Potential Benefits of Seasonal Inflow Prediction Uncertainty for Reservoir Release Decisions

Transcription

1 MAY 2008 G E ORGAKAKOS AND GRAHAM 1297 Potential Benefits of Seasonal Inflow Prediction Uncertainty for Reservoir Release Decisions KONSTANTINE P. GEORGAKAKOS* AND NICHOLAS E. GRAHAM* Hydrologic Research Center, San Diego, California (Manuscript received 19 December 2006, in final form 21 August 2007) ABSTRACT This paper examines the conditions for which beneficial use of forecast uncertainty may be made for improved reservoir release decisions. It highlights the parametric dependencies of the effects of uncertainty in seasonal inflow volumes on the optimal release and objective function of a single reservoir operated to meet a single volume target at the end of the season under volume and release constraints. The duration of the season may be one or several months long. The analysis invokes the application of Kuhn Tucker theory, and it shows that the presence of uncertainty introduces complex dependence of the optimal release and objective function on the reservoir parameters and uncertain inflow forcing. The seasonal inflow volume uncertainty is represented by a bounded symmetric beta distribution with a given mean, which is considered to be unbiased, and a half-range Q R. The authors find that the use of predicted inflow uncertainty is particularly beneficial during operation with a volume target that is either near reservoir capacity or near zero reservoir volume, with the optimal release being directly dependent on Q R in these situations. This positive finding is moderated by the additional finding that errors in the estimation of predicted Q R can result in significant operation losses (larger deviations from the target volume) that are due to suboptimal release decisions. Furthermore, the presence of binding release constraints leads to loss of optimal release and objective function benefits due to the seasonal inflow uncertainty predictions, suggesting less rigid release policies for improved operations under uncertain forecasts. It is also shown that the reservoir capacity values for which optimal reservoir operations are most sensitive to seasonal inflow uncertainty predictions are found to be at most 5 times the uncertainty range of the predicted seasonal inflow volume and to be at least as large as the uncertainty range of predicted inflow volumes. Suggestions for continued research in this area are offered. 1. Introduction a. Motivation It is commonly stated that even though seasonal climate predictions have skill and there is good potential identified for their application to the improvement of management of climate-sensitive sectors of the economy there is no widespread routine use of such forecasts for increasing management benefits (e.g., Weiss 1982; Greis 1982; Livezey 1990; Pulwarty and Redmond 1997; Lemos et al. 2002; Goddard et al. 2001; * Additional affiliation: Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California. Corresponding author address: K. P. Georgakakos, High Bluff Dr., Suite 250, San Diego, CA kgeorgakakos@hrc-lab.org Chiew et al. 2003; Garbrecht et al. 2005). Among the often-stated reasons for this unexpected status quo are the low skill of the predictions and the lack of generalizable methods or procedures for relating the uncertain forecasts with the decision variables of the management problem. For water resources management accomplished through the control of reservoir facilities, an important regional economic management (and planning) issue and the focus of this paper, the difficulty in developing generalizable methods may be further traced to the large uncertainties associated with predicted seasonal upstream inflow volumes and the case-by-case nature of the successful management applications. Be that as it may, the use of seasonal climate prediction information has indeed been proven to be useful for operational reservoir management in a number of recent applications (e.g., A. P. Georgakakos et al. 1998; K. P. Georgakakos et al. 1998, 2005; Graham et al. 2006), and in DOI: /2007JAMC American Meteorological Society

2 1298 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 addition there is evidence (e.g., Yao and Georgakakos 2001) that supports the assertion that using seasonal climate forecasts with adaptive management schemes may even be a means for mitigating adverse regional effects of projected climatic changes. Furthermore, there are significant recent national and international studies that advocate the improvement of reservoir operations on a global scale (Dynesius and Nilsson 1994; Graf 1999; World Commission on Dams 2000) and the development of approaches for the improved incorporation of forecast uncertainty in decision making (National Research Council 2006). It is thus desired to develop general methods for screening candidate application sites for which seasonal climate predictions may be expected to improve reservoir management operations. This paper represents a first step toward addressing this goal. b. Purpose The primary purpose of this paper is to develop objective methods for identifying the conditions among forecast uncertainty characteristics and reservoir parameters for which the use of uncertain seasonal climate predictions is expected to be beneficial for reservoir management. A secondary purpose of the paper is to gain insight into the behavior of the management objective values and attendant release decisions through sensitivity analyses. The target of analysis is an idealized single reservoir facility operated for a single purpose or for multiple purposes, which may be embodied in a single objective of meeting a desired volume target at the end of the season under consideration, under volume and release constraints. c. Approach The problem formulation represents a certain class of reservoir management problems that allows the development of volume targets (at the end of the season of interest) that can be functionally related to seasonal benefits from one or more of the following: water conservation, recreation, flood control, and hydroelectric energy production. These relationships may be obtained with a number of estimation methods that are available for the development of rule curves of this type. Examples of simulation or optimization methods may be found in the water resources research literature (e.g., Karamouz and Houck 1987; Oliveira and Loucks 1997; Peng and Buras 2000; Basson and Van Rooyen 2001). In this context, a rule curve specifies the most desirable (target) reservoir storage volume at the end of a given season given an initial storage volume and possible constraints on the reservoir release. The economic and other benefits that are associated with the rule curve as a result of the application of the estimation methods are used to select the most desirable rule curve. To arrive at quantitative relationships, we first cast the problem as a mathematical programming problem with uncertain inflows, and then we solve it using analytical methods tailored to the mathematical formulation. For apparently the first time, the analytical solutions are obtained as functions of the reservoir parameters and of the uncertainty characteristics of the seasonal inflow volume predictions. The emphasis in the development is on analytical solution methods because they preserve the parametric dependence of the solution without the need for specifying parameter values, thus allowing generalization. It is emphasized at the outset that the mathematical programming problem that is formulated and solved is a means to obtain relationships between uncertainty characteristics and reservoir parameters for optimal solutions to a fixed objective, and not to dictate operating policies for particular reservoirs that are typically dictated by comprehensive negotiations and agreements and with much more comprehensive simulation/optimization models (e.g., Oliveira and Loucks 1997). As such, this paper contributes an analytical screening procedure for identifying situations for which the uncertainty in seasonal climate predictions is expected to be most useful for improving reservoir management. In the identified cases, the investment in more detailed studies that are tailored to the specific operating conditions is warranted for developing templates for improved reservoir management under uncertain seasonal inflow predictions. Such studies should consider, among other issues, hourly-to-daily-to-seasonal operation (e.g., Yao and Georgakakos 2001) and the economic objectives in the form of trade-off functions (e.g., Georgakakos 1993). d. Outline The next section presents the mathematical formulation of the nonlinear optimization problem (its objective and constraints) and discusses the assumptions made to allow the development of tractable analytical solutions. The solutions are derived in section 3 as functions of the uncertainty characteristics of the predicted inflow and the reservoir structure and operation parameters. The appropriate section of the parameter space for which each solution is optimal is also derived in section 3 as a function of the inflow uncertainty and reservoir parameters. Section 4 uses the mathematical solutions to assert the importance of the inflow volume uncertainty in each solution case and for the seasonal reservoir management decision problem as a whole.

