## 1. Introduction

[2] An important consideration in hedging operations is how long hedging should be performed for particular operational frameworks. This problem is related to the question of how far into the future forecasting should be performed. In stochastic dynamic models of reservoir operations, decisions in the initial periods are not affected by forecast data beyond a certain point, which is known as the *forecast horizon* (FH), and the number of initial periods defines the *decision horizon* (DH) [*You and Cai*, 2008a]. In a previous study, we presented theoretical analysis applicable to determining the ideal FH for a given DH under the optimality condition of dynamic programming [*You and Cai*, 2008a]. However, from a practical viewpoint, two major concerns have to be addressed. The first is the assumption of a reliable probabilistic forecast of reservoir inflow. Current forecasting technology is unable to provide reliable hydrologic forecasts even for a few months, and a reliable long-range hydrological forecast is nearly impossible [*Somerville*, 1987]. Another concern is the sensitivity of information in the decision-making process. Operational policy in the current period may be affected by information in the distant future but is not necessarily sensitive to it [*You and Cai*, 2008a].

[3] A number of models with limited foresight have been applied to the operation of real-world systems. For example, the rolling-horizon model with limited foresight is a popular tool used in the analysis of dynamic operations, taking into consideration both operational efficiency and practicality [*Chand et al*., 2002]. As shown in Figure 1(a), implementing a rolling horizon begins with the establishment of a -horizon problem according to current conditions, such as initial storage and forecasted inflow, with release decisions made for the current period. The results of decisions made in period one are taken as the initial condition of period two (i.e., the ending storage of period one is the starting storage of period two). Observations and forecasts related to the system are updated up to period two, whereupon the model can be reformulated with an updated horizon , which is not necessarily equal to , as shown in Figure 1(b). This procedure is repeated with updated horizons from one period to the next. FHs that roll over periods are called *rolling horizons* or *study horizons* (SHs), representing a FH that is actually available for use in the analysis of any dynamic system [*Bean et al*., 1987].

[4] The rolling-horizon procedure is practical for reservoir operations, provided that weather forecasts and streamflow predictions are updated from one period to the next. Since December 1994, the Climate Prediction Center of the U.S. National Oceanic and Atmospheric Administration (NOAA) has been issuing forecasts of temperature and precipitation for the following three calendar months. The forecast is updated each month, and the 3 month window is shifted ahead month by month. Reservoir managers approve operational plans at the beginning of each period according to available forecast data. This process is usually performed on a weekly or monthly basis using rolling weather and hydrologic forecasts. A similar process was proposed by *Shiau* [2009], in which hedging for reservoir operations was performed for multiple periods in advance. In that study, hedging is rolled over each period and includes updated inflow information. As long as the forecasts are reliable, this kind of early hedging can be useful for the conservation of water for future use [*Shiau and Lee*, 2005; *Shiau*, 2009]. Nonetheless, selecting the length of the periods is crucial, inevitably involving a trade-off between the reliability of available information and the benefits of hedging.

[5] Given these concerns, FHs based on the horizon theory are probably too distant to be of practical use [*Chand et al*., 2002]. A SH that is shorter than the ideal FH required by optimal decision making may lead to suboptimal decisions due to limited foresight. In this study, however, the authors took a realistic approach to current forecasting technology, in which the adoption of a myopic SH with an acceptable degree of error could be a useful exercise. In other words, it was assumed that distant information could be neglected, if a reasonable level of error were permissible. Thus, rather than trying to determine the ideal FH [*You and Cai*, 2008a], this study adopted a dual approach, determining how far the actual decision is from optimality using a limited but realistic foresight horizon. This is a more practical approach for real-world applications because current weather forecasts and climate predictions do not necessarily extend as far as the “perfect foresight” (FH), which would be required to obtain the optimal solution.

[6] Decisions based on the rolling horizons are often suboptimal due to the current limitations of forecasting. This study defines *error* as the distance between solutions with a limited and the ideal forecast and attempts to identify a realistic FH with an acceptable level of error (i.e., *error bound*) by considering the trade-off among cost, data reliability, and decision accuracy. *Lundin and Morton* [1975] introduced the concept of error/error bound by defining the -optimality of a dynamic decision problem. Despite the wide acceptance of this concept, a common definition with illustrative details is still required. *Morton*'s [1981] first outline of the importance of applying the error bound concept was followed by other studies such as *Kleindorfer and Kunreuther* [1978] and *Sethi and Sorger* [1991]. Nonetheless, the error bound problem has yet to be studied systematically from a theoretical perspective, due to the complexity involved in developing a general analytical framework with which to evaluate rolling-horizon procedures [*Sethi and Sorger*, 1991]. Recently, *Chand et al*. [2002] revisited the error bound problem, which was proposed as a promising research topic deserving further attention.

[7] Although the issue of error resulting from limited foresight is important to reservoir operations, relatively few studies have addressed this problem. *Zhao et al*. [2012] utilized numerical simulation to evaluate release decisions under a limited forecast. This study adopted a different analytical viewpoint to deal with a similar problem; the error bound of rolling-horizon optimality was analyzed for problems associated with reservoir operation based on the concept proposed by *Lundin and Morton* [1975]. The generality of such analysis can provide a more complete understanding of the problem. The authors have modified the original concept of horizon theory presented by *You and Cai* [2008a] and examined the properties of error bound to deal with problems associated with the dynamic operation of reservoirs. This study introduces a method of measuring error and error bound according to terminal stage boundary conditions. Under the assumption of an interior solution, the optimal strategy for operations will not lead to a situation in which the reservoir is empty or full. Thus, operations are affected by forecasting and can be improved by the extension of the FH [*Morel-Seytoux*, 1999; *You and Cai*, 2008a]. This enables the determination of a theoretical rate of convergence related to horizon extension for further application during operations.