is sufficiently high, the release policy tends to be SOP regardless of stationary future inflow or nonstationary inflow (Vicuna 2007) and forecast is less important (Zhao et al. 2012).
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To demonstrate the impacts of the discount rate on total benefits, the ratio is defined aswhere = total benefits of HRs (HR-1, HR-R5, and HR-R10) calculated with different discount rates (, , and ), which are then compared with the total benefits of AO [] with a corresponding discount rate. HR-1, HR-R5, and HR-R10 result in an of 1.031, 1.014, and 1.005, respectively, which indicate that an increase in discount rate brings the hedging rule closer to AO. HR-R10 and AO confer nearly the same total benefit because they both chose to release additional water during the early stages.
(6)
Combined Effects of Discount Rate and Prediction Variance
As discussed previously, the discount rate and prediction variance are two important factors that exert opposite effects on the water release policy. An increased discount rate leads to greater water releases for current use, whereas an increased prediction variance results in more water storage to buffer against future drought risk. The combined effects of these two factors are represented analytically using the risk term in RAMB.
In the Miyun case, the can be represented with the following function:where is the third-order coefficient of a cubic function. Thus, the risk term is clearly influenced by both the discount rate and prediction variance. With the fixed total water availability, changes in the risk term affect the water allocation at different stages. If , i.e., , the prediction variance dominates and more water is stored for future stages compared with the deterministic case. This phenomenon occurs when prediction variances increase rapidly over time and the discount rate is relatively small. If , i.e., , the discount rate dominates and more water should be released in near-term stages. This phenomenon occurs when the prediction variances increase slowly over time and the discount rate is relatively large.
(7)
The risk term in the hedging rule is characterized by both the hydrologic risk (prediction uncertainty) and an economic factor (discount rate). According to the above comparison, risk aversion does not always lead to the conservation of more water for future use. In fact, under some situations, more water is released in the current stage due to a large benefit discount over time. Given that the traditional interpretation of HR considers the hydrologic risk (i.e., reserving water for future shortage) but not the discount rate, this conclusion can provide a more comprehensive insight into HR. If the third-order coefficient in Eq. (7) is lower, the effect of prediction variance is reduced. In particular, if (linear or quadratic function), the prediction variance has no impact on the water release policy.
Conclusion
The emergence of hydrological nonstationarity has generated unavoidable challenges to reservoir operations. Moreover, it requires hydrologic prediction as well as hedging policies to be coupled to support adaptive operations. This study presents a multistage hedging rule framework for the interannual operation of large reservoirs and merges hydrologic prediction and its uncertainty with a rolling optimization process. An algorithm based on the conditions of general nonlinear optimization is developed to solve the model. Using up-to-date predictions based on historical and observed data, the RHDP framework is similar to that of the actual operation; thus, it can be applied to adaptive operation in practice.
The Miyun Reservoir case yields useful insights into the operational policies of reservoirs with nonstationary inflow, which is practically important for adapting to climate change. In a nonstationary world with a decreasing inflow trend, using nonstationary prediction and considering prediction variance lead to the reservation policy for future water use, which improves total benefits.
Finally, this study provides a specific application of the derived framework. Other promising applications include long-term planning of water resources under different climate projections, for which policies can be optimized according to the HR framework by considering economic utilities and projection uncertainties. Given the complexity of real-world operations, numerous technical issues still need to be addressed, such as predicting the horizon and determining the ending storage of the last stage ().
Acknowledgments
This research was supported by the Ministry of Science and Technology of China (Grant Nos. 2013BAB05B03 and 2011BAC09B07) and the National Natural Science Foundation of China (Grant Nos. 51179085 and 91125018).
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Journal of Water Resources Planning and Management
Volume 141 • Issue 8 • August 2015
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This work is made available under the terms of the Creative Commons Attribution 4.0 International license, http://creativecommons.org/licenses/by/4.0/.
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Received: Mar 7, 2014
Accepted: Nov 20, 2014
Published online: Dec 31, 2014
Discussion open until: May 31, 2015
Published in print: Aug 1, 2015
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