3 MAY 2008 G E ORGAKAKOS AND GRAHAM 1299 Sensitivity analysis is employed in that section to identify regions in parameter space for which the optimal solutions are particularly sensitive to the inflow uncertainty. Last, relationships between the capacity of the reservoir active storage and inflow characteristics are derived for cases in which the prediction uncertainty estimates are expected to be particularly beneficial for reservoir management. Incidentally and throughout the discussion section 4, we also address the impact of misspecification of seasonal inflow prediction uncertainty on reservoir release and operational management objective. Concluding remarks and suggested future analyses are presented in section 5, and an appendix with notation is included. 2. Mathematical formulation a. Objective Consider a single reservoir of capacity C and initial storage volume V I at time t. At this time t, a noisy but unbiased prediction of the reservoir seasonal inflow volume Q I for the management (or planning) time horizon (t, t T] is available and may be characterized by a given probability distribution defined in a bounded domain of the inflow variable (e.g., a uniform distribution with mean volume Q m and half-range volume Q R ). The objective of reservoir operation during the interval (t, t T] (which does not include present time t) isto achieve the target storage volume V* att T, subject to 1) the reservoir storage V T at t T being bounded by reservoir capacity and zero (dead storage) and 2) the release volume R T in (t, t T] being between two prespecified bounds R TL (lower) and R TU (upper). Here, T signifies the duration of the season, which in this work may be one several months long. The decision to release a certain volume R T in the interval (t, t T] to meet target V* is taken at time t when the uncertain seasonal inflow volume prediction is available and prior to observing the actual seasonal inflow volume in (t, t T]. Of particular interest for this analysis is the effect of the inflow prediction uncertainty on the controlled release and on the degree to which the resultant storage volume at time t T will meet the target V*. The latter concern is tantamount to the estimation of the expected economic losses (e.g., due to unmet water supply or unrealized downstream flow augmentation). b. Assumptions The analysis concerns a single-season release decision to meet a certain desired (target) reservoir volume at the end of the season. As mentioned earlier, the emphasis in casting this problem as a mathematical programming problem is not to develop specific release policies but instead to derive analytical expressions for the dependence of optimal solutions on problem parameters for determining the sensitivity of optimal policies to inflow uncertainty. The former development would require the evaluation of performance of the decision sequence that would necessarily involve observed reservoir inflow data and multiple seasons of operation and would render the results specific to the case study (e.g., Yao and Georgakakos 2001; Graham et al. 2006). The second development, followed in this work, is a general sensitivity problem that does not require actual data, and we can proceed with a singleseason decision formulation accounting for various initial reservoir volumes at the beginning of the decision season. The analysis concerns a single multiobjective reservoir, but it is also applicable to cases with a main large reservoir that is downstream of small upstream reservoirs when these only influence the target end-of-season volume V* of the large downstream reservoir. In our recommendations for further research, we address the issue of several comparable reservoirs in series and in parallel by discussing possible extensions of our approach for these cases. To facilitate the development of a reduced parametric space for the optimal solutions, simple prespecified upper or lower bounds are imposed on the seasonal controlled release volume. This assumes that all the reservoir releases are controllable and that the restrictions in seasonal release volumes (e.g., for downstream flooding concerns or minimum flow requirements) may be simply represented by prespecified constants, which may be specific for the season of operation we consider. Our sensitivity analysis of the problem parameters also allows the consideration of any uncertainties in the release bounds and the volume target, as long as such uncertainties are independent of the dominant reservoir inflow prediction uncertainty (as expected in most cases). For this analysis, any reservoir evaporation loss during (t, t T] is considered to be a function of the initial reservoir surface area, which in turn is a monotonic function of the reservoir volume at time t, and is incorporated in V I. This tacitly assumes that reservoir level changes are relatively small in the interval (t, t T] and that the potential evaporation forcing carries low uncertainty relative to the seasonal inflow volume prediction (which includes the more significant upstream basin evapotranspiration and its uncertainty) so that reservoir evaporation volume uncertainty contributes little to the analytical results of this work. The reservoir in-

4 1300 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 1. Symmetric beta distributions defined in the interval [0, 1] for various values of c. The value of c 1 yields the uniform distribution. flow volume is balanced by the controlled release volume R T and the change in storage volume (V T V I ). We characterize the uncertainty in the seasonal inflow volume predictions by a beta probability distribution with the seasonal inflow volume rescaled from an interval with certain lower and upper bounds to the interval [0, 1]. The bounds of the predicted inflow may be season dependent, and they are imposed as anticipated natural limits to possible predicted seasonal inflow volumes at the initial time t. Such bounds may also be specified from an ensemble forecast of inflow volumes as quantiles of the sample distribution (e.g., 1st and 99th quantiles). The beta distribution (defined in the interval [0, 1]) has two parameters that are associated with the moments of the distribution, and it is general enough to accommodate skewed or symmetric distributions within a bounded domain (e.g., Abramowitz and Stegun 1970). The analytical framework of this work can accommodate general beta distributions specific to situations in which all of the characteristics of the inflow prediction uncertainty are known. To keep the following treatment parsimonious, in this analysis we use symmetric beta distributions, which can be characterized fully by three parameters. Two of the parameters are the mean (and median) Q m and the half-range Q R of the predicted inflow, which uniquely define the two bounds of symmetric beta distributions. The mean of the distribution is Q m, the variance is Q 2 R/(2c 1), with c being the third parameter with c 1, and the skewness is 0. The range of possible predicted inflows is bounded to be in the interval [Q m Q R, Q m Q R ]. This class of symmetric beta distributions includes the uniform distribution (c 1), and, as shown in Fig. 1 for symmetric beta distributions in the interval [0, 1], its variance decreases as c increases (c 1). In several examples of the analysis we use the uniform distribution as a conservative choice with respect to variance, but we also discuss the sensitivity of the results with respect to various values of parameter c. This treatment is restricted to cases in which the halfrange of the seasonal inflow prediction is not greater than the mean inflow (Q R Q m ), thus ignoring cases of very weak signal predictions. As will be seen later, such cases lead to no feasible solutions. In addition, unless specifically discussed in the text, it is assumed that the seasonal inflow prediction is unbiased. The latter condition is expected to hold when historical databases of past predictions and observations are available and have been used to adjust the seasonal inflow predictions (e.g., Carpenter and Georgakakos 2001; Graham et al. 2006). c. Mathematical programming problem This decision problem under uncertainty will be solved using stochastic optimization theory. The objective function F T to be minimized is defined as F T E{ V T V* 2 }, with E{} denoting the expectation (or mean value) operator with respect to the uncertain predicted inflow volume that infuses uncertainty into V T. The volume constraints are formulated as chance constraints (e.g., Loucks et al. 1981): 1 Pr V T C 1 and 2 Pr V T 0 1, whereas the release constraints are deterministic: R TL R T R TU. In the previous inequalities, Pr[] signifies the probability of the event in brackets. In addition to the presented constraints, the reservoir storage accounting equation holds: V T V I Q I R T. To reduce the analysis parameters, we divide V I, V T, Q I, Q m, Q R, R T, R TL, R TU, and V* byc to obtain the normalized quantities V o, V, q, q m, Q, R, R L, R U, and V T, respectively. (Note that V T represents the normalized target volume, whereas V T is the reservoir storage volume at time t T.) After substitution of (5) in the objective function and constraints, the normalized objective function F and the problem constraints may be written as follows: 3 4 5

5 MAY 2008 G E ORGAKAKOS AND GRAHAM 1301 subject to F E{ V o q R V T 2 }, Pr V o q R 1 1, 6 7 Pr V o q R 0 1, and 8 R L R R U. We expand the quadratic form in (6) and take the expected value of the resultant expression using the symmetric beta distribution of the inflow volume to obtain the deterministic expression for the objective function in (10). Use of the beta distribution may also be made to translate the probability statements of (7) and (8) in equivalent quantile deterministic statements of the constraints shown in (11) (14). The mathematical program is then written as subject to F V o q m R V T 2 Q 2 2c 1, R 1 V o q m Q, R V o q m Q, R R L, and 13 R R U. 14 The optimal release R* may be obtained by minimizing the objective function F (or maximizing F) subject to the (now deterministic) constraints (11) (14). Note that the parameter c that specifies the particular symmetric beta distribution used appears only in the objective function, and the optimal release and constraints only depend on the half-range Q as a measure of the inflow uncertainty and are thus independent of the particular symmetric beta distribution used. 3. Optimal solution Numerical methods may be employed to solve the nonlinear optimization problem posed in the previous section (e.g., Loucks et al. 1981). However, because of the functional form of this problem, the low dimensionality of the constraints and optimization variables, and our interest in parametric analysis, an analytical method is used next. It is well known that for a convex objective function and a convex set of constraints, the optimal solution of the nonlinear optimization problem is the solution of the Kuhn Tucker conditions (e.g., Simmons 1975). The reader may think of these conditions as being analogous to the ones obtained by setting the first derivative of the objective function to zero for unconstrained optimization problems. These conditions are shown next for the optimization problem posed, which satisfies the aforementioned requirements. Note that generalized Lagrange multipliers ( i ) are used with each of the four constraints of the problem to provide solutions for cases in which such constraints are binding for the optimal solution [i.e., the optimal solution is obtained from the equality sign of one of the constraints in the set (11) (14)]: 2 V o q m R* V T , R* 1 V o q m Q, R* V o q m Q, 17 R* R L, 18 R* R U, , 2 0, 3 0, 4 0, 1 1 V o q m Q R* 0, 2 V o q m Q R* 0, R L R* 0, and 26 4 R* R U 0, 27 where R* is the optimal release volume and 1, 2, 3, and 4 are the generalized Lagrange multipliers such that (15) (27) are satisfied. To find solutions for various combinations of parameter values, we proceed by identifying the applicability regions in parameter space (V o q m, Q, R L, R U, and V T ), which are defined by setting the generalized Lagrange multipliers equal to zero or greater than zero in all possible combinations. For the two values of i (i 1,..., 4) possible, this yields 2 4 (16) possible parameter regions for which there may be distinct solutions. Analysis of each of these solutions, however, shows that there are only five distinct parameter regions and that the rest may be described by one or more of these five. For each of these five distinct cases, the optimal release R*, the optimal value of the objective function F* and the constraints in parameter space that define the region of applicability of the solution have been obtained. These five solution cases are shown next. In the following, recall that the applicability regions in parameter space are further constrained in all cases by the parameter constraints 0 V o 1, 0 V T 1, q m 0, Q 0, and q m Q.

6 1302 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 a. Case I: Unconstrained solution ( ) For this case, R* V o q m V T and 28 F* Q 2 2c The applicability region in parameter space is Q V T 1, V T Q, V o q m V T R L, and 32 V o q m V T R U. b. Case II: Reservoir capacity constraint binding ( 1 0; ) For this case, 33 R* V o q m Q 1 and 34 F* Q 2 2c 1 1 Q V T The applicability region in parameter space is Q V T 1, Q 0.5, V o q m Q 1 R L, and 38 V o q m Q 1 R U. c. Case III: Storage nonnegativity constraint binding ( 2 0; ) For this case, 39 R* V o q m Q and 40 F* Q 2 2c 1 Q V T The applicability region in parameter space is Q V T, Q 0.5, V o q m Q R L, and 44 V o q m Q R U. d. Case IV: Release lower constraint binding ( 3 0; ) For this case, 45 R* R L and 46 F* Q 2 2c 1 V o q m R L V T The applicability region in parameter space is V T R L V o q m, V o q m Q 1 R L, V o q m Q R L, and 50 R L R U. e. Case V: Release upper constraint binding ( 4 0; ) For this case, 51 R* R U and 52 F* Q 2 2c 1 V o q m R U V T The applicability region in parameter space is 4. Discussion V T R U V o q m, V o q m Q 1 R U, V o q m Q R U, and 56 R L R U. 57 a. Predictions with very high uncertainty We start the interpretation of the mathematical results by noting that the applicability regions in parameter space in all of the above cases necessitate that Q 0.5. For cases II and III this constraint appears explicitly in the parameter constraint set [i.e., (37) and (43)]. For cases I, IV, and V it is easily inferred by combining the pair of inequalities (30) and (31), the pair (49) and (50), and the pair (55) and (56), respectively. Thus, for the optimization problem as formulated, Q 0.5 is infeasible. That is, no solution exists that will satisfy constraints (7) and (8) simultaneously under condition Q 0.5. Recall that Q represents the half-range of the seasonal inflow prediction uncertainty Q R normalized by the reservoir capacity C. Therefore, if the half-range of the inflow prediction uncertainty is greater than onehalf of the reservoir capacity, there can be no guarantee that the reservoir volume constraints can be enforced in the optimal solution with probability 1. This condition limits the applicability of the analysis to prediction lead times for which the inflow volume uncertainty would be small relative to the reservoir capacity. Alternatively stated, the analysis is valid for reservoirs of a capacity that is at least as large as the full range of the inflow volume uncertainty of the forecast period (i.e., season ) of this analysis. Note that in all of these situations

7 MAY 2008 G E ORGAKAKOS AND GRAHAM 1303 Q remains less than or equal to q m, as mentioned earlier. In addition to the relationship of reservoir capacity to inflow uncertainty inferred for this formulation of the optimization problem, these results reveal the importance of the type of probabilistic description imposed on the constraints in planning decision problems under uncertainty. For instance, had the expected-value operator been used in place of the probability in constraints (7) and (8) (e.g., 0 E{V o q R} 1), no such restriction with respect to Q would exist. However, use of the expectation operator in constraints containing random variables is only reasonable when low uncertainty situations exist (i.e., Q K q m ; e.g., Loucks et al. 1981). To support analysis for a wider range of uncertainty, the expectation operation was not used here in association with the reservoir volume constraints. b. Release constraints relaxed We first examine the case in which R L 0 and R U is very large. The situations discussed in this section are expected when the reservoir outlet works are adequate to pass very high flows and when there are no release impacts downstream due to flooding or low flow augmentation. 1) LOW EXPECTED INFLOWS We start the interpretation of the mathematical results focusing on case IV (nonnegativity release constraint is binding). In this case, the volume consisting of the initial reservoir volume and the mean predicted inflow (V o q m ) is not enough to meet the target V T (always at time t T). The optimal release is zero, and the optimal value of the objective function is the minimum of the unconstrained solution (case I) plus a term that depends on the expected deficit in reservoir volume from the target at time t T. Apart from the dependence of the unconstrained minimum of the objective function on Q, no additional dependence of F* and R* onq exists for this Case IV. This is a direct result of our choice of objective function in the optimization problem. We chose to minimize the expected value of the deviation from the reservoir volume target at t T rather than minimize or maximize some other probabilistic measure of the deviation (e.g., minimize Pr[ V V T ], with being a given small volume). In the latter case, which resembles a minimum maximum approach (e.g., Nardini and Montoya 1995), not only the mean but the range of the uncertainty in normalized storage V due to the uncertain inflow prediction would be part of the optimal objective function F*. This highlights the importance of the choice of the probabilistic description of the objective for these decision problems under uncertainty. For this analysis, we chose the expected-value operator for the objective function as most commonly done to assure convex nonlinear objective functions (e.g., Loucks et al. 1981; Peng and Buras 2000). Under the relaxed release constraint situation, case IV has no lessons to offer with respect to any direct uncertainty influences on optimal release and objective function (beyond that of the case I minimum), and we continue the discussion by focusing on cases I III. We do note that the parameter constraint set for case IV indicates that this solution is only applicable for expected reservoir volume at time t T (for R* 0) that is located sufficiently far from the reservoir volume bounds (Q V o q m 1 Q). 2) ADEQUATE EXPECTED INFLOWS (i) Unconstrained operation Case I represents the optimal solution for the unconstrained case. This case is characterized by 1) reservoir volume at time t T that is well within the reservoir storage bounds, including a margin of uncertainty in the inflow, 2) enough expected water volume at time t T to meet the target V T, and 3) an optimal release that is between the release bounds imposed. In this case, the optimal release is the difference between the expected available reservoir volume (V o q m ) and the volume target (V T )att T. The objective function is a quadratic function of the normalized inflow uncertainty half-range Q. This is the minimum optimal objective function for all cases examined, indicating that when the parameters are such that either the volume or the release constraints are binding, the resulting optimal value of the objective function is greater (worse) than that for case I [cf. (29) with (35), (41), (47), and (53)]. Furthermore, it also indicates that no matter what release policy is selected there is irreducible expected error in meeting the volume target because of the presence of nonzero uncertainty in seasonal inflow predictions. Although reservoir release planners do not need to worry about the inflow uncertainty in this case for deciding on the release, their ability to meet the target volume directly depends on the uncertainty of the predicted inflow. For the same predicted mean inflow, but for greater uncertainty, the objective function becomes substantially worse for the same optimal release decision. That is, greater predicted inflow uncertainty even for unbiased predictions will compromise reservoir op-

8 1304 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 eration benefits for activities that depend on meeting the target volume V T (water conservation, water supply, down-streamflow augmentation, and others, all in the interval from t to t T). Under the parametric constraints of case I, even when the prediction uncertainty is reliable and known, there is no better decision the decision maker can make in the face of increasing prediction uncertainty that incorporates this uncertainty in the release decision; the optimization formalism only offers the decision maker the ability to anticipate the increasing level of expected losses. (ii) Constrained operations for near-full or near-empty reservoirs Contrary to case I, if the parameters of a particular reservoir operation problem satisfy the parametric constraints of cases II and III, the optimal release becomes a function of the inflow uncertainty. In the first case (case II) the reservoir capacity constraint is binding, and in the second case (case III) the reservoir volume nonnegativity constraint is binding. In such cases, it behooves the decision maker to take into consideration the inflow uncertainty in making a release decision. Both the optimal release [see (34) and (40)] and the optimal objective function [see (35) and (41)] depend on the uncertainty measure Q; dependence is linear for the former while it is quadratic for the latter. In this case F* is the sum of the unconstrained minimum and of an additional term, both exhibiting dependence on Q. It is during reservoir operation near the boundaries of the reservoir volume (C and 0), when the release constraints are relaxed, that the decision maker can most benefit from knowledge of the uncertainty in inflow predictions. Furthermore, the reliability of the estimates of inflow prediction uncertainty to represent the actual inflow-prediction uncertainty matters most during these situations. The corollary from this analysis is that reservoir operation for reservoirs with capacity that is relatively small in comparison with average seasonal inflow uncertainty (but is greater than 2 times the half-range Q as mentioned earlier) stand to gain most from reliable predictions of seasonal inflow uncertainty. For larger, well-designed reservoirs in moist climates, events in which the sum of the initial plus the expected inflow volume and the target volume are either near the reservoir capacity or near the zero limit of active storage volume are rare. For such large reservoirs, to discern the benefits of using the uncertainty in reliable seasonal inflow forecasts would require study of such rare events alone; otherwise, the benefits of using the uncertainty in seasonal inflow forecasts would be diluted in the majority of events for which the inflow forecast uncertainty does not enter in the optimal release decision. (iii) Examples To bring the previous general discussion to focus and to reveal better the optimal solution behavior under cases I III, we next discuss examples of behavior for a few possible parameter sets. In these examples, the parameter V o q m is specified to be equal to a certain value, R L 0 and R U is large and nonbinding, and R* and F* are examined as functions of Q and V T. The first example EX1 has V o q m 0.5, and the second example EX2 has V o q m 1.0. The constraint sets of cases I and III have been used for EX1, and the constraint sets of cases I and II have been used for EX2. The values of V T used in each example obey the constraints of the applicable constraint set. We first discuss the results for the uniform distribution, and then we discuss the influence of using more general symmetric beta distributions. The results for EX1 appear in Fig. 2. The upper part of the figure displays the optimal release, and the lower part displays the optimal objective function. It is apparent that as V T becomes increasingly lower than the prespecified V o q m (0.5), the range of Q for which R* is independent of Q is limited to a decreasing range of small Q values. For larger Q values, the dependence of R* onq is linear, with R* diminishing as Q approaches 0.5. As the region of R* independence on Q decreases, the optimal objective function value increases, or becomes worse (Fig. 2. bottom). Figure 2 shows that larger uncertainty in predicted inflows affects situations with V T close to V o q m most. Consider the specific case V T 0.1. For a value of Q of 0.05, R* 0.4 and F* 0.0; for a value of Q of 0.3, R* 0.2 and F* Had the decision maker not considered the uncertainty of the inflow prediction for the higher Q value (i.e., the decision maker implemented the optimal release R* 0.4 in this case, too), the implemented decision would not be feasible with probability 1 (i.e., in all seasons for which V o q m 0.5). Furthermore, were the estimate Q of uncertainty in the seasonal inflow prediction 50% higher than the true value (e.g., Q TRUE 0.2 vs 0.3) because of an unreliable prediction system, the second case of larger Q would yield a suboptimal decision of R* 0.2 instead of R* 0.3, with higher-than-anticipated losses from reservoir operation (larger discrepancy from the target volume V T at time t T). Analogous inferences may be drawn from Fig. 3 that are relevant to decisions when the expected volume at

9 MAY 2008 G E ORGAKAKOS AND GRAHAM 1305 FIG. 2. Optimal (top) release R* and (bottom) objective function F* as functions of the inflow prediction uncertainty Q for various values of reservoir volume target V T, and for V o q m 0.5, with operation near the reservoir storage lower bound for nonbinding release constraints and c 1. time t T is near the reservoir capacity (EX2). In this case, failure of the decision maker to consider the uncertainty in the inflow prediction for significant values of Q will yield lower releases than are required for optimality and would also render the release decision infeasible to implement with probability 1 in the interval t T. It is noted that for this example too, with volumes near the reservoir capacity, larger uncertainty FIG. 3. As in Fig. 2, but for V o q m 1, with operation near reservoir capacity for nonbinding release constraints; c 1. Large circles and associated arrows indicate values for R* and F* that correspond to Q being assigned a value of 0.2 when its true value is 0.3 as discussed in the text.

10 1306 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 4. Sensitivity of the results of examples (top) EX1 and (bottom) EX2 to the value of parameter c of the symmetric beta distribution; c 1 corresponds to the uniform distribution used in Figs. 2 and 3. The values of V o q m and V T used are shown on the panels. in predicted inflows affects situations with V T close to V o q m most. In these situations, the sensitivity of both R* and F* with respect to Q is highest (e.g., cf. the lines corresponding to V T 1.0 with those of V T 0.7 in both panels of Fig. 3). For operation near reservoir capacity, the reliability of the predicted uncertainty becomes very significant for determining the optimal release and anticipating the optimal objective function value. For instance, for a target volume of V T 0.9, a 33% underestimation of Q, from a true value of 0.3 to a predicted value of 0.2, yields a 33% reduction in the optimal release, with an attendant 67% (erroneous) reduction in anticipated optimal expected quadratic deviation from the target volume (from a value of corresponding to the true optimal release to a value of corresponding to the nonoptimal release driven by the underdisperse seasonal inflow volume). This is very significant for anticipating losses from the reservoir operation. The influence of using a value of c greater than 1 (symmetric beta distributions other than the uniform distribution) may be seen in Fig. 4. It is apparent that, for operation near capacity or near zero volume and for large values of Q, the use of a symmetric beta distribution of lower variance (increasing c) reduces the anticipated optimal objective significantly (and therefore the anticipated losses from operation). In such situations, even though the optimal release will remain unaffected, it is then important to invest in some effort to estimate the shape of the seasonal inflow prediction distribution accurately instead of using the less-informative (conservative) uniform distribution function. (iv) Two-dimensional dependencies and normalized sensitivities The landscapes of the two-dimensional dependence of the optimal release and objective function on Q and V T are shown in Fig. 5 when the release constraints are not binding. The optimal release (Fig. 5, top) shows linear dependence on Q and piecewise linear dependence on V T. It is now clearly shown that the strongest dependence on Q is for V T near the (normalized by capacity) reservoir volume bounds (0 and 1). The optimal objective function (Fig. 5, bottom) exhibits a stingray shape with maximal losses expected for large Q and when V T is near the reservoir-normalized storage volume bounds. The results have been obtained with c 1, but other values of c produce qualitatively similar results for the optimal objective (see a more detailed discussion at the end of this section). The considerable region of insensitivity to Q covers a wide range of V T for low Q values, but the sensitivity region begins relatively abruptly with sharp loss increases expected (up to an order of magnitude greater) for increasing Q and V T approaching the normalized storage bounds 0 and 1. To determine the aforementioned sensitivity quantitatively, the normalized sensitivities of R* (R R ) and F*

11 MAY 2008 G E ORGAKAKOS AND GRAHAM 1307 FIG. 5. Optimal (top) release R* and (bottom) objective function F* as functions of Q and V T, for V o q m 1 and for nonbinding release constraints (cases I, II, and III); c 1. (R F ) with respect to Q have been computed based on Rabitz (1989): R J J* J* Q Q, 58 with J representing either R or F. Figure 6 shows R R and R F as functions of Q and V T ; R R (Fig. 6, top) is a discontinuous function of Q and V T, exhibiting zero normalized sensitivity for values of V T and Q in the closed triangle bounded by the constraints (Q V T

12 1308 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 6. Normalized sensitivities of optimal (top) release R R and (bottom) objective function R F as functions of Q and V T, for V o q m 1 and for nonbinding release constraints (cases I, II and III); c 1. 1 Q and Q 0). For the upper triangle of the upper panel of Fig. 6, R R is 1, indicating 100% positive sensitivity for all combinations of V T and Q in this region (it excludes the line V T 1 Q). For values of Q and V T in the lower triangle of the upper panel of Fig. 6, the normalized sensitivity to Q is negative, increasing nonlinearly from values of near zero for small values of Q to 1 ( 100%) for values of Q near 0.5. These results indicate that, for V o q m 1 and nonbinding release constraints, the importance of knowing Q reliably in-

13 MAY 2008 G E ORGAKAKOS AND GRAHAM 1309 creases for the release decision 1) for values of V T near 1 and for increasing Q and 2) for low target values as Q approaches 0.5. In the former case, the effect of knowing Q is increasing is to increase the release; in the latter case the effect is to decrease the release. The imposed reservoir storage volume bounds effect such dependence. The normalized sensitivity R F exhibits complex structure in the Q V T space, with two lines of local maxima and sharp falloff away from these lines. By setting the derivative R F / Q equal to zero, the lines of local maxima may be obtained as follows: Q c 1 1 VT and 59 Q c 1 VT. 60 These lines are associated with the sharp increases of the function F*(Q, V T ) for increasing Q and for V T near the storage bounds, as shown in the lower panel of Fig. 6 for the case c 1. Note also that the normalized sensitivity values obtained are very high, ranging from 2 to almost 3, or, expressed in percent, ranging from 200% to 300%. Sensitivity of 200% is associated with the unconstrained optimum of case I. These results highlight the importance of the predicted inflow uncertainty in estimating objective function values at the optimum or in anticipating the losses from not meeting the target volume. They also suggest the conditions for which reliability in the predicted uncertainty values becomes of paramount importance for reservoir operations. Figure 7 highlights the significant dependence of the normalized sensitivity R R on q m for a given V o and for a range of Q values. The results are shown for the feasible values of V T [that satisfy constraints (36) and (42), as appropriate] and address the case of operation near reservoir capacity (case II; Fig. 7, top) and near zero reservoir storage (case III; Fig. 7, bottom). The minimum of the normalized sensitivity function (zero normalized sensitivity) may be found to be the locus of points V o q m 1 in the upper panel of Fig. 7 and the locus of points Q 0 in the lower panel of Fig. 7. The maximum (which tends to infinity) normalized sensitivity is at the parametric constraint lines [see (38) and (44), with R L 0]: V o q m Q 1 and 61 V o q m Q. 62 The near-capacity-operation results (Fig. 7, top) show that, for values of Q near the expected deficit from reservoir capacity at time t T (1 V o q m ), the optimal release carries the highest normalized sensitivity to Q. Approaching this condition, the normalized sensitivity tends to infinity, and, even close to this condition, normalized sensitivities exceed 500%. In addition, the results show that, for larger values of Q, the normalized sensitivity is lowest for values of the expected volume at t T near capacity. It is also noted for this case that biases in q m have the potential for moving the normalized sensitivity from regions of very high values to regions of insensitive behavior. This, in turn, will produce significant errors in release decisions for feasible values of V T, unless the volume at t T is near capacity and Q is large (i.e., the half-range of predicted seasonal inflow uncertainty is a significant fraction of reservoir capacity). The near-zero storage results (Fig. 7, bottom) show that, for a fixed value of (V o q m ), increasing Q increases the normalized sensitivity up to the value of 100% at Q 0.5, for Q and V o q m values away from the line of equality. Most of the feasible region in the Q (V o q m ) domain has values between 0% and 100% normalized sensitivity, apart from the region very close to the line V o q m Q, which has very high negative sensitivities. In this case too (as in the case of the upper panel of Fig. 7), biases in q m will lead to significant release decision errors for large Q and for volume (V o q m ) near capacity. The sensitivity of the F* and R F results on the value of parameter c may be discerned by comparing the results of the upper and lower panels of Fig. 8 with the lower panels of Figs. 5 and 6. In Fig. 8, a sharper symmetric distribution is used with c 7 (see Fig. 1). The main features remain the same in the two figures with a lower value of F* for c 7 for operation near the reservoir volume bounds (top of Fig. 8 vs bottom of Fig. 5), and a much higher value of R F at the lines of maximum in the lower panel of Fig. 8 (R Fmax 5) than in the lower panel of Fig. 6 (R Fmax 3). Reducing the variance of the symmetric beta distribution increases the normalized sensitivity of the optimal objective to Q but also decreases the maximum deviations of the end-of-season reservoir volume to the target volume (with attendant reduced losses from reservoir operation). Knowing the shape of the distribution is clearly advantageous for anticipating reservoir operation losses. (v) Reservoir capacity and value of prediction uncertainty We now examine the relative importance of the reservoir capacity C. In particular, we consider three ques-

14 1310 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 7. Normalized sensitivity R R with respect to Q and (V o q m ) for operation (top) near reservoir capacity and (bottom) near zero storage. The lines of maximum (tends to infinity) R R are shown on the Q (V o q m ) plane. tions: 1) What is the sensitivity of the optimal decision release and the optimal objective function value on the reservoir capacity? 2) How is this sensitivity affected by Q? 3) What is the range of reservoir capacity for which predictions of the uncertainty in seasonal forecasts may be most beneficial? To start, we note that among cases I V, case II alone contains dependence of the optimal release and objective function on C [see (34) and (35) in

15 MAY 2008 G E ORGAKAKOS AND GRAHAM 1311 FIG. 8. (top) Optimal objective function F* and (bottom) normalized sensitivity R F as functions of Q and V T, for V o q m 1 and for nonbinding release constraints (cases I, II and III); c 7. which 1 is the normalized value of the capacity C]. We can thus shed light on the influence of the seasonal inflow uncertainty on the optimal release and objective sensitivities with respect to C by examining the parametric dependence of normalized sensitivities R RC and R FC for this case. Analogous to previous normalized sensitivities, these new sensitivities are defined by

16 1312 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 9. Normalized sensitivities (top) R RC and (bottom) R FC as functions of Q for various values of (top) V o q m and (bottom) V T. with R RC R* T R* T C C R FC F* F* C C, and R* T V I Q m Q R C 65 and with F* given by (35). Note that (65) is obtained from (34) by multiplication with C. Figure 9 shows the results for R RC (Fig. 9, top) and R FC (Fig. 9, bottom) as functions of Q and for V o q m and V T in the applicability region defined by constraints (36) (39). The results correspond to R L 0 and very high R U (nonbinding release constraints). The results in the bottom panel of Fig. 9 correspond to c 1, but other values of c gave similar behavior. Negative normalized sensitivities greater than 200% in absolute value are displayed with very sharp gradients of the sensitivities as functions of Q. Negative R RC and R FC imply that larger capacity reservoirs generally have reduced both optimal release and objective value (better ability to meet set targets). The upper panel of Fig. 9 shows that for Q values decreasing in the domain of applicability of case II, the sensitivity of the optimal release decision to the reservoir capacity C increases very substantially for all of the mean inflow and initial volumes shown. Large uncertainty makes dependence of the release on reservoir capacity lower when this volume constraint is binding (but at no time less than 200%). Reduction of V o q m from the value of 1 exaggerates this relationship, making dependence of the sensitivity on Q more significant for higher values of Q. The normalized sensitivity of the optimal objective function with respect to C shows more complex dependence on Q, including the existence of minima for certain values of Q and for values of the target volume V T that are less than 1. For certain values of Q there is maximum absolute normalized sensitivity R FC of the optimal expected deviation from the target volume (and, consequently, of the expected losses from reservoir operation). The maxima are sharper for values of V T near 1 (target volume near reservoir capacity), and for increasing V T toward 1 they occur at lower values of Q. Two different values of Q in such cases yield identical R FC values (Fig. 9, bottom) with very different R RC values (Fig. 9, top). These results corroborate those of Fig. 6 (bottom panel). The locus of maximum absolute normalized sensitivity is Q 1.5(1 V T ); see also (59) for c 1. The results of the lower panel of Fig. 9 also indicate that in general the normalized sensitivity R FC is lower for V T decreasing and V T 1. We now turn to the third question pertaining to the range of C for which optimal decisions and objectives benefit most from knowing seasonal inflow uncertainty. Multiplying the constraints of case II [(36) (38)] by the positive constant C yields Q R V* C, 66 Q R 0.5C, and 67 V I Q m Q R C. 68 We set target V* equal to C and the initial reservoir volume V I equal to C, with and being two nonnegative constants that are less than 1. Substituting these typical values in (66) (68), we obtain 2Q R C Q m Q R 1 and 69 2Q R C Q R 1, with 0.5, 70

17 MAY 2008 G E ORGAKAKOS AND GRAHAM 1313 where the additional constraint on is to produce a feasible region for C in (70). It is noted that, for (69) and (70) to produce a substantial range of C for which operation near capacity can benefit from knowledge of Q R [or Q, as in (34) and (35)], it is required that constants and be as close to 1 as possible without actually being equal to 1. It is anticipated that 1) it will be rare and unintentional (for the risk of overtopping) to reach values of V I greater than 0.90 of capacity and that 2) target volumes would rarely exceed 0.9C because they are average measures produced from results of simulation analysis with historical data involving multiple seasons. For 0.9, we obtain the nominal ranges 2Q R C 10 Q m Q R and 71 2Q R C 10Q R. 72 These results indicate the range of C for which knowledge of the inflow prediction uncertainty can benefit most reservoir operations near capacity. This range is for the seasons for which initial volume and target volume are very close to reservoir capacity. The range of capacities is considerably reduced for other seasons when V I and V* are substantially lower than reservoir capacity. Reservoir operations with larger reservoir capacities than those indicated by the upper limits in (71) and (72) would only rarely benefit from knowing the uncertainty in seasonal predictions of inflow, whereas values of the capacity lower than the lower limit in (71) and (72) will render the decision problem infeasible as mentioned earlier (Q 0.5). As expected, the range of uncertainty value diminishes to zero as Q R tends to zero. Larger C ranges are generally obtained for higher Q R and Q m values, and the largest range is obtained in the unusual situation in which V* C [in this case (66) is always satisfied] and the range is defined by (71). Three reservoir examples for which the authors have enough information to estimate the bounds indicated above are discussed next. The first two are in the continental United States in an extratropical climate, and the third one is in Panama in a tropical climate. The Folsom Lake on the American River (Carpenter and Georgakakos 2001) and the Saylorville Reservoir on the upper Des Moines River (A. P. Georgakakos et al. 1998) constitute the two U.S. examples, and the Gatun Madden Lake system (Graham et al. 2006) is the Panama example. For Folsom Lake, ensemble monthly inflow forecasts conditional on climate model forecasts have a typical range of m 3 month 1 and the reservoir active volume is m 3. For Saylorville, ensemble streamflow forecasts have a range of m 3 month 1 and the reservoir has an active capacity of m 3. In both cases and for monthly decision and prediction horizons, the reservoir capacity is between the bounds indicated by (72). In the first example, the capacity is in the middle of the range, and in the second example it is somewhat close to the lower bound of (72). This finding indicates that in the months with a higher-than-average range of the ensemble inflows the prediction uncertainty may be too high to be useful for monthly management involving target volumes, and a shorter time interval for management may be necessary. For the tropical Gatun Madden system of lakes, the reservoir active volume capacity is m 3 and the range of maximum monthly inflow uncertainty occurs in October December, ranging from m 3 month 1 (October) to m 3 month 1 (December). It is apparent that, in some Decembers with high inflow uncertainty, prediction uncertainty information may not be useful for management to attain end-of-month target volumes for a prediction lead time (and decision interval) of 1 month. Shorter time intervals and reduced prediction uncertainty of inflow volumes are necessary for the uncertainty to be useful for this example. c. Effects of release constraints During the previous discussion, the release constraints were not binding: R L was set equal to zero and R U was set to a very large value. In this section we impose prespecified positive bounds on the release that may render the release constraints binding for certain regions of the parameter space (cases IV and V). The primary effect is to reduce the parametric region for which the unconstrained optimum applies, with attendant larger expected deviations of the predicted volume at time t T from the target volume. This may be discerned by comparing the results shown in Fig. 10 (derived for R L 0.3 and R U 0.8 and c 1) with those in Fig. 5 for cases I, II, and III and for V o q m 1. The upper panels of Figs. 10 and 5 make it apparent that the effect of the upper release constraint is to curtail the high releases that would have been optimal for low values of Q and V T, and the effect of the lower release constraint is to increase the releases that would have been optimal for values of V T near 1 and for low values of Q. The attendant increases in optimal objective function value for the new case of release constraints in these two aforementioned parametric regions are seen by comparing the lower panels of Figs. 10 and 5. Whereas for low values of Q and values of V T

18 1314 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 10. As in Fig. 5, but for R L 0.3 and R U 0.8. near 1 the optimal objective function was very near zero (being equal to Q 2 /3) without release constraints, it is now near 0.1 [being equal to the sum of the terms shown in (47)]. The introduction of release constraints inhibits the achievement of the best possible performance near the bounds of the V T range. The presence of the release bounds also renders both the optimal release and the optimal objective function less dependent on the uncertainty in the seasonal inflow predictions. This may be discerned by the large flat regions in the normalized sensitivity functions R R and R F shown in the upper and lower panels of Fig. 11,

19 MAY 2008 G E ORGAKAKOS AND GRAHAM 1315 FIG. 11. As in Fig. 6, but for R L 0.3 and R U 0.8. respectively. The release constraints render R R equal to zero (Fig. 11, top) and R F equal to 2 (Fig. 11, bottom, for c 1) for a very significant part of the parametric domain, in comparison with these normalized sensitivity functions of Fig. 6 which were obtained without the release constraints. Examination of several pairs of release constraints indicates that the narrower the range [R L, R U ] is, the less sensitive the optimal reservoir operation becomes to the uncertainty in Q. The effects of introducing the release constraints also

20 1316 J O U R N A L O F A P P L I E D M E T E O R O L O G Y A N D C L I M A T O L O G Y VOLUME 47 FIG. 12. As in Fig. 5, but for V o q m 0.5. include reduction of the region in parameter space for which optimal solutions may be found to the reservoir operation problem. Consider the case with V o q m 0.5 and c 1. Figure 12 shows the optimal release (Fig. 12, top) and objective function (Fig. 12, bottom) for this case without release constraints, and the corresponding panels of Fig. 13 show the optimal release and objective of this case with the release constraints (R L 0.3 and R U 0.8). Only low values of uncertainty (Q 0.2) produce feasible results in the release-constrained case.